0 and sufficiently small, *P**e*(N) ≤ *ε* as N → ∞. Furthermore, for the channel C3, the following constraints on the conditional entropy must be satisfied for (*R*1, *R*2) to be considered achievable:\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_{1} - H({\\mathssf{M}}\\_{1}|{\\mathbsf{Y}}\\_2) \\leq \\mathrm{N}\\epsilon, \\label{eq:security1} \\\\\n\\mathrm{N}R\\_{2} - H({\\mathssf{M}}\\_{2}|{\\mathbsf{Y}}\\_1) \\leq \\mathrm{N}\\epsilon. \\label{eq:security2}\\end{aligned}$$ \nThe capacity region is defined as the closure of the set of all achievable rate pairs (*R*1, *R*2).\nMain results\n============\nIn this section, we state the achievability and converse theorems for all the channel models considered in this paper, and provide related discussion. Let Cc denote the capacity region of the channel Cc; c = 1, 2, 3. We use the following auxiliary RVs defined on finite sets: ${\\mathssf{U}} \\in \\mathcal{U}$, ${\\mathssf{V}}\\_1 \\in \\mathcal{V}\\_1$ and ${\\mathssf{V}}\\_2 \\in \\mathcal{V}\\_2$.\nClass~I channels\n----------------\nFor the channel C1, we consider the set P1 of all joint probability distributions *p*1(.) that can be factored as $p({\\mathssf{w}})p({\\mathssf{v}}\\_1,{\\mathssf{v}}\\_2|{\\mathssf{w}})p({\\mathssf{x}}|{\\mathssf{w}},{\\mathssf{v}}\\_1,{\\mathssf{v}}\\_2)\np({\\mathssf{y}}\\_1,{\\mathssf{y}}\\_2|{\\mathssf{x}})$. For a given *p*1(.) ∈ P1, a lower bound on the capacity region for C1 is described by the set R1, in(*p*1), which is defined as the union over all distributions *p*1(.) of the convex hull of the set of all rate pairs (*R*1, *R*2) that simultaneously satisfy $(\\ref{eq:rateregionC1R1})$ - $(\\ref{eq:rateregionC1R1plusR2})$.\n \n$$\\begin{aligned}\nR\\_1 &\\leq& I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1) - I({\\mathssf{V}}\\_1;{\\mathssf{W}}),\\label{eq:rateregionC1R1}\\\\\nR\\_2 &\\leq& I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2) - I({\\mathssf{V}}\\_2;{\\mathssf{W}}),\\label{eq:rateregionC1R2}\\\\\nR\\_1 + R\\_2 &\\leq& I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1) + I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2) - I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2) - I({\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2;{\\mathssf{W}}), \\label{eq:rateregionC1R1plusR2}\\end{aligned}$$ \nwhere ${\\mathssf{V}}\\_1$ and ${\\mathssf{V}}\\_2$ are constrained to satisfy the Markov chain $({\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2) \\rightarrow ({\\mathssf{X}},{\\mathssf{W}}) \\rightarrow ({\\mathssf{Y}}\\_1,{\\mathssf{Y}}\\_2)$.\n[thm:achievethmC1] Let R1, in = ⋃*p*1(.) ∈ P1R1, in(*p*1). Then, R1, in ⊆ C1.\nFor proof, see Section [subsec:achieveproofC1].\nFor a given *p*1(.) ∈ P1, an outer bound for C1 is described by the set R1, out(*p*1), which is defined as the union of all rate pairs (*R*1, *R*2) that simultaneously satisfy $(\\ref{eq:outboundC1R1})$ - $(\\ref{eq:outboundC1R2})$.\n \n$$\\begin{aligned}\nR\\_1 &\\leq& I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1) - I({\\mathssf{V}}\\_1;{\\mathssf{W}}), \\label{eq:outboundC1R1}\\\\\nR\\_2 &\\leq& I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2) - I({\\mathssf{V}}\\_2;{\\mathssf{W}}), \\label{eq:outboundC1R2},\\end{aligned}$$ \nwhere $({\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2) \\rightarrow ({\\mathssf{X}},{\\mathssf{W}}) \\rightarrow ({\\mathssf{Y}}\\_1,{\\mathssf{Y}}\\_2)$.\n[thm:conversethmC1] Let R1, out = ⋃*p*1(.) ∈ P1R1, out(*p*1). Then, C1 ⊆ R1, out.\nThe proof of Theorem [thm:conversethmC1] can be found in Section [subsec:converseproofC1]. However, this outer bound does not include a bound on the sum-rates. To explicitly bound the sum-rate, we provide the following alternative outer bound for the channel C1. We consider the set P1\\* of all joint probability distributions *p*1\\*(.) that can be factorized as follows: *p*(*w*)*p*(*u*, *v*1, *v*2∣*w*)*p*(*x*∣*w*, *u*, *v*1, *v*2)*p*(*y*1, *y*2∣*x*). For a given *p*1\\*(.) ∈ P1\\*, an outer bound for C1 is described by the set R1, out\\*(*p*1\\*), which is defined as the union of all rate pairs (*R*1, *R*2) that simultaneously satisfy -.\n \n$$\\begin{aligned}\nR\\_1 &\\leq& I({\\mathssf{U}},{\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1) - I({\\mathssf{V}}\\_1;{\\mathssf{W}}|{\\mathssf{U}}), \\label{eq:oboundnewC1R1}\\\\\nR\\_2 &\\leq& I({\\mathssf{U}},{\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2) - I({\\mathssf{V}}\\_2;{\\mathssf{W}}|{\\mathssf{U}}), \\label{eq:oboundnewC1R2}\\\\\nR\\_1 + R\\_2 &\\leq& I({\\mathssf{U}},{\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1) - I({\\mathssf{V}}\\_1;{\\mathssf{W}}|{\\mathssf{U}}) + I({\\mathssf{U}},{\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2|{\\mathssf{V}}\\_1) - I({\\mathssf{V}}\\_2;{\\mathssf{W}}|{\\mathssf{U}},{\\mathssf{V}}\\_1),\\label{eq:oboundnewC1R1plusR2\\_1}\\\\\nR\\_1 + R\\_2 &\\leq& I({\\mathssf{U}},{\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2) - I({\\mathssf{V}}\\_2;{\\mathssf{W}}|{\\mathssf{U}}) + I({\\mathssf{U}},{\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1|{\\mathssf{V}}\\_2) - I({\\mathssf{V}}\\_1;{\\mathssf{W}}|{\\mathssf{U}},{\\mathssf{V}}\\_2),\\label{eq:oboundnewC1R1plusR2\\_2}\\end{aligned}$$ \nwhere the following Markov chain is satisfied: $({\\mathssf{U}},{\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2) \\rightarrow ({\\mathssf{X}},{\\mathssf{W}}) \\rightarrow ({\\mathssf{Y}}\\_1,{\\mathssf{Y}}\\_2)$.\n[thm:conversethmC12] Let R1, out\\* = ⋃*p*1\\*(.) ∈ P1\\*R1, out\\*(*p*1\\*). Then, C1 ⊆ R1, out\\*.\nSection [subsec:converseproofC12] contains the proof of Theorem [thm:conversethmC12].\nClass~II channels\n-----------------\nFor the channel C2, we consider the set P2 of all joint probability distributions *p*2(.) of the form $p({\\mathssf{w}})p({\\mathssf{u}}|{\\mathssf{w}})p({\\mathssf{x}}|{\\mathssf{w}},{\\mathssf{u}})p({\\mathssf{y}}\\_1,{\\mathssf{y}}\\_2|{\\mathssf{x}})$. For a given *p*2(.) ∈ P2, a lower bound on the capacity region for C2 is described by the set R2, in(*p*2), which is defined as the union over all distributions *p*2(.) of the convex-hull of the set of all rate pairs (*R*1, *R*2) that simultaneously satisfy $(\\ref{eq:rateregionC2R1})$ - $(\\ref{eq:rateregionC2R2})$.\n \n$$\\begin{aligned}\nR\\_1 &\\leq& I({\\mathssf{U}};{\\mathssf{Y}}\\_1) - I({\\mathssf{U}};{\\mathssf{W}}),\\label{eq:rateregionC2R1}\\\\\nR\\_2 &\\leq& I({\\mathssf{U}};{\\mathssf{Y}}\\_2) - I({\\mathssf{U}};{\\mathssf{W}}),\\label{eq:rateregionC2R2}\\end{aligned}$$ \nwhere the Markov chain ${\\mathssf{U}} \\rightarrow ({\\mathssf{X}},{\\mathssf{W}}) \\rightarrow ({\\mathssf{Y}}\\_1,{\\mathssf{Y}}\\_2)$ holds.\n[thm:achievethmC2] Let R2, in = ⋃*p*2(.) ∈ P2R2, in(*p*2). Then, R2, in ⊆ C2.\nThe proof of Theorem [thm:achievethmC2] is relegated to Section [subsec:achieveproofC2].\nFor a given *p*2(.) ∈ P2, an outer bound for C2 is described by the set R2, out(*p*2), which is defined as the union of all rate pairs (*R*1, *R*2) that simultaneously satisfy $(\\ref{eq:outboundC2R1})$ - $(\\ref{eq:outboundC2R2})$.\n \n$$\\begin{aligned}\nR\\_1 &\\leq& I({\\mathssf{U}};{\\mathssf{Y}}\\_1) - I({\\mathssf{U}};{\\mathssf{W}}) + H({\\mathssf{U}}), \\label{eq:outboundC2R1}\\\\\nR\\_2 &\\leq& I({\\mathssf{U}};{\\mathssf{Y}}\\_2) - I({\\mathssf{U}};{\\mathssf{W}}) + H({\\mathssf{U}}), \\label{eq:outboundC2R2}\\end{aligned}$$ \nwith ${\\mathssf{U}} \\rightarrow ({\\mathssf{X}},{\\mathssf{W}}) \\rightarrow ({\\mathssf{Y}}\\_1,{\\mathssf{Y}}\\_2)$.\n[thm:conversethmC2] Let R2, out = ⋃*p*2(.) ∈ P2R2, out(*p*2). Then, C2 ⊆ R2, out.\nThe proof of Theorem [thm:conversethmC2] can be found in Section [subsec:converseproofC2].\nClass~III channels\n------------------\nFor the channel C3, we consider the set P3 of all joint probability distributions *p*3(.) that can be written as $p({\\mathssf{w}})p({\\mathssf{u}})p({\\mathssf{v}}\\_1,{\\mathssf{v}}\\_2|{\\mathssf{w}},{\\mathssf{u}})\np({\\mathssf{x}}|{\\mathssf{w}},{\\mathssf{v}}\\_1,{\\mathssf{v}}\\_2)p({\\mathssf{y}}\\_1,{\\mathssf{y}}\\_2|{\\mathssf{x}})$. For a given *p*3(.) ∈ P3, an inner bound on the capacity region for C3 is described by the set R3, in(*p*3), which is defined as the union over all distributions *p*3(.) of the convex-hull of the set of all rate pairs (*R*1, *R*2) that simultaneously satisfy $(\\ref{eq:rateregionC3R1})$ - $(\\ref{eq:rateregionC3R1plusR2})$.\n \n$$\\begin{aligned}\nR\\_1 &\\leq& I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1|{\\mathssf{U}}) - \\max [I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_2|{\\mathssf{U}},{\\mathssf{V}}\\_2),I({\\mathssf{V}}\\_1;{\\mathssf{W}}|\n{\\mathssf{U}})],\\label{eq:rateregionC3R1}\\\\\nR\\_2 &\\leq& I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2|{\\mathssf{U}}) - \\max [ I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_1|{\\mathssf{U}},{\\mathssf{V}}\\_1),I({\\mathssf{V}}\\_2;{\\mathssf{W}}|\n{\\mathssf{U}})],\\label{eq:rateregionC3R2}\\\\\n\\nonumber R\\_1 + R\\_2 &\\leq& I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1|{\\mathssf{U}}) + I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2|{\\mathssf{U}}) - I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_2|{\\mathssf{U}},{\\mathssf{V}}\\_2) - I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_1|{\\mathssf{U}},{\\mathssf{V}}\\_1)\\\\\n &&- I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2|{\\mathssf{U}}) - I({\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2;{\\mathssf{W}}|{\\mathssf{U}}), \\label{eq:rateregionC3R1plusR2}\\end{aligned}$$ \nwhere the following Markov chain is satisfied: ${\\mathssf{U}}\\rightarrow ({\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2) \\rightarrow ({\\mathssf{X}},{\\mathssf{W}}) \\rightarrow ({\\mathssf{Y}}\\_1,{\\mathssf{Y}}\\_2)$.\n[thm:achievethmC3] Let R3, in = ⋃*p*3(.) ∈ P3R3, in(*p*3). Then, R3, in ⊆ C3.\nSection [subsec:achieveproofC3] contains the proof of Theorem [thm:achievethmC3].\nFor a given *p*3(.) ∈ P3, an outer bound for C3 is described by the set R3, out(*p*3), which is defined as the union of all rate pairs (*R*1, *R*2) that simultaneously satisfy $(\\ref{eq:minoutboundC3R1})$ - $(\\ref{eq:minoutboundC3R2})$.\n \n$$\\begin{aligned}\nR\\_1 &\\leq& \\min[I\\_1, I^{\\ast}\\_1], \\label{eq:minoutboundC3R1}\\\\\nR\\_2 &\\leq& \\min[I\\_2, I^{\\ast}\\_2], \\label{eq:minoutboundC3R2},\\end{aligned}$$ \nwhere *I*1, …, *I*2\\* are given by $(\\ref{eq:outboundC3R1})$ - $(\\ref{eq:outboundC3R2genie})$, respectively.\n \n$$\\begin{aligned}\nI\\_1 &\\triangleq& I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1|{\\mathssf{U}}) - I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_2|{\\mathssf{U}}) + H({\\mathssf{W}}|{\\mathssf{U}},V\\_1), \\label{eq:outboundC3R1}\\\\\nI\\_2 &\\triangleq& I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2|{\\mathssf{U}}) - I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_1|{\\mathssf{U}}) + H({\\mathssf{W}}|{\\mathssf{U}},V\\_2), \\label{eq:outboundC3R2}\\\\\nI^{\\ast}\\_1 &\\triangleq& I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1|{\\mathssf{U}},{\\mathssf{V}}\\_2) - I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_2|{\\mathssf{U}},{\\mathssf{V}}\\_2) + H({\\mathssf{W}}|{\\mathssf{U}},{\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2), \\label{eq:outboundC3R1genie}\\\\\nI^{\\ast}\\_2 &\\triangleq& I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2|{\\mathssf{U}},{\\mathssf{V}}\\_1) - I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_1|{\\mathssf{U}},{\\mathssf{V}}\\_1) + H({\\mathssf{W}}|{\\mathssf{U}},{\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2), \\label{eq:outboundC3R2genie}\\end{aligned}$$ \nwhere ${\\mathssf{U}}\\rightarrow ({\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2) \\rightarrow ({\\mathssf{X}},{\\mathssf{W}}) \\rightarrow ({\\mathssf{Y}}\\_1,{\\mathssf{Y}}\\_2)$. The expressions $(\\ref{eq:outboundC3R1genie})$ - $(\\ref{eq:outboundC3R2genie})$ are obtained by letting a genie give D1 message ${\\mathssf{M}}\\_2$, while D2 computes the equivocation using ${\\mathssf{M}}\\_2$ as side-information.\n[thm:conversethmC31] Let R3, out = ⋃*p*3(.) ∈ P3R3, out(*p*3). Then, C3 ⊆ R3, out.\nThe proof of Theorem [thm:conversethmC31] can be found in Section [subsec:converseproofC31]. We also provide the following outer bound for the channel C3, which explicitly characterizes the sum-rates. Consider the set P3\\* of all joint probability distributions *p*3\\*(.) that can be factorized as follows: $p(w)p(u,v\\_1,v\\_2|w)p(x|w,u,v\\_1,v\\_2)\\\\p(y\\_1,y\\_2|x)$. For a given *p*3\\*(.) ∈ P3\\*, an outer bound for C3 is described by the set R3, out\\*(*p*3\\*), which is defined as the union of all rate pairs (*R*1, *R*2) that simultaneously satisfy -.\n \n$$\\begin{aligned}\nR\\_1 &\\leq& I({\\mathssf{U}},{\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1) - I({\\mathssf{V}}\\_1;{\\mathssf{W}}|{\\mathssf{U}}) - I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_2),\\label{eq:oboundnewC3R1} \\\\\nR\\_2 &\\leq& I({\\mathssf{U}},{\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2) - I({\\mathssf{V}}\\_2;{\\mathssf{W}}|{\\mathssf{U}}) - I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_1),\\label{eq:oboundnewC3R2} \\\\\n\\nonumber R\\_1 + R\\_2 &\\leq& I({\\mathssf{U}},{\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1) - I({\\mathssf{V}}\\_1;{\\mathssf{W}}|{\\mathssf{U}}) + I({\\mathssf{U}},{\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2|{\\mathssf{V}}\\_1) \\\\ && - I({\\mathssf{V}}\\_2;{\\mathssf{W}}|{\\mathssf{U}},{\\mathssf{V}}\\_1) - I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_2),\\label{eq:oboundnewC3R1plusR2\\_1}\\\\\n\\nonumber R\\_1 + R\\_2 &\\leq& I({\\mathssf{U}},{\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2) - I({\\mathssf{V}}\\_2;{\\mathssf{W}}|{\\mathssf{U}}) + I({\\mathssf{U}},{\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1|{\\mathssf{V}}\\_2) \\\\ && - I({\\mathssf{V}}\\_1;{\\mathssf{W}}|{\\mathssf{U}},{\\mathssf{V}}\\_2) - I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_1),\\label{eq:oboundnewC3R1plusR2\\_2}\\end{aligned}$$ \nwhere $({\\mathssf{U}},{\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2) \\rightarrow ({\\mathssf{X}},{\\mathssf{W}}) \\rightarrow ({\\mathssf{Y}}\\_1,{\\mathssf{Y}}\\_2)$.\n[thm:conversethmC32] Let R3, out\\* = ⋃*p*3\\*(.) ∈ P3\\*R3, out\\*(*p*3\\*). Then, C3 ⊆ R3, out\\*.\nThe proof of Theorem [thm:conversethmC32] can be found in Section [subsec:converseproofC32].\nDiscussion\n----------\nA pictorial representation of the rate region for the channel C1 is shown in Fig. [fig:C1bc1]. When *R*2 = 0, the channel resembles a single-user channel (S, D1) with side-information (the Gel’fand-Pinsker’s (GP) channel ) and S can transmit at the maximum achievable *R*1 given by $(\\ref{eq:rateregionC1R1})$, denoted by point the H. At the point H, the maximum achievable *R*2 is given by the point $\\mathrm{E}\\_1 \\equiv I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2) - I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2) - I({\\mathssf{W}};{\\mathssf{V}}\\_2)$; this is obtained by treating the channel (S, D2) as a single-user channel with side-information. Therefore, the rectangle OHGE1 is achievable. By exchanging *R*1 and *R*2 and following similar arguments the points E, given by $(\\ref{eq:rateregionC1R2})$, and $\\mathrm{F}\\_1 \\equiv I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1) - I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2|{\\mathssf{U}}) - I({\\mathssf{W}};{\\mathssf{V}}\\_1)$ are achievable. Hence, the rectangle OEFF1 is also achievable. Since the points F and G are shown to be achievable, any point which lies on the line FG can also be achieved by deriving a bound on the binning rates (see $(\\ref{eq:appenderrorencodeC1R1'})$ - $(\\ref{eq:appenderrorencodeC1R1plusR2'})$, Appendix [appendix:errorencodeC1]). This leads to a sum rate bound given by $(\\ref{eq:rateregionC1R1plusR2})$. Finally, owing to convexity of the rate region, any point in the interior of the line FG is also achievable. Therefore, an achievable rate region for C1 is described by the pentagon OEFGH.\nIn the absence of side-information, *i.e.*, W = {*ϕ*}, the channel reduces to the classical two-user BC whose rate region is described by the convex-hull of the set of all rate pairs (*R*1, *R*2) that satisfy the following inequalities:\n \n$$\\begin{aligned}\nR\\_1 &\\leq& I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1),\\label{eq:rateregionBCR1}\\\\\nR\\_2 &\\leq& I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2),\\label{eq:rateregionBCR2}\\\\\nR\\_1 + R\\_2 &\\leq& I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1) + I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2) - I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2). \\label{eq:rateregionBCR1plusR2}\\end{aligned}$$ \nFor channels of Class~II, each bound in $\\eqref{eq:rateregionC2R1}$ - $\\eqref{eq:rateregionC2R2}$ is the capacity of GP’s single-user channel with noncausal side-information. In the absence of side-information, *i.e.*, W = {*ϕ*}, we get *R**t* ≤ *I*(*U*; *Y**t*) = *I*(*X*; *Y**t*), which represents the capacity region of BC when each receiver is given the message it need not decode. Furthermore, the outer bounds $\\eqref{eq:outboundC2R1}$ - $\\eqref{eq:outboundC2R2}$ is within a fixed gap, $H({\\mathssf{U}})$, from the achievable region, where $H({\\mathssf{U}})$ is independent of the distribution characterizing this class of channels.\nFor Class~III channels, the terms $I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_2|{\\mathssf{U}},{\\mathssf{V}}\\_2)$ and $I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_1|{\\mathssf{U}},{\\mathssf{V}}\\_1)$ quantify the rate-penalty for having to deal with confidentiality constraints on the messages, while the terms $I({\\mathssf{V}}\\_1;{\\mathssf{W}}|{\\mathssf{U}})$ and $I({\\mathssf{V}}\\_2;{\\mathssf{W}}|{\\mathssf{U}})$ quantify the rate-penalty for having to deal with side-information.\nUsing a combination of results from GP’s channel and wiretap channels with side-information, we obtain a pictorial representation of the rate region for the channel C3 as shown in Fig. [fig:C1C3bc1]. The arguments used to obtain this schematic are similar to those used for the channel C1; therefore, we briefly explain the construction of Fig. [fig:C1C3bc1]. The point A1 corresponds to the maximum achievable *R*1 (when *R*2 = 0) and is given by $(\\ref{eq:rateregionC3R1})$. Exchanging *R*1 and *R*2 we get the point C1 given by $(\\ref{eq:rateregionC3R2})$. The points $\\mathrm{B}\\_1 \\equiv I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2|{\\mathssf{U}}) - I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_1|{\\mathssf{U}},{\\mathssf{V}}\\_1) - \\max [I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2|{\\mathssf{U}}),I({\\mathssf{W}};{\\mathssf{V}}\\_2|{\\mathssf{U}})]$ and $\\mathrm{D}\\_1 \\equiv I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1|{\\mathssf{U}}) - I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_2|{\\mathssf{U}},{\\mathssf{V}}\\_2) - \\max [I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2|{\\mathssf{U}}),I({\\mathssf{W}};{\\mathssf{V}}\\_1|{\\mathssf{U}})]$ are achievable by treating channels (S, D2) and (S, D1), respectively, as wiretap channels with side-information. The line E1F1 corresponds to the sum rate bound given by $(\\ref{eq:rateregionC3R1plusR2})$. Finally, owing to convexity of the rate region, any point in the interior of the line E1F1 is also achievable. Therefore, an achievable rate region for C3 is described by the pentagon OA1F1E1C1.\nIf the confidentiality constraints $(\\ref{eq:security1})$ - $(\\ref{eq:security2})$ are relaxed, the channel C3 reduces to the channel C1, whose rate region is described by $(\\ref{eq:rateregionC1R1})$ - $(\\ref{eq:rateregionC1R1plusR2})$. Further, in the absence of side-information, *i.e.*, W = {*ϕ*}, the channel reduces to the classical two-user BC whose rate region is described by $\\eqref{eq:rateregionBCR1}$ - $\\eqref{eq:rateregionBCR1plusR2}$. Lastly, if the encoder satisfies confidentiality constraints in the absence of side-information, the channel C3 reduces to BC with two independent and confidential messages whose rate region was first characterized by Liu *et. al*. It is described by the convex-hull of the set of all rate pairs (*R*1, *R*2) that satisfy the following inequalities:\n \n$$\\begin{aligned}\nR\\_1 \\leq I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_1|{\\mathssf{U}}) - I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_2|{\\mathssf{U}}) - I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2|{\\mathssf{U}}),\\label{eq:rateregionBCconfR1}\\\\\nR\\_2 \\leq I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_2|{\\mathssf{U}}) - I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_1|{\\mathssf{U}}) - I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2|{\\mathssf{U}}).\\label{eq:rateregionBCconfR2}\\end{aligned}$$ \nRelation to past work\n---------------------\nFor Class~I channels, an inner bound was presented in by extending Marton’s achievability scheme for the classical two-user BC to include noncausal side-information at the encoder. In this paper, we employ Marton’s technique and use results from the second moment method to derive the inner bound which matches with the results presented in. However, our method is simpler and generalizes well for obtaining inner bounds with other channel models, *e.g.*, for channels of Class~III considered in this paper. For the outer bound (specifically, for the sum-rate), we generalize the technique presented in, to handle side-information at the encoder. When the side-information constraint is relaxed, our result reduces to the one presented for the classical two-user BC.\nClass~II channels were also addressed in, where an inner bound was derived by employing Marton’s achievability scheme. An outer bound was also suggested in, but without a formal proof. In this paper, we derive an inner bound by generalizing the method suggested in by incorporating noncausal side-information at the encoder. Our inner bound coincides with the one presented in, but once again the proof technique is much simpler. Furthermore, for the outer bounds, we explicitly address the problem of dealing with the two-dimensional rate region with a single auxiliary random variable.\nFor Class~III channels, we show that when the confidentiality constraints are relaxed, our achievable rate region reduces to region presented for the Class~I channels, and hence to the one presented in. On the other hand, in the absence of side-information, our achievable region includes an explicit bound on the sum-rate for the two-user BC with confidentiality constraints (a model considered in ). This further strengthens the generalization of our proof technique.\nProofs of achievability theorems\n================================\nIn this section, we prove Theorem [thm:achievethmC1], Theorem [thm:achievethmC2] and Theorem [thm:achievethmC3]. For any *ε* > 0, we denote by $A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{X}}})$ an *ε*-typical set comprising sequences picked from the distribution $p({\\mathbsf{x}})$. For all the channel models, the encoder is given an *ε* − typical sequence ${\\mathbsf{W}} \\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{W}}})$ in a noncausal manner.\nProof of Theorem [thm:achievethmC1]\n-----------------------------------\nFor the channel C1, generate 2N[*R**t* + *R**t*ʹ] independent typical sequences ${\\mathbsf{V}}\\_t(i\\_t,j\\_t) \\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{V}}\\_t}); t = 1,2$. Here, *i**t* ∈ {1, …, 2N*R**t*}; *j**t* ∈ {1, …, 2N*R**t*ʹ}. Uniformly distribute 2N[*R**t* + *R**t*ʹ] sequences into 2N*R**t* bins, so that each bin, indexed by *i**t*, comprises 2N*R**t*ʹ sequences. To send the message pair $({\\mathssf{m}}\\_1 = i\\_1, {\\mathssf{m}}\\_2 = i\\_2)$, the encoder at S looks for a pair (*j*1, *j*2) that satisfies the following joint typicality condition: $E\\_{\\mathrm{S}} \\triangleq \\{({\\mathbsf{W}},{\\mathbsf{V}}\\_1(i\\_1,j\\_1),{\\mathbsf{V}}\\_2(i\\_2,j\\_2))\\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{W}},{\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2})\\}$. An error is declared at the encoder of S, if it is not possible to find the (*j*1, *j*2) − pair to satisfy the condition *E*S. The encoder error analysis can be found in Appendix [appendix:errorencodeC1]. The channel input sequence is ${\\mathbsf{X}} \\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{X|W,V\\_1,V\\_2}}})$.\nAt the destination D*t*, the decoder looks for (*î**t*, *ĵ**t*) that satisfies the following joint typicality condition: $E\\_{\\mathrm{D}\\_t} \\triangleq \\{({\\mathbsf{V}}\\_t(\\hat{i}\\_t,\\hat{j}\\_t),{\\mathbsf{Y}}\\_t)\\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{V}}\\_t,{\\mathssf{Y}}\\_t})\\}$. An error is declared at decoder of D*t*, if it not possible to find a unique integer *î**t* to satisfy the condition *E*D*t*. From the union of events bound, the probability of decoder error at D*t* can be upper bounded as follows: *P**e*, D*t*(N) ≤ Pr(*E*D*t**c*∣*E*S) + ∑*î**t* ≠ *i**t*∑*j**t*Pr(*E*D*t*∣*E*S). From the asymptotic equipartition property (AEP), ∀*ε* > 0 and sufficiently small; and for large N, Pr(*E*D*t**c*∣*E*S) ≤ *ε*. Further, for *î**t* ≠ *i**t*, Pr(*E*D*t*∣*E*S) ≤ 2− N[*I*(*V**t*; *Y**t*) − *ε*]. Therefore, we have *P**e*, D*t*(N) ≤ *ε* + 2N[*R**t* + *R**t*ʹ]2− N[*I*(*V**t*; *Y**t*) − *ε*], leading us to conclude that, for any *ε*0 > 0 and sufficiently small; and for large N, *P**e*, D*t*(N) ≤ *ε*0 if\n \n$$\\begin{aligned}\nR\\_t+R^{'}\\_t < I({\\mathssf{V}}\\_t;{\\mathssf{Y}}\\_t). \\label{eq:achieveproofC1rates}\\end{aligned}$$ \nFor the channel C1, the rate inequalities $(\\ref{eq:achieveproofC1rates})$ and the bounds on the binning rates $(\\ref{eq:appenderrorencodeC1R1'})$ - $(\\ref{eq:appenderrorencodeC1R1plusR2'})$ (see Appendix [appendix:errorencodeC1]) are combined to obtain an achievable rate region given by $(\\ref{eq:rateregionC1R1})$ - $(\\ref{eq:rateregionC1R1plusR2})$. This completes the proof of Theorem [thm:achievethmC1].\nProof of Theorem [thm:achievethmC2]\n-----------------------------------\nFor the channel C2, we consider the following two cases.\n1. When *R*1 ≤ *R*2: Generate 2N(*R*2 + *R*\\*) typical sequences ${\\mathbsf{U}}(i,j) \\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{U}}}); i \\in \\{1,\\dots,2^{\\mathrm{N}R\\_2}\\}; \\\\ j \\in \\{1,\\dots,2^{\\mathrm{N}R^{\\ast}}\\}$. Uniformly distribute these sequences into 2N*R*2 bins, so that each bin comprises 2N*R*\\* sequences. The bins are indexed by *i*. Define now the following mappings:\n \n$$\\begin{aligned}\n {\\mathssf{m}}\\_t \\in \\{1,\\dots,2^{\\mathrm{N}R\\_t}\\} &\\mapsto& \\mathrm{Int}({\\mathssf{m}}\\_t) \\in \\{0,\\dots,2^{\\mathrm{N}R\\_2}-1\\}; t=1,2,\n \\end{aligned}$$ \nwhere Int(*α*) denotes an integer to represent *α*. To transmit the message pair $({\\mathssf{m}}\\_1,{\\mathssf{m}}\\_2)$, compute $\\left(\\mathrm{Int}({\\mathssf{m}}\\_1)+\\mathrm{Int}({\\mathssf{m}}\\_2) \\mod{2^{\\mathrm{N}R\\_2}}\\right)$. By construction, the bin index \n$i \\triangleq \\mathrm{Int}^{-1}\\left(\\mathrm{Int}({\\mathssf{m}}\\_1)+\\mathrm{Int}({\\mathssf{m}}\\_2) \\mod{2^{\\mathrm{N}R\\_2}}\\right)$. Given the sequence ${\\mathbsf{W}}$, the encoder looks for an integer *j* to satisfy the following joint typicality condition:\n \n$$\\begin{aligned}\n ({\\mathbsf{U}}(i,j), {\\mathbsf{W}})\\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{W}},{\\mathssf{U}}}).\n \\end{aligned}$$ \nFinally, ${\\mathbsf{X}} \\triangleq {\\mathbsf{f}}({\\mathbsf{U}}(i,j),{\\mathbsf{W}})$ is transmitted in N channel uses.\nAt receiver D1, given ${\\mathssf{m}}\\_2$, the decoder looks for the pair $(\\hat{i}\\triangleq \\hat{{\\mathssf{m}}}\\_1,\\hat{j})$ such that the following joint typicality condition is satisfied:\n \n$$\\begin{aligned}\n E\\_{\\mathrm{D}\\_1}\\triangleq \\{({\\mathbsf{U}}(\\mathrm{Int}^{-1}\\left(\\mathrm{Int}(\\hat{{\\mathssf{m}}}\\_1)+\\mathrm{Int}({\\mathssf{m}}\\_2) \\mod{2^{\\mathrm{N}R\\_2}}\\right),j), {\\mathbsf{Y}}\\_1)\\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{U}},{\\mathssf{Y}}\\_1})\\}.\n \\end{aligned}$$ \nFrom AEP, it can be shown that Pr(*E*D1*c*) ≤ *δ*1; ∀*δ*1 > 0 and sufficiently small; and for large N, if $R\\_1 + R^{\\ast} \\leq I({\\mathssf{U}};{\\mathssf{Y}}\\_1)$. Similarly, it can be shown that Pr(*E*D2*c*) ≤ *δ*2; ∀*δ*2 > 0 and sufficiently small; and for large N, if $R\\_2 + R^{\\ast} \\leq I({\\mathssf{U}};{\\mathssf{Y}}\\_2)$. Additionally, by following a procedure similar to the one presented in Appendix [appendix:errorencodeC1], we bound the binning rate as follows: $R^{\\ast} > I({\\mathssf{U}};{\\mathssf{W}})$. Therefore, ${\\mathssf{m}}\\_1$ (resp. ${\\mathssf{m}}\\_2$) can be reliably decoded at D1 (resp. D2) if\n \n$$\\begin{aligned}\n R\\_1 \\leq I({\\mathssf{U}};{\\mathssf{Y}}\\_1) - I({\\mathssf{U}};{\\mathssf{W}}), \\label{eq:achieveproofC2R1}\\\\\n R\\_2 \\leq I({\\mathssf{U}};{\\mathssf{Y}}\\_2) - I({\\mathssf{U}};{\\mathssf{W}}). \\label{eq:achieveproofC2R2}\n \\end{aligned}$$\n2. When *R*2 ≤ *R*1: By symmetry, we get the same rate bounds as in $\\eqref{eq:achieveproofC2R1}$ and $\\eqref{eq:achieveproofC2R2}$.\nThis completes the proof of Theorem [thm:achievethmC2].\nProof of Theorem [thm:achievethmC3]\n-----------------------------------\nFor the channel C3, generate a typical sequence ${\\mathbsf{U}} \\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{U}}})$, known to all nodes in the network. Generate 2N[*R**t* + *R**t*ʹ + *R**t*\\*] independent typical sequences ${\\mathbsf{V}}\\_t(i\\_t,j\\_t,k\\_t) \\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{V}}\\_t})$; *i**t* ∈ {1, …, 2N*R**t*}; *j**t* ∈ {1, …, 2N*R**t*ʹ}; *k**t* ∈ {1, …, 2N*R**t*\\*}. Uniformly distribute 2N[*R**t* + *R**t*ʹ + *R**t*\\*] sequences into 2N*R**t* bins, so that each bin, indexed by *i**t*, comprises 2N[*R**t*ʹ + *R**t*\\*] sequences. Uniformly distribute 2N[*R**t*ʹ + *R**t*\\*] sequences into 2N*R**t*ʹ sub-bins indexed by (*i**t*, *j**t*), so that each sub-bin comprises 2N*R**t*\\* sequences.\nTo send the message pair $({\\mathssf{m}}\\_1,{\\mathssf{m}}\\_2)$, S employs a stochastic encoder. In the bin indexed by *i**t*, *randomly* pick a sub-bin indexed (*i**t*, *j**t*). The encoder then looks for a pair (*k*1, *k*2) that satisfies the following joint typicality condition: $({\\mathbsf{W}},{\\mathbsf{V}}\\_1(i\\_1,j\\_1,k\\_1),{\\mathbsf{V}}\\_2(i\\_2,j\\_2,k\\_2))\\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{W}},{\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2|{\\mathssf{U}}})$. The channel input sequence ${\\mathbsf{X}} \\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{X}}|{\\mathssf{W}},{\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2})$ is transmitted in N uses of the channel.\nAt the destination D*t*, given ${\\mathbsf{U}}$, the decoder picks *k**t* that satisfies the following joint typicality condition: $E\\_{\\mathrm{D}\\_t} \\triangleq \\{({\\mathbsf{V}}\\_t(i\\_t,j\\_t,k\\_t),{\\mathbsf{Y}}\\_t)\\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{V}}\\_t,{\\mathssf{Y}}\\_t|{\\mathssf{U}}})\\}$. An error is declared at the decoder of D*t* if it not possible to find an integer *î**t* satisfying *E*D*t*. From union of events bound, the probability of decoder error at D*t* can be upper bounded as follows: *P**e*, D*t*(N) ≤ Pr(*E*D*t**c*∣*E*S) + ∑*î**t* ≠ *i**t*∑*j**t*, *k**t*Pr(*E*D*t*∣*E*S). From AEP, ∀*ε* > 0 and sufficiently small; and for large N, Pr(*E*D*t**c*∣*E*S) ≤ *ε* and for *î**t* ≠ *i**t*, we have $\\text{Pr}(E\\_{\\mathrm{D}\\_t}|E\\_{\\mathrm{S}}) \\leq 2^{-\\mathrm{N}[I({\\mathssf{V}}\\_t;{\\mathssf{Y}}\\_t|U)-\\epsilon]}$. Therefore, $P^{(\\mathrm{N})}\\_{e,\\mathrm{D}\\_t} \\leq \\epsilon + 2^{\\mathrm{N}[R\\_t+R^{'}\\_t+R\\_t^{\\ast}]}2^{-\\mathrm{N}[I({\\mathssf{V}}\\_t;{\\mathssf{Y}}\\_t|{\\mathssf{U}})-\\epsilon]}$. For any *ε*0 > 0 and sufficiently small; and for large N, *P**e*, D*t*(N) ≤ *ε*0 if\n \n$$\\begin{aligned}\nR\\_t+R^{'}\\_t+R\\_t^{\\ast} < I({\\mathssf{V}}\\_t;{\\mathssf{Y}}\\_t|{\\mathssf{U}}). \\label{eq:achieveproofC3rates}\\end{aligned}$$ \nThe equivocation at the decoder of D2 is calculated by first considering the following lower bound: $H(M\\_1|{\\mathbsf{Y}}^{\\mathrm{N}}\\_2) \\geq H(M\\_1|{\\mathbsf{Y}}^{\\mathrm{N}}\\_2,{\\mathbsf{U}}^{\\mathrm{N}},{\\mathbsf{V}}^{\\mathrm{N}}\\_2)$. Following the procedure in and using the fact that $M\\_1 \\rightarrow ({\\mathbsf{U}}^{\\mathrm{N}},{\\mathbsf{V}}^{\\mathrm{N}}\\_1,{\\mathbsf{V}}^{\\mathrm{N}}\\_2) \\rightarrow {\\mathbsf{Y}}^{\\mathrm{N}}\\_2$ forms a Markov chain, we get\n \n$$\\begin{aligned}\nH(M\\_1|{\\mathbsf{Y}}^{\\mathrm{N}}\\_2) \\geq H({\\mathbsf{V}}^{\\mathrm{N}}\\_1|{\\mathbsf{U}}^{\\mathrm{N}}) - I({\\mathbsf{V}}^{\\mathrm{N}}\\_1;{\\mathbsf{V}}^{\\mathrm{N}}\\_2|{\\mathbsf{U}}^{\\mathrm{N}})\n- H({\\mathbsf{V}}^{\\mathrm{N}}\\_1|M\\_1,{\\mathbsf{U}}^{\\mathrm{N}},{\\mathbsf{V}}^{\\mathrm{N}}\\_2,{\\mathbsf{Y}}^{\\mathrm{N}}\\_2) -\nI({\\mathbsf{V}}^{\\mathrm{N}}\\_1;{\\mathbsf{Y}}^{\\mathrm{N}}\\_2|{\\mathbsf{U}}^{\\mathrm{N}},{\\mathbsf{V}}^{\\mathrm{N}}\\_2). \\label{eq:achieveproofC3equiv2}\\end{aligned}$$ \nNow, ∀*ε**l* > 0; *l* = 4, …, 10 and sufficiently small; and for large N, the terms in $(\\ref{eq:achieveproofC3equiv2})$ become\n \n$$\\begin{aligned}\n\\nonumber H({\\mathbsf{V}}^{\\mathrm{N}}\\_1|{\\mathbsf{U}}^{\\mathrm{N}}) \\stackrel{(a)}{=} \\mathrm{N}[R\\_1+R'\\_1+R^{\\ast}\\_1];\nI({\\mathbsf{V}}^{\\mathrm{N}}\\_1;{\\mathbsf{V}}^{\\mathrm{N}}\\_2|{\\mathbsf{U}}^{\\mathrm{N}})\\stackrel{(b)}{=}\\mathrm{N}\nI({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2|{\\mathssf{U}})\n+\\mathrm{N}\\epsilon\\_4; \\\\\nH({\\mathbsf{V}}^{\\mathrm{N}}\\_1|M\\_1,{\\mathbsf{U}}^{\\mathrm{N}},{\\mathbsf{V}}^{\\mathrm{N}}\\_2,{\\mathbsf{Y}}^{\\mathrm{N}}\\_2) \\stackrel{(c)}{\\leq} \\mathrm{N}\\epsilon\\_5;\nI({\\mathbsf{V}}^{\\mathrm{N}}\\_1;{\\mathbsf{Y}}^{\\mathrm{N}}\\_2|{\\mathbsf{U}}^{\\mathrm{N}},{\\mathbsf{V}}^{\\mathrm{N}}\\_2) \\stackrel{(d)}{=}\\mathrm{N}I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_2|{\\mathssf{U}},{\\mathssf{V}}\\_2)+\\mathrm{N}\\epsilon\\_6.\n\\label{eq:achieveproofC3equiv3}\\end{aligned}$$ \nIn, (*a*) follows from the codebook construction; (*b*) and (*d*) follow from standard techniques (for *e.g.*, see ); and (*c*) is proved in. A similar procedure is followed to calculate the equivocation at the decoder at D1. Finally, the security constraints $(\\ref{eq:security1})$ and $(\\ref{eq:security2})$ are satisfied by letting\n \n$$\\begin{aligned}\nR'\\_1 &=& I({\\mathssf{V}}\\_1;{\\mathssf{Y}}\\_2|{\\mathssf{U}},{\\mathssf{V}}\\_2) - \\epsilon\\_7;R\\_1^{\\ast} = I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2|{\\mathssf{U}}) - \\epsilon\\_8;\\label{eq:achieveproofC3R1\\*}\\\\\nR'\\_2 &=& I({\\mathssf{V}}\\_2;{\\mathssf{Y}}\\_1|{\\mathssf{W}},{\\mathssf{U}},{\\mathssf{V}}\\_1) - \\epsilon\\_9;R\\_2^{\\ast} = I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2|{\\mathssf{W}},{\\mathssf{U}}) - \\epsilon\\_{10}. \\label{eq:achieveproofC3R2\\*}\\end{aligned}$$ \nFor the channel C3, rate inequalities $(\\ref{eq:achieveproofC3rates})$, constraints $(\\ref{eq:achieveproofC3R1\\*})$ - $(\\ref{eq:achieveproofC3R2\\*})$ and bounds on the binning rates $\\eqref{eq:appenderrorencodeC3R1\\*}$ - $\\eqref{eq:appenderrorencodeC3R1plusR2\\*}$ (see Appendix [appendix:errorencodeC1]) are combined to obtain the rate region described by $(\\ref{eq:rateregionC3R1})$ - $(\\ref{eq:rateregionC3R1plusR2})$. This completes the proof of Theorem [thm:achievethmC3].\nProofs of converse theorems\n===========================\nIn this section, we prove Theorem [thm:conversethmC1], Theorem [thm:conversethmC12], Theorem [thm:conversethmC2], Theorem [thm:conversethmC31] and Theorem [thm:conversethmC32].\nProof of Theorem [thm:conversethmC1]\n------------------------------------\nFor the channel C1, ∀*ε* > 0 and sufficiently small; and for large N, *R*1 can be bounded as follows:\n \n$$\\begin{aligned}\n\\nonumber \\mathrm{N}R\\_1 &=& H({\\mathssf{M}}\\_1) = I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N})+H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_1^\\mathrm{N})\\\\\n\\nonumber &\\stackrel{(a)}\\leq& I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N}) + \\mathrm{N}\\epsilon\n\\stackrel{(b)}= \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[H({\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1) - H({\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathssf{M}}\\_1)] + \\mathrm{N}\\epsilon \\\\\n\\nonumber &\\stackrel{(c)}\\leq& \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[H({\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) - H({\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathssf{M}}\\_1)] + \\mathrm{N}\\epsilon\n= \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1;{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) + \\mathrm{N}\\epsilon \\\\\n\\nonumber &=& \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) - I({\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1)] + \\mathrm{N}\\epsilon \\\\\n\\nonumber &\\stackrel{(d)}=& \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) - I({\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1;W\\_{\\mathrm{n}}|{\\mathssf{M}}\\_1,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1})] + \\mathrm{N}\\epsilon \\\\\n\\nonumber &\\stackrel{(e)}=& \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) - I({\\mathssf{M}}\\_1,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1},{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1;W\\_{\\mathrm{n}})] + \\mathrm{N}\\epsilon,\\end{aligned}$$ \nwhere (*a*) follows from Fano’s inequality, (*b*) follows from the chain rule, (*c*) follows from the fact that conditioning reduces entropy, (*d*) follows from Csiszár’s sum identity and (*e*) is due to the fact that $({\\mathssf{M}}\\_1,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1})$ is independent of ${\\mathssf{W}}\\_{\\mathrm{n}}$. We let ${\\mathssf{V}}\\_{1,{\\mathrm{n}}}=({\\mathssf{M}}\\_1,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1},{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1)$ and note that this choice satisfies the Markov chain requirement ${\\mathssf{V}}\\_1 \\rightarrow ({\\mathssf{X}},{\\mathssf{W}}) \\rightarrow ({\\mathssf{Y}}\\_1,{\\mathssf{Y}}\\_2)$, specified in Section [sec:statementofresults] for the channel C1. Thus, we get\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_1 \\leq \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathssf{V}}\\_{1,{\\mathrm{n}}};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) - I({\\mathssf{V}}\\_{1,{\\mathrm{n}}};{\\mathssf{W}}\\_{\\mathrm{n}}) + \\mathrm{N}\\epsilon. \\label{eq:converseproofC1R1}\\end{aligned}$$ \nProceeding in a similar manner and letting ${\\mathssf{V}}\\_{2,{\\mathrm{n}}}=({\\mathssf{M}}\\_2,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1},{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_2)$, we get\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_2 \\leq \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathssf{V}}\\_{2,{\\mathrm{n}}};{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}) - I({\\mathssf{V}}\\_{2,{\\mathrm{n}}};{\\mathssf{W}}\\_{\\mathrm{n}}) + \\mathrm{N}\\epsilon. \\label{eq:converseproofC1R2}\\end{aligned}$$ \nProof of Theorem [thm:conversethmC12]\n-------------------------------------\nFor the channel C1, ∀*ε* > 0 and sufficiently small; and for large N, *R*1 can be bounded as\n \n$$\\begin{aligned}\n\\nonumber \\mathrm{N}R\\_1 &=& H({\\mathssf{M}}\\_1) = I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N})+H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_1^\\mathrm{N})\\\\\n\\nonumber &\\stackrel{(a)}\\leq& I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N}) + \\mathrm{N}\\epsilon,\\end{aligned}$$ \nwhere (*a*) follows from Fano’s inequality. Proceeding in a manner similar to the proof of Theorem [thm:conversethmC1] (see Section [subsec:converseproofC1]), and letting ${\\mathssf{U}}\\_{\\mathrm{n}}=({\\mathbsf{W}}^{\\mathrm{N}}\\_{\\mathrm{n}+1},{\\mathbsf{Y}}^{\\mathrm{n}-1}\\_1,\n{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,\\mathrm{n}+1})$ and ${\\mathssf{V}}\\_{1,\\mathrm{n}} = {\\mathssf{M}}\\_1$.\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_1 \\leq \\sum^{\\mathrm{N}}\\_{\\mathrm{n}=1}I({\\mathssf{U}}\\_{\\mathrm{n}}, {\\mathssf{V}}\\_{1,\\mathrm{n}};{\\mathssf{Y}}\\_{1,\\mathrm{n}}) - I({\\mathssf{V}}\\_{1,\\mathrm{n}};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{U}}\\_{\\mathrm{n}}) + \\mathrm{N}\\epsilon. \\label{eq:outboundnewC1R1}\\end{aligned}$$ \nSimilarly, letting ${\\mathssf{V}}\\_{2,\\mathrm{n}} = {\\mathssf{M}}\\_2$, *R*2 can be upper bounded as follows:\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_2 \\leq \\sum^{\\mathrm{N}}\\_{\\mathrm{n}=1}I({\\mathssf{U}}\\_{\\mathrm{n}}, {\\mathssf{V}}\\_{2,\\mathrm{n}};{\\mathssf{Y}}\\_{2,\\mathrm{n}}) - I({\\mathssf{V}}\\_{2,\\mathrm{n}};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{U}}\\_{\\mathrm{n}}) + \\mathrm{N}\\epsilon. \\label{eq:outboundnewC1R2}\\end{aligned}$$ \nWe next upper bound *R*1 + *R*2 as follows. ∀*ε* > 0 and sufficiently small; and for large N, we have\n \n$$\\begin{aligned}\n\\nonumber \\mathrm{N}(R\\_1+R\\_2) &=& H({\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2) = H({\\mathssf{M}}\\_1) + H({\\mathssf{M}}\\_2|{\\mathssf{M}}\\_1)\\\\\n\\nonumber &=& I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^{\\mathrm{N}})+H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_1^{\\mathrm{N}}) +I({\\mathssf{M}}\\_2;{\\mathbsf{Y}}\\_2^{\\mathrm{N}}|{\\mathssf{M}}\\_1)+H({\\mathssf{M}}\\_2|{\\mathbsf{Y}}\\_2^{\\mathrm{N}},{\\mathssf{M}}\\_1)\\\\\n\\nonumber &\\stackrel{(a)}\\leq& \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1)+ \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_2;{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{N}}}\\_{2,{\\mathrm{n}}+1},{\\mathssf{M}}\\_1)+ 2{\\mathrm{N}}\\epsilon,\\end{aligned}$$ \nwhere (*a*) follows from Fano’s inequality. Consider\n \n$$\\begin{aligned}\n\\nonumber \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1) &\\leq& \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1;{\\mathbsf{Y}}\\_{1,{\\mathrm{n}}})\\\\\n\\nonumber &=& \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{Y}}^{{\\mathrm{N}}}\\_{2,{\\mathrm{n}}+1};{\\mathbsf{Y}}\\_{1,{\\mathrm{n}}})\n- \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}I({\\mathbsf{Y}}^{{\\mathrm{N}}}\\_{2,{\\mathrm{n}}+1};{\\mathbsf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1)\\\\\n\\nonumber &=& \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{Y}}^{{\\mathrm{N}}}\\_{2,{\\mathrm{n}}+1},{\\mathbsf{W}}^{{\\mathrm{N}}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) - I({\\mathbsf{W}}^{{\\mathrm{N}}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{Y}}^{{\\mathrm{N}}}\\_{2,{\\mathrm{n}}+1})] \\\\\n\\nonumber &-& \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}I({\\mathbsf{Y}}^{{\\mathrm{N}}}\\_{2,{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1)\\\\\n\\nonumber &\\stackrel{(b)}=& \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{Y}}^{{\\mathrm{N}}}\\_{2,{\\mathrm{n}}+1},{\\mathbsf{W}}^{{\\mathrm{N}}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) - I({\\mathssf{M}}\\_1;{\\mathssf{W}}\\_{{\\mathrm{n}}}|{\\mathbsf{W}}^{{\\mathrm{N}}}\\_{{\\mathrm{n}}+1},{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{Y}}^{{\\mathrm{N}}}\\_{2,{\\mathrm{n}}+1})] \\\\\n&-& \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}I({\\mathbsf{Y}}^{{\\mathrm{N}}}\\_{2,{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1)\\label{eq:sumrateout1}\\end{aligned}$$ \nNext consider\n \n$$\\begin{aligned}\n\\nonumber \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}I(M\\_2;{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{N}}}\\_{2,{\\mathrm{n}}+1},{\\mathssf{M}}\\_1) \\leq \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_2,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1;{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{N}}}\\_{2,{\\mathrm{n}}+1},{\\mathssf{M}}\\_1)\\\\ \\nonumber\n= \\sum^{{\\mathrm{N}}}\\_{{\\mathrm{n}}=1}I({\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1;{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,{\\mathrm{n}}+1},{\\mathssf{M}}\\_1) +\n\\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_2;{\\mathbsf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,\n{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,{\\mathrm{n}}+1},{\\mathssf{M}}\\_1)\\\\ \\nonumber\n= \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1;{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,{\\mathrm{n}}+1},{\\mathssf{M}}\\_1) +\n\\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_2,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,\n{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,{\\mathrm{n}}+1},{\\mathssf{M}}\\_1)\\\\ \\nonumber - \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,\n{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,{\\mathrm{n}}+1},{\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2)\\\\ \\nonumber = \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1;{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,{\\mathrm{n}}+1},{\\mathssf{M}}\\_1) +\n\\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_2,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,{\\mathrm{n}}+1},\n{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{2,n}|{\\mathssf{M}}\\_1)\\\\ \\nonumber - \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_2;{\\mathssf{W}}\\_{{\\mathrm{n}}}|{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1},\n{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,{\\mathrm{n}}+1},{\\mathssf{M}}\\_1)\\\\\n\\nonumber \\stackrel{(c)}= \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1;{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,{\\mathrm{n}}+1},{\\mathssf{M}}\\_1) +\n\\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_2,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,{\\mathrm{n}}+1},\n{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathssf{M}}\\_1)\\\\ - \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_2;{\\mathssf{W}}\\_{{\\mathrm{n}}}|{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1},\n{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,{\\mathrm{n}}+1},{\\mathssf{M}}\\_1) \\label{eq:sumrateout2}\\end{aligned}$$ \nwhere (*b*) and (*c*) follow from Csiszár’s sum identity. With ${\\mathssf{U}}\\_{{\\mathrm{n}}}=({\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1},{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,{\\mathrm{n}}+1})$; ${\\mathssf{V}}\\_{1,{\\mathrm{n}}} = {\\mathssf{M}}\\_1$; and ${\\mathssf{V}}\\_{2,{\\mathrm{n}}} = {\\mathssf{M}}\\_2$, from $(\\ref{eq:sumrateout1})$ and $(\\ref{eq:sumrateout2})$, we get\n \n$$\\begin{aligned}\n\\nonumber \\mathrm{N}(R\\_1+R\\_2) \\leq \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{U}}\\_{\\mathrm{n}}, {\\mathssf{V}}\\_{1,{\\mathrm{n}}};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) - I({\\mathssf{V}}\\_{1,{\\mathrm{n}}};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{U}}\\_{\\mathrm{n}})]\\\\ + \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{U}}\\_{\\mathrm{n}}, {\\mathssf{V}}\\_{2,{\\mathrm{n}}};{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathssf{V}}\\_{1,{\\mathrm{n}}}) - I({\\mathssf{V}}\\_{2,{\\mathrm{n}}};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{V}}\\_{1,{\\mathrm{n}}},{\\mathssf{U}}\\_{\\mathrm{n}})] + 2\\mathrm{N}\\epsilon. \\label{eq:outboundnewC1R1plusR21}\\end{aligned}$$ \nSimilarly, it can be shown that\n \n$$\\begin{aligned}\n\\nonumber \\mathrm{N}(R\\_1+R\\_2) \\leq \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{U}}\\_{\\mathrm{n}}, {\\mathssf{V}}\\_{2,{\\mathrm{n}}};{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}) - I({\\mathssf{V}}\\_{2,{\\mathrm{n}}};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{U}}\\_{\\mathrm{n}})]\\\\ + \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{U}}\\_{\\mathrm{n}}, {\\mathssf{V}}\\_{1,{\\mathrm{n}}};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathssf{V}}\\_{2,{\\mathrm{n}}}) - I({\\mathssf{V}}\\_{1,{\\mathrm{n}}};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{V}}\\_{2,{\\mathrm{n}}},{\\mathssf{U}}\\_{\\mathrm{n}})] + 2\\mathrm{N}\\epsilon. \\label{eq:outboundnewC1R1plusR22}\\end{aligned}$$ \nProof of Theorem [thm:conversethmC2]\n------------------------------------\nFor the channel C2, ∀*ε* > 0 and sufficiently small; and for large N, *R*1 can be bounded as follows:\n \n$$\\begin{aligned}\n\\nonumber \\mathrm{N}R\\_1 &=& H({\\mathssf{M}}\\_1) = I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N})+H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_1^\\mathrm{N})\\\\\n\\nonumber &\\stackrel{(a)}\\leq& I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N}) + \\mathrm{N}\\epsilon\n\\stackrel{(b)}\\leq I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N},{\\mathssf{M}}\\_2) + \\mathrm{N}\\epsilon\n= I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N}|{\\mathssf{M}}\\_2) + \\mathrm{N}\\epsilon \\\\\n\\nonumber &\\stackrel{(c)}=& \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[H({\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathssf{M}}\\_2) - H({\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2)] + \\mathrm{N}\\epsilon \\\\\n\\nonumber &\\stackrel{(d)}\\leq& \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[H({\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) - H({\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2)] + \\mathrm{N}\\epsilon \\\\\n\\nonumber &=& \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}I({\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1;{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) + \\mathrm{N}\\epsilon \\\\\n\\nonumber &=& \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) -I({\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1)] + \\mathrm{N}\\epsilon \\\\\n\\nonumber &\\stackrel{(e)}=& \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) -I({\\mathssf{M}}\\_1;{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{M}}\\_2,{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1, {\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1})] + \\mathrm{N}\\epsilon \\\\\n\\nonumber &\\stackrel{(f)}=& \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) -I({\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathbsf{Y}}^{{\\mathrm{n}}-1}\\_1)] + \\mathrm{N}\\epsilon \\\\\n&\\stackrel{(g)}\\leq& \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) -I({\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1};{\\mathssf{W}}\\_{\\mathrm{n}}) + H({\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2,{\\mathbsf{W}}^{\\mathrm{N}}\\_{{\\mathrm{n}}+1})] + \\mathrm{N}\\epsilon.\\end{aligned}$$ \nwhere (*a*) follows from Fano’s inequality; (*b*) follows from the data-processing inequality; (*c*) follows from chain rule; (*d*) follows from the fact that conditioning reduces entropy; (*e*) follows from Csiszár’s sum identity; (*f*) is due to the memoryless nature of the channel; and (*g*) is obtained after simple calculations. We let ${\\mathssf{U}}\\_{n}\\triangleq ({\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2,{\\mathbsf{W}}^{\\mathrm{N}}\\_{n+1})$ and note that this choice satisfies the Markov chain requirement ${\\mathssf{U}} \\rightarrow ({\\mathssf{X}},{\\mathssf{W}}) \\rightarrow ({\\mathssf{Y}}\\_1,{\\mathssf{Y}}\\_2)$ specified in Section [sec:statementofresults] for the channel C2 to get\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_1 \\leq \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{U}}\\_{n};{\\mathssf{Y}}\\_{1,n}) - I({\\mathssf{U}}\\_{n};{\\mathssf{W}}\\_n) + H({\\mathssf{U}}\\_{n})] + \\mathrm{N}\\epsilon. \\label{eq:converseproofC2R1}\\end{aligned}$$ \nBy symmetry, we get the following bound on *R*2:\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_2 \\leq \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{U}}\\_{n};{\\mathssf{Y}}\\_{2,n}) - I({\\mathssf{U}}\\_{n};{\\mathssf{W}}\\_n) + H({\\mathssf{U}}\\_{n})] + \\mathrm{N}\\epsilon. \\label{eq:converseproofC2R2}\\end{aligned}$$ \nWe note that the factor $H({\\mathssf{U}}\\_{n})$ is independent of the distribution characterizing the channel C2.\nProof of Theorem [thm:conversethmC31]\n-------------------------------------\nFor the channel C3, ∀*ε* > 0 and sufficiently small; and for large N, *R*1 can be bounded as follows:\n \n$$\\begin{aligned}\n\\nonumber \\mathrm{N}R\\_1 &=& H({\\mathssf{M}}\\_1) = I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N})+H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_1^\\mathrm{N})\\\\\n\\nonumber &\\stackrel{(a)}\\leq& I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N}) + \\mathrm{N}\\epsilon\n\\stackrel{(b)}\\leq I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N}) - I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_2^\\mathrm{N}) + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &=& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1}) - I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{n-1}\\_2)] + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &\\stackrel{(c)}=& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{n-1}\\_2;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1}) - I({\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1};{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{n-1}\\_2)] + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &\\stackrel{(d)}=& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2) - I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2)] + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &\\leq& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1,{\\mathssf{W}}\\_n;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2) - I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2)] + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &\\stackrel{(e)}=& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2)+ I({\\mathssf{W}}\\_n;{\\mathssf{Y}}\\_{1,n}|{\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2)\\\\\n\\nonumber &&-I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2)] + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &=& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2) + H({\\mathssf{W}}\\_n|{\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2)\\\\\n\\nonumber &&- H({\\mathssf{W}}\\_n|{\\mathssf{M}}\\_1,{\\mathssf{Y}}\\_{1,n},{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2) - I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2)] + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &\\leq& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2) + H({\\mathssf{W}}\\_n|{\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2)\\\\\n\\nonumber &&-I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2)] + 2\\mathrm{N}\\epsilon,\\end{aligned}$$ \nwhere (*a*) is from Fano’s inequality, (*b*) is from confidentiality constraints, (*c*) and (*d*) follow from Csiszár’s sum identity and (*e*) is the chain rule for mutual information. Letting ${\\mathssf{U}}\\_n \\triangleq ({\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2)$; and ${\\mathssf{V}}\\_{1,1} = \\dots = {\\mathssf{V}}\\_{1,\\mathrm{N}} \\triangleq {\\mathssf{M}}\\_1$, where ${\\mathssf{U}}$ and ${\\mathssf{V}}\\_{1}$ satisfy the Markov chain ${\\mathssf{U}}\\rightarrow {\\mathssf{V}}\\_1 \\rightarrow {\\mathssf{X}}$ specified in Section [sec:statementofresults] for the channel C3, we get\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_1 &\\leq& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{V}}\\_{1,n};{\\mathssf{Y}}\\_{1,n}|{\\mathssf{U}}\\_n)+H({\\mathssf{W}}\\_n|{\\mathssf{U}}\\_n,{\\mathssf{V}}\\_{1,n}) - I({\\mathssf{V}}\\_{1,n};{\\mathssf{Y}}\\_{2,n}|{\\mathssf{U}}\\_n)] + 2\\mathrm{N}\\epsilon. \\label{eq:converseproofC3R1}\\end{aligned}$$ \nProceeding in a similar fashion and letting ${\\mathssf{V}}\\_{2,1} = \\dots = {\\mathssf{V}}\\_{2,\\mathrm{N}} \\triangleq {\\mathssf{M}}\\_2$,\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_2 &\\leq& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{V}}\\_{2,n};{\\mathssf{Y}}\\_{2,n}|{\\mathssf{U}}\\_n)+H({\\mathssf{W}}\\_n|{\\mathssf{U}}\\_n,{\\mathssf{V}}\\_{2,n}) - I({\\mathssf{V}}\\_{2,n};{\\mathssf{Y}}\\_{1,n}|U\\_n)] + 2\\mathrm{N}\\epsilon. \\label{eq:converseproofC3R2}\\end{aligned}$$ \nFor the channel C3, we also derive a genie-aided outer bound by letting a hypothetical genie give D1 message ${\\mathssf{M}}\\_2$, while D2 computes the equivocation using ${\\mathssf{M}}\\_2$ as side-information. ∀*ε* > 0 and sufficiently small; and for large N, *R*1 can be upper bounded as follows:\n \n$$\\begin{aligned}\n\\nonumber \\mathrm{N}R\\_1 &=& H({\\mathssf{M}}\\_1) \\leq H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_2^\\mathrm{N}) + \\mathrm{N}\\epsilon\n\\leq H({\\mathssf{M}}\\_1,{\\mathssf{M}}\\_2|{\\mathbsf{Y}}\\_2^\\mathrm{N}) + \\mathrm{N}\\epsilon\\\\\n\\nonumber &=& H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_2^\\mathrm{N},{\\mathssf{M}}\\_2) + H({\\mathssf{M}}\\_2|{\\mathbsf{Y}}\\_2^\\mathrm{N}) + \\mathrm{N}\\epsilon\n\\leq H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_2^\\mathrm{N},{\\mathssf{M}}\\_2) + \\mathrm{N}\\epsilon\\\\\n\\nonumber &\\leq& H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_2^\\mathrm{N},{\\mathssf{M}}\\_2) - H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_1^\\mathrm{N}) + \\mathrm{N}\\epsilon\n\\stackrel{(a)}\\leq H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_2^\\mathrm{N},{\\mathssf{M}}\\_2) - H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_1^\\mathrm{N},{\\mathssf{M}}\\_2) + \\mathrm{N}\\epsilon\\\\\n\\nonumber &\\leq& I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N}|{\\mathssf{M}}\\_2) - I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_2^\\mathrm{N}|{\\mathssf{M}}\\_2) + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &=& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathssf{M}}\\_2) - I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2)] + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &\\stackrel{(b)}=& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{n-1}\\_2;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathssf{M}}\\_2) - I({\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1};{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2)] + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &\\stackrel{(c)}=& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2) - I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2)] + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &\\leq& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1,{\\mathssf{W}}\\_n;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2) - I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2)] + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &=& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2) + I({\\mathssf{W}}\\_n;{\\mathssf{Y}}\\_{1,n}|{\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2)\\\\\n\\nonumber &&-I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2)] + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &=& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2) + H({\\mathssf{W}}\\_n|{\\mathssf{M}}\\_1,Y^{\\mathrm{N}}\\_{n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2)\\\\\n\\nonumber &&- H({\\mathssf{W}}\\_n|{\\mathssf{M}}\\_1,{\\mathssf{Y}}\\_{1,n},{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2) - I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2)] + 2\\mathrm{N}\\epsilon \\\\\n\\nonumber &\\leq& \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{1,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2) + H({\\mathssf{W}}\\_n|{\\mathssf{M}}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2)\\\\\n\\nonumber &&-I({\\mathssf{M}}\\_1;{\\mathssf{Y}}\\_{2,n}|{\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2,{\\mathssf{M}}\\_2)] + 2\\mathrm{N}\\epsilon,\\end{aligned}$$ \nwhere (*a*) follows since the genie gives D1 message ${\\mathssf{M}}\\_2$, (*b*) and (*c*) follow from Csiszár’s sum identity. Letting ${\\mathssf{U}}\\_n \\triangleq ({\\mathbsf{Y}}^{\\mathrm{N}}\\_{1,n+1},{\\mathbsf{Y}}^{n-1}\\_2)$, ${\\mathssf{V}}\\_{1,1} = \\dots = {\\mathssf{V}}\\_{1,\\mathrm{N}} \\triangleq {\\mathssf{M}}\\_1$ and ${\\mathssf{V}}\\_{2,1} = \\dots = {\\mathssf{V}}\\_{2,\\mathrm{N}} \\triangleq {\\mathssf{M}}\\_2$, where ${\\mathssf{U}}$, ${\\mathssf{V}}\\_{1}$ and ${\\mathssf{V}}\\_{2}$ satisfy the Markov chains ${\\mathssf{U}}\\rightarrow {\\mathssf{V}}\\_1 \\rightarrow {\\mathssf{X}}$ and ${\\mathssf{U}}\\rightarrow {\\mathssf{V}}\\_2 \\rightarrow {\\mathssf{X}}$ specified in Section [sec:statementofresults] for the channel C3, *R*1 can be bounded as\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_1 \\leq \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{V}}\\_{1,n};{\\mathssf{Y}}\\_{1,n}|{\\mathssf{U}}\\_n,{\\mathssf{V}}\\_{2,n}) + H({\\mathssf{W}}\\_n|{\\mathssf{U}}\\_n,{\\mathssf{V}}\\_{1,n},{\\mathssf{V}}\\_{2,n})\n- I({\\mathssf{V}}\\_{1,n};{\\mathssf{Y}}\\_{2,n}|{\\mathssf{U}}\\_n,{\\mathssf{V}}\\_{2,n})] + 2\\mathrm{N}\\epsilon. \\label{eq:converseproofC3R1genie}\\end{aligned}$$ \nSimilarly,\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_1 \\leq \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{V}}\\_{2,n};{\\mathssf{Y}}\\_{2,n}|{\\mathssf{U}}\\_n,{\\mathssf{V}}\\_{1,n}) + H({\\mathssf{W}}\\_n|{\\mathssf{U}}\\_n,{\\mathssf{V}}\\_{1,n},{\\mathssf{V}}\\_{2,n})\n- I({\\mathssf{V}}\\_{2,n};{\\mathssf{Y}}\\_{1,n}|{\\mathssf{U}}\\_n,{\\mathssf{V}}\\_{1,n})] + 2\\mathrm{N}\\epsilon. \\label{eq:converseproofC3R2genie}\\end{aligned}$$ \nFor the channel C3, the outer bound on *R*1 + *R*2 can be made tighter by the following procedure. From $(\\ref{eq:minoutboundC3R1})$ - $(\\ref{eq:minoutboundC3R2})$, we see that\n \n$$\\begin{aligned}\nR\\_1+R\\_2 \\leq I\\_1 + I\\_2, \\label{eq:sumratebound1}\\\\\nR\\_1+R\\_2 \\leq I^{\\ast}\\_1 + I^{\\ast}\\_2. \\label{eq:sumratebound2}\\end{aligned}$$ \nTherefore,\n \n$$\\begin{aligned}\nR\\_1+R\\_2 \\leq \\min [I\\_1 + I^{\\ast}\\_2, I\\_2 + I^{\\ast}\\_1]. \\label{eq:sumratebound}\\end{aligned}$$ \nWe show now that the bound $(\\ref{eq:sumratebound})$ is a tighter bound than $(\\ref{eq:sumratebound1})$ and $(\\ref{eq:sumratebound2})$. It is easy to see that\n \n$$\\begin{aligned}\n\\nonumber I\\_1 + I\\_2 = I^{\\ast}\\_1 + I^{\\ast}\\_2 + I(W;V\\_1|U,V\\_2) + I(W;V\\_2|U,V\\_1).\\end{aligned}$$ \nConsider 2(*I*1 + *I*2) = 2[*I*1\\* + *I*2\\* + *I*(*W*; *V*1∣*U*, *V*2) + *I*(*W*; *V*2∣*U*, *V*1)], which implies the following:\n \n$$\\begin{aligned}\n\\nonumber \\min[I\\_1 + I^{\\ast}\\_2, I\\_2 + I^{\\ast}\\_1] &\\leq& I\\_1 + I\\_2,\\\\\n\\nonumber \\min[I\\_1 + I^{\\ast}\\_2, I\\_2 + I^{\\ast}\\_1] &\\leq& I^{\\ast}\\_1 + I^{\\ast}\\_2.\\end{aligned}$$ \nTherefore, the sum rate bound given by $(\\ref{eq:sumratebound})$ is tighter than $(\\ref{eq:sumratebound1})$ and $(\\ref{eq:sumratebound2})$.\nProof of Theorem [thm:conversethmC32]\n-------------------------------------\nFor the channel C3, ∀*ε* > 0 and sufficiently small; and for large N, *R*1 can be bounded as follows:\n \n$$\\begin{aligned}\n\\nonumber \\mathrm{N}R\\_1 &=& H({\\mathssf{M}}\\_1) = I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N})+H({\\mathssf{M}}\\_1|{\\mathbsf{Y}}\\_1^\\mathrm{N})\\\\\n\\nonumber &\\stackrel{(a)}\\leq& I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N}) + \\mathrm{N}\\epsilon\n\\stackrel{(b)}\\leq I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_1^\\mathrm{N}) - I({\\mathssf{M}}\\_1;{\\mathbsf{Y}}\\_2^\\mathrm{N}) + 2\\mathrm{N}\\epsilon,\\end{aligned}$$ \nwhere (*a*) follows from Fano’s inequality; and (*b*) follows from confidentiality constraints. Following the procedure used to prove Theorem [thm:conversethmC12] (see Section [subsec:converseproofC12]) and letting ${\\mathssf{U}}\\_{\\mathrm{n}}=({\\mathbsf{W}}^{\\mathrm{N}}\\_{\\mathrm{n}+1},{\\mathbsf{Y}}^{\\mathrm{n}-1}\\_1,{\\mathbsf{Y}}^{\\mathrm{N}}\\_{2,\\mathrm{n}+1})$ and ${\\mathssf{V}}\\_{1,\\mathrm{n}} = {\\mathssf{M}}\\_1$,\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_1 \\leq \\sum^{\\mathrm{N}}\\_{\\mathrm{n}=1}I({\\mathssf{U}}\\_{\\mathrm{n}}, {\\mathssf{V}}\\_{1,\\mathrm{n}};{\\mathssf{Y}}\\_{1,\\mathrm{n}}) - I({\\mathssf{V}}\\_{1,\\mathrm{n}};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{U}}\\_{\\mathrm{n}}) -\nI({\\mathssf{V}}\\_{1,\\mathrm{n}};{\\mathssf{Y}}\\_{2,\\mathrm{n}}) + 2\\mathrm{N}\\epsilon.\\end{aligned}$$ \nSimilarly, letting ${\\mathssf{V}}\\_{2,\\mathrm{n}} = {\\mathssf{M}}\\_2$, we get\n \n$$\\begin{aligned}\n\\mathrm{N}R\\_2 \\leq \\sum^{\\mathrm{N}}\\_{\\mathrm{n}=1}I({\\mathssf{U}}\\_{\\mathrm{n}}, {\\mathssf{V}}\\_{2,\\mathrm{n}};{\\mathssf{Y}}\\_{2,\\mathrm{n}}) - I({\\mathssf{V}}\\_{2,\\mathrm{n}};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{U}}\\_{\\mathrm{n}}) - I({\\mathssf{V}}\\_{2,\\mathrm{n}};{\\mathssf{Y}}\\_{1,\\mathrm{n}}) + 2\\mathrm{N}\\epsilon,\\end{aligned}$$ \nand the following bounds on the sum-rate *R*1 + *R*2:\n \n$$\\begin{aligned}\n\\nonumber \\mathrm{N}(R\\_1+R\\_2) \\leq \\sum^{\\mathrm{N}}\\_{n=1}[I({\\mathssf{U}}\\_{\\mathrm{n}}, {\\mathssf{V}}\\_{1,{\\mathrm{n}}};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) - I({\\mathssf{V}}\\_{1,{\\mathrm{n}}};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{U}}\\_{\\mathrm{n}})]\\\\ + \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{U}}\\_{\\mathrm{n}}, {\\mathssf{V}}\\_{2,{\\mathrm{n}}};{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}|{\\mathssf{V}}\\_{1,{\\mathrm{n}}}) - I({\\mathssf{V}}\\_{2,{\\mathrm{n}}};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{V}}\\_{1,{\\mathrm{n}}},{\\mathssf{U}}\\_{\\mathrm{n}})] - I({\\mathssf{V}}\\_{1,{\\mathrm{n}}};{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}) + 2\\mathrm{N}\\epsilon,\\\\\n\\nonumber \\mathrm{N}(R\\_1+R\\_2) \\leq \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{U}}\\_{\\mathrm{n}}, {\\mathssf{V}}\\_{2,{\\mathrm{n}}};{\\mathssf{Y}}\\_{2,{\\mathrm{n}}}) - I({\\mathssf{V}}\\_{2,{\\mathrm{n}}};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{U}}\\_{\\mathrm{n}})]\\\\ + \\sum^{\\mathrm{N}}\\_{{\\mathrm{n}}=1}[I({\\mathssf{U}}\\_{\\mathrm{n}}, {\\mathssf{V}}\\_{1,{\\mathrm{n}}};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}|{\\mathssf{V}}\\_{2,{\\mathrm{n}}}) - I({\\mathssf{V}}\\_{1,{\\mathrm{n}}};{\\mathssf{W}}\\_{\\mathrm{n}}|{\\mathssf{V}}\\_{2,{\\mathrm{n}}},{\\mathssf{U}}\\_{\\mathrm{n}})] - I({\\mathssf{V}}\\_{2,{\\mathrm{n}}};{\\mathssf{Y}}\\_{1,{\\mathrm{n}}}) + 2\\mathrm{N}\\epsilon.\\end{aligned}$$ \nA time sharing RV ${\\mathssf{Q}}$, which is uniformly distributed over N symbols and independent of the RVs ${\\mathssf{M}}\\_1$, ${\\mathssf{M}}\\_2$, ${\\mathssf{W}}$, ${\\mathssf{U}}$, ${\\mathssf{V}}\\_1$, ${\\mathssf{V}}\\_2$, ${\\mathssf{X}}$, ${\\mathssf{Y}}\\_1$ and ${\\mathssf{Y}}\\_2$ is introduced for the single letter characterization of the above derived outer bounds. Applying the procedure similar to the one presented in on the N-letter expressions obtained in the above stated theorems, we get the outer bounds presented in Section [sec:statementofresults]. This completes the proofs of Theorem [thm:conversethmC1], Theorem [thm:conversethmC12], Theorem [thm:conversethmC2], Theorem [thm:conversethmC31] and Theorem [thm:conversethmC32].\nConclusions\n===========\nWe presented inner and outer bounds on the capacity region of three classes of two-user discrete memoryless broadcast channels, with noncausal side-information at the encoder. We generalized existing approaches to prove the achievability theorems, and characterized the rate penalties for having to deal with side-information at the encoder. For channels with confidentiality constraints, we showed that rate penalties exist for dealing with both side-information and confidentiality constraints. In the case of outer bounds, we focus on the explicit characterization of the sum-rate bounds. For channels where each receiver has *a priori* knowledge of the message of the other receiver, we showed that the outer bounds are only a factor away from the achievable region, where the factor is independent of the channel distribution.\nEncoder error analysis\n======================\nHere, we upper bound the probability of encoder error for the channel C1, by using results from the second moment method. This method was also employed in and to provide an alternative proof of Marton’s achievability scheme. An error is declared at the encoder of S if it is not possible to find a pair (*i*1, *i*2) to satisfy the condition $E\\_{\\mathrm{S}} \\triangleq \\{({\\mathbsf{W}},{\\mathbsf{V}}\\_1(i\\_1,j\\_1),{\\mathbsf{V}}\\_2(i\\_2,j\\_2))\\in A^{(\\mathrm{N})}\\_{\\epsilon}(P\\_{{\\mathssf{W}},{\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2})\\}$. Let *P**e*, *E*S denote the probability of error at the encoder, *i.e.*, $P\\_{e,E\\_{\\mathrm{S}}} \\triangleq \\text{Pr}(E\\_{\\mathrm{S}}^c)$. Let ${\\mathssf{I}}$ be an indicator RV that the event *E*S has occurred. Let ${\\mathssf{Q}} = \\sum\\_{j\\_1,j\\_2} {\\mathssf{I}}$; $\\bar{{\\mathssf{Q}}} = \\mathbb{E}[{\\mathssf{Q}}]$; and $\\text{Var}[{\\mathssf{Q}}] = \\mathbb{E}[({\\mathssf{Q}}-\\bar{{\\mathssf{Q}}})^2]$, where E(.) denotes the expectation operator. *P**e*, *E*S can be upper bounded as follows:\n \n$$\\begin{aligned}\nP\\_{e,E\\_{\\mathrm{S}}} = \\text{Pr}({\\mathssf{Q}}=0) \\stackrel{(i)}{\\leq} \\text{Var}[{\\mathssf{Q}}]/\\bar{{\\mathssf{Q}}}^2, \\label{eq:appenderrorencodeC1PeS}\\end{aligned}$$ \nwhere (*i*) follows from Markov’s inequality for non-negative RVs. Consider now\n \n$$\\begin{aligned}\n\\bar{{\\mathssf{Q}}} &=& \\sum\\_{j\\_1,j\\_2}\\mathbb{E}({\\mathssf{I}}) \\geq \\sum\\_{j\\_1,j\\_2}(1-\\delta^{(\\mathrm{N})})2^{-\\mathrm{N}[I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2)+I({\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2;{\\mathssf{W}})\n+4\\epsilon]}\\\\\n&=& (1-\\delta^{(\\mathrm{N})})2^{-\\mathrm{N}[R^{\\ast}\\_1+R^{\\ast}\\_2 - I({\\mathssf{V}}\\_1;{\\mathssf{V}}\\_2) - I({\\mathssf{V}}\\_1,{\\mathssf{V}}\\_2;{\\mathssf{W}}) - 4\\epsilon]}.\\end{aligned}$$ \nNext, consider $\\text{Var}[{\\mathssf{Q}}] = \\sum\\_{j\\_1,j\\_2}\\sum\\_{j'\\_1,j'\\_2}\\{\\mathbb{E}[{\\mathssf{I}}(j\\_1,j\\_2){\\mathssf{I}}(j'\\_1,j'\\_2)] - \\mathbb{E}[{\\mathssf{I}}(j\\_1,j\\_2)]\\mathbb{E}{\\mathssf{I}}(j'\\_1,j'\\_2)]\\}$. We have the following four cases:\n1. If *j*ʹ1 ≠ *j*1 and *j*ʹ2 ≠ *j*2, then ${\\mathssf{I}}(j\\_1,j\\_2)$ and ${\\mathssf{I}}(j'\\_1,j'\\_2)$ are independent and $\\text{Var}[{\\mathssf{Q}}]=0$.\n2. If *j*ʹ1 = *j*1 and *j*ʹ2 = *j*2, then"},"data_id":{"kind":"string","value":"arxiv_0183472"}}},{"rowIdx":45893,"cells":{"content":{"kind":"string","value":"Hospital management in the COVID-19 emergency: Abelian Sandpile paradigmand beyond\n==================================================================================\nIn this article, we propose a mathematical model –based on a cellular automaton– for the redistribution of patients within a network of hospitals with limited available resources, in order to reduce the risks of a local/global collapse of the healthcare system. We attempt at developing a conceptual tool to support making rational decisions relevant to the optimisation of the allocation of patients into accessible medical facilities. The strategy is based on a version of the Abelian Sandpile model for the Self-Organised Criticality, with the idea of testing the paradigm for the management of patients among the COVID-19 hospitals in Italian regions. In particular, we compare the novel proposal to the standard management of connections between hospitals, showing a number of advantages at a local and global level, by means of a reliable indicator function introduced for measuring the effectiveness of the allocation strategies.\nIntroduction\n============\nSince the beginning of 2020, people from all parts of the world are struggling with the COVID-19 coronavirus pandemic, which has led to a hundred million of infections and over 2 million deaths worldwide . The health emergency is not over yet and it will take months before being under control, possibly becoming endemic among most of the world population . \nThe current data suggest the paramount importance of acting promptly, and especially for trying to reorganize the medical facilities in order to avoid saturation of the critical care capacity in the various territories (refer to Figure [figure:TerapieIntensive] for the evolution of employment of the intensive care units in Italy). Indeed, several countries have been experiencing recurring epidemic waves, with a consequent worsening of the social and economic conditions, thus increasing dramatically the pressure on the healthcare systems.\n![Occupancy level (aggregate data) of intensive care units in the Italian healthcare system.](TerapieIntensive \"fig:\") [figure:TerapieIntensive]\nThe emergence of infectious diseases has to be considered as an inherent property of human and animal populations, which cannot be generally avoided and/or foreseen: before the present COVID-19 emergency, SARS (2002-2004) and influenza A/H1N1 (2009-2010) are other recent notable incidents . \nMost epidemics are characterised by outbreaks in localized geographical areas, and there may be no pharmaceutical solutions already available to safeguard the people’s health when the disease begins to spread largely around. Moreover, the development of effective vaccines typically takes a rather long time, and therefore temporary alternative strategies must be implemented (quarantine, travel restrictions, activities closing, social distancing, face masks obligation,...) However, the demand for specific healthcare facilities such as hospitals consistently increases, reaching and sometimes exceeding critical levels. For example, in the case of COVID-19, the saturation of intensive care units depends on the amount of patients requiring mechanical support for ventilation and proper equipment, so that any healthcare system can be overwhelmed if the number of severe cases becomes very high (refer to Figure [figure:saturazione] for the present-day situation of intensive care units in Italian regions).\n![Saturation of intensive care units in the Italian regions (updated to January 23, 2021). Source: Il Sole 24 Ore – elaboration on data from the Italian Civil Protection Department](saturazione \"fig:\") [figure:saturazione]\nIn that context, the rapid filling of the so-called *COVID-19 hospitals* in the areas most affected by the disease represents a serious danger while dealing with sudden emergencies at different scales (villages, cities, regions, countries). Hence, it is compelling to develop new arrangements of the healthcare system, for optimizing the overall structure especially in terms of an efficient distribution of patients among the most suitable facilities.\nIn this article, we present innovative intervention options, by comparing the standard organisation of the Italian healthcare system with a novel proposal based on the *Abelian Sandpile paradigm*. We provide a systematic approach for an improved planning of the healthcare system by means of a mathematical model for the *self-organisation of critical issues*, which is capable to achieve some specific optimization inside the hospital management. In particular, we aim at reducing the risks of a local/global collapse of hospitals in times of crisis, while improving their functioning in normal times. More specifically, the basic model is grounded in the framework of *cellular automata* and postulates a large network of links between hospitals in a cooperative style of communication. Unfortunately, such connections are currently very limited, which drastically restricts the possibilities of reallocation for the supernumerary patients, and the strong urgency to enhance the hospital network is an important conclusion of the present analysis.\nThe content of this article is organized as follows. \nIn Section [section:abelian], we introduce the Abelian Sandpile paradigm, with a short description of its basic concepts and properties. Section [section:healthcare] focuses on the adaptation of such paradigm to build up a novel proposal for the organisation of the healthcare system. In Section [section:examples], we provide a gallery of examples aiming at illustrating how the proposed model works in the present context of COVID-19 epidemic. More specifically, we analyze two cases relevant to realistic applications: the central and peripheral outbreaks. The article is concluded by Section [section:beyond] which includes some generalisations of the model and (provisional) conclusions.\nAbbreviations\n=============\n– Abelian Sandpile Model \nSOC – Self-Organized Criticality \nCAM – Cellular Automaton Model \nSRH – Sandpile with Redistribution to the Hub \nSID – Sandpiles with Internal Dissipation\nAbelian Sandpile paradigm\n=========================\nThe notion of Self-Organized Criticality (SOC) was originally introduced by Bak, Tang and Wiesenfeld starting from a basic example proposed as a model for sandpiles (we refer to for a general introduction to this broad subject). Since then, the concept has been expanded in many different directions, spanning from classical topics of physics (sandpile avalanches , distribution of earthquakes , amplitude of solar flares ) to less standard economic and socio-political contexts , passing through computer networks and biological applications . At the same time, a huge effort to extend the mathematical tools to deal with theoretical questions has been made, thus contributing to drive SOC into an extraordinary crossroads of probabilistic approaches, graph theory, algebraic geometry, mathematical analysis and optimisation .\nSOC and Sandpiles\n-----------------\nThe original application concerns with modeling of sandpiles, which are regarded as a manageable prototype of SOC, and fundamental contributions have been made by Dhar notably in showing the crucial property of *commutativity*. On this account, the adjective Abelian has since been added to the technical terminology, leading to the actual meaning of Abelian Sandpile Model (ASM). Nevertheless, as usual in the most active fields of science, the terminology is not unique, and indeed similar topics are explored under different names (avalanches, chip-firing games, forest-fire models, parking functions, probabilistic abacus, Rotor-Router or Eulerian-walker models,...) with more or less the same meaning. We refer to for introductory presentations of the ASM and its various applications.\nWe are particularly interested to the *load balancing* property, which denotes the method of distributing a set of tasks across a group of resources with the purpose of making their overall processing more efficient by balancing the workload of each operating unit. In the abstract formulation, units are represented by vertexes/nodes of a graph/network, with the corresponding connections represented by edges/links. The objective is to balance the loading process by allowing nodes to exchange *particles* with their neighbors through the incident edges. \nFor the problem covered in this article, we build a specific ASM with *critical height* provided by the capacity of the medical facilities, and we propose to apply the Abelian Sandpile paradigm to achieve a methodical and efficient distribution of patients across the hospitals, in order to maintain the occupancy below the saturation level for dealing with eventual sudden and unexpected emergencies.\nA sandpile is also a type of Cellular Automaton Model (CAM), namely a discrete mathematical system fulfilling the following essential conditions:\n* the evolution takes place without external interventions;\n* the overall structural development depends only on local rules.\nTo a large extent, the CAM is capable of simulating the dynamics of complex systems which are disposed to organize themselves through unstable critical states until reaching a stable configuration. In addition, the CAM implementation makes it possible to generate global coherent patterns starting from suitable local instructions, without any external supervisor in charge for understanding the process in its entirety . \nIn practical applications, each restricted portion of space contains a finite number of cells, which assume a finite set of (time-dependent) states. The initial configuration $\\bar{\\mathbf{z}}^0$ is the combination of a *ground state* configuration **z**0 with some perturbation **w**, and for instance\n \n$$\\bar{\\mathbf{z}}^0 = \\mathbf{z}^0 + \\mathbf{w}\\,.$$ \nAfter a given time interval Δ*t* > 0, the system typically comes to a new state **z**1, designated *final configuration*, which is determined by the changes of state of the single cell together with all the others, according to the preliminarily established series of (local) rules. Therefore, the evolution is described by means of a function Φ mapping some input $\\bar{\\mathbf{z}}^0$ defined (from **z**0 and **w**) over the graph/network Γ to an output **z**1, as specified by the simple formula $\\mathbf{z}^1=\\Phi(\\bar{\\mathbf{z}}^0)$.\nIt is worth stressing that, in order to properly manage the patients between hospitals of the healthcare system, an optimal underlying ethical-cooperative paradigm has to be stipulated .\nSandpiles on a general network\n------------------------------\nWe start by recollecting some basic definitions in *graph theory* .\nA **graph** (or **network**) is a pair Γ = (*X*, *E*) where *X* is a set whose elements are called **vertexes** (or **nodes**) and *E* is a set of paired vertexes called **edges** (or **links**). A **rooted graph** (or **pointed graph**) is a graph in which one vertex, denoted by *x*\\*, is distinguished as the **root** of the graph.\nIn what follows, the root *x*\\* of the graph is also referred to as the **hub**, coherently with the subsequent meaning of the specific application to the healthcare system.\nAn **adjacent vertex** *y* ∈ *X* of *x* ∈ *X* is a vertex which is connected to *x* by an edge, so that (*x*, *y*) ∈ *E*. The **neighbourhood** *I*(*x*) of a vertex *x* is the subgraph composed of the vertexes adjacent to *x* and all the edges connecting vertexes adjacent to *x*. The **degree** (or **valency**) of a vertex *x*, denoted by \\rm deg(*x*), is the number of edges which are incident to the vertex *x*.\nLet the graph Γ be connected, finite (with a finite number of vertexes denoted by *x*1, *x*2, …, *x**p*), simple (i.e. there are no loops connecting any vertex with itself) and undirected (i.e. the edges are bidirectional). We assume that a stock of identical particles is initially allocated at any vertex *x**i* of the graph Γ. The configurations **z**0, $\\bar{\\mathbf{z}}^0$ and **z**1, collecting the number of elements located at *x**i* for any *i*, are natural-valued functions defined on the graph. The rules of the evolution are established to guarantee *stability* of the sandpile dynamics, starting from the definition of the **height function** given by **z** : *X* → N*p* (refer to Figure [figure:sandpile] for the graphical representation of a sandpile on a two-dimensional Cartesian grid).\n![A simple sandpile with variable height (on a two-dimensional Cartesian grid)](sandpile \"fig:\") [figure:sandpile]\nThe height *z**i* = **z**(*x**i*) is **unstable** (at *x**i* ∈ *X*) if *z**i* ≥ \\rm deg(*x**i*), where \\rm deg(*x**i*) is the degree of the vertex *x**i*. Otherwise, the vertex is **stable**. \nA **stable configuration** is a configuration of vertexes which are stable for any index *i*. A **maximum stable configuration** (or **minimally stable state**) is a configuration in which all the vertexes have a height *z**i* of one unit lower than the corresponding threshold value \\rm deg(*x**i*). An **almost stable configuration** is a configuration of vertexes which are stable for any index *i* different from the root, and a **maximum almost stable configuration** is defined accordingly.\nIn that framework, an unstable vertex is forced to *topple* over its neighbouring vertexes, thus causing a change of state in the entire configuration.\nA **toppling** (or **firing**) from the vertex *x**i* is the mapping Ξ from the graph Γ to itself determined by the following (local) rules:\n \n$$\\Xi(\\mathbf{z})\\_j = z\\_j - \\Delta\\_{i\\to j} \\qquad \\textrm{with} \\quad\n \\Delta\\_{i\\to j} := \\left\\{\\begin{aligned}\n & \\textrm{\\rm deg}(x\\_i) && \\textrm{if}\\;\\;x\\_j=x\\_i\\\\\n & -1 && \\textrm{if}\\;\\; x\\_j\\in I(x\\_i)\\\\\n & \\;\\; 0 && \\textrm{otherwise}\n \\end{aligned}\\right.$$ \nand Δ*i* → *j* is called the **toppling matrix** .\nIn principle, different stability criteria could be proposed involving values other than the degree \\rm deg(*x**i*) of the vertex *x**i*, and modified redistribution criteria could also be considered, but we focus on the above definitions for the sake of simplicity in presentation.\nThe addition of a particle to the sandpile is schematized by summing to the ground state configuration **z**0 = (*z*10, *z*20, …, *z**p*0) the vector $\\boldsymbol{\\delta}\\_i$ with 1 at some fixed index *i* and zeros elsewhere. By iterating this procedure *m* times for possibly different choices of the index, the initial configuration $\\bar{\\mathbf{z}}^0$ is finally given by **z**0 + **w** with $\\mathbf{w}=\\boldsymbol{\\delta}\\_{i\\_1}+\\boldsymbol{\\delta}\\_{i\\_2}+\\dots+\\boldsymbol{\\delta}\\_{i\\_m}$. It is a straightforward consequence that the operation of adding particles to a sandpile is associative. Moreover, the final configuration **z**1 resulting from the evolution is independent of the order in which the topplings are performed, and therefore the operation of toppling is commutative . \nAs an illustrative case, we assume that \\rm deg(*x**i*) = \\rm deg(*x**j*) for two adjacent vertexes *x**i* and *x**j*. Then, we consider a sandpile in which both *x**i* and *x**j* are at their critical height, that is *z**i* ≥ \\rm deg(*x**i*) and *z**j* ≥ \\rm deg(*x**j*). According to the dynamics described above, a toppling from the vertex *x**i* causes the vertex *x**j* to become unstable, and viceversa, and the sandpile comes to a configuration in which the height *z**k* for some *k* ∈ {1, 2, …, *p*} increases by a value △ *i* → *k* + △ *j* → *k*. Such procedure being symmetrical, by applying this argument repeatedly, we conclude that the same stable configuration is reached irrespective of whether *x**i* or *x**j* is toppled first, and in general regardless of the sequence in which unstable vertexes are toppled.\nTwo-dimensional Cartesian grids\n-------------------------------\nWe focus on the simple case of a two-dimensional Cartesian grid, with two standard schemes for adjacent vertexes given by the *von Neumann neighbourhood*, consisting of the four cells obtained moving one step towards North/East/South/West (refer to Figure [figure:neighbourhood](left)), and the *Moore neighbourhood*, which includes also the four diagonal cells (refer to Figure [figure:neighbourhood](right)). Of course, selecting a more general network provides a higher degree of flexibility and allows to closely represent the connections between various medical facilities of the healthcare system (refer to Figure [figure:network]).\n![The von Neumann neighbourhood (left) of the (blue) central cell (an extended neighbourhood includes also the pink cells) and the Moore neighbourhood (right)](CA-Moore.png \"fig:\") [figure:neighbourhood]\nWe consider a special grid of *p* = *n*2 cells organized over a squared matrix of size *n* × *n*, where the particles are dropped randomly being allowed to stack on top of each other, so that a configuration of the sandpile is described by an element of the space of natural-valued matrices\n \n$$\\mathcal{M}\\_{n}(\\mathbb{N}) := \\bigl\\{ A\\in \\mathbb{R}^{n\\times n} \\,:\\, a\\_{ij}\\in \\mathbb{N} \\;\\; \\textrm{for any} \\;\\; i,j=1,2,\\dots,n \\bigr\\}.$$ \nThe position of each vertex *x**i* (with its corresponding height *z**i* ∈ N) is determined by the index *i* ∈ {1, 2, …, *p* = *n*2} according to the following common practise:\n \n$$\\begin{pmatrix}\n 1 & 2 & 3 & \\dots & n \\\\\n n+1 & n+2 & n+3 & \\dots & 2n \\\\\n 2n+1 & 2n+2 & 2n+3 & \\dots & 3n \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n \\; n(n-1)+1 & n(n-1)+2 & \\dots & \\hdots & n^2\n \\end{pmatrix}$$ \nFor the von Neumann neighborhood, we have \\rm deg(*x**i*) = 4 for any index *i* associated with an element of the bulk of the matrix, and for instance\n \n$$I(x\\_{n+2}) = \\bigl\\{ x\\_2 \\,\\textrm{(North)}, \\,x\\_{n+3} \\,\\textrm{(East)}, \\,x\\_{2n+2} \\,\\textrm{(South)}, \\,x\\_{n+1} \\,\\textrm{(West)} \\bigr\\}.$$ \nSimilarly, for the Moore neighborhood, we have \\rm deg(*x**i*) = 8 for any index *i* associated with an element of the bulk of the matrix, and for instance\n \n$$\\begin{split}\n I(x\\_{n+2}) = \\bigl\\{ & x\\_2 \\,\\textrm{(North)}, \\,x\\_3 \\,\\textrm{(North-East)}, \\,x\\_{n+3} \\,\\textrm{(East)}, \\,x\\_{2n+3} \\,\\textrm{(South-East)},\\\\\n & \\,x\\_{2n+2} \\,\\textrm{(South)}, \\,x\\_{2n+1} \\,\\textrm{(South-West)}, \\,x\\_{n+1} \\,\\textrm{(West)}, \\,x\\_1 \\,\\textrm{(North-West)} \\bigr\\}.\n \\end{split}$$ \nWe analyze the simple case *n* = 3 (*p* = *n*2 = 9) with the von Neumann neighbourhood. By combining the perturbation $\\mathbf{w}=4\\,\\boldsymbol{\\delta}\\_5$ with the ground state configuration **z**0 = **0**, we obtain the following initial and final configurations\n \n$$\\bar{\\mathbf{z}}^0 = \\mathbf{0} + 4\\,\\boldsymbol{\\delta}\\_5\n = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & \\mathbf{4} & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}\n \\quad \\longrightarrow \\quad\n \\mathbf{z}^1 = \\Xi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix}$$ \nAs expected, the threshold value \\rm deg(*x*5) = 4 is reached at the central cell, thus inducing a destabilisation of the corresponding vertex, with the consequence that four particles are poured into the adjacent vertexes –the elements of the von Neumann neighbourhood– to compose a stable final configuration.\nAs a matter of fact, the toppling from an unstable vertex changes the state of the adjacent vertexes, sometimes causing the appearance of instabilities at some other vertex. Then, subsequent topplings over the adjacent vertexes are generated, triggering a sequence of events which are evocative of *sandpile avalanches* and stopping only when all the cells return strictly below their threshold capacity. \nIf the graph is infinite and connected, with a finite number of particles, all vertexes become stable after a finite number of topplings; moreover, it can be proven that the final (stable) configuration **z**1 depends solely on the initial configuration $\\bar{\\mathbf{z}}^0$, independently of the order in which the topplings are performed . \nOn the other hand, if the graph is finite, appropriate boundary conditions must be implemented: assuming that the particles are evacuated from the boundaries, an analogous result of stability holds , but other types of boundary conditions may not guarantee the same property. During the operation of toppling, no particles are created as they are redistributed to neighbouring cells. For dissipative boundary conditions, the toppling can actually lead to the loss of particles if it occurs on a boundary cell, and we refer to this case as *open boundary conditions*.\nApplication to healthcare system management\n===========================================\nIn this article, we pursue the idea of applying the Abelian Sandpile paradigm to the management of patients among the COVID-19 hospitals in Italy. \nThe current organisation assigns responsibility for the healthcare system to the regions, which are restricted territorial bodies with their own statutes, powers and functions established by the Italian Constitution. We refer to the Lazio region when selecting (the order of magnitude of) the number of medical facilities relevant to the numerical simulations. More specifically, the healthcare system of the Lazio region is composed of about 100 hospitals, mostly located in the metropolitan area of Rome, which are structured within a network of reciprocal connections .\n![Arrangement of hospitals on a general unstructured and undirected network.](network \"fig:\") [figure:network]\nIn that framework, the **height function** of the nodes (hospitals) indicates the number of hospitalised patients, and the links between different nodes represent the possibility of a direct exchange of patients (that is subject to constraints of geographical proximity, together with inherent organisational requirements of the connecting medical facilities). The node of the network exhibiting the largest number of links is the **hub**, which usually takes on a strategic function for the whole system, whilst the nodes with fewer links identify the hospitals with unfavorable geographical location and reduced capacity/functionality (each hospital having different resources, in the general case the threshold value changes from node to node).\nReassignment of outgoing particles\n----------------------------------\nWe have already discussed about the crucial role played by the boundary conditions associated with the ASM settled on finite networks, and indeed the existence of stable configurations is strongly influenced by the presence of a *dissipation mechanism* such as the open boundary conditions. \nBecause the loss of particles through the edge of the network and the hypothesis of an infinite network are unrealistic assumptions for practical applications, we have to implement a suitable *reassignment law* for particles generating a critical height at some boundary cells. Hence, we are induced to redistributing the outgoing particles to the root of the network, namely the hub, which is typically a collecting node of the healthcare system. Therefore, we call the resulting process a Sandpile with Redistribution to the Hub (SRH), and we notice that actually the SRH is equivalent to the ASM when there are no topplings occurring at the edge of the network.\nIn particular, for two-dimensional Cartesian grids with odd size, we incorporate the reassignment of outgoing particles from cells toppling at the border of the matrix by proposing that they are reallocated to the central cell. Furthermore, in case the threshold capacity is reached in several cells at the same time, we impose that the first cell to topple is always the hub, and then additional toppling is performed from the other nodes. In particular, the hub is allowed to topple only once, at the very beginning of the toppling procedure.\nWe summarize the essential steps of the *CAM workflow* –which turns out to be also the implementation scheme of the numerical codes– as follows.\n* *Initialisation*. We choose a (stable) ground state **z**0 = (*z*10, *z*20, …, *z**p*0) which satisfies the condition that *z**i*0 < \\rm deg(*x**i*) for any index *i*;\n* *Inflow*. We combine the ground state **z**0 with a perturbation **w** = (*w*1, *w*2, …, *w**p*) which indicates the number of new patients requiring hospitalization, and we obtain the initial configuration $\\bar{\\mathbf{z}}^0=\\mathbf{z}^0+\\mathbf{w}$;\n* *Hub toppling*. If *z*\\*0 ≥ \\rm deg(*x*\\*), the hub performs a toppling towards the adjacent nodes of its neighbourhood *I*(*x*\\*);\n* *Additional toppling*. If *z**i*0 ≥ \\rm deg(*x**i*) for some *x**i* ≠ *x*\\*, these nodes perform (a sequence of) topplings until reaching an almost stable configuration $\\mathbf{z}^1=\\Phi(\\bar{\\mathbf{z}}^0)$;\n* *Iteration*. We reinitialize the ground state with **z**0 equal to **z**1, and we restart from 2.\nComparison with the standard healthcare system management\n---------------------------------------------------------\nIn order to to compare the efficiency of the novel proposal with the standard organisation of the Italian healthcare system, we provide a reformulation of the current management of connections between hospitals in terms of the CAM paradigm. \nPerhaps surprisingly, the connection between medical facilities is presently mainly determined by their geographical proximity, corresponding to the basic *adjacency* of nodes. If a patient comes to a hospital where there are no places available, because the threshold capacity has been reached, then the single patient is reallocated to the nearest hospital with available places, and the reassignment is limited to the patients exceeding the threshold value. That being the case, the redistribution of patients is handled manually at each hospitalization, without foreseeing the possibility of repeated similar events which are instead highly probable during sudden and unexpected emergencies.\nWe summarize the essential steps of the *standard workflow* as follows, by assuming the reassignement law from the edge of the network at the hub as in Section [section:outgoing].\n* *Initialisation/Inflow*. We proceed as in Section [section:outgoing];\n* *Hub redistribution*. If *z*\\*0 ≥ \\rm deg(*x*\\*), the hub reallocates only the exceeding patients to the adjacent medical facilities, starting from the less crowded ones (in case of equal crowding, a random choice is operated);\n* *Additional redistribution*. If *z**i*0 ≥ \\rm deg(*x**i*) for some *x**i* ≠ *x*\\*, the redistribution procedure is repeated for these nodes, until reaching an almost stable configuration $\\mathbf{z}^1=\\Psi(\\bar{\\mathbf{z}}^0)$;\n* *Iteration*. We reinitialize the ground state with **z**0 equal to **z**1, and we restart from 2.\nWe illustrate the comparison by considering the following simple example.\nLet the healthcare system be represented by a two-dimensional Cartesian grid of size 3 × 3, with the von Neumann neighbourhood (so that \\rm deg(*x**i*) = 4 for any index *i*) and reassignment of outgoing particles to the hub located at the central cell. We choose\n \n$$\\mathbf{z}^0 = \\begin{pmatrix} 2 & 1 & 3 \\\\ 1 & 3 & 1 \\\\ 1 & 0 & 2 \\end{pmatrix}\n \\quad\\textrm{and}\\quad\n \\mathbf{w} = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix},$$ \nso that\n \n$$\\bar{\\mathbf{z}}^0 = \\mathbf{z}^0+\\mathbf{w}\n = \\begin{pmatrix} 2 & 1 & 3 \\\\ 1 & \\mathbf{4} & 1 \\\\ 1 & 0 & 2 \\end{pmatrix}$$ \nand the initial configuration is unstable. The standard organization manages this critical situation by moving a single particle (patient) from the central cell towards one in the von Neumann neighbourhood, and preferably the South cell, to obtain\n \n$$\\Psi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix} 2 & 1 & 3 \\\\ 1 & 3 & 1 \\\\ 1 & 1 & 2 \\end{pmatrix},$$ \nwhere Ψ denotes the *evolution mapping* of the grid according to the standard approach. Then, we reinitialize the ground state with $\\Psi(\\bar{\\mathbf{z}}^0)$ and the perturbation **w** is chosen as above, so that the configuration $\\Psi(\\bar{\\mathbf{z}}^0)+\\mathbf{w}$ is also unstable with respect to the central cell. By selecting randomly one of the neighbouring cells, another single particle is transferred, and after two additional iterations, the system reaches the final configuration\n \n$$\\Psi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix} 2 & 2 & 3 \\\\ 2 & 3 & 2 \\\\ 1 & 1 & 2 \\end{pmatrix}.$$ \nOn the other hand, the ASP paradigm suggests to transfer at the same time all the four patients initially allocated to the hub towards the von Neumann neighbourhood, to obtain\n \n$$\\Phi(\\mathbf{z}^0) = \\begin{pmatrix} 2 & 2 & 3 \\\\ 2 & 0 & 2 \\\\ 1 & 1 & 2 \\end{pmatrix}.$$ \nFinally, under the assumption that new patients are always collected at the hub, the system can accept exactly 3 more particles from the additional iterations, before reaching a critical situation as before. Of course, it could be argued that, being the final configurations the same, there would be no obvious evidence to prefer the novel approach to the standard organization. However, the deciding factor is that we have unified within a single time-step the efforts for reallocating patients among the medical facilities, which is usually regarded as a source of stress for the overall healthcare system. Hence, we have left the structure time to reorganize without suffering a constant state of emergency, which is precisely the issue of the so-called *predictive logistics*.\nExamples and numerical simulations\n==================================\nWe collect numerical results from a selection of preliminary cases, which are nevertheless useful to understand the alternative methods of healthcare system management provided by the two approaches described in Section [section:healthcare], and their inherent dynamics. We consider two-dimensional Cartesian grids, which are arranged into *n* × *n* (squared) matrices with the Moore neighbourhood, so that \\rm deg(*x**i*) = 8 for any index *i* ∈ {1, 2, …, *p* = *n*2}, and we assume the reassignment to the hub of outgoing particles from the edge of the network as in Section [section:outgoing]. \nWe recall that each cell/node of the grid represents a hospital, and the height function reproduces the number of patients in the medical facilities, expressed as the percentage of capacity already achieved.\nWe start by focusing on the possible outputs of a basic case consisting in a single hospital originally attaining its threshold capacity, which is located at the hub (central cell), with a (randomly chosen) stable ground state configuration. To highlight possibly critical situations, we boldmark all values above the threshold capacity, and we encircle the values 5, 6 and 7 dangerously close to the saturation level.\nFor *n* = 3, the hospitals are *p* = *n*2 = 9, and we choose an initial configuration with all new patients coming to the hub *x*5, given by the perturbation $\\mathbf{w}=4\\,\\boldsymbol{\\delta}\\_5$, so that\n \n$$\\mathbf{z}^0 = \\begin{pmatrix} 2 & 3 & 1 \\\\ \\circled{5} & \\circled{7} & 2 \\\\ 4 & 3 & 3 \\end{pmatrix}\n \\quad\\textrm{and}\\quad\n \\bar{\\mathbf{z}}^0 = \\begin{pmatrix} 2 & 3 & 1 \\\\ \\circled{5} & {\\bf 11} & 2 \\\\ 4 & 3 & 3 \\end{pmatrix}.$$ \nBecause the hub has to manage a number of hospitalisations greater than its threshold capacity, some patients must be transferred to adjacent facilities. \nAccording to the standard strategy, only four patients need a different allocation to be found among the neighbourhing cells, ending at the (stable) configuration\n \n$$\\Psi(\\bar{\\mathbf{z}}^0)\n = \\begin{pmatrix} 2 & 4 & 1 \\\\\n \\circled{5} & \\circled{7} & 4 \\\\\n 4 & 4 & 3 \\end{pmatrix}.$$ \nAs it is clearly seen, an additional iteration with $\\mathbf{w}=4\\,\\boldsymbol{\\delta}\\_5$ determines a new unstable/critical configuration at the central node (with again four patients in excess), demanding for a further reorganisation of the hub by means of another transfer operation. On the other hand, by employing the SRH approach, we obtain a different outcome given by\n \n$$\\Phi(\\bar{\\mathbf{z}}^0)\n = \\begin{pmatrix} 3 & 4 & 2 \\\\\n \\circled{6} & 3 & 3 \\\\\n \\circled{5} & 4 & 4 \\end{pmatrix},$$ \nsuch situation being more favorable in terms of load-balancing since an additional iteration with perturbation $\\mathbf{w}=4\\,\\boldsymbol{\\delta}\\_\\ast$ does not produce any unstable configuration. Moreover, there is still a certain amount of available places at the hub, and therefore the novel proposal solves the criticalities possibly generated by an emergency.\nDespite its simplicity, the above example suggests to attribute to each configuration the value of a given functional, aiming to provide an easily readable *indicator* of the effectiveness of the allocation strategy. This is necessarily a very delicate issue. \nAs a first attempt, a possible choice is to count the total number of medical facilities attaining a given fraction of their threshold capacity, which are denoted by *critical points*. Such choice is actually quite questionable, since it does not take keep memory of the value of incoming patients at the beginning of the iteration. \nA different (rough) quantitative measure of the system management efficiency is\n \n$$\\mathcal{F}(\\mathbf{w},\\mathbf{z}) := \\frac{\\mathbf{w}}{\\sum\\_{j} w\\_j}\\cdot \\mathbf{z}\\,,$$ \nwhich intends to evaluate the risk that a new patient comes to a given location, taking into account the previous history of the system (in this case, the effect of the perturbation **w**).\nFor the case illustrated in the Example 3, the function F equals the state of the hub *x*5, so that\n \n$$\\mathcal{F}(\\mathbf{w},\\bar{\\mathbf{z}}^0) = \\bar z^0\\_5 = 11,\\quad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Psi(\\bar{\\mathbf{z}}^0)\\bigr) = \\Psi(\\bar{\\mathbf{z}}^0)\\_5 = 7,\\quad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Phi(\\bar{\\mathbf{z}}^0)\\bigr) = \\Phi(\\bar{\\mathbf{z}}^0)\\_\\ast = 3\\,.$$ \nThe minimum value is achieved for the configuration $\\Phi(\\bar{\\mathbf{z}}\\_0)$ corresponding to the SRH approach (or, equivalently, to ASM).\nFor *n* = 5, the hospitals are *p* = *n*2 = 25, and we choose an initial configuration with all new patients coming to the hub *x*13, given by the perturbation $\\mathbf{w}=\\boldsymbol{\\delta}\\_{9}+\\boldsymbol{\\delta}\\_{12}+4\\boldsymbol{\\delta}\\_13\n+2\\boldsymbol{\\delta}\\_{14}+\\boldsymbol{\\delta}\\_{18}+\\boldsymbol{\\delta}\\_{19}$, so that\n \n$$\\mathbf{z}^0 = \\begin{pmatrix}\n 1 & 2 & 4 & 2 & \\circled{5} \\\\\n 2 & 4 & 2 & 3 & 1 \\\\\n 3 & 2 & \\circled{7} & 2 & 3 \\\\\n 2 & 1 & 4 & 2 & 2 \\\\\n 4 & 2 & 1 & \\circled{5} & 4\n \\end{pmatrix}\n \\quad\\textrm{and}\\quad\n \\bar{\\mathbf{z}}^0 = \\begin{pmatrix}\n 1 & 2 & 4 & 2 & \\circled{5} \\\\\n 2 & 4 & 2 & 4 & 1 \\\\\n 3 & 3 & {\\bf 11} & 4 & 3 \\\\\n 2 & 1 & \\circled{5} & 3 & 2 \\\\\n 4 & 2 & 1 & \\circled{5} & 4\n \\end{pmatrix}.$$ \nAccording to the standard strategy, an admissible solution could be\n \n$$\\Psi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix}\n 1 & 2 & 4 & 2 & \\circled{5} \\\\\n 2 & 4 & 4 & 4 & 1 \\\\\n 3 & 3 & \\circled{7} & 4 & 3 \\\\\n 2 & 3 & \\circled{5} & 3 & 2 \\\\\n 4 & 2 & 1 & \\circled{5} & 4\n \\end{pmatrix},$$ \nand the same observation previously done holds, that is such configuration likely determines an instability located at the hub after a subsequent process iteration, since the number of hospitalised patients is very close to the critical threshold. \nOn the other hand, by employing the SRH approach, we obtain\n \n$$\\Phi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix}\n 1 & 2 & 4 & 2 & \\circled{5} \\\\\n 2 & \\circled{5} & 3 & \\circled{5} & 1 \\\\\n 3 & 4 & 3 & \\circled{5} & 3 \\\\\n 2 & 2 & \\circled{6} & 4 & 2 \\\\\n 4 & 2 & 1 & \\circled{5} & 4\n \\end{pmatrix},$$ \nwhere the number of patients hospitalised at the hub is far away from the critical level. Indeed, by computing the indicator function F at the different outcomes, we deduce that\n \n$$\\mathcal{F}\\bigl(\\mathbf{w},\\bar{\\mathbf{z}}^0\\bigr) = 6.7,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Psi(\\bar{\\mathbf{z}}^0)\\bigr) = 5.1,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Phi(\\bar{\\mathbf{z}}^0)\\bigr) = 4.1\\,,$$ \nsince ∑*j**w**j* = 10, and the smallest value is achieved by the configuration resulting from the SRH approach; thus, such strategy must be preferred to the standard one.\nNext, we explore a case where two subsequent topplings occur.\n[ex:10] For *n* = 5, we choose the ground state and the initial configuration as follows,\n \n$$\\mathbf{z}^0 = \\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 0 & \\circled{5} & 1 & 1 \\\\\n 1 & 2 & \\circled{7} & \\circled{7} & 4 \\\\\n \\circled{5} & \\circled{5} & 2 & 4 & \\circled{5} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix}\n \\quad\\textrm{and}\\quad\n \\bar{\\mathbf{z}}^0 = \\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 0 & \\circled{5} & 1 & 1 \\\\\n 1 & 2 & {\\bf 11} & \\circled{7} & 4 \\\\\n \\circled{5} & \\circled{5} & 2 & 4 & \\circled{5} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix},$$ \nwhere $\\mathbf{w}=4\\boldsymbol{\\delta}\\_{13}$. Then, according to the standard strategy, a final configuration is given by\n \n$$\\Psi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 3 & \\circled{5} & 2 & 1 \\\\\n 1 & 2 & \\circled{7} & \\circled{7} & 4 \\\\\n \\circled{5} & \\circled{5} & 2 & 4 & \\circled{5} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix}.$$ \nFor applying the SRH approach, we pass through an intermediate configuration\n \n$$\\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 0+1 & \\circled{5+1} & 1+1 & 1 \\\\\n 1 & 2+1 & 11-8 & \\mathbf{7+1} & 4 \\\\\n \\circled{5} & \\circled{5+1} & 2+1 & \\circled{4+1} & \\circled{5} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix}\n =\n \\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 1 & \\circled{6} & 2 & 1 \\\\\n 1 & 3 & 3 & \\mathbf{8} & 4 \\\\\n \\circled{5} & \\circled{6} & 3 & \\circled{5} & \\circled{5} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix},$$ \nin which the cell *x*12 is unstable, and finally we arrive at the final configuration\n \n$$\\Phi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 1 & \\circled{6+1} & 2+1 & 1+1 \\\\\n 1 & 3 & 3+1 & 8-8 & \\circled{4+1} \\\\\n \\circled{5} & \\circled{6} & 3+1 & \\circled{5+1} & \\circled{5+1} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix}\n = \\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 1 & \\circled{7} & 3 & 2 \\\\\n 1 & 3 & 4 & 0 & \\circled{5} \\\\\n \\circled{5} & \\circled{6} & 4 & \\circled{6} & \\circled{6} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix}.$$ \nThe matrices $\\Psi(\\bar{\\mathbf{z}}^0)$ and $\\Phi(\\bar{\\mathbf{z}}^0)$ are represented in Figure [figure:figure6] with different colors assigned to cells, corresponding to the degree of saturation achieved.\n![Model comparison from the Example [ex:10]: final configurations \\Psi(\\bar{\\mathbf{z}}^0) (left) and \\Phi(\\bar{\\mathbf{z}}^0) (right). The colors attached to each cell/node represent its corresponding relative capacity (0=green, 1/2=yellow, 3/4=magenta, 4/5=red, 6/7=black)](figure6 \"fig:\") [figure:figure6]\nAt the bottom of each subfigure, we report the counting of critical points, given by the number of cells with values 6 or 7. Although a quick glance can make us suspect that the left configuration is preferable, the situation is indeed different. In fact, first of all we distinguish the presence of two almost critical values 7 inside the configuration plotted on the left, against the presence of only one value 7 from the second approach. Furthermore, the hub –the facility most at risk taking into account recent history– has a value 7 in the first case, whilst a value 4 occurs in the second case, thus making the latter situation preferable.\nFinally, we compute the function F corresponding to the different configurations: since **w**/∑*j**w**j* is everywhere zero except at the hub, where it is equal to 1, the values of F coincide with the values at the hub, which are given by\n \n$$\\mathcal{F}(\\mathbf{w},\\bar{\\mathbf{z}}^0) = 11,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Psi(\\bar{\\mathbf{z}}^0)\\bigr) = 7,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Phi(\\bar{\\mathbf{z}}^0)\\bigr) = 4,$$ \nsuggesting again the advantages of configurations proposed by the SRH approach.\nSimulation of multiple central outbreaks\n----------------------------------------\nWe analyze the case with more than one hospital in the same restricted area (including the hub) which is concerned with new incoming patients. One might think of this event as the emergence of multiple outbreaks within some smaller district of a larger area. We consider a larger network, with *n* = 9, whose size is comparable to the capacity of the healthcare system in the Lazio region in Italy .\nWe consider the perturbation $\\mathbf{w}=2\\boldsymbol{\\delta}\\_{32}+\\boldsymbol{\\delta}\\_{33}+5\\boldsymbol{\\delta}\\_{41}+2\\boldsymbol{\\delta}\\_{42}$ and the ground state\n \n$$\\mathbf{z}^0 = \\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} & 2 & 3 & \\circled{6} & 2 \\\\\n 3 & 2 & \\circled{5} & 2 & 1 & 3 & 2 & 4 & 3 \\\\\n 3 & 3 & 1 & \\circled{5} & \\circled{6} & 2 & 3 & 1 & 3 \\\\\n \\circled{6} & 1 & \\circled{5} & 3 & 2 & \\circled{5} & 2 & 3 & 4 \\\\\n 1 & 3 & 2 & 1 & \\circled{6} & 3 & 4 & \\circled{5} & \\circled{5} \\\\\n 1 & 4 & 1 & 2 & 2 & \\circled{5} & 1 & 2 & 3 \\\\\n 4 & 1 & 3 & 4 & \\circled{5} & \\circled{6} & 2 & \\circled{5} & 3 \\\\\n 4 & 2 & 3 & \\circled{5} & 2 & 2 & \\circled{6} & 3 & 1 \\\\\n 1 & \\circled{6} & \\circled{5} & 2 & 4 & 4 & 2 & 1 & 2\n \\end{pmatrix},$$ \nso that ∑*j**z**j*0 = 250 and ∑*j**w**j* = 10, corresponding to the initial configuration\n \n$$\\begin{aligned}\n \\bar{\\mathbf{z}}^0 &= \\begin{pmatrix}\n 1 &2 & 1 &4 & \\circled{6} &2 &3 &\\circled{6} &2 \\\\\n 3 &2 & \\circled{5} &2 & 1 &3 &2 &4 &3 \\\\\n 3 &3 & 1 & \\circled{5} & \\circled{6} &2 &3 &1 &3 \\\\\n \\circled{6} &1 & \\circled{5} &3 & 2+2 & \\circled{5+1} &2 &3 &4 \\\\\n 1 &3 & 2 &1 & \\circled{6+5} & \\circled{3+2} &4 & \\circled{5} & \\circled{6} \\\\\n 1 &4 & 1 &2 & 2 & \\circled{5} &1 &2 &3 \\\\\n 4 &1 & 3 &4 & \\circled{5} & \\circled{6} &2 & \\circled{5} &3 \\\\\n 4 &2 & 3 & \\circled{5} & 2 &2 & \\circled{6} &3 &1 \\\\\n 1 & \\circled{6} & \\circled{5} &2 & 4 &4 &2 &1 &2\n \\end{pmatrix}\\\\ \\\\\n &= \\begin{pmatrix}\n 1 &2 &1 &4 & \\circled{6} &2 &3 & \\circled{6} &2 \\\\\n 3 &2 & \\circled{5} &2 &1 &3 &2 &4 & \\circled{6} \\\\\n 3 &3 &1 & \\circled{5} & \\circled{6} &2 &3 &1 &3 \\\\\n \\circled{6} &1 & \\circled{5} &3 &4 & \\circled{6} &2 &3 &4 \\\\\n 1 &3 &2 &1 &{\\bf 11} & \\circled{5} &4 & \\circled{5} & \\circled{6} \\\\\n 1 &4 &1 &2 &2 & \\circled{5} &1 &2 &3 \\\\\n 4 &1 &3 &4 & \\circled{5} & \\circled{6} &2 & \\circled{5} &3 \\\\\n 4 &2 &3 & \\circled{5} &2 &2 & \\circled{6} &3 &1 \\\\\n 1 &3 & \\circled{5} &2 &4 &4 &2 &1 &2\n \\end{pmatrix}.\n\\end{aligned}$$ \nThen, according to the standard strategy, a final configuration is given by\n \n$$\\Psi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} &2 &3 & \\circled{6} & 2 \\\\\n 3 & 2 & \\circled{5} & 2 &1 &3 &2 & 4 & 3 \\\\\n 3 & 3 & 1 & \\circled{5} & \\circled{6} &2 &3 & 1 & 3 \\\\\n \\circled{6} & 1 & \\circled{5} & 3+0 & 4+0 &\\circled{6+0} &2 & 3 & 4 \\\\\n 1 & 3 & 2 & 1+2 & \\circled{11-4} &\\circled{5+0} &4 & \\circled{5} & \\circled{6} \\\\\n 1 & 4 & 1 & 2+1 & 2+1 &\\circled{5+0} &1 & 2 & 3 \\\\\n 4 & 1 & 3 & 4 & \\circled{5} &6 &2 & \\circled{5} & 3 \\\\\n 4 & 2 & 3 & \\circled{5} &2 &2 &\\circled{6} & 3 & 1 \\\\\n 1 & \\circled{6} & \\circled{5} & 2 &4 &4 &2 & 1 & 2\n \\end{pmatrix}.$$ \nHence, the central subgraph around the hub is composed by the elements\n \n$$\\begin{pmatrix} 3 & 4 & \\circled{6} \\\\ 3 & \\circled{7} & \\circled{5} \\\\ 3 & 3 & \\circled{5} \\end{pmatrix}.$$ \nIn the subsequent steps, any additional patients coming to the hub destabilises the configuration. On the other hand, by employing the SRH approach, we obtain\n \n$$\\Phi(\\bar{\\mathbf{z}}^0)=\\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} & 2 & 3 & \\circled{6} & 2 \\\\\n 3 & 2 & \\circled{5} & 2 & 1 & 3 & 2 & 4 & 3 \\\\\n 3 & 3 & 1 & \\circled{5} & \\circled{6} & 2 & 3 & 1 & 3 \\\\\n \\circled{6} & 1 & \\circled{5} & 3+1 & \\circled{4+1} & \\circled{6+1} & 2 & 3 & 4 \\\\\n 1 & 3 & 2 & 1+1 & 11-8 & \\circled{5+1} & 4 & \\circled{5} & \\circled{6} \\\\\n 1 & 4 & 1 & 2+1 & 2+1 & \\circled{5+1} & 1 & 2 & 3 \\\\\n 4 & 1 & 3 & 4 & \\circled{5} & \\circled{6} & 2 &\\circled{5} & 3 \\\\\n 4 & 2 & 3 & \\circled{5} & 2 & 2 & \\circled{6} & 3 & 1 \\\\\n 1 & \\circled{6} & \\circled{5} & 2 & 4 & 4 & 2 & 1 & 2\n \\end{pmatrix},$$ \nand the corresponding central subgraph around the hub is\n \n$$\\begin{pmatrix} 4 & \\circled{5} & \\circled{7} \\\\ 2 & 3 & \\circled{6} \\\\ 3 & 3 & \\circled{6} \\end{pmatrix}.$$ \nThe difference between $\\Psi(\\bar{\\mathbf{z}}^0)$ and $\\Phi(\\bar{\\mathbf{z}}^0)$ is transparent regarding, specifically, the number of patients hospitalised in the hub. A further iteration with the same perturbation **w** would lead to a new critical situation for the hub in the first case, but it does not in the second.\nFigure [figure:figure7] provides a representation of the two configurations $\\Psi(\\bar{\\mathbf{z}}^0)$ and $\\Phi(\\bar{\\mathbf{z}}^0\\bigr)$.\n![Model comparison from the case of central areas outbreaks: final configurations \\Psi(\\bar{\\mathbf{z}}^0) (left) and \\Phi(\\bar{\\mathbf{z}}^0) (right). The colors attached to each cell/node represent its corresponding relative capacity (0=green, 1/2=yellow, 3/4=magenta, 4/5=red, 6/7=black)](figure7 \"fig:\") [figure:figure7]\nCounting the critical points in Figure [figure:figure7] could give the (incorrect) impression that the standard approach has to be preferred with respect to the novel proposal. But considering the indicator function F applied to $\\bar{\\mathbf{z}}^0$, $\\Psi(\\bar{\\mathbf{z}}^0)$ and $\\Phi(\\bar{\\mathbf{z}}^0)$, we obtain\n \n$$\\mathcal{F}\\bigl(\\mathbf{w},\\bar{\\mathbf{z}}^0\\bigr) =7.9,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Psi(\\bar{\\mathbf{z}}^0)\\bigr)=5.9,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Phi(\\bar{\\mathbf{z}}^0\\bigr))=4.5,$$ \nas a consequence of the fact that memory of the dynamics –represented by **w**– is now considered, and again the minimum value is achieved for the SRH strategy.\nSimulation of peripheral outbreaks\n----------------------------------\nNext, we focus on the case of outbreaks occurring in the vicinity of a node located at the edge of the Cartesian grid. Specifically, we take *n* = 9 and the same ground state **z**0 of the previous example, and an *inflow matrix* of patients **w** given by the submatrix\n \n$$\\begin{pmatrix} \\circled{5} & 2 \\\\ 2 & 1 \\end{pmatrix}$$ \nlocated at the top-right corner of the null matrix. Then, the initial configuration $\\bar{\\mathbf{z}}^0=\\mathbf{z}^0+\\mathbf{w}$ is given by\n \n$$\\bar{\\mathbf{z}}^0 = \\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} & 2 & 3 & \\mathbf{11} & 4 \\\\\n 3 & 2 & \\circled{5} & 2 & 1 & 3 & 2 & \\circled{6} & 4 \\\\\n 3 & 3 & 1 & \\circled{5} & \\circled{6} & 2 & 3 & 1 & 3 \\\\\n \\circled{6} & 1 & \\circled{5} & 3 & 2 & \\circled{5} & 2 & 3 & 4 \\\\\n 1 & 3 & 2 & 1 & \\circled{6} & 3 & 4 & \\circled{5} & \\circled{6} \\\\\n 1 & 4 & 1 & 2 & 2 & \\circled{5} & 1 & 2 & 3 \\\\\n 4 & 1 & 3 & 4 & \\circled{5} & \\circled{6} & 2 & \\circled{5} & 3 \\\\\n 4 & 2 & 3 & \\circled{5} & 2 & 2 & \\circled{6} & 3 & 1 \\\\\n 1 & \\circled{6} & \\circled{5} & 2 & 4 & 4 & 2 & 1 & 2\n \\end{pmatrix},$$ \nwith the cell above the threshold capacity located at position (1, 8). Therefore, four patients are in excess and require a different reallocation. Among others, a possible output of the standard strategy is the final configuration given by\n \n$$\\Psi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} & 2 & \\circled{5} & \\circled{7} & 4 \\\\\n 3 & 2 & \\circled{5} & 2 & 1 & 3 & 4 & \\circled{6} & 4 \\\\\n 3 & 3 & 1 & \\circled{5} & \\circled{6} & 2 & 3 & 1 & 3 \\\\\n \\circled{6} & 1 & \\circled{5} & 3 & 2 & \\circled{5} & 2 & 3 & 4 \\\\\n 1 & 3 & 2 & 1 & \\circled{6} & 3 & 4 & \\circled{5} & \\circled{6} \\\\\n 1 & 4 & 1 & 2 & 2 & \\circled{5} & 1 & 2 & 3 \\\\\n 4 & 1 & 3 & 4 & \\circled{5} & \\circled{6} & 2 & \\circled{5} & 3 \\\\\n 4 & 2 & 3 & \\circled{5} & 2 & 2 & \\circled{6} & 3 & 1 \\\\\n 1 & \\circled{6} & \\circled{5} & 2 & 4 & 4 & 2 & 1 & 2\n \\end{pmatrix}.$$ \nOn the other hand, by employing the SRH approach, we pass through an intermediate state\n \n$$\\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} & 2 & 4 & 3 & \\circled{5} \\\\\n 3 & 2 & \\circled{5} & 2 & 1 & 3 & \\circled{3} & \\circled{7} & \\circled{5} \\\\\n 3 & 3 & 1 & \\circled{5} & \\circled{6} & 2 & 3 & 1 & 3 \\\\\n \\circled{6} & 1 & \\circled{5} & 3 & 2 & \\circled{5} & 2 & 3 & 4 \\\\\n 1 & 3 & 2 & 1 & \\mathbf{9} & 3 & 4 & \\circled{5} & \\circled{6} \\\\\n 1 & 4 & 1 & 2 & 2 & \\circled{5} & 1 & 2 & 3 \\\\\n 4 & 1 & 3 & 4 & \\circled{5} & \\circled{6} & 2 & \\circled{5} & 3 \\\\\n 4 & 2 & 3 & \\circled{5} & 2 & 2 & \\circled{6} & 3 & 1 \\\\\n 1 & \\circled{6} & \\circled{5} & 2 & 4 & 4 & 2 & 1 & 2\n \\end{pmatrix},$$ \nto reach the final configuration\n \n$$\\Phi(\\bar{\\mathbf{z}}^0)\n =\\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} & 2 & 4 & 3 & \\circled{5} \\\\\n 3 & 2 & \\circled{5} & 2 & 1 & 3 & \\circled{3} & \\circled{7} & \\circled{5} \\\\\n 3 & 3 & 1 & \\circled{5} & \\circled{6} & 2 & 3 & 1 & 3 \\\\\n \\circled{6} & 1 & \\circled{5} & 4 & 3 & \\circled{6} & 2 & 3 & 4 \\\\\n 1 & 3 & 2 & 2 & 1 & 4 & 4 & \\circled{5} & \\circled{6} \\\\\n 1 & 4 & 1 & 3 & 3 & \\circled{6} & 1 & 2 & 3 \\\\\n 4 & 1 & 3 & 4 & \\circled{5} & \\circled{6} & 2 & \\circled{5} & 3 \\\\\n 4 & 2 & 3 & \\circled{5} & 2 & 2 & \\circled{6} & 3 & 1 \\\\\n 1 & \\circled{6} & \\circled{5} & 2 & 4 & 4 & 2 & 1 & 2\n \\end{pmatrix}.$$ \nFigure [figure:figure8] provides a representation of the two configurations $\\Psi(\\bar{\\mathbf{z}}^0)$ and $\\Phi(\\bar{\\mathbf{z}}^0\\bigr)$, by indicating the level of saturation reached in each hospital.\n![Model comparison from the case of peripheral outbreaks: final configurations \\Psi(\\bar{\\mathbf{z}}^0) (left) and \\Phi(\\bar{\\mathbf{z}}^0) (right). The colors attached to each cell/node represent its corresponding relative capacity (0=green, 1/2=yellow, 3/4=magenta, 4/5=red, 6/7=black)](figure8 \"fig:\") [figure:figure8]\nFinally, the indicator function F defined at the beginning of Section [section:examples] takes the following values for the different configurations,\n \n$$\\mathcal{F}\\bigl(\\mathbf{w},\\bar{\\mathbf{z}}^0\\bigr)=7.9,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Psi(\\bar{\\mathbf{z}}^0)\\bigr)=5.9,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Phi(\\bar{\\mathbf{z}}^0\\bigr))=4.4\\,,$$ \nand this fact contributes to the advantages of the SHR approach.\nBeyond the Abelian Sandpile paradigm\n====================================\nThe mathematical model introduced in this article lays the foundation for an optimisation of the healthcare system management. A notable feature of the novel proposal is its *scalability* to various levels of description, and also the possibility of improving the experimental simulation by including more realistic situations and different types of medical facilities (emergency rooms, external care points,...) into an integrated dynamical system.\nSandpiles with internal dissipation\n-----------------------------------\nIn order to incorporate other relevant elements into the SHR model, such as recovery or (unfortunately) death of patients, we postulate an *elimination mechanism* inherent to the system, which is translated in mathematical formalism by considering the presence of some *dissipation* during the evolution.\nFor instance, we describe the effect of removing particles from the network by a simple subtraction of a (randomly chosen, but possibly measurable) distribution $\\boldsymbol{\\zeta}=(\\zeta\\_1,\\zeta\\_2,\\dots,\\zeta\\_p)$ with the obvious constraint that\n \n0 ≤ *ζ**i* ≤ Φ(**z**0)*i* for any index *i* = 1, 2, …, *p* . \nThe modified model workflow is divided into the following steps.\n* *Initialisation/Inflow*. We proceed as in Section [section:outgoing];\n* *Hub/Additional toppling*. We proceed as in Section [section:outgoing];\n* *Internal dissipation*. We subtract the distribution $\\boldsymbol{\\zeta}$ to the intermediate \nconfiguration **z**1, leading to the final configuration $\\Phi(\\bar{\\mathbf{z}}^0) = \\mathbf{z}^1-\\boldsymbol{\\zeta}$;\n* *Iteration*. We reinitialize the ground state with **z**0 equal to $\\Phi(\\bar{\\mathbf{z}}^0)$, \nand we restart from 2.\nWe refer to this modified model as *Sandpile with Internal Dissipation* (SID).\nIn the long run, after a large but finite number of iterations, a balance between the inflow and outflow steps has also to be incorporated in order to guarantee conservation of the total number of patients. In particular, it could be useful to add the hypothesis that\n \n∑*n* = 1*N*∑*i* = 1*p**w**i**n* = ∑*n* = 1*N*∑*i* = 1*p**ζ**i**n* , \nwhere *N* is the total number of iterations, with **w***n* and $\\boldsymbol{\\zeta}^n$ denoting the inflow and outflow contributions at the time-step *n*, respectively. Indeed, we notice that if the lefthand side is larger than the righthand side, the whole system risks to undergo a finite time collapse, by reaching its *total capability* –sum of the capacities of each single structure– in finite time.\nOther interesting features can be added to provide the model with a higher level of realism: for instance, the presence of some inertia to the transfer process , giving preference to structures with a certain level of hospitalized patients , or constrain additional bulk dissipation .\nConclusion and perspetives\n--------------------------\nFrom the analysis developed in this article, we observe that the standard healthcare system management typically generates highly unstable and unbalanced configurations, where specific geographical areas with semi-empty hospitals alternate with others where all medical facilities are saturated, especially during sudden and unforeseen events like the spread of epidemics. Instead, following the SRH strategy for optimized management of connections between hospitals, seems to produce more sparse allocation of patients, which has to be considered as a preferable configuration in terms of load-balancing.\nOn the other hand, it is crucial to improve the exchange of information and to provide decision-making tools to the local structures, in order to optimise the healthcare response in normal times and to avoid the collapse of individual hospitals in times of crisis. There are many relevant consequences in the socio-economic field: among others, we stress the riveting possibility of the automation of health protocols, meaning to build an application capable of learning something from the data autonomously, without receiving explicit instructions from the outside.\nSuch conceptual experimentation could create a learning environment in which policy makers may gain a better understanding of how the system responds to their decisions, providing forecasts of potential different choices and strategies. We are aware that a paradigm shift is required and we hope to give a contribution to this respect. It is worth stressing that the agreement with realistic experimental data is presently very limited. However, the conceptual framework we have proposed in this article applies in principle to many different contexts, and these research directions are currently being explored.\nAcknowledgements\n================\nThe authors thank the Department of Mathematics G. Castelnuovo, Sapienza University of Rome, for hosting the electronic workshop *COVID-19 calls for Mathematics* – <www1.mat.uniroma1.it/ricerca/ convegni/2020/COVIDchiamaMAT> – \nwhich has motivated this article. \nThe authors thank prof. Ferdinando Romano (Department of Public Health and Infectious Diseases, Sapienza University of Rome) for the stimulating discussions about the logistic issues of the healthcare system organisation. \nThe authors also thank prof. Stefano Finzi Vita (Department of Mathematics G. 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We attempt at developing a conceptual tool to support making rational decisions relevant to the optimisation of the allocation of patients into accessible medical facilities. The strategy is based on a version of the Abelian Sandpile model for the Self-Organised Criticality, with the idea of testing the paradigm for the management of patients among the COVID-19 hospitals in Italian regions. In particular, we compare the novel proposal to the standard management of connections between hospitals, showing a number of advantages at a local and global level, by means of a reliable indicator function introduced for measuring the effectiveness of the allocation strategies.\nIntroduction\n============\nSince the beginning of 2020, people from all parts of the world are struggling with the COVID-19 coronavirus pandemic, which has led to a hundred million of infections and over 2 million deaths worldwide . The health emergency is not over yet and it will take months before being under control, possibly becoming endemic among most of the world population . \nThe current data suggest the paramount importance of acting promptly, and especially for trying to reorganize the medical facilities in order to avoid saturation of the critical care capacity in the various territories (refer to Figure [figure:TerapieIntensive] for the evolution of employment of the intensive care units in Italy). Indeed, several countries have been experiencing recurring epidemic waves, with a consequent worsening of the social and economic conditions, thus increasing dramatically the pressure on the healthcare systems.\n![Occupancy level (aggregate data) of intensive care units in the Italian healthcare system.](TerapieIntensive \"fig:\") [figure:TerapieIntensive]\nThe emergence of infectious diseases has to be considered as an inherent property of human and animal populations, which cannot be generally avoided and/or foreseen: before the present COVID-19 emergency, SARS (2002-2004) and influenza A/H1N1 (2009-2010) are other recent notable incidents . \nMost epidemics are characterised by outbreaks in localized geographical areas, and there may be no pharmaceutical solutions already available to safeguard the people’s health when the disease begins to spread largely around. Moreover, the development of effective vaccines typically takes a rather long time, and therefore temporary alternative strategies must be implemented (quarantine, travel restrictions, activities closing, social distancing, face masks obligation,...) However, the demand for specific healthcare facilities such as hospitals consistently increases, reaching and sometimes exceeding critical levels. For example, in the case of COVID-19, the saturation of intensive care units depends on the amount of patients requiring mechanical support for ventilation and proper equipment, so that any healthcare system can be overwhelmed if the number of severe cases becomes very high (refer to Figure [figure:saturazione] for the present-day situation of intensive care units in Italian regions).\n![Saturation of intensive care units in the Italian regions (updated to January 23, 2021). Source: Il Sole 24 Ore – elaboration on data from the Italian Civil Protection Department](saturazione \"fig:\") [figure:saturazione]\nIn that context, the rapid filling of the so-called *COVID-19 hospitals* in the areas most affected by the disease represents a serious danger while dealing with sudden emergencies at different scales (villages, cities, regions, countries). Hence, it is compelling to develop new arrangements of the healthcare system, for optimizing the overall structure especially in terms of an efficient distribution of patients among the most suitable facilities.\nIn this article, we present innovative intervention options, by comparing the standard organisation of the Italian healthcare system with a novel proposal based on the *Abelian Sandpile paradigm*. We provide a systematic approach for an improved planning of the healthcare system by means of a mathematical model for the *self-organisation of critical issues*, which is capable to achieve some specific optimization inside the hospital management. In particular, we aim at reducing the risks of a local/global collapse of hospitals in times of crisis, while improving their functioning in normal times. More specifically, the basic model is grounded in the framework of *cellular automata* and postulates a large network of links between hospitals in a cooperative style of communication. Unfortunately, such connections are currently very limited, which drastically restricts the possibilities of reallocation for the supernumerary patients, and the strong urgency to enhance the hospital network is an important conclusion of the present analysis.\nThe content of this article is organized as follows. \nIn Section [section:abelian], we introduce the Abelian Sandpile paradigm, with a short description of its basic concepts and properties. Section [section:healthcare] focuses on the adaptation of such paradigm to build up a novel proposal for the organisation of the healthcare system. In Section [section:examples], we provide a gallery of examples aiming at illustrating how the proposed model works in the present context of COVID-19 epidemic. More specifically, we analyze two cases relevant to realistic applications: the central and peripheral outbreaks. The article is concluded by Section [section:beyond] which includes some generalisations of the model and (provisional) conclusions.\nAbbreviations\n=============\n– Abelian Sandpile Model \nSOC – Self-Organized Criticality \nCAM – Cellular Automaton Model \nSRH – Sandpile with Redistribution to the Hub \nSID – Sandpiles with Internal Dissipation\nAbelian Sandpile paradigm\n=========================\nThe notion of Self-Organized Criticality (SOC) was originally introduced by Bak, Tang and Wiesenfeld starting from a basic example proposed as a model for sandpiles (we refer to for a general introduction to this broad subject). Since then, the concept has been expanded in many different directions, spanning from classical topics of physics (sandpile avalanches , distribution of earthquakes , amplitude of solar flares ) to less standard economic and socio-political contexts , passing through computer networks and biological applications . At the same time, a huge effort to extend the mathematical tools to deal with theoretical questions has been made, thus contributing to drive SOC into an extraordinary crossroads of probabilistic approaches, graph theory, algebraic geometry, mathematical analysis and optimisation .\nSOC and Sandpiles\n-----------------\nThe original application concerns with modeling of sandpiles, which are regarded as a manageable prototype of SOC, and fundamental contributions have been made by Dhar notably in showing the crucial property of *commutativity*. On this account, the adjective Abelian has since been added to the technical terminology, leading to the actual meaning of Abelian Sandpile Model (ASM). Nevertheless, as usual in the most active fields of science, the terminology is not unique, and indeed similar topics are explored under different names (avalanches, chip-firing games, forest-fire models, parking functions, probabilistic abacus, Rotor-Router or Eulerian-walker models,...) with more or less the same meaning. We refer to for introductory presentations of the ASM and its various applications.\nWe are particularly interested to the *load balancing* property, which denotes the method of distributing a set of tasks across a group of resources with the purpose of making their overall processing more efficient by balancing the workload of each operating unit. In the abstract formulation, units are represented by vertexes/nodes of a graph/network, with the corresponding connections represented by edges/links. The objective is to balance the loading process by allowing nodes to exchange *particles* with their neighbors through the incident edges. \nFor the problem covered in this article, we build a specific ASM with *critical height* provided by the capacity of the medical facilities, and we propose to apply the Abelian Sandpile paradigm to achieve a methodical and efficient distribution of patients across the hospitals, in order to maintain the occupancy below the saturation level for dealing with eventual sudden and unexpected emergencies.\nA sandpile is also a type of Cellular Automaton Model (CAM), namely a discrete mathematical system fulfilling the following essential conditions:\n* the evolution takes place without external interventions;\n* the overall structural development depends only on local rules.\nTo a large extent, the CAM is capable of simulating the dynamics of complex systems which are disposed to organize themselves through unstable critical states until reaching a stable configuration. In addition, the CAM implementation makes it possible to generate global coherent patterns starting from suitable local instructions, without any external supervisor in charge for understanding the process in its entirety . \nIn practical applications, each restricted portion of space contains a finite number of cells, which assume a finite set of (time-dependent) states. The initial configuration $\\bar{\\mathbf{z}}^0$ is the combination of a *ground state* configuration **z**0 with some perturbation **w**, and for instance\n \n$$\\bar{\\mathbf{z}}^0 = \\mathbf{z}^0 + \\mathbf{w}\\,.$$ \nAfter a given time interval Δ*t* > 0, the system typically comes to a new state **z**1, designated *final configuration*, which is determined by the changes of state of the single cell together with all the others, according to the preliminarily established series of (local) rules. Therefore, the evolution is described by means of a function Φ mapping some input $\\bar{\\mathbf{z}}^0$ defined (from **z**0 and **w**) over the graph/network Γ to an output **z**1, as specified by the simple formula $\\mathbf{z}^1=\\Phi(\\bar{\\mathbf{z}}^0)$.\nIt is worth stressing that, in order to properly manage the patients between hospitals of the healthcare system, an optimal underlying ethical-cooperative paradigm has to be stipulated .\nSandpiles on a general network\n------------------------------\nWe start by recollecting some basic definitions in *graph theory* .\nA **graph** (or **network**) is a pair Γ = (*X*, *E*) where *X* is a set whose elements are called **vertexes** (or **nodes**) and *E* is a set of paired vertexes called **edges** (or **links**). A **rooted graph** (or **pointed graph**) is a graph in which one vertex, denoted by *x*\\*, is distinguished as the **root** of the graph.\nIn what follows, the root *x*\\* of the graph is also referred to as the **hub**, coherently with the subsequent meaning of the specific application to the healthcare system.\nAn **adjacent vertex** *y* ∈ *X* of *x* ∈ *X* is a vertex which is connected to *x* by an edge, so that (*x*, *y*) ∈ *E*. The **neighbourhood** *I*(*x*) of a vertex *x* is the subgraph composed of the vertexes adjacent to *x* and all the edges connecting vertexes adjacent to *x*. The **degree** (or **valency**) of a vertex *x*, denoted by \\rm deg(*x*), is the number of edges which are incident to the vertex *x*.\nLet the graph Γ be connected, finite (with a finite number of vertexes denoted by *x*1, *x*2, …, *x**p*), simple (i.e. there are no loops connecting any vertex with itself) and undirected (i.e. the edges are bidirectional). We assume that a stock of identical particles is initially allocated at any vertex *x**i* of the graph Γ. The configurations **z**0, $\\bar{\\mathbf{z}}^0$ and **z**1, collecting the number of elements located at *x**i* for any *i*, are natural-valued functions defined on the graph. The rules of the evolution are established to guarantee *stability* of the sandpile dynamics, starting from the definition of the **height function** given by **z** : *X* → N*p* (refer to Figure [figure:sandpile] for the graphical representation of a sandpile on a two-dimensional Cartesian grid).\n![A simple sandpile with variable height (on a two-dimensional Cartesian grid)](sandpile \"fig:\") [figure:sandpile]\nThe height *z**i* = **z**(*x**i*) is **unstable** (at *x**i* ∈ *X*) if *z**i* ≥ \\rm deg(*x**i*), where \\rm deg(*x**i*) is the degree of the vertex *x**i*. Otherwise, the vertex is **stable**. \nA **stable configuration** is a configuration of vertexes which are stable for any index *i*. A **maximum stable configuration** (or **minimally stable state**) is a configuration in which all the vertexes have a height *z**i* of one unit lower than the corresponding threshold value \\rm deg(*x**i*). An **almost stable configuration** is a configuration of vertexes which are stable for any index *i* different from the root, and a **maximum almost stable configuration** is defined accordingly.\nIn that framework, an unstable vertex is forced to *topple* over its neighbouring vertexes, thus causing a change of state in the entire configuration.\nA **toppling** (or **firing**) from the vertex *x**i* is the mapping Ξ from the graph Γ to itself determined by the following (local) rules:\n \n$$\\Xi(\\mathbf{z})\\_j = z\\_j - \\Delta\\_{i\\to j} \\qquad \\textrm{with} \\quad\n \\Delta\\_{i\\to j} := \\left\\{\\begin{aligned}\n & \\textrm{\\rm deg}(x\\_i) && \\textrm{if}\\;\\;x\\_j=x\\_i\\\\\n & -1 && \\textrm{if}\\;\\; x\\_j\\in I(x\\_i)\\\\\n & \\;\\; 0 && \\textrm{otherwise}\n \\end{aligned}\\right.$$ \nand Δ*i* → *j* is called the **toppling matrix** .\nIn principle, different stability criteria could be proposed involving values other than the degree \\rm deg(*x**i*) of the vertex *x**i*, and modified redistribution criteria could also be considered, but we focus on the above definitions for the sake of simplicity in presentation.\nThe addition of a particle to the sandpile is schematized by summing to the ground state configuration **z**0 = (*z*10, *z*20, …, *z**p*0) the vector $\\boldsymbol{\\delta}\\_i$ with 1 at some fixed index *i* and zeros elsewhere. By iterating this procedure *m* times for possibly different choices of the index, the initial configuration $\\bar{\\mathbf{z}}^0$ is finally given by **z**0 + **w** with $\\mathbf{w}=\\boldsymbol{\\delta}\\_{i\\_1}+\\boldsymbol{\\delta}\\_{i\\_2}+\\dots+\\boldsymbol{\\delta}\\_{i\\_m}$. It is a straightforward consequence that the operation of adding particles to a sandpile is associative. Moreover, the final configuration **z**1 resulting from the evolution is independent of the order in which the topplings are performed, and therefore the operation of toppling is commutative . \nAs an illustrative case, we assume that \\rm deg(*x**i*) = \\rm deg(*x**j*) for two adjacent vertexes *x**i* and *x**j*. Then, we consider a sandpile in which both *x**i* and *x**j* are at their critical height, that is *z**i* ≥ \\rm deg(*x**i*) and *z**j* ≥ \\rm deg(*x**j*). According to the dynamics described above, a toppling from the vertex *x**i* causes the vertex *x**j* to become unstable, and viceversa, and the sandpile comes to a configuration in which the height *z**k* for some *k* ∈ {1, 2, …, *p*} increases by a value △ *i* → *k* + △ *j* → *k*. Such procedure being symmetrical, by applying this argument repeatedly, we conclude that the same stable configuration is reached irrespective of whether *x**i* or *x**j* is toppled first, and in general regardless of the sequence in which unstable vertexes are toppled.\nTwo-dimensional Cartesian grids\n-------------------------------\nWe focus on the simple case of a two-dimensional Cartesian grid, with two standard schemes for adjacent vertexes given by the *von Neumann neighbourhood*, consisting of the four cells obtained moving one step towards North/East/South/West (refer to Figure [figure:neighbourhood](left)), and the *Moore neighbourhood*, which includes also the four diagonal cells (refer to Figure [figure:neighbourhood](right)). Of course, selecting a more general network provides a higher degree of flexibility and allows to closely represent the connections between various medical facilities of the healthcare system (refer to Figure [figure:network]).\n![The von Neumann neighbourhood (left) of the (blue) central cell (an extended neighbourhood includes also the pink cells) and the Moore neighbourhood (right)](CA-Moore.png \"fig:\") [figure:neighbourhood]\nWe consider a special grid of *p* = *n*2 cells organized over a squared matrix of size *n* × *n*, where the particles are dropped randomly being allowed to stack on top of each other, so that a configuration of the sandpile is described by an element of the space of natural-valued matrices\n \n$$\\mathcal{M}\\_{n}(\\mathbb{N}) := \\bigl\\{ A\\in \\mathbb{R}^{n\\times n} \\,:\\, a\\_{ij}\\in \\mathbb{N} \\;\\; \\textrm{for any} \\;\\; i,j=1,2,\\dots,n \\bigr\\}.$$ \nThe position of each vertex *x**i* (with its corresponding height *z**i* ∈ N) is determined by the index *i* ∈ {1, 2, …, *p* = *n*2} according to the following common practise:\n \n$$\\begin{pmatrix}\n 1 & 2 & 3 & \\dots & n \\\\\n n+1 & n+2 & n+3 & \\dots & 2n \\\\\n 2n+1 & 2n+2 & 2n+3 & \\dots & 3n \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n \\; n(n-1)+1 & n(n-1)+2 & \\dots & \\hdots & n^2\n \\end{pmatrix}$$ \nFor the von Neumann neighborhood, we have \\rm deg(*x**i*) = 4 for any index *i* associated with an element of the bulk of the matrix, and for instance\n \n$$I(x\\_{n+2}) = \\bigl\\{ x\\_2 \\,\\textrm{(North)}, \\,x\\_{n+3} \\,\\textrm{(East)}, \\,x\\_{2n+2} \\,\\textrm{(South)}, \\,x\\_{n+1} \\,\\textrm{(West)} \\bigr\\}.$$ \nSimilarly, for the Moore neighborhood, we have \\rm deg(*x**i*) = 8 for any index *i* associated with an element of the bulk of the matrix, and for instance\n \n$$\\begin{split}\n I(x\\_{n+2}) = \\bigl\\{ & x\\_2 \\,\\textrm{(North)}, \\,x\\_3 \\,\\textrm{(North-East)}, \\,x\\_{n+3} \\,\\textrm{(East)}, \\,x\\_{2n+3} \\,\\textrm{(South-East)},\\\\\n & \\,x\\_{2n+2} \\,\\textrm{(South)}, \\,x\\_{2n+1} \\,\\textrm{(South-West)}, \\,x\\_{n+1} \\,\\textrm{(West)}, \\,x\\_1 \\,\\textrm{(North-West)} \\bigr\\}.\n \\end{split}$$ \nWe analyze the simple case *n* = 3 (*p* = *n*2 = 9) with the von Neumann neighbourhood. By combining the perturbation $\\mathbf{w}=4\\,\\boldsymbol{\\delta}\\_5$ with the ground state configuration **z**0 = **0**, we obtain the following initial and final configurations\n \n$$\\bar{\\mathbf{z}}^0 = \\mathbf{0} + 4\\,\\boldsymbol{\\delta}\\_5\n = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & \\mathbf{4} & 0 \\\\ 0 & 0 & 0 \\end{pmatrix}\n \\quad \\longrightarrow \\quad\n \\mathbf{z}^1 = \\Xi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix}$$ \nAs expected, the threshold value \\rm deg(*x*5) = 4 is reached at the central cell, thus inducing a destabilisation of the corresponding vertex, with the consequence that four particles are poured into the adjacent vertexes –the elements of the von Neumann neighbourhood– to compose a stable final configuration.\nAs a matter of fact, the toppling from an unstable vertex changes the state of the adjacent vertexes, sometimes causing the appearance of instabilities at some other vertex. Then, subsequent topplings over the adjacent vertexes are generated, triggering a sequence of events which are evocative of *sandpile avalanches* and stopping only when all the cells return strictly below their threshold capacity. \nIf the graph is infinite and connected, with a finite number of particles, all vertexes become stable after a finite number of topplings; moreover, it can be proven that the final (stable) configuration **z**1 depends solely on the initial configuration $\\bar{\\mathbf{z}}^0$, independently of the order in which the topplings are performed . \nOn the other hand, if the graph is finite, appropriate boundary conditions must be implemented: assuming that the particles are evacuated from the boundaries, an analogous result of stability holds , but other types of boundary conditions may not guarantee the same property. During the operation of toppling, no particles are created as they are redistributed to neighbouring cells. For dissipative boundary conditions, the toppling can actually lead to the loss of particles if it occurs on a boundary cell, and we refer to this case as *open boundary conditions*.\nApplication to healthcare system management\n===========================================\nIn this article, we pursue the idea of applying the Abelian Sandpile paradigm to the management of patients among the COVID-19 hospitals in Italy. \nThe current organisation assigns responsibility for the healthcare system to the regions, which are restricted territorial bodies with their own statutes, powers and functions established by the Italian Constitution. We refer to the Lazio region when selecting (the order of magnitude of) the number of medical facilities relevant to the numerical simulations. More specifically, the healthcare system of the Lazio region is composed of about 100 hospitals, mostly located in the metropolitan area of Rome, which are structured within a network of reciprocal connections .\n![Arrangement of hospitals on a general unstructured and undirected network.](network \"fig:\") [figure:network]\nIn that framework, the **height function** of the nodes (hospitals) indicates the number of hospitalised patients, and the links between different nodes represent the possibility of a direct exchange of patients (that is subject to constraints of geographical proximity, together with inherent organisational requirements of the connecting medical facilities). The node of the network exhibiting the largest number of links is the **hub**, which usually takes on a strategic function for the whole system, whilst the nodes with fewer links identify the hospitals with unfavorable geographical location and reduced capacity/functionality (each hospital having different resources, in the general case the threshold value changes from node to node).\nReassignment of outgoing particles\n----------------------------------\nWe have already discussed about the crucial role played by the boundary conditions associated with the ASM settled on finite networks, and indeed the existence of stable configurations is strongly influenced by the presence of a *dissipation mechanism* such as the open boundary conditions. \nBecause the loss of particles through the edge of the network and the hypothesis of an infinite network are unrealistic assumptions for practical applications, we have to implement a suitable *reassignment law* for particles generating a critical height at some boundary cells. Hence, we are induced to redistributing the outgoing particles to the root of the network, namely the hub, which is typically a collecting node of the healthcare system. Therefore, we call the resulting process a Sandpile with Redistribution to the Hub (SRH), and we notice that actually the SRH is equivalent to the ASM when there are no topplings occurring at the edge of the network.\nIn particular, for two-dimensional Cartesian grids with odd size, we incorporate the reassignment of outgoing particles from cells toppling at the border of the matrix by proposing that they are reallocated to the central cell. Furthermore, in case the threshold capacity is reached in several cells at the same time, we impose that the first cell to topple is always the hub, and then additional toppling is performed from the other nodes. In particular, the hub is allowed to topple only once, at the very beginning of the toppling procedure.\nWe summarize the essential steps of the *CAM workflow* –which turns out to be also the implementation scheme of the numerical codes– as follows.\n* *Initialisation*. We choose a (stable) ground state **z**0 = (*z*10, *z*20, …, *z**p*0) which satisfies the condition that *z**i*0 < \\rm deg(*x**i*) for any index *i*;\n* *Inflow*. We combine the ground state **z**0 with a perturbation **w** = (*w*1, *w*2, …, *w**p*) which indicates the number of new patients requiring hospitalization, and we obtain the initial configuration $\\bar{\\mathbf{z}}^0=\\mathbf{z}^0+\\mathbf{w}$;\n* *Hub toppling*. If *z*\\*0 ≥ \\rm deg(*x*\\*), the hub performs a toppling towards the adjacent nodes of its neighbourhood *I*(*x*\\*);\n* *Additional toppling*. If *z**i*0 ≥ \\rm deg(*x**i*) for some *x**i* ≠ *x*\\*, these nodes perform (a sequence of) topplings until reaching an almost stable configuration $\\mathbf{z}^1=\\Phi(\\bar{\\mathbf{z}}^0)$;\n* *Iteration*. We reinitialize the ground state with **z**0 equal to **z**1, and we restart from 2.\nComparison with the standard healthcare system management\n---------------------------------------------------------\nIn order to to compare the efficiency of the novel proposal with the standard organisation of the Italian healthcare system, we provide a reformulation of the current management of connections between hospitals in terms of the CAM paradigm. \nPerhaps surprisingly, the connection between medical facilities is presently mainly determined by their geographical proximity, corresponding to the basic *adjacency* of nodes. If a patient comes to a hospital where there are no places available, because the threshold capacity has been reached, then the single patient is reallocated to the nearest hospital with available places, and the reassignment is limited to the patients exceeding the threshold value. That being the case, the redistribution of patients is handled manually at each hospitalization, without foreseeing the possibility of repeated similar events which are instead highly probable during sudden and unexpected emergencies.\nWe summarize the essential steps of the *standard workflow* as follows, by assuming the reassignement law from the edge of the network at the hub as in Section [section:outgoing].\n* *Initialisation/Inflow*. We proceed as in Section [section:outgoing];\n* *Hub redistribution*. If *z*\\*0 ≥ \\rm deg(*x*\\*), the hub reallocates only the exceeding patients to the adjacent medical facilities, starting from the less crowded ones (in case of equal crowding, a random choice is operated);\n* *Additional redistribution*. If *z**i*0 ≥ \\rm deg(*x**i*) for some *x**i* ≠ *x*\\*, the redistribution procedure is repeated for these nodes, until reaching an almost stable configuration $\\mathbf{z}^1=\\Psi(\\bar{\\mathbf{z}}^0)$;\n* *Iteration*. We reinitialize the ground state with **z**0 equal to **z**1, and we restart from 2.\nWe illustrate the comparison by considering the following simple example.\nLet the healthcare system be represented by a two-dimensional Cartesian grid of size 3 × 3, with the von Neumann neighbourhood (so that \\rm deg(*x**i*) = 4 for any index *i*) and reassignment of outgoing particles to the hub located at the central cell. We choose\n \n$$\\mathbf{z}^0 = \\begin{pmatrix} 2 & 1 & 3 \\\\ 1 & 3 & 1 \\\\ 1 & 0 & 2 \\end{pmatrix}\n \\quad\\textrm{and}\\quad\n \\mathbf{w} = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix},$$ \nso that\n \n$$\\bar{\\mathbf{z}}^0 = \\mathbf{z}^0+\\mathbf{w}\n = \\begin{pmatrix} 2 & 1 & 3 \\\\ 1 & \\mathbf{4} & 1 \\\\ 1 & 0 & 2 \\end{pmatrix}$$ \nand the initial configuration is unstable. The standard organization manages this critical situation by moving a single particle (patient) from the central cell towards one in the von Neumann neighbourhood, and preferably the South cell, to obtain\n \n$$\\Psi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix} 2 & 1 & 3 \\\\ 1 & 3 & 1 \\\\ 1 & 1 & 2 \\end{pmatrix},$$ \nwhere Ψ denotes the *evolution mapping* of the grid according to the standard approach. Then, we reinitialize the ground state with $\\Psi(\\bar{\\mathbf{z}}^0)$ and the perturbation **w** is chosen as above, so that the configuration $\\Psi(\\bar{\\mathbf{z}}^0)+\\mathbf{w}$ is also unstable with respect to the central cell. By selecting randomly one of the neighbouring cells, another single particle is transferred, and after two additional iterations, the system reaches the final configuration\n \n$$\\Psi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix} 2 & 2 & 3 \\\\ 2 & 3 & 2 \\\\ 1 & 1 & 2 \\end{pmatrix}.$$ \nOn the other hand, the ASP paradigm suggests to transfer at the same time all the four patients initially allocated to the hub towards the von Neumann neighbourhood, to obtain\n \n$$\\Phi(\\mathbf{z}^0) = \\begin{pmatrix} 2 & 2 & 3 \\\\ 2 & 0 & 2 \\\\ 1 & 1 & 2 \\end{pmatrix}.$$ \nFinally, under the assumption that new patients are always collected at the hub, the system can accept exactly 3 more particles from the additional iterations, before reaching a critical situation as before. Of course, it could be argued that, being the final configurations the same, there would be no obvious evidence to prefer the novel approach to the standard organization. However, the deciding factor is that we have unified within a single time-step the efforts for reallocating patients among the medical facilities, which is usually regarded as a source of stress for the overall healthcare system. Hence, we have left the structure time to reorganize without suffering a constant state of emergency, which is precisely the issue of the so-called *predictive logistics*.\nExamples and numerical simulations\n==================================\nWe collect numerical results from a selection of preliminary cases, which are nevertheless useful to understand the alternative methods of healthcare system management provided by the two approaches described in Section [section:healthcare], and their inherent dynamics. We consider two-dimensional Cartesian grids, which are arranged into *n* × *n* (squared) matrices with the Moore neighbourhood, so that \\rm deg(*x**i*) = 8 for any index *i* ∈ {1, 2, …, *p* = *n*2}, and we assume the reassignment to the hub of outgoing particles from the edge of the network as in Section [section:outgoing]. \nWe recall that each cell/node of the grid represents a hospital, and the height function reproduces the number of patients in the medical facilities, expressed as the percentage of capacity already achieved.\nWe start by focusing on the possible outputs of a basic case consisting in a single hospital originally attaining its threshold capacity, which is located at the hub (central cell), with a (randomly chosen) stable ground state configuration. To highlight possibly critical situations, we boldmark all values above the threshold capacity, and we encircle the values 5, 6 and 7 dangerously close to the saturation level.\nFor *n* = 3, the hospitals are *p* = *n*2 = 9, and we choose an initial configuration with all new patients coming to the hub *x*5, given by the perturbation $\\mathbf{w}=4\\,\\boldsymbol{\\delta}\\_5$, so that\n \n$$\\mathbf{z}^0 = \\begin{pmatrix} 2 & 3 & 1 \\\\ \\circled{5} & \\circled{7} & 2 \\\\ 4 & 3 & 3 \\end{pmatrix}\n \\quad\\textrm{and}\\quad\n \\bar{\\mathbf{z}}^0 = \\begin{pmatrix} 2 & 3 & 1 \\\\ \\circled{5} & {\\bf 11} & 2 \\\\ 4 & 3 & 3 \\end{pmatrix}.$$ \nBecause the hub has to manage a number of hospitalisations greater than its threshold capacity, some patients must be transferred to adjacent facilities. \nAccording to the standard strategy, only four patients need a different allocation to be found among the neighbourhing cells, ending at the (stable) configuration\n \n$$\\Psi(\\bar{\\mathbf{z}}^0)\n = \\begin{pmatrix} 2 & 4 & 1 \\\\\n \\circled{5} & \\circled{7} & 4 \\\\\n 4 & 4 & 3 \\end{pmatrix}.$$ \nAs it is clearly seen, an additional iteration with $\\mathbf{w}=4\\,\\boldsymbol{\\delta}\\_5$ determines a new unstable/critical configuration at the central node (with again four patients in excess), demanding for a further reorganisation of the hub by means of another transfer operation. On the other hand, by employing the SRH approach, we obtain a different outcome given by\n \n$$\\Phi(\\bar{\\mathbf{z}}^0)\n = \\begin{pmatrix} 3 & 4 & 2 \\\\\n \\circled{6} & 3 & 3 \\\\\n \\circled{5} & 4 & 4 \\end{pmatrix},$$ \nsuch situation being more favorable in terms of load-balancing since an additional iteration with perturbation $\\mathbf{w}=4\\,\\boldsymbol{\\delta}\\_\\ast$ does not produce any unstable configuration. Moreover, there is still a certain amount of available places at the hub, and therefore the novel proposal solves the criticalities possibly generated by an emergency.\nDespite its simplicity, the above example suggests to attribute to each configuration the value of a given functional, aiming to provide an easily readable *indicator* of the effectiveness of the allocation strategy. This is necessarily a very delicate issue. \nAs a first attempt, a possible choice is to count the total number of medical facilities attaining a given fraction of their threshold capacity, which are denoted by *critical points*. Such choice is actually quite questionable, since it does not take keep memory of the value of incoming patients at the beginning of the iteration. \nA different (rough) quantitative measure of the system management efficiency is\n \n$$\\mathcal{F}(\\mathbf{w},\\mathbf{z}) := \\frac{\\mathbf{w}}{\\sum\\_{j} w\\_j}\\cdot \\mathbf{z}\\,,$$ \nwhich intends to evaluate the risk that a new patient comes to a given location, taking into account the previous history of the system (in this case, the effect of the perturbation **w**).\nFor the case illustrated in the Example 3, the function F equals the state of the hub *x*5, so that\n \n$$\\mathcal{F}(\\mathbf{w},\\bar{\\mathbf{z}}^0) = \\bar z^0\\_5 = 11,\\quad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Psi(\\bar{\\mathbf{z}}^0)\\bigr) = \\Psi(\\bar{\\mathbf{z}}^0)\\_5 = 7,\\quad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Phi(\\bar{\\mathbf{z}}^0)\\bigr) = \\Phi(\\bar{\\mathbf{z}}^0)\\_\\ast = 3\\,.$$ \nThe minimum value is achieved for the configuration $\\Phi(\\bar{\\mathbf{z}}\\_0)$ corresponding to the SRH approach (or, equivalently, to ASM).\nFor *n* = 5, the hospitals are *p* = *n*2 = 25, and we choose an initial configuration with all new patients coming to the hub *x*13, given by the perturbation $\\mathbf{w}=\\boldsymbol{\\delta}\\_{9}+\\boldsymbol{\\delta}\\_{12}+4\\boldsymbol{\\delta}\\_13\n+2\\boldsymbol{\\delta}\\_{14}+\\boldsymbol{\\delta}\\_{18}+\\boldsymbol{\\delta}\\_{19}$, so that\n \n$$\\mathbf{z}^0 = \\begin{pmatrix}\n 1 & 2 & 4 & 2 & \\circled{5} \\\\\n 2 & 4 & 2 & 3 & 1 \\\\\n 3 & 2 & \\circled{7} & 2 & 3 \\\\\n 2 & 1 & 4 & 2 & 2 \\\\\n 4 & 2 & 1 & \\circled{5} & 4\n \\end{pmatrix}\n \\quad\\textrm{and}\\quad\n \\bar{\\mathbf{z}}^0 = \\begin{pmatrix}\n 1 & 2 & 4 & 2 & \\circled{5} \\\\\n 2 & 4 & 2 & 4 & 1 \\\\\n 3 & 3 & {\\bf 11} & 4 & 3 \\\\\n 2 & 1 & \\circled{5} & 3 & 2 \\\\\n 4 & 2 & 1 & \\circled{5} & 4\n \\end{pmatrix}.$$ \nAccording to the standard strategy, an admissible solution could be\n \n$$\\Psi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix}\n 1 & 2 & 4 & 2 & \\circled{5} \\\\\n 2 & 4 & 4 & 4 & 1 \\\\\n 3 & 3 & \\circled{7} & 4 & 3 \\\\\n 2 & 3 & \\circled{5} & 3 & 2 \\\\\n 4 & 2 & 1 & \\circled{5} & 4\n \\end{pmatrix},$$ \nand the same observation previously done holds, that is such configuration likely determines an instability located at the hub after a subsequent process iteration, since the number of hospitalised patients is very close to the critical threshold. \nOn the other hand, by employing the SRH approach, we obtain\n \n$$\\Phi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix}\n 1 & 2 & 4 & 2 & \\circled{5} \\\\\n 2 & \\circled{5} & 3 & \\circled{5} & 1 \\\\\n 3 & 4 & 3 & \\circled{5} & 3 \\\\\n 2 & 2 & \\circled{6} & 4 & 2 \\\\\n 4 & 2 & 1 & \\circled{5} & 4\n \\end{pmatrix},$$ \nwhere the number of patients hospitalised at the hub is far away from the critical level. Indeed, by computing the indicator function F at the different outcomes, we deduce that\n \n$$\\mathcal{F}\\bigl(\\mathbf{w},\\bar{\\mathbf{z}}^0\\bigr) = 6.7,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Psi(\\bar{\\mathbf{z}}^0)\\bigr) = 5.1,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Phi(\\bar{\\mathbf{z}}^0)\\bigr) = 4.1\\,,$$ \nsince ∑*j**w**j* = 10, and the smallest value is achieved by the configuration resulting from the SRH approach; thus, such strategy must be preferred to the standard one.\nNext, we explore a case where two subsequent topplings occur.\n[ex:10] For *n* = 5, we choose the ground state and the initial configuration as follows,\n \n$$\\mathbf{z}^0 = \\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 0 & \\circled{5} & 1 & 1 \\\\\n 1 & 2 & \\circled{7} & \\circled{7} & 4 \\\\\n \\circled{5} & \\circled{5} & 2 & 4 & \\circled{5} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix}\n \\quad\\textrm{and}\\quad\n \\bar{\\mathbf{z}}^0 = \\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 0 & \\circled{5} & 1 & 1 \\\\\n 1 & 2 & {\\bf 11} & \\circled{7} & 4 \\\\\n \\circled{5} & \\circled{5} & 2 & 4 & \\circled{5} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix},$$ \nwhere $\\mathbf{w}=4\\boldsymbol{\\delta}\\_{13}$. Then, according to the standard strategy, a final configuration is given by\n \n$$\\Psi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 3 & \\circled{5} & 2 & 1 \\\\\n 1 & 2 & \\circled{7} & \\circled{7} & 4 \\\\\n \\circled{5} & \\circled{5} & 2 & 4 & \\circled{5} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix}.$$ \nFor applying the SRH approach, we pass through an intermediate configuration\n \n$$\\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 0+1 & \\circled{5+1} & 1+1 & 1 \\\\\n 1 & 2+1 & 11-8 & \\mathbf{7+1} & 4 \\\\\n \\circled{5} & \\circled{5+1} & 2+1 & \\circled{4+1} & \\circled{5} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix}\n =\n \\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 1 & \\circled{6} & 2 & 1 \\\\\n 1 & 3 & 3 & \\mathbf{8} & 4 \\\\\n \\circled{5} & \\circled{6} & 3 & \\circled{5} & \\circled{5} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix},$$ \nin which the cell *x*12 is unstable, and finally we arrive at the final configuration\n \n$$\\Phi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 1 & \\circled{6+1} & 2+1 & 1+1 \\\\\n 1 & 3 & 3+1 & 8-8 & \\circled{4+1} \\\\\n \\circled{5} & \\circled{6} & 3+1 & \\circled{5+1} & \\circled{5+1} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix}\n = \\begin{pmatrix}\n 4 & 1 & 0 & 1 & 3 \\\\\n \\circled{5} & 1 & \\circled{7} & 3 & 2 \\\\\n 1 & 3 & 4 & 0 & \\circled{5} \\\\\n \\circled{5} & \\circled{6} & 4 & \\circled{6} & \\circled{6} \\\\\n 3 & \\circled{5} & 4 & \\circled{5} & 3\n \\end{pmatrix}.$$ \nThe matrices $\\Psi(\\bar{\\mathbf{z}}^0)$ and $\\Phi(\\bar{\\mathbf{z}}^0)$ are represented in Figure [figure:figure6] with different colors assigned to cells, corresponding to the degree of saturation achieved.\n![Model comparison from the Example [ex:10]: final configurations \\Psi(\\bar{\\mathbf{z}}^0) (left) and \\Phi(\\bar{\\mathbf{z}}^0) (right). The colors attached to each cell/node represent its corresponding relative capacity (0=green, 1/2=yellow, 3/4=magenta, 4/5=red, 6/7=black)](figure6 \"fig:\") [figure:figure6]\nAt the bottom of each subfigure, we report the counting of critical points, given by the number of cells with values 6 or 7. Although a quick glance can make us suspect that the left configuration is preferable, the situation is indeed different. In fact, first of all we distinguish the presence of two almost critical values 7 inside the configuration plotted on the left, against the presence of only one value 7 from the second approach. Furthermore, the hub –the facility most at risk taking into account recent history– has a value 7 in the first case, whilst a value 4 occurs in the second case, thus making the latter situation preferable.\nFinally, we compute the function F corresponding to the different configurations: since **w**/∑*j**w**j* is everywhere zero except at the hub, where it is equal to 1, the values of F coincide with the values at the hub, which are given by\n \n$$\\mathcal{F}(\\mathbf{w},\\bar{\\mathbf{z}}^0) = 11,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Psi(\\bar{\\mathbf{z}}^0)\\bigr) = 7,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Phi(\\bar{\\mathbf{z}}^0)\\bigr) = 4,$$ \nsuggesting again the advantages of configurations proposed by the SRH approach.\nSimulation of multiple central outbreaks\n----------------------------------------\nWe analyze the case with more than one hospital in the same restricted area (including the hub) which is concerned with new incoming patients. One might think of this event as the emergence of multiple outbreaks within some smaller district of a larger area. We consider a larger network, with *n* = 9, whose size is comparable to the capacity of the healthcare system in the Lazio region in Italy .\nWe consider the perturbation $\\mathbf{w}=2\\boldsymbol{\\delta}\\_{32}+\\boldsymbol{\\delta}\\_{33}+5\\boldsymbol{\\delta}\\_{41}+2\\boldsymbol{\\delta}\\_{42}$ and the ground state\n \n$$\\mathbf{z}^0 = \\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} & 2 & 3 & \\circled{6} & 2 \\\\\n 3 & 2 & \\circled{5} & 2 & 1 & 3 & 2 & 4 & 3 \\\\\n 3 & 3 & 1 & \\circled{5} & \\circled{6} & 2 & 3 & 1 & 3 \\\\\n \\circled{6} & 1 & \\circled{5} & 3 & 2 & \\circled{5} & 2 & 3 & 4 \\\\\n 1 & 3 & 2 & 1 & \\circled{6} & 3 & 4 & \\circled{5} & \\circled{5} \\\\\n 1 & 4 & 1 & 2 & 2 & \\circled{5} & 1 & 2 & 3 \\\\\n 4 & 1 & 3 & 4 & \\circled{5} & \\circled{6} & 2 & \\circled{5} & 3 \\\\\n 4 & 2 & 3 & \\circled{5} & 2 & 2 & \\circled{6} & 3 & 1 \\\\\n 1 & \\circled{6} & \\circled{5} & 2 & 4 & 4 & 2 & 1 & 2\n \\end{pmatrix},$$ \nso that ∑*j**z**j*0 = 250 and ∑*j**w**j* = 10, corresponding to the initial configuration\n \n$$\\begin{aligned}\n \\bar{\\mathbf{z}}^0 &= \\begin{pmatrix}\n 1 &2 & 1 &4 & \\circled{6} &2 &3 &\\circled{6} &2 \\\\\n 3 &2 & \\circled{5} &2 & 1 &3 &2 &4 &3 \\\\\n 3 &3 & 1 & \\circled{5} & \\circled{6} &2 &3 &1 &3 \\\\\n \\circled{6} &1 & \\circled{5} &3 & 2+2 & \\circled{5+1} &2 &3 &4 \\\\\n 1 &3 & 2 &1 & \\circled{6+5} & \\circled{3+2} &4 & \\circled{5} & \\circled{6} \\\\\n 1 &4 & 1 &2 & 2 & \\circled{5} &1 &2 &3 \\\\\n 4 &1 & 3 &4 & \\circled{5} & \\circled{6} &2 & \\circled{5} &3 \\\\\n 4 &2 & 3 & \\circled{5} & 2 &2 & \\circled{6} &3 &1 \\\\\n 1 & \\circled{6} & \\circled{5} &2 & 4 &4 &2 &1 &2\n \\end{pmatrix}\\\\ \\\\\n &= \\begin{pmatrix}\n 1 &2 &1 &4 & \\circled{6} &2 &3 & \\circled{6} &2 \\\\\n 3 &2 & \\circled{5} &2 &1 &3 &2 &4 & \\circled{6} \\\\\n 3 &3 &1 & \\circled{5} & \\circled{6} &2 &3 &1 &3 \\\\\n \\circled{6} &1 & \\circled{5} &3 &4 & \\circled{6} &2 &3 &4 \\\\\n 1 &3 &2 &1 &{\\bf 11} & \\circled{5} &4 & \\circled{5} & \\circled{6} \\\\\n 1 &4 &1 &2 &2 & \\circled{5} &1 &2 &3 \\\\\n 4 &1 &3 &4 & \\circled{5} & \\circled{6} &2 & \\circled{5} &3 \\\\\n 4 &2 &3 & \\circled{5} &2 &2 & \\circled{6} &3 &1 \\\\\n 1 &3 & \\circled{5} &2 &4 &4 &2 &1 &2\n \\end{pmatrix}.\n\\end{aligned}$$ \nThen, according to the standard strategy, a final configuration is given by\n \n$$\\Psi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} &2 &3 & \\circled{6} & 2 \\\\\n 3 & 2 & \\circled{5} & 2 &1 &3 &2 & 4 & 3 \\\\\n 3 & 3 & 1 & \\circled{5} & \\circled{6} &2 &3 & 1 & 3 \\\\\n \\circled{6} & 1 & \\circled{5} & 3+0 & 4+0 &\\circled{6+0} &2 & 3 & 4 \\\\\n 1 & 3 & 2 & 1+2 & \\circled{11-4} &\\circled{5+0} &4 & \\circled{5} & \\circled{6} \\\\\n 1 & 4 & 1 & 2+1 & 2+1 &\\circled{5+0} &1 & 2 & 3 \\\\\n 4 & 1 & 3 & 4 & \\circled{5} &6 &2 & \\circled{5} & 3 \\\\\n 4 & 2 & 3 & \\circled{5} &2 &2 &\\circled{6} & 3 & 1 \\\\\n 1 & \\circled{6} & \\circled{5} & 2 &4 &4 &2 & 1 & 2\n \\end{pmatrix}.$$ \nHence, the central subgraph around the hub is composed by the elements\n \n$$\\begin{pmatrix} 3 & 4 & \\circled{6} \\\\ 3 & \\circled{7} & \\circled{5} \\\\ 3 & 3 & \\circled{5} \\end{pmatrix}.$$ \nIn the subsequent steps, any additional patients coming to the hub destabilises the configuration. On the other hand, by employing the SRH approach, we obtain\n \n$$\\Phi(\\bar{\\mathbf{z}}^0)=\\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} & 2 & 3 & \\circled{6} & 2 \\\\\n 3 & 2 & \\circled{5} & 2 & 1 & 3 & 2 & 4 & 3 \\\\\n 3 & 3 & 1 & \\circled{5} & \\circled{6} & 2 & 3 & 1 & 3 \\\\\n \\circled{6} & 1 & \\circled{5} & 3+1 & \\circled{4+1} & \\circled{6+1} & 2 & 3 & 4 \\\\\n 1 & 3 & 2 & 1+1 & 11-8 & \\circled{5+1} & 4 & \\circled{5} & \\circled{6} \\\\\n 1 & 4 & 1 & 2+1 & 2+1 & \\circled{5+1} & 1 & 2 & 3 \\\\\n 4 & 1 & 3 & 4 & \\circled{5} & \\circled{6} & 2 &\\circled{5} & 3 \\\\\n 4 & 2 & 3 & \\circled{5} & 2 & 2 & \\circled{6} & 3 & 1 \\\\\n 1 & \\circled{6} & \\circled{5} & 2 & 4 & 4 & 2 & 1 & 2\n \\end{pmatrix},$$ \nand the corresponding central subgraph around the hub is\n \n$$\\begin{pmatrix} 4 & \\circled{5} & \\circled{7} \\\\ 2 & 3 & \\circled{6} \\\\ 3 & 3 & \\circled{6} \\end{pmatrix}.$$ \nThe difference between $\\Psi(\\bar{\\mathbf{z}}^0)$ and $\\Phi(\\bar{\\mathbf{z}}^0)$ is transparent regarding, specifically, the number of patients hospitalised in the hub. A further iteration with the same perturbation **w** would lead to a new critical situation for the hub in the first case, but it does not in the second.\nFigure [figure:figure7] provides a representation of the two configurations $\\Psi(\\bar{\\mathbf{z}}^0)$ and $\\Phi(\\bar{\\mathbf{z}}^0\\bigr)$.\n![Model comparison from the case of central areas outbreaks: final configurations \\Psi(\\bar{\\mathbf{z}}^0) (left) and \\Phi(\\bar{\\mathbf{z}}^0) (right). The colors attached to each cell/node represent its corresponding relative capacity (0=green, 1/2=yellow, 3/4=magenta, 4/5=red, 6/7=black)](figure7 \"fig:\") [figure:figure7]\nCounting the critical points in Figure [figure:figure7] could give the (incorrect) impression that the standard approach has to be preferred with respect to the novel proposal. But considering the indicator function F applied to $\\bar{\\mathbf{z}}^0$, $\\Psi(\\bar{\\mathbf{z}}^0)$ and $\\Phi(\\bar{\\mathbf{z}}^0)$, we obtain\n \n$$\\mathcal{F}\\bigl(\\mathbf{w},\\bar{\\mathbf{z}}^0\\bigr) =7.9,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Psi(\\bar{\\mathbf{z}}^0)\\bigr)=5.9,\\qquad\n \\mathcal{F}\\bigl(\\mathbf{w},\\Phi(\\bar{\\mathbf{z}}^0\\bigr))=4.5,$$ \nas a consequence of the fact that memory of the dynamics –represented by **w**– is now considered, and again the minimum value is achieved for the SRH strategy.\nSimulation of peripheral outbreaks\n----------------------------------\nNext, we focus on the case of outbreaks occurring in the vicinity of a node located at the edge of the Cartesian grid. Specifically, we take *n* = 9 and the same ground state **z**0 of the previous example, and an *inflow matrix* of patients **w** given by the submatrix\n \n$$\\begin{pmatrix} \\circled{5} & 2 \\\\ 2 & 1 \\end{pmatrix}$$ \nlocated at the top-right corner of the null matrix. Then, the initial configuration $\\bar{\\mathbf{z}}^0=\\mathbf{z}^0+\\mathbf{w}$ is given by\n \n$$\\bar{\\mathbf{z}}^0 = \\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} & 2 & 3 & \\mathbf{11} & 4 \\\\\n 3 & 2 & \\circled{5} & 2 & 1 & 3 & 2 & \\circled{6} & 4 \\\\\n 3 & 3 & 1 & \\circled{5} & \\circled{6} & 2 & 3 & 1 & 3 \\\\\n \\circled{6} & 1 & \\circled{5} & 3 & 2 & \\circled{5} & 2 & 3 & 4 \\\\\n 1 & 3 & 2 & 1 & \\circled{6} & 3 & 4 & \\circled{5} & \\circled{6} \\\\\n 1 & 4 & 1 & 2 & 2 & \\circled{5} & 1 & 2 & 3 \\\\\n 4 & 1 & 3 & 4 & \\circled{5} & \\circled{6} & 2 & \\circled{5} & 3 \\\\\n 4 & 2 & 3 & \\circled{5} & 2 & 2 & \\circled{6} & 3 & 1 \\\\\n 1 & \\circled{6} & \\circled{5} & 2 & 4 & 4 & 2 & 1 & 2\n \\end{pmatrix},$$ \nwith the cell above the threshold capacity located at position (1, 8). Therefore, four patients are in excess and require a different reallocation. Among others, a possible output of the standard strategy is the final configuration given by\n \n$$\\Psi(\\bar{\\mathbf{z}}^0) = \\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} & 2 & \\circled{5} & \\circled{7} & 4 \\\\\n 3 & 2 & \\circled{5} & 2 & 1 & 3 & 4 & \\circled{6} & 4 \\\\\n 3 & 3 & 1 & \\circled{5} & \\circled{6} & 2 & 3 & 1 & 3 \\\\\n \\circled{6} & 1 & \\circled{5} & 3 & 2 & \\circled{5} & 2 & 3 & 4 \\\\\n 1 & 3 & 2 & 1 & \\circled{6} & 3 & 4 & \\circled{5} & \\circled{6} \\\\\n 1 & 4 & 1 & 2 & 2 & \\circled{5} & 1 & 2 & 3 \\\\\n 4 & 1 & 3 & 4 & \\circled{5} & \\circled{6} & 2 & \\circled{5} & 3 \\\\\n 4 & 2 & 3 & \\circled{5} & 2 & 2 & \\circled{6} & 3 & 1 \\\\\n 1 & \\circled{6} & \\circled{5} & 2 & 4 & 4 & 2 & 1 & 2\n \\end{pmatrix}.$$ \nOn the other hand, by employing the SRH approach, we pass through an intermediate state\n \n$$\\begin{pmatrix}\n 1 & 2 & 1 & 4 & \\circled{6} & 2 & 4 & 3 & \\circled{5} \\\\\n 3 & 2 & \\circled{5} & 2 & 1 & 3 & \\circled{3} & \\circled{7} & \\circled{5} \\\\\n 3 & 3 &"},"data_id":{"kind":"string","value":"arxiv_0183474"}}},{"rowIdx":45894,"cells":{"content":{"kind":"string","value":" by a clump of 16 sungrazers discovered over 12 days, December 9–20. This rate is equivalent to nearly 500 objects per year. A third, double spike is located symmetrically to the largest one, in late 2009. Thus, comparing only December spikes in Figure 3 — in order to eliminate much of the annual selection effect — we are finding a *pattern that is strongly resembling that exhibited by the brighter sungrazers* in Figure 2.\n[b]\n \nOrbital Similarity for 22 Members of December 2010 \nClump of SOHO/STEREO Kreutz Sungrazers with \n2010/2011 Swarm of Bright Dwarf Comets. \nl@\nc@\nc@\nc@\nc@\nc@\nc\n \n& & & & \n& Number & Corona- & & \n& of used & graph(s) & Best & C/1843 & C/2011 & Part of \nObject$^{\\rm a}$ & positions & used & fit$^{\\rm b}$ & D1 & W3 & swarm? \n \nC/2010 X11 & 5 & C2 & ± 0′.05 &\n ± 5 ′.24 & ± 0′.81 & No?? \nC/2010 X12 & 26 & C2,C3 & ± 0.46 & ± 0.62 & ± 7.08 & Yes \nC/2010 X13 & 9 & C2 & ± 0.29 & ± 14.63 & ± 20.46 & Yes?? \nC/2010 X14 & 49 & C2,C3 & ± 0.69 & ± 15.58 & ± 24.87 & Yes \nC/2010 X15 & 17 & C2,C3 & ± 0.52 & ± 5.51 & ± 4.84 &? \nC/2010 X16 & 27 & C2,C3 & ± 0.47 & ± 29.65 & ± 38.11 & Yes \nC/2010 X17 & 18 & C2,C3 & ± 0.42 & ± 0.56 & ± 8.71 & Yes \nC/2010 Y1 & 11 & C2 & ± 0.27 & ± 0.79 & ± 7.80 & Yes \nC/2010 Y2 & 18 & C2 & ± 0.28 & ± 14.04 & ± 23.02 & Yes \nC/2010 Y3 & 24 & C2,C3 & ± 0.80 & ± 1.34 & ± 8.73 & Yes \nC/2010 Y4 & 12 & C2 & ± 0.40 & ± 3.20 & ± 6.17 & Yes \nC/2010 Y5 & 27 & C2,C3 & ± 0.62 & ± 1.40 & ± 10.36 & Yes \nC/2010 Y6 & 19 & C2 & ± 0.42 & ± 3.22 & ± 7.34 & Yes \nC/2010 Y7 & 27 & C2,C3 & ± 0.47 & ± 20.77 & ± 30.07 & Yes \nC/2010 Y8 & 16 & C2 & ± 0.41 & ± 0.81 & ± 10.18 & Yes \nC/2010 Y9 & 25 & C3 & ± 0.44 & ± 8.58 & ± 17.64 & Yes \nC/2010 Y10 & 88 & C2,C3 & ± 0.53 & ± 5.88 & ± 51.64 & Yes \nC/2010 Y11 & 19 & C2,C3 & ± 0.69 & ± 7.71 & ± 18.42 & Yes \nC/2010 Y12 & 16 & C2 & ± 0.19 & ± 5.65 & ± 6.50 & Yes? \nC/2010 Y13 & 14 & C2 & ± 0.28 & ± 2.73 & ± 9.68 & Yes \nC/2010 Y14 & 13 & C2 & ± 0.53 & ± 5.76 & ± 5.91 &? \nC/2010 Y15 & 59 & C2,C3 & ± 0.36 & ± 4.10 & ± 19.23 & Yes \n \n \n \n \nGray’s (2013) very recent publication of the orbital elements of the SOHO/STEREO sungrazers from the second half of 2010 includes all 22 objects of the December 2010 clump. Their definitive designations are C/2010 X11 through C/2010 X17 and C/2010 Y1 through C/2010 Y15. Their astrometric observations, now available as well, serve here to examine the similarity of their orbits to those of C/1843 D1 and C/2011 W3. For each sungrazer, Table 5 provides the number of observations used, the coronagraphs employed, and the RMS residuals from Gray’s best-fit parabolic solution and from the elliptical orbits for C/1843 D1 (Sekanina & Chodas 2008) and C/2011 W3 (Sekanina & Chodas 2012) in columns 2 to 6, respectively. The last column conveys our conclusion whether or not each of the objects is likely to be associated with C/1843 D1 and therefore with the swarm of bright dwarf sungrazers.\nA cursory glance through Gray’s list of orbits suggests at once that up to five of them have angular orbital elements that resemble those of C/1843 D1. They include the three most extensively observed among the 22 sungrazers — C/2010 Y10, C/2010 Y15, and C/2010 X14, which are in Table 5 labeled with a star. The other two sungrazers are C/2010 X16 and C/2010 Y12. The number of observations, on which their orbits are based, are comparable to those for the orbits of C/2010 X12 and C/2010 X17, which do not resemble that of C/1843 D1, yet the residuals from their forced solutions in colmn 5 are nearly as good as those from the best-fit solutions. Thus, for C/2010 X16 and C/2010 Y12 the orbital agreement with C/1843 D1 does not offer much confidence in the quality of the published solutions and a comparison of the residuals in columns 5 and 6 should be the decisive criterion, just as is the case with the rest of Table 5.\nOverall, the mean residuals in columns 5 and 6 suggest that the orbits of 19 of the 22 objects are better, usually much better, fitted by the orbit of C/1843 D1 than C/2011 W3. The association with the swarm of bright dwarf sungrazers is well established for 17, a little less so for C/2010 Y12 (because of the competing residuals in columns 5 and 6), and problematic but still possible for C/2010 X13 (because of the very small number of used observations).\nWe have serious doubts about the direct relationship to the swarm of bright dwarf sungrazers only for three members of the clump in Table 5: C/2010 X11, C/2010 X15, and C/2010 Y14. Information on the first object is based on only five observations, while other two show nearly equal mean residuals from the two reference orbits; there is thus very little evidence on which a judgment could be predicated.\nAs a general comment on column 5 of Table 5, the high mean residuals from the orbit of comet C/1843 D1 displayed by C/2010 X14, C/2010 X16, C/2010 Y2, and C/2010 Y7 are a product of these objects’ perihelion distances appreciably exceeding that of C/1843 D1, and should not alarm the reader.\nWe conclude that a *great majority of the clump of faint Kreutz sungrazers from the period of December 13–20, 2010 was associated with C/1843 D1 and therefore with the swarm of bright dwarf sungrazers*.\nNothing definite can at present be concluded about the clump of faint SOHO/STEREO sungrazers from December 2011. This problem must wait until the astrometric positions and orbits of these sungrazers become available.\nThe spike-like features in the distribution of arrivals of faint SOHO/STEREO Kreutz minicomets have their counterparts among the bright dwarf sungrazers as well (step-like features; see Sec. 2 and Figure 1). The four objects brighter at maximum than magnitude 3 that arrived between 2010.00 and 2010.27 (C/2009 Y4, C/2010 B3, C/2010 E6, and C/2010 G4) fit a slope distinctly steeper than the polynomial’s slope and so do the three objects between 2010.80 and 2010.88 (C/2010 U8, C/2010 V8, and C/2010 W2). The equivalent rates are, respectively, ∼ 15 and nearly 40 per year, several times greater than the peak rate of the smoothed distribution.\nWe detect a yet another effect that shows that during the few years around 2010 the distribution in Figure 3 behaves untypically. It is the relation between the May and the October-December peaks that we already mentioned in Sec. 2. While the May spikes are never smaller than the October-December spikes in 2004–2007 and 2012, the former are outperformed by the latter in 2008–2011.\nThe cumulative distribution of arrival times of all Kreutz sungrazers seen with the SOHO and STEREO coronagraphs between 2004 and 2013 is presented in Figure 4, in which we plot the number of objects after each six-day interval of time. The averaged slope of the distribution is practically constant until about the end of 2008, then it picks up a little, as can be expected from the distribution in Figure 3.\nPeculiar features on the cumulative-distribution curve in Figure 4 are flat segments or “flats,” *fairly prolonged* intervals of time during which the distribution curve essentially levels off, with a zero or near-zero arrival rate of the SOHO/STEREO sungrazers. Their times do not correlate well with those of SOHO’s roll and must be a true property of the Kreutz population’s distribution along the orbit. Because of the extremely compressed scale in Figure 4, the flats appear very short. The appearance of flats is in greater detail illustrated on a section of the cumulative distribution in Figure 5. Each flat was required to satisfy at least one of two conditions: either no sungrazers detected to arrive for a minimum of three six-day consecutive intervals, or their total number not exceeding four over a minimum of five six-day consecutive intervals. Statistically, one would expect the highest rates of flats during 2004–2005 because of lower arrival rates, no high spikes on the distribution curve in Figure 3, and the absence of bright SOHO/STEREO sungrazers. However, Figure 4 shows that this was not the case, suggesting instead a seemingly counterintuitive positive correlation between the bright dwarf sungrazers’ arrival rate and the rate of flats on the cumulative-distribution curve during the period ending with 2008: there were no flats between the start of 2004 and mid-2006, while during the next two years, between the arrivals of the bright sungrazers C/2006 A5, C/2006 V2, and C/2008 K4, the average rate was 3 flats per year. From mid-2009 to mid-2011 the rate of flats was nearly constant at 2 per year.\nDiscussion\n==========\nThe properties of the distribution of arrivals of the SOHO/STEREO sungrazers in the years 2004–2013 illustrate the morphological complexity of the Kreutz system, which is driven by — and can be understood qualitatively and, to a degree, even quantitatively, in terms of — a cascading fragmentation process, proposed by Sekanina (2002) and in greater detail by Sekanina and Chodas (2007). In this model, the fragments continue to break up over and over again at all heliocentric distances, so that an enormous and *essentially* continuous stream of subfragments (or high-generation fragments) arrives at perihelion at any time. This stream, which forms a ring along the orbit, is, however, not structureless, and the degree of its nonuniformity varies from place to place on many spatial scales that are equivalent to time scales from days to dozens of years and longer. The segment of the Kreutz system that arrived at perihelion in 2004–2005 was apparently characterized by a high degree of uniformity, while in the subsequent years the opposite was the case.\nThe existence of brighter objects ( in Figures 3 and 4) in the cloud of SOHO/STEREO minicomets suggests that, among the sungrazers, they may consist of a material of a greater than average cohesion, so that their fragmentation rate is slower than that of other fragments in the stream. Because their parent(s) broke up less often and probably also at later times than the rest, the spatial distribution of subfragments that derive from such objects is less uniform and more cluster-like, which necessarily leads to greater variations and relatively often to *gaps* in their distribution along the orbit. These gaps show up as the flat segments on the cumulative-distribution curve. Thus, to a point, the occurrence rate of flats should indeed vary in some proportion to the arrival rate of bright dwarf fragments. Only when this rate increases significantly is there much more debris around, so that the number of flats begins to drop, but even then they should not be entirely absent.\nThe orbital results for the bright dwarf sungrazers in Table 2, which completely rule out any potential relationship between their sharply elevated rate and comet C/2011 W3, are supported by the evidence of very few *faint* SOHO/STEREO Kreutz minicomets in orbits similar to that of C/2011 W3. Table 6 shows that there were only nine such objects in the set of more than 1000 SOHO/STEREO Kreutz sungrazers between 2004 and 2013 with currently known orbits. Faint companions in close proximity of C/2011 W3 were recent fragments whose existence does not change the overall picture.\n[ht]\n \nList of Orbital Elements and Apsidal Orientation for SOHO/STEREO Sungrazers with Available Orbits in 2004–2013 \nMoving in Paths Generally Similar to That of C/2011 W3 (Eq. J2000.0). \nl@\nl@\nc@\nr@\nc@\nc@\nc@\nc@\nc@\nc@\nc@\nc@\nc@\nc\n \n& & & & Number & & \n& & & Corona- & of used & & \nObject & & *ω* & Ω & *i* & *q*($R\\_{\\mbox{\\boldmath $$}}$) & *L**π* & *B**π* & graph(s) & positions & Author & Reference \n \nC/2004 A4 & 2004 & Jan. & 8.64 & 60∘.94 & 333∘.15 & 134∘.78 & 1.03 & 281∘.42 & +38∘.35 & C3 & 15 & Marsden & MPC60097 \nC/2004 P5 & 2004 & Aug. & 9.45 & 52.72 & 324.88 & 133.10 & 1.31 & 282.97 & +35.52 & C3 & 25 & Marsden & MPC52766 \nC/2005 U5 & 2005 & Oct. & 22.20 & 49.13 & 319.04 & 131.09 & 1.05 & 281.82 & +34.75 & C3 & 15 & Marsden & MPC55718 \nC/2006 U8 & 2006 & Oct. & 17.09 & 53.25 & 319.52 & 137.33 & 1.07 & 274.96 & +32.89 & C2,C3 & 10 & Marsden & MPC58538 \nC/2007 S4 & 2007 & Sept. & 24.31 & 57.15 & 325.85 & 138.08 & 1.07 & 276.80 & +34.14 & C2,C3 & 14 & Marsden & MPC60927 \nC/2010 C4 & 2010 & Feb. & 7.99 & 61.46 & 333.75 & 135.99 & 1.10 & 280.85 & +37.62 & C3 & 14 & Marsden & MPC72131 \nC/2010 Q2 & 2010 & Aug. & 24.79 & 51.38 & 325.82 & 133.48 & 1.73 & 285.08 & +34.54 & C2,C3 & 62 & Gray & MPC84618 \nC/2010 Y16 & 2010 & Dec. & 21.83 & 57.04 & 329.47 & 133.49 & 1.67 & 282.76 & +37.50 & C3 & 15 & Gray & MPC84623 \nC/2013 F4 & 2013 & Mar. & 27.51 & 60.71 & 336.38 & 138.47 & 1.79 & 283.22 & +35.33 & C3 & 26 & Gray & MPC84626 \n \n \n*What are these conflicting lines of evidence telling us?* By far the most plausible hypothesis is that *another Lovejoy-sized fragment*, closely related to C/1843 D1 and moving in a similar orbit but with its perihelion time shortly preceding that of C/2011 W3, *continued to break up during its orbital motion still at large heliocentric distances*. This hypothesis logically explains several major findings of our paper.\nFirst, the *16 tightly related bright dwarf sungrazers in the swarm*, identified in Sec. 3 and Table 2, are the *prime fragmentation products of the postulated object — their parent*. Indeed, adding up the brightness contributions from the 16 sungrazers in Table 1 leads to an equivalent of apparent magnitude − 2.0, not much fainter than was C/2011 W3 (whose apparent magnitude at maximum light was estimated at − 3; e.g., Green 2011).\nSecond, a population of fainter dwarf sungrazers must be, in addition to the 16 bright ones, also derived from the same parent. An example of direct evidence for this argument is the *clump* of 22 faint objects from December 2010, most of which were shown to be associated with the swarm of bright dwarf sungrazers, representing in this scenario a *debris of a sizable, but less cohesive member of this swarm that fragmented long before reaching perihelion*. This clump is probably the most extreme example of this advanced stage of cascading fragmentation but certainly not the only one.\nThird, a *hint of a hierarchy in the distribution of fragment dimensions is perceived in both the swarm and the clump*. The swarm’s brightest object, SOHO-2143, is in about the same position relative to the apparent ends of the swarm modeled in Figure 2, as is the clump’s most prominent object, C/2010 Y10, relative to the apparent ends of the clump. In both cases this “primary” fragment is in the feature’s second half. Its positioning may provide information on the vector field of separation velocities and on the fragmentation mechanism’s nature.\nFourth, this hypothesis is consistent with our explanation of the *gaps* in the distribution of arrival times of the SOHO/STEREO Kreutz population or, equivalently, the *flats* in its cumulative distribution, as described in Sec. 4 and earlier in this section.\nFifth, the presented hypothesis explains the *fairly well-defined limits of the tightly-knit swarm*. It is in fact remarkable that SOHO-2571, the last point on the cumulative distribution of bright dwarf sungrazers in Figure 1, has orbital elements so dramatically different from those of the swarm’s members. The future will show whether this comet is a “wanderer” (like bright SOHO sungrazers before 2004) or the first member of a new swarm.\n[b]\n \nPerturbation of Arrival Time (yr) at Next Perihelion by Separation Velocity Acquired in Fragmentation Event \nas Function of Heliocentric Distance. \nc@\nc@\nc@\nc@\nc@\nc\n \nDistance & Elapsed & & \nfrom Sun$^{\\rm a}$ & time$^{\\rm b}$ & & \n(AU) & (yr) & & & & \n \n10 & 2.42 & +3.6715 & − 3.6463 & +0.0891 & − 0.0888 \n20 & 6.96 & +2.5195 & − 2.5074 & +0.0446 & − 0.0443 \n30 & 13.01 & +1.9941 & − 1.9865 & +0.0297 & − 0.0294 \n50 & 29.10 & +1.4441 & − 1.4400 & +0.0178 & − 0.0175 \n75 & 56.47 & +1.0707 & − 1.0683 & +0.0117 & − 0.0115 \n00 & 92.79 & +0.8261 & − 0.8246 & +0.0086 & − 0.0083 \n30 & 51.88 & +0.6061 & − 0.6052 & +0.0063 & − 0.0060 \n60 & 40.25 & +0.4178 & − 0.4174 & +0.0045 & − 0.0043 \n86 & 50.00 & +0.1851 & − 0.1849 & +0.0025 & − 0.0023 \n60 & 59.75 & +0.0649 & − 0.0648 & +0.0012 & − 0.0011 \n30 & 48.12 & +0.0325 & − 0.0324 & +0.0007 & − 0.0007 \n00 & 07.21 & +0.0159 & − 0.0159 & +0.0004 & − 0.0004 \n75 & 43.53 & +0.0079 & − 0.0079 & +0.0002 & − 0.0002 \n50 & 70.90 & +0.0032 & − 0.0032 & +0.0001 & − 0.0001 \n30 & 86.99 & +0.0011 & − 0.0011 & +0.0001 & − 0.0001 \n20 & 93.04 & +0.0005 & − 0.0005 & 0.0000 & 0.0000 \n10 & 97.58 & +0.0001 & − 0.0001 & 0.0000 & 0.0000 \n \n \n \nMore broadly, the hypothesis of a broken-up parent sungrazer is closely linked to, and symptomatic of, the process of nucleus fragmentation at large heliocentric distance, whose important trait is its fairly insignificant effect on the orbital periods of the parent’s fragments (Sekanina 2002). Accordingly, they arrive at perihelion nearly simultaneously (on a scale of the orbital period) as a *true swarm*. For illustration, Table 7 lists the perturbations of the arrival time at next perihelion for a fragment that separated far from the Sun with a typical velocity of 1 m s− 1 in either of two cardinal directions in the plane of a sungrazing orbit with a perihelion distance of 1.2 $R\\_{\\mbox{\\scriptsize \\boldmath $$}}$ and an orbital period of 900 years (with aphelion at 186.43 AU). The radial component, *V**R*, is directed away from (+) or toward the Sun, the transverse component, *V**T*, is in the orbit plane and perpendicular to *V**R*, pointing in the general direction of the orbital motion (+) or in the opposite direction. There is no contribution from the normal component.\nTable 7 shows some noteworthy facts. First of all, the radial separation velocity is 10 to 100 times more efficient in triggering off the same change in the arrival time at next perihelion passage than is the transverse velocity. Second, the effect is approximately, but not exactly, symmetrical for positive and negative velocities. And third, a peak arrival rate of 4.6 per year for the bright dwarf sungrazers (Sec. 2) is equivalent to their minimum temporal separation of 0.22 yr, which in Table 7 matches the arrival-time perturbation for a separation near aphelion. If the separation velocity should be lower than 1 m s− 1, the assumed fragmentation event would have taken place before aphelion, and vice versa. The temporal relations among the bright dwarf sungrazers from are therefore on the right order of magnitude to be consistent with the process of cascading fragmentation.\n[t]\n \nPerturbations of Argument of Perihelion (*ω*), \nLongitude of Ascending Node (Ω), and Inclination (*i*) \nby Normal Component of Separation Velocity \nAcquired in Fragmentation Event as Function \nof Heliocentric Distance. \nc@\nc@\nc@\nc@\nc@\nc@\nc\n \nDistance & & & \nfrom Sun$^{\\rm a}$ & & & \n(AU) & & & & & & \n \n10 & − 0∘.25 & +0∘.25 & − 0∘.31 & +0∘.31 & − 0∘.03 & +0∘.03 \n20 & − 0.50 & +0.50 & − 0.62 & +0.62 & − 0.06 & +0.06 \n30 & − 0.75 & +0.75 & − 0.92 & +0.93 & − 0.09 & +0.08 \n50 & − 1.25 & +1.26 & − 1.54 & +1.55 & − 0.14 & +0.12 \n75 & − 1.87 & +1.90 & − 2.31 & +2.33 & − 0.21 & +0.17 \n00 & − 2.49 & +2.53 & − 3.07 & +3.11 & − 0.29 & +0.21 \n30 & − 3.23 & +3.30 & − 3.99 & +4.05 & − 0.38 & +0.25 \n60 & − 3.97 & +4.06 & − 4.90 & +4.99 & − 0.48 & +0.28 \n86 & − 4.61 & +4.73 & − 5.69 & +5.81 & − 0.56 & +0.29 \n60 & − 3.98 & +4.06 & − 4.91 & +4.99 & − 0.45 & +0.25 \n30 & − 3.24 & +3.30 & − 4.00 & +4.05 & − 0.35 & +0.21 \n00 & − 2.50 & +2.53 & − 3.08 & +3.11 & − 0.25 & +0.17 \n75 & − 1.88 & +1.90 & − 2.32 & +2.33 & − 0.18 & +0.13 \n50 & − 1.26 & +1.27 & − 1.55 & +1.56 & − 0.11 & +0.09 \n30 & − 0.76 & +0.76 & − 0.93 & +0.93 & − 0.06 & +0.05 \n20 & − 0.50 & +0.51 & − 0.62 & +0.62 & − 0.04 & +0.03 \n10 & − 0.25 & +0.25 & − 0.31 & +0.31 & − 0.02 & +0.01 \n \n \nThe argument of perihelion *ω* and the longitude of the ascending node Ω are both affected by fragmentation at large heliocentric distance rather strongly, while the inclination *i* to a much lesser degree, as is shown in Table 8. It is now the normal component, *V**N*, of the separation velocity that is primarily involved, pointing to the north orbital pole (+) or in the opposite direction. There is no contribution from the radial component and, only in *ω*, a very minor one, not exceeding 0∘.02 per 1 m s− 1 velocity from the transverse component. The normal-velocity effect is again nearly symmetrical relative to aphelion and also for positive and negative velocities. An exception is the inclination, with the negative velocity having a generally smaller effect, as small at aphelion as one half of that of the positive velocity of the same magnitude.\n[b]\n \nPerturbation of Perihelion Distance ($R\\_{\\mbox{\\scriptsize \\boldmath $$}}$) by Transverse \nComponent of Separation Velocity Acquired in \nFragmentation Event as Function \nof Heliocentric Distance. \nc@\nc@\nc@\nc@\nc@\nc@\nc\n \n& & & & & & \nDistance & & & Distance & \nfrom Sun & & & from Sun$^{\\rm a}$ & \n(AU) & & & & (AU) & & \n \n& & & & & & \n10 & +0.008 & − 0.008 & & 100 & +0.077 & − 0.075 \n20 & +0.015 & − 0.015 & & 130 & +0.101 & − 0.097 \n30 & +0.023 & − 0.023 & & 160 & +0.125 & − 0.119 \n50 & +0.038 & − 0.038 & & 186 & +0.146 & − 0.138 \n75 & +0.058 & − 0.057 & & & \n \nThe perturbations of the three angular elements, which always have the same sign, are only very approximately in constant proportions at different heliocentric distances. At aphelion, for example, the perturbation ratios are. Given the errors involved, these ratios are in good agreement (including the correct signs) with the relative ratios of of the average slopes in Table 3. This match is a very encouraging indication that the angular-element trends among the tightly-knit swarm’s sungrazers were indeed triggered by perturbations due to separation velocities acquired in fragmentation events involving these objects. The ratios also explain why in Table 2 the trends in *ω* and Ω are obvious but those in *i* are not.\nFragmentation always has an extremely small effect on the orientation of the line of apsides. The magnitude of these perturbations never exceeds 0∘.03 per 1 m s− 1 at > 10 AU from the Sun and they are caused by the transverse component of the separation velocity in the perihelion longitude *L**π* and by the normal component in the perihelion latitude *B**π*.\nThe perturbations of the perihelion distance for a fragment that broke off from its parent far from the Sun are dominated by the transverse component of the separation velocity, as shown in Table 9. There is a perfect symmetry relative to aphelion, so only the preaphelion numbers are listed. On the other hand, the effect’s symmetry due to a positive (in the general direction of the orbital motion) vs a negative velocity is only approximate. There is no contribution from the radial component and only a minor contribution, not exceeding 0.004 $R\\_{\\mbox{\\scriptsize \\boldmath $$}}$ per 1 m s− 1, from the normal component. It is noted that even a submeter-per-second separation velocity acquired during a *single* fragmentation event near aphelion can turn a fragment’s sungrazing orbit with a perihelion distance of 1.1 $R\\_{\\mbox{\\scriptsize \\boldmath $$}}$ into a sun-striking orbit. It is noted that two objects in Table 2 do indeed have their perihelion distances smaller than 1 $R\\_{\\mbox{\\scriptsize \\boldmath $$}}$.\nFinally, perturbations of the eccentricity due to fragmentation at large heliocentric distance are not discussed here in any detail because of the parabolic orbital approximation used for the swarm members. We only note that they are expressible in terms of the perturbations of the perihelion distance and the semimajor axis, whose reciprocal, 1/*a*, is affected by less than 0.00003 AU− 1 at > 10 AU from the Sun primarily by the radial component. Unlike near the Sun, the transverse component is much less important and there is no contribution to 1/*a* from the normal component.\nTo summarize up to this point, our discussion suggests that the orbital perturbations of fragments due to submeter- or meter-per-second separation velocities acquired during breakup events at large heliocentric distances are generally consistent with the systematic trends in the orbital elements in Table 2, supporting our hypothesis that the tightly-knit swarm of 16 bright dwarf sungrazers (as well as a debris of additional, fragmented members of the swarm), peaking in 2010/2011, consisted of fragments of an originally common parent, which itself must have been a fragment of a precursor that also gave birth to comet C/1843 D1. This precursor could have been the celebrated comet X/1106 C1, as suggested by Sekanina & Chodas (2007). The arrival time of the swarm, about 168 years after the appearance of C/1843 D1, requires a relative separation velocity of only about 1 m s− 1 between the swarm’s parent and this 19th-century sungrazer, if they detached from X/1106 C1 right at perihelion, or somewhat greater than 1 m s− 1, if a little off the perihelion point, but in any case a velocity typical for separations of sungrazer fragments near perihelion (Sekanina & Chodas 2007). It is possible that C/1843 D1, the swarm’s parent, many additional sizable sungrazers as well as huge amounts of debris had separated from X/1106 C1 as a single object, which became subject to sudden or gradual fragmentation only during its liftoff through the precursor’s atmosphere and/or soon afterwards.\nBecause SOHO-2143 was clearly the brightest among the swarm’s members (Table 1), it is plausible to deem this sungrazer the most massive fragment, located near the position that would have been occupied by the parent had it survived intact. The swarm’s 15 remaining objects are then products of one or more breakup events, in harmony with the conceptual paradigm of cascading fragmentation (Sekanina 2002, Sekanina & Chodas 2007). If the mechanism that accounts for the separation velocities causes them to prefer a certain direction — such as, for example, the parent’s spin — the contributions from the individual fragmentation events to the total perturbation effect would simply add up in some direction in an inertial coordinate system, to which the introduced coordinate system is, due to the orbit’s extreme elongation, practically equivalent at any point far enough from perihelion. The task is now reduced to finding out whether it is at all possible to choose a limited number of, and appropriate locations for, the breakup events in which a swarm’s fragment could have been reasonably involved, such that the respective orbital difference from Table 2 is in each of the five elements matched closely enough by the sum of perturbations from these events.\n[t]\n \nExample of Interpreting Orbital Difference Between Two Swarm Members in Terms of Perturbations \nAcquired During Series of Six Fragmentation Events.$^{\\rm a}$ \nl@\nc@\nc@\nc@\nc@\nc@\nc@\nc@\nc\n \nFragmentation event & #1 & #2 & #3 & #4 & #5 & #6 & Sum & Difference \nTime from previous perihelion (yr) & 2.4 & 9.8 & 29.1 & 69.8 & 151.9 & 50.0 & from & C/2006 A5 \nHeliocentric distance (AU) & 10.0 & 25.0 & 50.0 & 85.0 & 130.0 & 86.4 & the six & minus \nPerturbations: & & & & & & & events & SOHO-2143 \n \nArrival time at next perihelion (yr) & − 2.32 & − 1.40 & − 0.91 & − 0.61 & − 0.38 & − 0.12 & − 5.74& − 5.74 \nArgument of perihelion & − 0∘.11 & − 0∘.26 & − 0∘.51 & − 0∘.86 & − 1∘.30 & − 1∘.85 & − 4∘.89 & − 5∘.00 \nLongitude of ascending node & − 0∘.13 & − 0∘.31 & − 0∘.62 & − 1∘.05 & − 1∘.60 & − 2∘.28 & − 5∘.99 & − 6∘.08 \nInclination & − 0∘.01 & − 0∘.03 & − 0∘.05 & − 0∘.09 & − 0∘.15 & − 0∘.22 & − 0∘.55 & − 0∘.43 \nPerihelion distance ($R\\_{\\mbox{\\scriptsize \\boldmath $$}}$) & − 0.005 & − 0.011 & − 0.023 & − 0.039 & − 0.059 & − 0.083 & − 0.220 & − 0.22 \n \n \nThere exists far too little information to derive the unique solution. We offer one example of a greater than average difficulty to outline the issues involved and to illustrate the plausibility of our swarm’s fragmentation hypothesis. The first issues are whether in our scenario we should (a) presume more than one original parent and (b) consider more than a single orbit about the Sun. The answer is no on both counts. One, the lifetimes of all SOHO/STEREO Kreutz sungrazers observed to disintegrate just before perihelion are shorter than one revolution, as otherwise these objects would have disintegrated just before their previous perihelion passage and could not be seen nowadays. And two, even if fragmentation began before previous perihelion, the secondary parents, which would have to have been large enough objects to survive, and their own subsequently generated swarms would be separated from each other by such huge gaps in time that their association could not easily be recognized. Thus, our task is indeed confined to one parent and a single revolution about the Sun.\nWith these constraints in mind, let us consider the first entry in Table 2, C/2006 A5. Its orbital elements suggest that it was a member of the tightly-knit swarm. Because it preceded SOHO-2143 (a proxy for the parent) by as much as 5.74 years, it represents a highly challenging case. In concert with our objectives, we ask: Could the early arrival of C/2006 A5 and the deviation of its orbital elements from those of SOHO-2143 be explained by perturbations due to separation velocities acquired in a sequence of fragmentation events?\nIn our scenario, the difference of almost 6 years is an *integrated* perturbation of the arrival time at next perihelion, that is, the sum of perturbations at *ν* fragmentation events in which this sungrazer was involved either as part of a larger object in each of the first *ν* − 1 events or as a stand-alone fragment produced by the last, *ν*-th event.\nTo estimate the number of fragmentation events, we use a simple model, with the magnitude at maximum light (normalized, strictly, to 1 AU from Earth, but differing very little from the apparent magnitude in close proximity of the Sun) as a proxy for the initial mass, since both quantities were shown to have been tightly correlated (Sekanina 2003). For C/2006 A5 the magnitude at maximum light was 2.9 (Table 1); for the parent we approximate the lower limit to its peak brightness by that of SOHO-2143, that is, by magnitude − 0.5. The upper limit to the peak brightness of the original parent is uncertain. The object may have been as bright as magnitude − 2.0 (see above) or even brighter; we adopt, rather arbitrarily, magnitude − 2.5. If the parent and its fragments are assumed to have split, in each step of the process, into two parts of equal mass (50 percent of their immediate parent’s mass), the mass of any fragment after *ν* events is a part of the original parent. Because the estimated mass of C/2006 A5 comes\nout on our assumptions to be\nbetween and times smaller than the original parent’s mass, we obtain a condition. To the extent that this scenario applies, C/2006 A5 should have been involved in fragmentation events during the 900 years, not counting the initial breakup of X/1106 C1 near perihelion.\nIn our example, we deem C/2006 A5 a final product of six fragmentation events and require that the magnitude of the separation velocity acquired by it during each event not exceed 1 m s− 1. Since this sungrazer preceded SOHO-2143, the parent’s proxy, the radial component, which dominates the perturbations of the arrival time at next perihelion, must have been directed toward the Sun. We also require, rather arbitrarily, that the separation velocity vector not vary from event to event and that the last event have occurred at aphelion. With these constraints, all that needs to be chosen in Table 7 (or its expanded version) is six entries such that their sum is greater in absolute value than 5.74 years. One of many solutions that fits well the differences between the orbital elements of C/2006 A5 and SOHO-2143 in Table 2 is presented in Table 10.\nThe choice of orbital locations for the fragmentation events is, in the absence of conditions constraining the fragment’s history, rather arbitrary. Whether the fragmentation events were a corollary of major thermal stresses that continued to ravage the body’s interior after perihelion (Sekanina & Chodas 2012) or more or less spontaneous, the example in Table 10 conforms to an expectation that they became gradually less frequent and eventually ceased. Other scenarios are of course possible, but, as seen from Table 7, postaphelion fragmentation fits only differences of < 1 year or so in the arrival times at next perihelion, unless ∣*V**N*∣ ≫ 1ms− 1 or the number of fragmentation events is extraordinarily large.\nThe separation velocity that fits the fragmentation sequence in Table 10 is 0.96 m s− 1 per event, with the components,, and, all in m s− 1. The perfect agreement between the perturbation sum and the orbital difference is to be expected for the arrival time and the perihelion distance, because the perturbations of these elements are dominated by a single component of the separation velocity, the radial and the transverse one, respectively. The agreement between the perturbation sums and the orbital differences in the three angular elements is another matter, because the perturbations of all three are dominated by the normal component of the separation velocity. Also, when the radial and transverse components exceed in magnitude the normal component, the contributions from the transverse velocity to the arrival-time perturbation and from the normal component to the perihelion-distance perturbation are not entirely negligible and they impact the balance of the result. The residuals in Table 10 of about 0∘.1 in each of the three angular elements suggest that they are within the errors of observation, thus acceding to the plausibility of our perturbation scenario and to the status of the bright dwarf sungrazers in the swarm as fragments of a common parent, sharing orbits very similar to that of comet C/1843 D1 and being closely related to this spectacular object. In the light of this compelling evidence, the remarkable coincidence in timing between the arrival of the swarm and the appearance of comet C/2011 W3 is found to be purely fortuitous.\nThe clump of nearly two dozen faint SOHO/STEREO Kreutz minicomets from mid-December 2010 is shown to be a debris of the swarm’s another member that fragmented far from the Sun on its way to perihelion. For the clump’s duration of about eight days, Table 7 suggests that the parent’s fragmentation most probably began more recently than some 150 years ago and less than ∼ 130 AU from the Sun after aphelion. When orbits of all SOHO/STEREO sungrazers from 2011 are available, the origin of the second major clump, from December of that year, can likewise be examined.\nThe situation with the swarm of bright dwarf sungrazers is reminiscent of that with the temporal distribution of the Solar Maximum Mission (SMM) sungrazers, whose orbits were studied extensively by Marsden (1989). He found that, like the swarm members in our investigation, the objects belonged to what he called *Subgroup I*, which includes C/1843 D1, and that two pairs of them arrived only 12–13 days apart. Although the SMM sungrazers’ orbits were determined less accurately than those of the brighter SOHO/STEREO sungrazers, a general tendency is apparent among the SMM objects’ orbits (Marsden 1989, Marsden & Williams 2008) for the argument of perihelion and the longitude of the ascending node to systematically increase with time in the course of that “swarm’s” observed duration of about two years.\nThe experience with the 2010/2011 swarm of bright dwarf sungrazers and comet C/2011 W3 shows that it is *not advisable to employ suddenly increasing rates of SOHO/STEREO Kreutz sungrazers*, especially the brighter ones, *as a token of an imminent* (on a time scale of a year or so) *arrival of a spectacular member of the Kreutz system*. Although it would have worked, because of a coincidence, in the case of C/2011 W3, a prediction of this kind — with no further supporting evidence — is in fact merely a product of wishful thinking and very risky. It appears that the process of cascading fragmentation, whose influence on the evolution of comets in general and of the Kreutz system in particular proves overwhelming, makes it virtually impossible to predict the arrival of a spectacular sungrazer on time scales of a year or so.\nIn evolutionary terms, the swarm of bright dwarf sungrazers provides useful information toward our understanding the Kreutz system’s morphology, just as does C/2011 W3. They both represent *warning signals* that the expected new, 21st-century cluster of spectacular members of the Kreutz system is approaching. Even though the swarm and C/2011 W3 are products of different evolutionary paths, the orbital periods — directly computed for C/2011 W3 and indirectly inferred for the swarm — consistently show that the 20th-century cluster, which was observed to extend from 1945 to 1970, did not represent the last “bead” of the “string of pearls” [to euphemistically express the extent of the protofragment’s perihelion breakup in analogy to the appearance of comet D/1993 F2 (Shoemaker-Levy)], but that one or more “beads” are yet to come. The alert to the expected new cluster is the only parallel link that we find in our effort to decode the meaning of the two remarkable, nearly-simultaneous events.\nConclusions\n===========\nOur study of the population of SOHO/STEREO’s Kreutz sungrazers and their rate of arrival in the years 2004–2013 offers the following conclusions on the Kreutz system and on the potential temporal relationship with comet C/2011 W3:\n(1) In addition to its usual annual periodicity, caused by selection effects, the overall arrival-time distribution displays a number of prominent spikes, whose magnitude peaked approximately one year before the appearance of comet C/2011 W3, in December 2010.\n(2) Comet C/2011 W3 arrived also about one year after the sharply elevated arrival rate of the SOHO/STEREO sungrazers brighter at maximum light than apparent magnitude 3, which are free from major selection effects. A polynomial fitted through a set of 19 objects of this swarm showed a peak rate of 4.6 per year centered on 2010.88, while a similar polynomial fitted to a subset of 13 sungrazers brighter at maximum light than apparent magnitude 2 indicated a peak rate of 4.3 per year centered on 2011.35, compared with a pre-2006 average rate of ∼ 1 per year. The estimated uncertainty in the times of peak arrival rate is a few months.\n(3) A potential physical correlation between comet C/2011 W3 and the elevated rate of the distribution of bright SOHO/STEREO sungrazers is strongly contradicted by dynamical evidence, which shows no orbital similarity between these bright objects and C/2011 W3, with deviations of 10∘ or more in each of the three angular elements.\n(4) On the other hand, fully 18 of the 19 bright SOHO/STEREO sungrazers were found to move in orbits very similar to that of the spectacular Kreutz comet C/1843 D1, and 16 of them made up a swarm of tightly related objects, with clear systematic trends in the argument of perihelion, the longitude of the ascending node, and the perihelion distance.\n(5) The cumulative distribution of SOHO/STEREO sungrazers’ arrivals exhibits a series of flat segments or “flats,” relatively prolonged intervals of time (3–5 weeks) during which the arrival rate of the population of Kreutz minicomets was nil or extremely low (gaps); they do not correlate well with the instances of SOHO’s roll.\n(6) The temporal distribution of the flats is strongly nonuniform:no flat is located from the beginning of 2004 to mid-2006, while their rate is the highest and equal to 3 per year between mid-2006 and mid-2008; from mid-2009 to mid-2011 the rate is nearly constant at 2 per year.\n(7) The number of flats appears to correlate with the arrival rate of SOHO/STEREO sungrazers brighter at maximum light than magnitude 3; this correlation appears counterintuitive in that there are no flats in a segment of the distribution curve that is not populated by any bright dwarf sungrazer while the temporal rate of flats peaks in a segment of the curve that is marked by the arrivals of three such sungrazers.\n(8) This peculiar property is understood in the framework of cascading fragmentation, as gaps in the temporal distribution of subfragments are byproducts of relatively more sizable and cohesive parent fragments; their breakup generally leads to more nonuniform, cluster-like temporal arrangement of the ensuing subfragments.\n(9) Overall, the results of this study vividly illustrate the immense morphological complexity of the Kreutz system. In connection with C/2011 W3 they not only show that no bright SOHO/STEREO sungrazer was moving in a similar orbit at least between 2004 and the end of 2012, but that even faint Kreutz minicomets in such orbits were very rare, a few close short-lived companions accompanying the comet notwithstanding.\n(10) As a plausible interpretation of the evidence, we propose that another Lovejoy-sized sungrazer, moving in an orbit similar to that of C/1843 D1 but with the perihelion time close to that of C/2011 W3, continued to break up at large heliocentric distances; and that the swarm of bright dwarf sungrazers with their arrival rate peaking some time in late 2010 or early 2011 was the prime product of this fragmentation process.\n(11) The prominently spike-like distribution of fainter SOHO/STEREO minicomets suggests that it contains a debris of prematurely fragmented bright dwarf sungrazers also belonging to the swarm. This scenario is shown to apply to the most prominent clump of 22 faint SOHO/STEREO sungrazers from mid-December 2010, up to 19 of which fit a test of their association with the swarm. This correlation should also be verified on another clump of faint sungrazers from December 2011, when their orbital data become available.\n(12) A decisive role of sheer chance in the timing of the appearance of comet C/2011 W3, on the one hand, and of the significantly elevated arrival rate of the population of SOHO/STEREO Kreutz minicomets and especially the swarm of bright dwarf sungrazers, on the other hand, is under the circumstances indisputable.\n(13) A common origin of all 16 members of the tightly-knit swarm is understood in terms of perturbations during their original parent’s fragmentation, which proceeded in a cascading fashion at large heliocentric distances. This scenario is strongly corroborated by comparison of systematic trends in the swarm members’ orbital elements with the perturbations expected to be caused by separation velocities acquired by the fragments during such breakup events.\n(14) Spikes in, and sharply elevated rates of, the temporal distribution of SOHO/STEREO Kreutz sungrazers, bright and faint alike, should not be mistaken for a token of an imminent (on a time scale of a year or so) arrival of a spectacular member of the Kreutz system. Because of an overwhelming effect of cascading fragmentation, it is practically impossible to predict the arrival of a major Kreutz sungrazer on such short time scales.\n(15) The swarm of bright dwarf sungrazers and the clumps of fainter sungrazers, causing prominent spikes in the SOHO/STEREO arrival-time distribution, represent another warning signal that the expected 21st-century cluster of spectacular Kreutz comets is on its way to perihelion, to arrive in the coming decades (Sekanina & Chodas 2012). This alert is the only common link that we find between the recent nearly-simultaneous appearance of C/2011 W3 and the elevated arrival rates of the SOHO/STEREO Kreutz population. \nThis research was carried out in part at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. \nREFERENCES\n \n \n \n \n \n \n \n \n \n \n \n \n \n---\n1. See\nhttp:/\n/sohodata.nascom.nasa.gov/cgi-bin/data\\_query.[↩](#fnref1)\n2. See\nhttp:/\n/www.rkracht.de/soho/bright/bright.htm.[↩](#fnref2)\n\nPopulation of SOHO/STEREO Kreutz Sungrazers andthe Arrival of Comet C/2011 W3 (Lovejoy)\n=======================================================================================\nWe examine properties of the population of SOHO/STEREO (dwarf) Kreutz sungrazing comets from 2004 to 2013, including the arrival rates, peculiar gaps, and a potential relationship to the spectacular comet C/2011 W3 (Lovejoy). Selection effects, influencing the observed distribution, are largely absent among bright dwarf sungrazers, whose temporal sequence implies the presence of a swarm, with the objects brighter at maximum than apparent magnitude 3 arriving at a peak rate of ∼ 4.6 per year in late 2010, while those brighter than magnitude 2 at a peak rate of ∼ 4.3 per year in early 2011, both a few times the pre-swarm rate. The entire population of SOHO/STEREO Kreutz sungrazers also peaked about one year before the appearance of C/2011 W3. Orbital data show, however, that a great majority of bright dwarf sungrazers moved in paths similar to that of comet C/1843 D1, deviating 10∘ or more from the orbit of C/2011 W3 in the angular elements. The evidence from the swarm and the overall elevated arrival rates suggests the existence of a fragmented sizable sungrazer that shortly preceded C/2011 W3 but was independent of it. On the other hand, these findings represent another *warning signal* that the expected 21st-century cluster of spectacular Kreutz comets is on its way to perihelion, to arrive during the coming decades. It is only in this sense that we find a parallel link between C/2011 W3 and the spikes in the population of SOHO/STEREO Kreutz sungrazers.\nIntroduction\n============\nThe osculating orbital period of about 700 years determined for comet C/2011 W3 (Lovejoy), the most recent spectacular member of the Kreutz system of sungrazers (Sekanina & Chodas 2012), rules out this comet’s identity with any known Kreutz sungrazer. In particular, it cannot represent the return of any member of this system that has arrived to perihelion since the 17th century (Kreutz 1901, Marsden 1967), thus providing another example of the immense complexity of this comet . With about 2000 minor members known, thanks to the ongoing vigorous search by amateur astronomers in images exposed with the coronagraphs on board the *and Heliospheric Observatory* (SOHO; see Brueckner et al. 1995) and, more recently, also with the coronagraphs on board the two spacecraft of the *Solar Terrestrial Relations Observatory* (STEREO; see Howard et al. 2008), the relationship between the minor sungrazers and comet C/2011 W3 needs to be addressed. For example, did the temporal distribution of the SOHO/STEREO Kreutz sungrazers change on account of the impending arrival of C/2011 W3? Is there any evidence for changes in the brightness distribution of these minor sungrazers? If yes, how could any such changes be interpreted? And could the results be employed in the future to forecast the *imminent* arrival of another spectacular sungrazer?\nBright SOHO/STEREO Sungrazers in 2004–2013\n==========================================\nThe prediction, published several years ago (Sekanina & Chodas 2007), of another cluster of bright sungrazers to appear in the coming decades was already commented on (Sekanina & Chodas 2012). Based on a study of the long-term evolution of the Kreutz system, this prediction was borne out by C/2011 W3, the expected cluster’s first apparent member. In comparison, a successful effort aimed at forecasting the arrival of a strikingly bright sungrazer we refer to at the end of Sec. 1 is meant to confine the object’s appearance to a temporally much more constrained window.\nTo address the raised questions, we begin with the distribution, between 2004 and 2013, of arrival times of the SOHO/STEREO Kreutz sungrazers that were at their maximum brighter than apparent magnitude 3; they are referred to hereafter as the *bright dwarf* sungrazers. Like the fainter Kreutz minicomets, they all failed to survive perihelion. We felt that the interval of nearly eight years is more than sufficient to cover all interesting objects in the SOHO/STEREO population that preceded, yet might still show a potential relationship to, C/2011 W3.\nWe should emphasize that because the definition of a bright dwarf sungrazer is entirely arbitrary, the limit at apparent magnitude 3 should be regarded only as a crude constraint. Indeed, because of SOHO’s and STEREO’s instrumental limitations and biases as well as operational practices and options (e.g., orange vs. clear filters, variable image cadence, uneven duty cycle, etc.; see Knight et al. 2010 for a comprehensive photometric analysis), it could be very difficult to determine a magnitude with much accuracy and virtually impossible to establish its equivalent value in any standard photometric band. For example, to avoid mixing the clear-filter and orange-filter magnitudes for objects observed with SOHO’s C2 and/or C3 coronagraphs is practically unfeasible, because this problem is intrinsic to the data-acquisition process. The resulting color index is known to show an enormous scatter of several magnitudes (Knight et al. 2010) and even though systematic trends are vaguely detected, a color correction inferred from the statistics can never be warranted to indiscriminately apply to any particular sungrazer’s magnitude and is not necessarily better than no correction. Among the brightest comets listed by Knight et al., such as C/1998 K10, C/2000 H2, or C/2003 K7, the color index is indeed near zero.\nThe objects that satisfy our definition of a bright dwarf sungrazer are summarized in Table 1, which presents 19 selected objects. The magnitude of most is at maximum brightness in the C2 coronagraph, even though some may have peaked in C3, with no such data being available. Only for a few listed sungrazers the brightness was measured in C3. The magnitudes were taken from various data sources, as indicated. Because of uncertainties, borderline cases are unavoidable. It is always possible that one or two listed objects may, in a uniform photometric system, be somewhat fainter than the chosen magnitude limit, while the brightness of a very few comets not included may just barely exceed it. In any case, we regard the set as a fairly representative sample.\n \nSOHO/STEREO Kreutz Sungrazers from 2004–2013 \nBrighter at Maximum Light Than Magnitude 3. \nl@\nl@\nc@\nl\n \n& Date of first & Apparent & \n& observation & magnitude & \nObject & in C3 (UT) & at max. & Reference(s)$^{\\rm a}$ \n \nC/2006 A5 & 2006 Jan.4 & 2.9 & IAUC 8694,[1] \nC/2006 V2 & 2006 Nov.1 & 1.3 & IAUC 8811,[1] \nC/2008 K4 & 2008 May 22 & 0.5 & IAUC 8982,[1] \nC/2009 C3 & 2009 Feb.4 & 2–3 & IAUC 9055 \nC/2009 D4 & 2009 Feb.21 & 2–3 & IAUC 9056 \nC/2009 Y4 & 2009 Dec.30 & 1.0 & IAUC 9117,[1],[2] \nC/2010 B3 & 2010 Jan.17 & ∼ 2$^{\\rm c}$ & [2] \nC/2010 E6 & 2010 Mar.10 & 1.1 & IAUC 9151,9157,[1],[2] \nC/2010 G4 & 2010 Apr.8 & 0 & [1],[2] \nC/2010 U8 & 2010 Oct.19 & 1.8 & [1],[2] \nC/2010 V8 & 2010 Nov.12 & 2.3 & [1],[2] \nC/2010 W2 & 2010 Nov.17 & 1.4 & [1],[2] \nSOHO-2062 & 2011 May 9 & 0.9 & [1],[2] \nSOHO-2072 & 2011 May 20 & 1.5 & [1] \nC/2011 N3 & 2011 Jul.4 & 1 & IAUC 9227,[1],[2] \nSOHO-2143 & 2011 Sept.30 & − 0.5 & [1],[2],[3] \nC/2012 E2 & 2012 Mar.13 & 1 & CBET 3047,[1],[3] \nSOHO-2505 & 2013 May 10 & 1.8 & [1] \nSOHO-2571 & 2013 Aug.18 & 2.6 & [1] \n \n \n \n \nWe took further measures to constrain the magnitudes for the individual entries in Table 1. Some of the objects were compared with other sungrazers, whose maximum brightness was reported to have been near apparent magnitude 3, by inspecting the SOHO’s C2 and C3 coronagraphic movies.[1](#fn1) Known magnitudes for these comparisons were taken primarily from Knight et al. (2010). The second author’s website blog,[2](#fn2) which provides a table of bright Kreutz sungrazers, proved especially useful in compiling Table 1. Also, his determination that the detector of the SOHO’s C2 coronagraph begins to saturate by objects of apparent magnitude ∼ 1.5 allows one to instantly set an upper or lower brightness limit by just looking at the image. The arguments for including some objects are presented in the table’s footnotes, which also provide information on the electronic blogs. Finally, the properties of the chosen set of SOHO/STEREO sungrazers brighter at maximum light than apparent magnitude 3 is in the following compared with the properties of a subset, the sungrazers brighter at maximum light than magnitude 2. The magnitudes of most of these objects were determined from C2 imaging and come from the second author’s website.2\nWhile the number of bright SOHO/STEREO sungrazers is limited, their advantage is that they are practically free from the selection effects that influence the detection of fainter Kreutz minicomets. In particular, strong annual double-peak variations in the discovery rate from images taken with the C2 coronagraph are due to periodic orientation changes in the Kreutz comets’ orbital plane relative to Earth (Sekanina 2002; Knight et al. 2010): the more prominent peak occurs in May, while the less pronounced but much more extended one covers October-December (e.g., Knight et al.’s Figure 13). One of the deep minima is in January and, accordingly, in a set of 19 objects there should be none during that month. Yet, in Table 1 we have two of them, consistent with an essentially random annual-distribution sample.\nFor the years 2004–2013, the cumulative distributions of arrival times of the SOHO/STEREO sungrazers brighter at their maximum than, respectively, apparent magnitude 3 and 2 are plotted in Figure 1, in which the entries from Table 1 are fitted with a polynomial of the 4th power modified at the ends. The arrival time is the calendar date of the first observation in SOHO’s C3 coronagraph, which is shown in column 2 of Table 1. Because no Kreutz sungrazer can enter the field of view of the SOHO’s C3 coronagraph earlier than less than 5 days — and the field of view of the C2 coronagraph earlier than less than 0.5 day — before perihelion, the difference between the arrival time and the perihelion time is negligible on the scale of Figure 1. The reason for choosing this approximation is the fact that perihelion times of nearly all sungrazers that have arrived since the beginning of 2011 are not yet available.\nThe upper curve in Figure 1 illustrates the rapidly increasing number of bright SOHO/STEREO sungrazers starting in 2008, while the lower curve, based on 13 objects, a year or two later. The overall shapes of the two distribution curves are similar, except for the slightly less steep rate of growth of the lower curve, as expected. This fact strengthens our confidence that we deal with representative sets of bright dwarf sungrazers. Both distributions — especially the upper one — do, however, display pronounced local deviations from the fitted polynomials, appearing as step-like features separated from each other by gaps extending over several months. Their nature will be addressed in Sec. 5.\nNo sungrazers were brighter at maximum light than magnitude 3 in 2004–2005, although in 2003, just off the scale of Figure 1, two were brighter:C/2003 F5 and C/2003 K7. From Table 2 of Knight et al. (2010), which includes only those Kreutz sungrazers whose brightness peaked in C3, it follows that during the decade 1996–2005 there was a total of eight SOHO sungrazers brighter at maximum than magnitude 3, suggesting an average rate of 0.8 per year. Knight et al.’s list does not though include C/1996 Y1 and C/2001 G2, both of which according to Kracht (see footnote 2) were brighter at maximum than magnitude 1. For the same period of time, Kracht lists eight sungrazers — or an average of 0.8 per year — that, mostly in C2, reached at maximum at least apparent magnitude 1, the same rate as that implied by Knight et al.’s data but for clearly brighter objects.\nBy differentiating the polynomial approximations to the curves in Figure 1, we derived the smoothed arrival-rate distributions of the bright SOHO/STEREO Kreutz sungrazers in Figure 2. The arrival-rate curves — poorly determined before and in early 2008 and again from late 2012 on — tend to converge to zero in 2006 and 2014.\nFrom Figure 2, the times of peak arrival rate are 2010.88 ( ∼ mid-November 2010) for the dwarf sungrazers brighter at maximum light than magnitude 3 and 2011.35 ( ∼ early May 2011) for those brighter at maximum light than magnitude 2. With the larger set assigned a greater weight and with an estimated uncertainty of two to four months in the times of peak arrival rate, we conclude that, on the average, this group of *bright dwarf sungrazers preceded the arrival of C/2011 W3 by just about one year, which suggests a potentially close relationship between both*.\nThe polynomial fitting indicates that the smoothed peak arrival rates were ∼ 4.6 and ∼ 4.3 objects per year, respectively, for the two sets of data, showing that the arrival rate of sungrazers with a magnitude at their maximum between 2 and 3 was only about 0.3 per year. At much lower arrival rates before 2009, the difference could not possibly have been any greater than this. For illustration, Figure 2 shows the pre-2008 constant rates of 1.3 per year for the sungrazers brighter at maximum light than magnitude 3 and 1.0 per year for those brighter than magnitude 2.\nThe relatively flat magnitude distribution function of the bright SOHO/STEREO Kreutz sungrazers is also documented by the surprising merger of the two curves in Figure 2 at times starting in the second half of 2011, suggesting a deficit of the sungrazers with magnitudes at maximum light between 2 and 3. Only to a degree may this property of the distribution function be affected by insufficient data. The same probably also applies to the precipitous dropoff rate during 2012, which in Figure 2 appears steeper than the rate of rise before 2011 and which is of course a product of a deficit of bright dwarf sungrazers during much of 2012.\n[ht]\n \nOrbital Elements and Apsidal Orientation of Bright SOHO/STEREO Kreutz Sungrazers from the Years 2004–2013 \nCompared with the Orbital Data for Spectacular Comets C/1843 D1 and C/2011 W3 (Eq.J2000.0). \nl@\nl@\nc@\nr@\nc@\nc@\nc@\nc@\nc@\nc@\nc@\nl\n \n& & & Number & \n& & & of used \nObject$^{\\rm a}$ & & *ω* & Ω & *i* & *q*($R\\_{\\mbox{\\boldmath $$}}$) & *L**π* & *B**π* & positions & Reference \n \nC/2006 A5\\* & 2006 & Jan. & 5.66 & 80∘.27 & 0∘.54 & 144∘.04 & 0.92 & 282∘.50 & +35∘.36 & 62 & MPC56609 \nC/2006 V2 & 2006 & Nov. & 3.78 & 84.76 & 5.89 & 144.58 & 1.07 & 282.31 & +35.25 & 48 & MPC58779 \nC/2008 K4\\* & 2008 & May & 23.84 & 82.63 & 3.57 & 144.57 & 0.99 & 282.59 & +35.09 & 39 & MPC63377 \nC/2009 C3\\* & 2009 & Feb. & 5.65 & 82.15 & 3.59 &144.82 & 1.01 & 283.16 & +34.80 & 17 & MPC66465 \nC/2009 D4\\* & 2009 & Feb. & 23.43 & 82.30 & 4.20 & 144.67 & 1.19 & 283.60 & +34.95 & 99 & MPC66465 \nC/2009 Y4\\* & 2010 & Jan. & 3.53 & 83.19 & 4.56 & 144.57 & 1.04 & 282.90 & +35.14 & 28 & MPC68393 \nC/2010 B3\\* & 2010 & Jan. & 21.78 & 83.17 & 4.42 & 144.57 & 1.03 & 282.78 & +35.13 & 98 & MPC72130 \nC/2010 E6\\* & 2010 & Mar. & 12.89 & 83.20 & 4.38 & 144.59 & 1.03 & 282.69 & +35.12 & 41 & MPC70816 \nC/2010 G4\\* & 2010 & Apr. & 10.24 & 83.40 & 4.39 & 144.66 & 1.01 & 282.45 & +35.06 & 07 & MPC72133 \nC/2010 U8\\* & 2010 & Oct. & 21.48 & 85.08 & 5.48 & 144.64 & 1.07 & 281.49 & +35.21 & 83 & This work \nC/2010 V8\\* & 2010 & Nov. & 14.50 & 85.18 & 6.57 & 144.49 & 1.15 & 282.48 & +35.36 & 61 & This work \nC/2010 W2\\* & 2010 & Nov. & 19.24 & 85.24 & 6.66 & 144.45 & 1.15 & 282.49 & +35.40 & 77 & This work \nSOHO-2062 & 2011 & May & 11.26 & 84.72 & 6.75 & 144.11 & 1.35 & 283.25 & +35.71 & 57 & This work \nSOHO-2072\\* & 2011 & May & 22.05 & 85.33 & 6.83 & 144.40 & 1.16 & 282.56 & +35.45 & 35 & This work \nC/2011 N3\\* & 2011 & Jul. & 6.00 & 85.10 & 6.41 & 144.41 & 1.14 & 282.43 & +35.44 & 02 & MPC75517 \nSOHO-2143\\* & 2011 & Oct. & 1.85 & 85.26 & 6.62 & 144.47 & 1.14 & 282.43 & +35.39 & 29 & This work \nC/2012 E2\\* & 2012 & Mar. & 15.00 & 85.72 & 7.46 & 144.34 & 1.15 & 282.71 & +35.54 & 61 & This work \nSOHO-2505\\* & 2013 & May & 12.31 & 85.78 & 7.13 & 144.62 & 1.10 & 282.30 & +35.26 & 28 & This work \nSOHO-2571 & 2013 & Aug. & 20.11 & 76.70 & 6.27 & 143.69 & 1.29 & 282.50 & +35.20 & 75 & This work \n \nC/1843 D1 & 1843 & Feb. & 27.91 & 82.75 & 3.69 & 144.38 & 1.17 & 282.58 & +35.28 & 37 & Sekanina & Chodas (2008) \nC/2011 W3 & 2011 & Dec. & 16.01 & 53.51 & 26.369 & 134.35 & 1.19 & 282.98 & +35.08 & 23 & Sekanina & Chodas (2012) \n \n \nA major conclusion from our analysis of the temporal distribution of bright dwarf sungrazers is that in the past several years we witnessed the *arrival of a swarm*, or, because of the local deviations from a smooth curve, the *arrival of a swarm of clumps* of these objects. An exciting possibility exists that all, or at least most, of these bright minicomets were closely related to each other, and possibly to C/2011 W3. First, however, this hypothesis needs to be tested dynamically.\nOrbital Data for Bright Dwarf Sungrazers\n========================================\nParabolic orbital elements for the first nine sungrazers in Table 1 were computed by B. G. Marsden and those for C/2011 N3 by G. V. Williams; they were published in various *Minor Planet Circulars* (MPC). A parabolic orbit determination for C/2012 E2 in MPC79023 rested on merely 39 observations of low accuracy from images taken with the HI1-B and C3 coronagraphs on board, respectively, STEREO-B and SOHO. With no orbits available for the more recent objects in Table 1 at the time we began to tackle this problem, the second author took up the task of measuring hundreds of their positions and computing their parabolic orbits, including the orbital re-determination of C/2012 E2 based, in addition, on more accurate positions from 19 images taken with the COR2-B coronagraph. Only after this work was completed did Gray (2013) independently publish the elements for comets C/2010 U8 (SOHO-1932), C/2010 V8 (SOHO-1948), and C/2010 W2 (SOHO-1954).\nThe parabolic orbits for the 19 sungrazers are, with the references, presented in Table 2; the second author’s results are listed for nine. For comparison, the last two lines contain the elements from elliptical solutions for two major sungrazers. The columns *L**π* and *B**π* provide, respectively, the longitude and latitude of perihelion.\nThe orbital data in Table 2 have major implications. They primarily show an utter lack of orbital similarity between the 19 bright dwarf sungrazers on the one hand and C/2011 W3 on the other. The discrepancies are ∼ 10∘ in the inclination and much more in the other angular elements. Thus, *the potential correlation suggested by the near-coincidence between the appearance of C/2011 W3 and the arrival-rate peak of the 19 bright dwarf sungrazers is strongly contradicted by this orbital evidence*.\nOur second finding is that except for the last entry, SOHO-2571, the bright dwarf sungrazers move in orbits that are *remarkably similar to that of C/1843 D1*, one of the two most spectacular Kreutz system’s members since 1800, just as do the orbits of most SOHO Kreutz sungrazers (sometimes called *Subgroup I*, e.g., Sekanina 2002).\nAnd third, with the exception of two more sungrazers, C/2006 V2 and SOHO-2062, three elements — the argument of perihelion *ω*, the longitude of the ascending node Ω, and the perihelion distance *q* — of the remaining objects’ orbits show a *strong tendency to increase systematically with time*, despite noise in the data. These *16 bright dwarf sungrazers*, marked with asterisks in Table 2, make up a *swarm of objects that are particularly tightly related to one another* (a tightly-knit swarm). The ranges spanned are about 5∘.5 in *ω*, 7∘ in Ω, and 0.24$R\\_{\\mbox{\\scriptsize \\boldmath $$}}$ in *q*. The range that measures a systematic trend in the inclination *i* must be less than 0∘.2. In fact, the inclination and the apsidal line exhibit no obvious trends.\n[t]\n \nAverage Slopes and Correlation Coefficients for \nOrbital Parameters of the 16 Tightly Related Members \nof a Swarm of Bright Dwarf Sungrazers in Table 2. \nl@\nc@\nc\n \nOrbital & Average & Correlation \nparameter & slope & coefficient \n \nArgument of perihelion (yr− 1) & +0∘.889 ± 0∘.089 & 0.94 \nLong. ascending node (yr− 1) & +1∘.010 ± 0∘.090 & 0.95 \nPerihelion distance ($R\\_{\\mbox{\\scriptsize \\boldmath $$}}$yr− 1) & +0.0322 ± 0.0088 & 0.70 \nInclination (yr− 1) & +0∘.024 ± 0∘.027 & 0.23 \nLongitude of perihelion (yr− 1) & − 0∘.078 ± 0∘.066 & 0.30 \nLatitude of perihelion (yr− 1) & +0∘.051 ± 0∘.029 & 0.42 \n \nTo confirm the veracity of these patterns, we offer two tests. The first is a comparison of the total range with the estimated uncertainty in each orbital element. No errors were published for the orbits in Table 2 taken from the MPCs. The solutions for the nine entries computed by the second author yielded, on the average, the RMS deviations of ± 0.0015 day in *t**π*, ± 0∘.19 in *ω*, ± 0∘.30 in Ω, ± 0∘.05 in *i*, and ± 0.011$R\\_{\\mbox{\\scriptsize \\boldmath $$}}$ in *q*. Equating the signal with the range and the noise with the sum of the RMS deviations for the 16 swarm’s members calculated as $\\sqrt{16} = 4$ times the average RMS deviation, we have a signal-to-noise ratio of 5–7 for *ω*, Ω, and *q*. On the other hand, the signal-to-noise ratio for *i* comes out to be < 1. This explains why the systematic trends are apparent in the first three elements but not in the inclination.\nThe differences between the behavior of *ω*, Ω, and *q* of the 16 swarm members, on the one hand, and their *i*, *L**π*, and *B**π*, on the other hand, are independently illustrated by comparing the relative errors of their average slopes, ⟨*d**ω*/*d**t*⟩, …, ⟨*d**B**π*/*d**t*⟩, and the correlation coefficients, which are summarized in Table 3. The results show huge gaps in both criteria between the two groups of orbital parameters. We will address the meaning of these effects and provide their interpretation in Sec. 5.\nThe trends in Table 2 are also corroborated by the object SOHO-2574, a sungrazer discovered in the second half of August 2013. From the brightness measurements by the second author, this comet never attained apparent magnitude 3, peaking at about 3.8 in C3, so it is not included in Tables 1 and 2. However, its orbit, in Table 4, further extends the time interval of the systematic pattern of the bright dwarf sungrazers. This fainter object probably is an outlying member of the swarm of tightly related sungrazers.\nTemporal Distribution of Arrivals of All SOHO/STEREO Sungrazers in 2004–2013\n============================================================================\n[b]\n \nParabolic Orbit of Comet SOHO-2574 (Eq. J2000.0). \nl@\nl@\nc@\nl@\nl@\nr\n \n*t**π* & 2013Aug.23.7201ET & *ω* & 85∘.817 ± 0∘.08 & *L**π* & 282∘.483 \n&\n ± 0.0005 & Ω & 7∘.356 ± 0∘.07 & *B**π* & +35∘.289 \n*q* & 1.123 ± 0.003$R\\_{\\mbox{\\scriptsize \\boldmath $$}}$ & *i* & 144∘.603 ± 0∘.017 & & 114 obs. \n \nThe spike-like distribution of arrival times of all Kreutz sungrazers detected with the coronagraphs on board the SOHO and STEREO spacecraft from the beginning of 2004 until the time of this paper’s submittal in August 2013 is shown in Figure 3. The plotted numbers are the totals of sungrazers with their arrival times within 12-day periods of time with a six-day overlap. The first interval (in UT) starts actually on December 29.0, 2003 and ends on January 10.0, 2004, the second interval extends from January 4.0 until January 16.0, 2004, etc. The overlap is intended to suppress sharp variations in the number of arriving sungrazers and makes thus a search for unexpected peaks deliberately more difficult.\nThe distribution clearly shows the annual variations due to the selection effects already noted in Sec. 2. However, superposed on these quasi-periodic fluctuations are, in spite of the employed overlapping technique, a clear systematic long-term variation, with the most conspicuous spike at 2010.97, almost exactly one year prior to the perihelion time of C/2011 W3. Inspection suggests that this spike was dominated by *a clump of 22 Kreutz sungrazers* detected during eight consecutive days, December 13–20, with an equivalent rate of an astonishing 1000 objects per year! The second highest spike in Figure 3 occurred one year later, at 2011.95, practically coinciding with the perihelion time of C/2011 W3, and dominated by a clump of 16 sungrazers discovered over 12 days, December 9–20. This rate is equivalent to nearly 500 objects per year. A third, double spike is located symmetrically to the largest one, in late 2009. Thus, comparing only December spikes in Figure 3 — in order to eliminate much of the annual selection effect — we are finding a *pattern that is strongly resembling that exhibited by the brighter sungrazers* in Figure 2.\n[b]\n \nOrbital Similarity for 22 Members of December 2010 \nClump of SOHO/STEREO Kreutz Sungrazers with \n2010/2011 Swarm of Bright Dwarf Comets. \nl@\nc@\nc@\nc@\nc@\nc@\nc\n \n& & & & \n& Number & Corona- & & \n& of used & graph(s) & Best & C/1843 & C/2011 & Part of \nObject$^{\\rm a}$ & positions & used & fit$^{\\rm b}$ & D1 & W3 & swarm? \n \nC/2010 X11 & 5 & C2 & ± 0′.05 &\n ± 5 ′.24 & ± 0′.81 & No?? \nC/2010 X12 & 26 & C2,C3 & ± 0.46 & ± 0.62 & ± 7.08 & Yes \nC/2010 X13 & 9 & C2 & ± 0.29 & ± 14.63 & ± 20.46 & Yes?? \nC/2010 X14 & 49 & C2,C3 & ± 0.69 & ± 15.58 & ± 24.87 & Yes \nC/2010 X15 & 17 & C2,C3 & ± 0.52 & ± 5.51 & ± 4.84 &? \nC/2010 X16 & 27 & C2,C3 & ± 0.47 & ± 29.65 & ± 38.11 & Yes \nC/2010 X17 & 18 & C2,C3 & ± 0.42 & ± 0.56 & ± 8.71 & Yes \nC/2010 Y1 & 11 & C2 & ± 0.27 & ± 0.79 & ± 7.80 & Yes \nC/2010 Y2 & 18 & C2 & ± 0.28 & ± 14.04 & ± 23.02 & Yes \nC/2010 Y3 & 24 & C2,C3 & ± 0.80 & ± 1.34 & ± 8.73 & Yes \nC/2010 Y4 & 12 & C2 & ± 0.40 & ± 3.20 & ± 6.17 & Yes \nC/2010 Y5 & 27 & C2,C3 & ± 0.62 & ± 1.40 & ± 10.36 & Yes \nC/2010 Y6 & 19 & C2 & ± 0.42 & ± 3.22 & ± 7.34 & Yes \nC/2010 Y7 & 27 & C2,C3 & ± 0.47 & ± 20.77 & ± 30.07 & Yes \nC/2010 Y8 & 16 & C2 & ± 0.41 & ± 0.81 & ± 10.18 & Yes \nC/2010 Y9 & 25 & C3 & ± 0.44 & ± 8.58 & ± 17.64 & Yes \nC/2010 Y10 & 88 & C2,C3 & ± 0.53 & ± 5.88 & ± 51.64 & Yes \nC/2010 Y11 & 19 & C2,C3 & ± 0.69 & ± 7.71 & ± 18.42 & Yes \nC/2010 Y12 & 16 & C2 & ± 0.19 & ± 5.65 & ± 6.50 & Yes? \nC/2010 Y13 & 14 & C2 & ± 0.28 & ± 2.73 & ± 9.68 & Yes \nC/2010 Y14 & 13 & C2 & ± 0.53 & ± 5.76 & ± 5.91 &? \nC/2010 Y15 & 59 & C2,C3 & ± 0.36 & ± 4.10 & ± 19.23 & Yes \n \n \n \n \nGray’s (2013) very recent publication of the orbital elements of the SOHO/STEREO sungrazers from the second half of 2010 includes all 22 objects of the December 2010 clump. Their definitive designations are C/2010 X11 through C/2010 X17 and C/2010 Y1 through C/2010 Y15. Their astrometric observations, now available as well, serve here to examine the similarity of their orbits to those of C/1843 D1 and C/2011 W3. For each sungrazer, Table 5 provides the number of observations used, the coronagraphs employed, and the RMS residuals from Gray’s best-fit parabolic solution and from the elliptical orbits for C/1843 D1 (Sekanina & Chodas 2008) and C/2011 W3 (Sekanina & Chodas 2012) in columns 2 to 6, respectively. The last column conveys our conclusion whether or not each of the objects is likely to be associated with C/1843 D1 and therefore with the swarm of bright dwarf sungrazers.\nA cursory glance through Gray’s list of orbits suggests at once that up to five of them have angular orbital elements that resemble those of C/1843 D1. They include the three most extensively observed among the 22 sungrazers — C/2010 Y10, C/2010 Y15, and C/2010 X14, which are in Table 5 labeled with a star. The other two sungrazers are C/2010 X16 and C/2010 Y12. The number of observations, on which their orbits are based, are comparable to those for the orbits of C/2010 X12 and C/2010 X17, which do not resemble that of C/1843 D1, yet the residuals from their forced solutions in colmn 5 are nearly as good as those from the best-fit solutions. Thus, for C/2010 X16 and C/2010 Y12 the orbital agreement with C/1843 D1 does not offer much confidence in the quality of the published solutions and a comparison of the residuals in columns 5 and 6 should be the decisive criterion, just as is the case with the rest of Table 5.\nOverall, the mean residuals in columns 5 and 6 suggest that the orbits of 19 of the 22 objects are better, usually much better, fitted by the orbit of C/1843 D1 than C/2011 W3. The association with the swarm of bright dwarf sungrazers is well established for 17, a little less so for C/2010 Y12 (because of the competing residuals in columns 5 and 6), and problematic but still possible for C/2010 X13 (because of the very small number of used observations).\nWe have serious doubts about the direct relationship to the swarm of bright dwarf sungrazers only for three members of the clump in Table 5: C/2010 X11, C/2010 X15, and C/2010 Y14. Information on the first object is based on only five observations, while other two show nearly equal mean residuals from the two reference orbits; there is thus very little evidence on which a judgment could be predicated.\nAs a general comment on column 5 of Table 5, the high mean residuals from the orbit of comet C/1843 D1 displayed by C/2010 X14, C/2010 X16, C/2010 Y2, and C/2010 Y7 are a product of these objects’ perihelion distances appreciably exceeding that of C/1843 D1, and should not alarm the reader.\nWe conclude that a *great majority of the clump of faint Kreutz sungrazers from the period of December 13–20, 2010 was associated with C/1843 D1 and therefore with the swarm of bright dwarf sungrazers*.\nNothing definite can at present be concluded about the clump of faint SOHO/STEREO sungrazers from December 2011. This problem must wait until the astrometric positions and orbits of these sungrazers become available.\nThe spike-like features in the distribution of arrivals of faint SOHO/STEREO Kreutz minicomets have their counterparts among the bright dwarf sungrazers as well (step-like features; see Sec. 2 and Figure 1). The four objects brighter at maximum than magnitude 3 that arrived between 2010.00 and 2010.27 (C/2009 Y4, C/2010 B3, C/2010 E6, and C/2010 G4) fit a slope distinctly steeper than the polynomial’s slope and so do the three objects between 2010.80 and 2010.88 (C/2010 U8, C/2010 V8, and C/2010 W2). The equivalent rates are, respectively, ∼ 15 and nearly 40 per year, several times greater than the peak rate of the smoothed distribution.\nWe detect a yet another effect that shows that during the few years around 2010 the distribution in Figure 3 behaves untypically. It is the relation between the May and the October-December peaks that we already mentioned in Sec. 2. While the May spikes are never smaller than the October-December spikes in 2004–2007 and 2012, the former are outperformed by the latter in 2008–2011.\nThe cumulative distribution of arrival times of all Kreutz sungrazers seen with the SOHO and STEREO coronagraphs between 2004 and 2013 is presented in Figure 4, in which we plot the number of objects after each six-day interval of time. The averaged slope of the distribution is practically constant until about the end of 2008, then it picks up a little, as can be expected from the distribution in Figure 3.\nPeculiar features on the cumulative-distribution curve in Figure 4 are flat segments or “flats,” *fairly prolonged* intervals of time during which the distribution curve essentially levels off, with a zero or near-zero arrival rate of the SOHO/STEREO sungrazers. Their times do not correlate well with those of SOHO’s roll and must be a true property of the Kreutz population’s distribution along the orbit. Because of the extremely compressed scale in Figure 4, the flats appear very short. The appearance of flats is in greater detail illustrated on a section of the cumulative distribution in Figure 5. Each flat was required to satisfy at least one of two conditions: either no sungrazers detected to arrive for a minimum of three six-day consecutive intervals, or their total number not exceeding four over a minimum of five six-day consecutive intervals. Statistically, one would expect the highest rates of flats during 2004–2005 because of lower arrival rates, no high spikes on the distribution curve in Figure 3, and the absence of bright SOHO/STEREO sungrazers. However, Figure 4 shows that this was not the case, suggesting instead a seemingly counterintuitive positive correlation between the bright dwarf sungrazers’ arrival rate and the rate of flats on the cumulative-distribution curve during the period ending with 2008: there were no flats between the start of 2004 and mid-2006, while during the next two years, between the arrivals of the bright sungrazers C/2006 A5, C/2006 V2, and C/2008 K4, the average rate was 3 flats per year. From mid-2009 to mid-2011 the rate of flats was nearly constant at 2 per year.\nDiscussion\n==========\nThe properties of the distribution of arrivals of the SOHO/STEREO sungrazers in the years 2004–2013 illustrate the morphological complexity of the Kreutz system, which is driven by — and can be understood qualitatively and, to a degree, even quantitatively, in terms of — a cascading fragmentation process, proposed by Sekanina (2002) and in greater detail by Sekanina and Chodas (2007). In this model, the fragments continue to break up over and over again at all heliocentric distances, so that an enormous and *essentially* continuous stream of subfragments (or high-generation fragments) arrives at perihelion at any time. This stream, which forms a ring along the orbit, is, however, not structureless, and the degree of its nonuniformity varies from place to place on many spatial scales that are equivalent to time scales from days to dozens of years and longer. The segment of the Kreutz system that arrived at perihelion in 2004–2005 was apparently characterized by a high degree of uniformity, while in the subsequent years the opposite was the case.\nThe existence of brighter objects ( in Figures 3 and 4) in the cloud of SOHO/STEREO minicomets suggests that, among the sungrazers, they may consist of a material of a greater than average cohesion, so that their fragmentation rate is slower than that of other fragments in the stream. Because their parent(s) broke up less often and probably also at later times than the rest, the spatial distribution of subfragments that derive from such objects is less uniform and more cluster-like, which necessarily leads to greater variations and relatively often to *gaps* in their distribution along the orbit. These gaps show up as the flat segments on the cumulative-distribution curve. Thus, to a point, the occurrence rate of flats should indeed vary in some proportion to the arrival rate of bright dwarf fragments. Only when this rate increases significantly is there much more debris around, so that the number of flats begins to drop, but even then they should not be entirely absent.\nThe orbital results for the bright dwarf sungrazers in Table 2, which completely rule out any potential relationship between their sharply elevated rate and comet C/2011 W3, are supported by the evidence of very few *faint* SOHO/STEREO Kreutz minicomets in orbits similar to that of C/2011 W3. Table 6 shows that there were only nine such objects in the set of more than 1000 SOHO/STEREO Kreutz sungrazers between 2004 and 2013 with currently known orbits. Faint companions in close proximity of C/2011 W3 were recent fragments whose existence does not change the overall picture.\n[ht]\n \nList of Orbital Elements and Apsidal Orientation for SOHO/STEREO Sungrazers with Available Orbits in 2004–2013 \nMoving in Paths Generally Similar to That of C/2011 W3 (Eq. J2000.0). \nl@\nl@\nc@\nr@\nc@\nc@\nc@\nc@\nc@\nc@\nc@\nc@\nc@\nc\n \n& & & & Number & & \n& & & Corona- & of used & & \nObject & & *ω* & Ω & *i* & *q*($R\\_{\\mbox{\\boldmath $$}}$) & *L**π* & *B**π* & graph(s) & positions & Author & Reference \n \nC/2004 A4 & 2004 & Jan. & 8.64 & 60∘.94 & 333∘.15 & 134∘.78 & 1.03 & 281∘.42 & +38∘.35 & C3 & 15 & Marsden & MPC60097 \nC/2004 P5 & 2004 & Aug. & 9.45 & 52.72 & 324.88 & 133.10 & 1.31 & 282.97 & +35.52 & C3 & 25 & Marsden & MPC52766 \nC/2005 U5 & 2005 & Oct. & 22.20 & 49.13 & 319.04 & 131.09 & 1.05 & 281.82 & +34.75 & C3 & 15 & Marsden & MPC55718 \nC/2006 U8 & 2006 & Oct. & 17.09 & 53.25 & 319.52 & 137.33 & 1.07 & 274.96 & +32.89 & C2,C3 & 10 & Marsden & MPC58538 \nC/2007 S4 & 2007 & Sept. & 24.31 & 57.15 & 325.85 & 138.08 & 1.07 & 276.80 & +34.14 & C2,C3 & 14 & Marsden & MPC60927 \nC/2010 C4 & 2010 & Feb. & 7.99 & 61.46 & 333.75 & 135.99 & 1.10 & 280.85 & +37.62 & C3 & 14 & Marsden & MPC72131 \nC/2010 Q2 & 2010 & Aug. & 24.79 & 51.38 & 325.82 & 133.48 & 1.73 & 285.08 & +34.54 & C2,C3 & 62 & Gray & MPC84618 \nC/2010 Y16 & 2010 & Dec. & 21.83 & 57.04 & 329.47 & 133.49 & 1.67 & 282.76 & +37.50 & C3 & 15 & Gray & MPC84623 \nC/2013 F4 & 2013 & Mar. & 27.51 & 60.71 & 336.38 & 138.47 & 1.79 & 283.22 & +35.33 & C3 & 26 & Gray & MPC84626 \n \n \n*What are these conflicting lines of evidence telling us?* By far the most plausible hypothesis is that *another Lovejoy-sized fragment*, closely related to C/1843 D1 and moving in a similar orbit but with its perihelion time shortly preceding that of C/2011 W3, *continued to break up during its orbital motion still at large heliocentric distances*. This hypothesis logically explains several major findings of our paper.\nFirst, the *16 tightly related bright dwarf sungrazers in the swarm*, identified in Sec. 3 and Table 2, are the *prime fragmentation products of the postulated object — their parent*. Indeed, adding up the brightness contributions from the 16 sungrazers in Table 1 leads to an equivalent of apparent magnitude − 2.0, not much fainter than was C/2011 W3 (whose apparent magnitude at maximum light was estimated at − 3; e.g., Green 2011).\nSecond, a population of fainter dwarf sungrazers must be, in addition to the 16 bright ones, also derived from the same parent. An example of direct evidence for this argument is the *clump* of 22 faint objects from December 2010, most of which were shown to be associated with the swarm of bright dwarf sungrazers, representing in this scenario a *debris of a sizable, but less cohesive member of this swarm that fragmented long before reaching perihelion*. This clump is probably the most extreme example of this advanced stage of cascading fragmentation but certainly not the only one.\nThird, a *hint of a hierarchy in the distribution of fragment dimensions is perceived in both the swarm and the clump*. The swarm’s brightest object, SOHO-2143, is in about the same position relative to the apparent ends of the swarm modeled in Figure 2, as is the clump’s most prominent object, C/2010 Y10, relative to the apparent ends of the clump. In both cases this “primary” fragment is in the feature’s second half. Its positioning may provide information on the vector field of separation velocities and on the fragmentation mechanism’s nature.\nFourth, this hypothesis is consistent with our explanation of the *gaps* in the distribution of arrival times of the SOHO/STEREO Kreutz population or, equivalently, the *flats* in its cumulative distribution, as described in Sec. 4 and earlier in this section.\nFifth, the presented hypothesis explains the *fairly well-defined limits of the tightly-knit swarm*. It is in fact remarkable that SOHO-2571, the last point on the cumulative distribution of bright dwarf sungrazers in Figure 1, has orbital elements so dramatically different from those of the swarm’s members. The future will show whether this comet is a “wanderer” (like bright SOHO sungrazers before 2004) or the first member of a new swarm.\n[b]\n \nPerturbation of Arrival Time (yr) at Next Perihelion by Separation Velocity Acquired in Fragmentation Event \nas Function of Heliocentric Distance. \nc@\nc@\nc@\nc@\nc@\nc\n \nDistance & Elapsed & & \nfrom Sun$^{\\rm a}$ & time$^{\\rm b}$ & & \n(AU) & (yr) & & & & \n \n10 & 2.42 & +3.6715 & − 3.6463 & +0.0891 & − 0.0888 \n20 & 6.96 & +2.5195 & − 2.5074 & +0.0446 & − 0.0443 \n30 & 13.01 & +1.9941 & − 1.9865 & +0.0297 & − 0.0294 \n50 & 29.10 & +1.4441 & − 1.4400 & +0.0178 & − 0.0175 \n75 & 56.47 & +1.0707 & − 1.0683 & +0.0117 & − 0.0115 \n00 & 92.79 & +0.8261 & − 0.8246 & +0.0086 & − 0.0083 \n30 & 51.88 & +0.6061 & − 0.6052 & +0.0063 & − 0.0060 \n60 & 40.25 & +0.4178 & − 0.4174 & +0.0045 & − 0.0043 \n86 & 50.00 & +0.1851 & − 0.1849 & +0.0025 & − 0.0023 \n60 & 59.75 & +0.0649 & − 0.0648 & +0.0012 & − 0.0011 \n30 & 48.12 & +0.0325 & − 0.0324 & +0.0007 & − 0.0007 \n00 & 07.21 & +0.0159 & − 0.0159 & +0.0004 & − 0.0004 \n75 & 43.53 & +0.0079 & − 0.0079 & +0.0002 & − 0.0002 \n50 & 70.90 & +0.0032 & − 0.0032 & +0.0001 & − 0.0001 \n30 & 86.99 & +0.0011 & − 0.0011 & +0.0001 & − 0.0001 \n20 & 93.04 & +0.0005 & − 0.0005 & 0.0000 & 0.0000 \n10 & 97.58 & +0.0001 & − 0.0001 & 0.0000 & 0.0000 \n \n \n \nMore broadly, the hypothesis of a broken-up parent sungrazer is closely linked to, and symptomatic of, the process of nucleus fragmentation at large heliocentric distance, whose important trait is its fairly insignificant effect on the orbital periods of the parent’s fragments (Sekanina 2002). Accordingly, they arrive at perihelion nearly simultaneously (on a scale of the orbital period) as a *true swarm*. For illustration, Table 7 lists the perturbations of the arrival time at next perihelion for a fragment that separated far from the Sun with a typical velocity of 1 m s− 1 in either of two cardinal directions in the plane of a sungrazing orbit with a perihelion distance of 1.2 $R\\_{\\mbox{\\scriptsize \\boldmath $$}}$ and an orbital period of 900 years (with aphelion at 186.43 AU). The radial component, *V**R*, is directed away from (+) or toward the Sun, the transverse component, *V**T*, is in the orbit plane and perpendicular to *V**R*, pointing in the general direction of the orbital motion (+) or in the opposite direction. There is no contribution from the normal component.\nTable 7 shows some noteworthy facts. First of all, the radial separation velocity is 10 to 100 times more efficient in triggering off the same change in the arrival time at next perihelion passage than is the transverse velocity. Second, the effect is approximately, but not exactly, symmetrical for positive and negative velocities. And third, a peak arrival rate of 4.6 per year for the bright dwarf sungrazers (Sec. 2) is equivalent to their minimum temporal separation of 0.22 yr, which in Table 7 matches the arrival-time perturbation for a separation near aphelion. If the separation velocity should be lower than 1 m s− 1, the assumed fragmentation event would have taken place before aphelion, and vice versa. The temporal relations among the bright dwarf sungrazers from are therefore on the right order of magnitude to be consistent with the process of cascading fragmentation.\n[t]\n \nPerturbations of Argument of Perihelion (*ω*), \nLongitude of Ascending Node (Ω), and Inclination (*i*) \nby Normal Component of Separation Velocity \nAcquired in Fragmentation Event as Function \nof Heliocentric Distance. \nc@\nc@\nc@\nc@\nc@\nc@\nc\n \nDistance & & & \nfrom Sun$^{\\rm a}$ & & & \n(AU) & & & & & & \n \n10 & − 0∘.25 & +0∘.25 & − 0∘.31 & +0∘.31 & − 0∘.03 & +0∘.03 \n20 & − 0.50 & +0.50 & − 0.62 & +0.62 & − 0.06 & +0.06 \n30 & − 0.75 & +0.75 & − 0.92 & +0.93 & − 0.09 & +0.08 \n50 & − 1.25 & +1.26 & − 1.54 & +1.55 & − 0.14 & +0.12 \n75 & − 1.87 & +1.90 & − 2.31 & +2.33 & − 0.21 & +0.17 \n00 & − 2.49 & +2.53 & − 3.07 & +3.11 & − 0.29 & +0.21 \n30 & − 3.23 & +3.30 & − 3.99 & +4.05 & − 0.38 & +0.25 \n60 & − 3.97 & +4.06 & − 4.90 & +4.99 & − 0.48 & +0.28 \n86 & − 4.61 & +4.73 & − 5.69 & +5.81 & − 0.56 & +0.29 \n60 & − 3.98 & +4.06 & − 4.91 & +4.99 & − 0.45 & +0.25 \n30 & − 3.24 & +3.30 & − 4.00 & +4.05 & − 0.35 & +0.21 \n00 & − 2.50 & +2.53 & − 3.08 & +3.11 & − 0.25 & +0.17 \n75 & − 1.88 & +1.90 & − 2.32 & +2.33 & − 0.18 & +0.13 \n50 & − 1.26 & +1.27 & − 1.55 & +1.56 & − 0.11 & +0.09 \n30 & − 0.76 & +0.76 & − 0.93 & +0.93 & − 0.06 & +0.05 \n20 & − 0.50 & +0.51 & − 0.62 & +0.62 & − 0.04 & +0.03 \n10 & − 0.25 & +0.25 & − 0.31 & +0.31 & − 0.02 & +0.01 \n \n \nThe argument of perihelion *ω* and the longitude of the ascending node Ω are both affected by fragmentation at large heliocentric distance rather strongly, while the inclination *i* to a much lesser degree, as is shown in Table 8. It is now the normal component, *V**N*, of the separation velocity that is primarily involved, pointing to the north orbital pole (+) or in the opposite direction. There is no contribution from the radial component and, only in *ω*, a very minor one, not exceeding 0∘.02 per 1 m s− 1 velocity from the transverse component. The normal-velocity effect is again nearly symmetrical relative to aphelion and also for positive and negative velocities. An exception is the inclination, with the negative velocity having a generally smaller effect, as small at aphelion as one half of that of the positive velocity of the same magnitude.\n[b]\n \nPerturbation of Perihelion Distance ($R\\_{\\mbox{\\scriptsize \\boldmath $$}}$) by Transverse \nComponent of Separation Velocity Acquired in \nFragmentation Event as Function \nof Heliocentric Distance. \nc@\nc@\nc@\nc@\nc@\nc@\nc\n \n& & & & & & \nDistance & & & Distance & \nfrom Sun & & & from Sun$^{\\rm a}$ & \n(AU) & & & & (AU) & & \n \n& & & & & & \n10 & +0.008 & − 0.008 & & 100 & +0.077 & − 0.075 \n20 & +0.015 & − 0.015 & & 130 & +0.101 & − 0.097 \n30 & +0.023 & − 0.023 & & 160 & +0.125 & − 0.119 \n50 & +0.038 & − 0.038 & & 186 & +0.146 & − 0.138 \n75 & +0.058 & − 0.057 & & & \n \nThe perturbations of the three angular elements, which always have the same sign, are only very approximately in constant proportions at different heliocentric distances. At aphelion, for example, the perturbation ratios are. Given the errors involved, these ratios are in good agreement (including the correct signs) with the relative ratios of of the average slopes in Table 3. This match is a very encouraging indication that the angular-element trends among the tightly-knit swarm’s sungrazers were indeed triggered by perturbations due to separation velocities acquired in fragmentation events involving these objects. The ratios also explain why in Table 2 the trends in *ω* and Ω are obvious but those in *i* are not.\nFragmentation always has an extremely small effect on the orientation of the line of apsides. The magnitude of these perturbations never exceeds 0∘.03 per 1 m s− 1 at > 10 AU from the Sun and they are caused by the transverse component of the separation velocity in the perihelion longitude *L**π* and by the normal component in the perihelion latitude *B**π*.\nThe perturbations of the perihelion distance for a fragment that broke off from its parent far from the Sun are dominated by the transverse component of the separation velocity, as shown in Table 9. There is a perfect symmetry relative to aphelion, so only the preaphelion numbers are listed. On the other hand, the effect’s symmetry due to a positive (in the general direction of the orbital motion) vs a negative velocity is only approximate. There is no contribution from the radial component and only a minor contribution, not exceeding 0.004 $R\\_{\\mbox{\\scriptsize \\boldmath $$}}$ per 1 m s− 1, from the normal component. It is noted that even a submeter-per-second separation velocity acquired during a *single* fragmentation event near aphelion can turn a fragment’s sungrazing orbit with a perihelion distance of 1.1 $R\\_{\\mbox{\\scriptsize \\boldmath $$}}$ into a sun-striking orbit. It is noted that two objects in Table 2 do indeed have their perihelion distances smaller than 1 $R\\_{\\mbox{\\scriptsize \\boldmath $$}}$.\nFinally, perturbations of the eccentricity due to fragmentation at large heliocentric distance are not discussed here in any detail because of the parabolic orbital approximation used for the swarm members. We only note that they are expressible in terms of the perturbations of the perihelion distance and the semimajor axis, whose reciprocal, 1/*a*, is affected by less than 0.00003 AU− 1 at > 10 AU from the Sun primarily by the radial component. Unlike near the Sun, the transverse component is much less important and there is no contribution to 1/*a* from the normal component.\nTo summarize up to this point, our discussion suggests that the orbital perturbations of fragments due to submeter- or meter-per-second separation velocities acquired during breakup events at large heliocentric distances are generally consistent with the systematic trends in the orbital elements in Table 2, supporting our hypothesis that the tightly-knit swarm of 16 bright dwarf sungrazers (as well as a debris of additional, fragmented members of the swarm), peaking in 2010/2011, consisted of fragments of an originally common parent, which itself must have been a fragment of a precursor that also gave birth to comet C/1843 D1. This precursor could have been the celebrated comet X/1106 C1, as suggested by Sekanina & Chodas (2007). The arrival time of the swarm, about 168 years after the appearance of C/1843 D1, requires a relative separation velocity of only about 1 m s− 1 between the swarm’s parent and this 19th-century sungrazer, if they detached from X/1106 C1 right at perihelion, or somewhat greater than 1 m s− 1, if a little off the perihelion point, but in any case a velocity typical for separations of sungrazer fragments near perihelion (Sekanina & Chodas 2007). It is possible that C/1843 D1, the swarm’s parent, many additional sizable sungrazers as well as huge amounts of debris had separated from X/1106 C1 as a single object, which became subject to sudden or gradual fragmentation only during its liftoff through the precursor’s atmosphere and/or soon afterwards.\nBecause SOHO-2143 was clearly the brightest among the swarm’s members (Table 1), it is plausible to deem this sungrazer the most massive fragment, located near the position that would have been occupied by the parent had it survived intact. The swarm’s 15 remaining objects are then products of one or more breakup events, in harmony with the conceptual paradigm of cascading fragmentation (Sekanina 2002, Sekanina & Chodas 2007). If the mechanism that accounts for the separation velocities causes them to prefer a certain direction — such as, for example, the parent’s spin — the contributions from the individual fragmentation events to the total perturbation effect would simply add up in some direction in an inertial coordinate system, to which the introduced coordinate system is, due to the orbit’s extreme elongation, practically equivalent at any point far enough from perihelion. The task is now reduced to finding out whether it is at all possible to choose a limited number of, and appropriate locations for, the breakup events in which a swarm’s fragment could have been reasonably involved, such that the respective orbital difference from Table 2 is in each of the five elements matched closely enough by the sum of perturbations from these events.\n[t]\n \nExample of Interpreting Orbital Difference Between Two Swarm Members in Terms of Perturbations \nAcquired During Series of Six Fragmentation Events.$^{\\rm a}$ \nl@\nc@\nc@\nc@\nc@\nc@\nc@\nc@\nc\n \nFragmentation event & #1 & #2 & #3 & #4 & #5 & #6 & Sum & Difference \nTime from previous perihelion (yr) & 2.4 & 9.8 & 29.1 & 69.8 & 151.9 & 50.0 & from & C/2006 A5 \nHeliocentric distance (AU) & 10.0 & 25.0 & 50.0 & 85.0 & 130.0 & 86.4 & the six & minus \nPerturbations: & & & & & & & events & SOHO-2143 \n \nArrival time at next perihelion (yr) & − 2.32 & − 1.40 & − 0.91 & − 0.61 & − 0.38 & − 0.12 & − 5.74& − 5.74 \nArgument of perihelion & − 0∘.11 & − 0∘.26 & − 0∘.51 & − 0∘.86 & − 1∘.30 & − 1∘.85 & − 4∘.89 & − 5∘.00 \nLongitude of ascending node & − 0∘.13 & − 0∘.31 & − 0∘.62 & − 1∘.05 & − 1∘.60 & − 2∘.28 & − 5∘.99 & − 6∘.08 \nInclination & − 0∘.01 & − 0∘.03 & − 0∘.05 & − 0∘.09 & − 0∘.15 & − 0∘.22 & − 0∘.55 & − 0∘.43 \nPerihelion distance ($R\\_{\\mbox{\\scriptsize \\boldmath $$}}$) & − 0.005 & − 0.011 & − 0.023 & − 0.039 & − 0.059 & − 0.083 & − 0.220 & − 0.22 \n \n \nThere exists far too little information to derive the unique solution. We offer one example of a greater than average difficulty to outline the issues involved and to illustrate the plausibility of our swarm’s fragmentation hypothesis. The first issues are whether in our scenario we should (a) presume more than one original parent and (b) consider more than a single orbit about the Sun. The answer is no on both counts. One, the lifetimes of all SOHO/STEREO Kreutz sungrazers observed to disintegrate just before perihelion are shorter than one revolution, as otherwise these objects would have disintegrated just before their previous perihelion passage and could not be seen nowadays. And two, even if fragmentation began before previous perihelion, the secondary parents, which would have to have been large enough objects to survive, and their own subsequently generated swarms would be separated from each other by such huge gaps in time that their association could not easily be recognized. Thus, our task is indeed confined to one parent and a single revolution about the Sun.\nWith these constraints in mind, let us consider the first entry in Table 2, C/2006 A5. Its orbital elements suggest that it was a member of the tightly-knit swarm. Because it preceded SOHO-2143 (a proxy for the parent) by as much as 5.74 years, it represents a highly challenging case. In concert with our objectives, we ask: Could the early arrival of C/2006 A5 and the deviation of its orbital elements from those of SOHO-2143 be explained by perturbations due to separation velocities acquired in a sequence of fragmentation events?\nIn our scenario, the difference of almost 6 years is an *integrated* perturbation of the arrival time at next perihelion, that is, the sum of perturbations at *ν* fragmentation events in which this sungrazer was involved either as part of a larger object in each of the first *ν* − 1 events or as a stand-alone fragment produced by the last, *ν*-th event.\nTo estimate the number of fragmentation events, we use a simple model, with the magnitude at maximum light (normalized, strictly, to 1 AU from Earth, but differing very little from the apparent magnitude in close proximity of the Sun) as a proxy for the initial mass, since both quantities were shown to have been tightly correlated (Sekanina 2003). For C/2006 A5 the magnitude at maximum light was 2.9 (Table 1); for the parent we approximate the lower limit to its peak brightness by that of SOHO-2143, that is, by magnitude − 0.5. The upper limit to the peak brightness of the original parent is uncertain. The object may have been as bright as magnitude − 2.0 (see above) or even brighter; we adopt, rather arbitrarily, magnitude − 2.5. If the parent and its fragments are assumed to have split, in each step of the process, into two parts of equal mass (50 percent of their immediate parent’s mass), the mass of any fragment after *ν* events is a part of the original parent. Because the estimated mass of C/2006 A5 comes\nout on our assumptions to be\nbetween and times smaller than the original parent’s mass, we obtain a condition. To the extent that this scenario applies, C/2006 A5 should have been involved in fragmentation events during the 900 years, not counting the initial breakup of X/1106 C1 near perihelion.\nIn our example, we deem C/2006 A5 a final product of six fragmentation events and require that the magnitude of the separation velocity acquired by it during each event not exceed 1 m s− 1. Since this sungrazer preceded SOHO-2143, the parent’s proxy, the radial component, which dominates the perturbations of the arrival time at next perihelion, must have been directed toward the Sun. We also require, rather arbitrarily, that the separation velocity vector not vary from event to event and that the last event have occurred at aphelion. With these constraints, all that needs to be chosen in Table 7 (or its expanded version) is six entries such that their sum is greater in absolute value than 5.74 years. One of many solutions that fits well the differences between the orbital elements of C/2006 A5 and SOHO-2143 in Table 2 is presented in Table 10.\nThe choice of orbital locations for the fragmentation events is, in the absence of conditions constraining the fragment’s history, rather arbitrary. Whether the fragmentation events were a corollary of major thermal stresses that continued to ravage the body’s interior after perihelion (Sekanina & Chodas 2012) or more or less spontaneous, the example in Table 10 conforms to an expectation that they became gradually less frequent and eventually ceased. Other scenarios are of course possible, but, as seen from Table 7, postaphelion fragmentation fits only differences of < 1 year or so in the arrival times at next perihelion, unless ∣*V**N*∣ ≫ 1ms− 1 or the number of fragmentation events is extraordinarily large.\nThe separation velocity that fits the fragmentation sequence in Table 10 is 0.96 m s− 1 per event, with the components,, and, all in m s− 1. The perfect agreement between the perturbation sum and the orbital difference is to be expected for the arrival time and the perihelion distance, because the perturbations of these elements are dominated by a single component of the separation velocity, the radial and the transverse one, respectively. The agreement between the perturbation sums and the orbital differences in the three angular elements is another matter, because the perturbations of all three are dominated by the normal component of the separation velocity. Also, when the radial and transverse components exceed in magnitude the normal component, the contributions from the transverse velocity to the arrival-time perturbation and from the normal component to the perihelion-distance perturbation are not entirely negligible and they impact the balance of the result. The residuals in Table 10 of about 0∘.1 in each of the three angular elements suggest that they are within the errors of observation, thus acceding to the plausibility of our perturbation scenario and to the status of the bright dwarf sungrazers in the swarm as fragments of a common parent, sharing orbits very similar to that of comet C/1843 D1 and being closely related to this spectacular object. In the light of this compelling evidence, the remarkable coincidence in timing between the arrival of the swarm and the appearance of comet C/2011 W3 is found to be purely fortuitous.\nThe clump of nearly two dozen faint SOHO/STEREO Kreutz minicomets from mid-December 2010 is shown to be a debris of the swarm’s another member that fragmented far from the Sun on its way to perihelion. For the clump’s duration of about eight days, Table 7 suggests that the parent’s fragmentation most probably began more recently than some 150 years ago and less than ∼ 130 AU from the Sun after aphelion. When orbits of all SOHO/STEREO sungrazers from 2011 are available, the origin of the second major clump, from December of that year, can likewise be examined.\nThe situation with the swarm of bright dwarf sungrazers is reminiscent of that with the temporal distribution of the Solar Maximum Mission (SMM) sungrazers, whose orbits were studied extensively by Marsden (1989). He found that, like the swarm members in our investigation, the objects belonged to what he called *Subgroup I*, which includes C/1843 D1, and that two pairs of them arrived only 12–13 days apart. Although the SMM sungrazers’ orbits were determined less accurately than those of the brighter SOHO/STEREO sungrazers, a general tendency is apparent among the SMM objects’ orbits (Marsden 1989, Marsden & Williams 2008) for the argument of perihelion and the longitude of the ascending node to systematically increase with time in the course of that “swarm’s” observed duration of about two years.\nThe experience with the 2010/2011 swarm of bright dwarf sungrazers and comet C/2011 W3 shows that it is *not advisable to employ suddenly increasing rates of SOHO/STEREO Kreutz sungrazers*, especially the brighter ones, *as a token of an imminent* (on a time scale of a year or so) *arrival of a spectacular member of the Kreutz system*. Although it would have worked, because of a coincidence, in the case of C/2011 W3, a prediction of this kind — with no further supporting evidence — is in fact merely a product of wishful thinking and very risky. It appears that the process of cascading fragmentation, whose influence on the evolution of comets in general and of the Kreutz system in particular proves overwhelming, makes it virtually impossible to predict the arrival of a spectacular sungrazer on time scales of a year or so.\nIn evolutionary terms, the swarm of bright dwarf sungrazers provides useful information toward our understanding the Kreutz system’s morphology, just as does C/2011 W3. They both represent *warning signals* that the expected new, 21st-century cluster of spectacular members of the Kreutz system is approaching. Even though the swarm and C/2011 W3 are products of different evolutionary paths, the orbital periods — directly computed for C/2011 W3 and indirectly inferred for the swarm — consistently show that the 20th-century cluster, which was observed to extend from 1945 to 1970, did not represent the last “bead” of the “string of pearls” [to euphemistically express the extent of the protofragment’s perihelion breakup in analogy to the appearance of comet D/1993 F2 (Shoemaker-Levy)], but that one or more “beads” are yet to come. The alert to the expected new cluster is the only parallel link that we find in our effort to decode the meaning of the two remarkable, nearly-simultaneous events.\nConclusions\n===========\nOur study of the population of SOHO/STEREO’s Kreutz sungrazers and their rate of arrival in the years 2004–2013 offers the following conclusions on the Kreutz system and on the potential temporal relationship with comet C/2011 W3:\n(1) In addition to its usual annual periodicity, caused by selection effects, the overall arrival-time distribution displays a number of prominent spikes, whose magnitude peaked approximately one year before the appearance of comet C/2011 W3, in December 2010.\n(2) Comet C/2011 W3 arrived also about one year after the sharply elevated arrival rate of the SOHO/STEREO sungrazers brighter at maximum light than apparent magnitude 3, which are free from major selection effects. A polynomial fitted through a set of 19 objects of this swarm showed a peak rate of 4.6 per year centered on 2010.88, while a similar polynomial fitted to a subset of 13 sungrazers brighter at maximum light than apparent magnitude 2 indicated a peak rate of 4.3 per year centered on 2011.35, compared with a pre-2006 average rate of ∼ 1 per year. The estimated uncertainty in the times of peak arrival rate is a few months.\n(3) A potential physical correlation between comet C/2011 W3 and the elevated rate of the distribution of bright SOHO/STEREO sungrazers is strongly contradicted by dynamical evidence, which shows no orbital similarity between these bright objects and C/2011 W3, with deviations of 10∘ or more in each of the three angular elements.\n(4) On the other hand, fully 18 of the 19 bright SOHO/STEREO sungrazers were found to move in orbits very similar to that of the spectacular Kreutz comet C/1843 D1, and 16 of them made up a swarm of tightly related objects, with clear systematic trends in the argument of perihelion, the longitude of the ascending node, and the perihelion distance.\n(5) The cumulative distribution of SOHO/STEREO sungrazers’ arrivals exhibits a series of flat segments or “flats,” relatively prolonged intervals of time (3–5 weeks) during which the arrival rate of the population of Kreutz minicomets was nil or extremely low (gaps); they do not correlate well with the instances of SOHO’s roll.\n(6) The temporal distribution of the flats is strongly nonuniform:no flat is located from the beginning of 2004 to mid-2006, while their rate is the highest and equal to 3 per year between mid-2006 and mid-2008; from mid-2009 to mid-2011 the rate is nearly constant at 2 per year.\n(7) The number of flats appears to correlate with the arrival rate of SOHO/STEREO sungrazers brighter at maximum light than magnitude 3; this correlation appears counterintuitive in that there are no flats in a segment of the distribution curve that is not populated by any bright dwarf sungrazer while the temporal rate of flats peaks in a segment of the curve that is marked by the arrivals of three such sungrazers.\n(8) This peculiar property is understood in the framework of cascading fragmentation, as gaps in the temporal distribution of subfragments are byproducts of relatively more sizable and cohesive parent fragments; their breakup generally leads to more nonuniform, cluster-like temporal arrangement of the ensuing subfragments.\n(9) Overall, the results of this study vividly illustrate the immense morphological complexity of the Kreutz system. In connection with C/2011 W3 they not only show that no bright SOHO/STEREO sungrazer was moving in a similar orbit at least between 2004 and the end of 2012, but that even faint Kreutz minicomets in such orbits were very rare, a few close short-lived companions accompanying the comet notwithstanding.\n(10) As a plausible interpretation of the evidence, we propose that another Lovejoy-sized sungrazer, moving in an orbit similar to that of C/1843 D1 but with the perihelion time close to that of C/2011 W3, continued to break up at large heliocentric distances; and that the swarm of bright dwarf sungrazers with their arrival rate peaking some time in late 2010 or early 2011 was the prime product of this fragmentation process.\n(11) The prominently spike-like distribution of fainter SOHO/STEREO minicomets suggests that it contains a debris of prematurely fragmented bright dwarf sungrazers also belonging to the swarm. This scenario is shown to apply to the most prominent clump of 22 faint SOHO/STEREO sungrazers from mid-December 2010, up to 19 of which fit a test of their association with the swarm. This correlation should also be verified on another clump of faint sungrazers from December 2011, when their orbital data become available.\n(12) A decisive role of sheer chance in the timing of the appearance of comet C/2011 W3, on the one hand, and of the significantly elevated arrival rate of the population of SOHO/STEREO Kreutz minicomets and especially the swarm of bright dwarf sungrazers, on the other hand, is under the circumstances indisputable.\n(13) A common origin of all 16 members of the tightly-knit swarm is understood in terms of perturbations during their original parent’s fragmentation, which proceeded in a cascading fashion at large heliocentric distances. This scenario is strongly corroborated by comparison of systematic trends in the swarm members’ orbital elements with the perturbations expected to be caused by separation velocities acquired by the fragments during such breakup events.\n(14) Spikes in, and sharply elevated rates of, the temporal distribution of SOHO/STEREO Kreutz sungrazers, bright and faint alike, should not be mistaken for a token of an imminent (on a time scale of a year or so) arrival of a spectacular member of the Kreutz system. Because of an overwhelming effect of cascading fragmentation, it is practically impossible to predict the arrival of a major Kreutz sungrazer on such short time scales.\n(15) The swarm of bright dwarf sungrazers and the clumps of fainter sungrazers, causing prominent spikes in the SOHO/STEREO arrival-time distribution, represent another warning signal that the expected 21st-century cluster of spectacular Kreutz comets is on its way to perihelion, to arrive in the coming decades (Sekanina & Chodas 2012). This alert is the only common link that we find between the recent nearly-simultaneous appearance of C/2011 W3 and the elevated arrival rates of the SOHO/STEREO Kreutz population. \nThis research was carried out in part at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. \nREFERENCES\n \n \n \n \n \n \n \n \n \n \n \n \n \n---\n1. See\nhttp:/\n/sohodata.nascom.nasa.gov/cgi-bin/data\\_query.[↩](#fnref1)\n2. See\nhttp:/\n/www.rkracht.de/soho/bright/bright.htm.[↩](#fnref2)\n"},"data_id":{"kind":"string","value":"arxiv_0183476"}}},{"rowIdx":45895,"cells":{"content":{"kind":"string","value":"Nonparametric Bayesian estimation of a Hölder continuous diffusion coefficient\n==============================================================================\nWe consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a fixed time interval. As a prior on the diffusion coefficient, we employ a histogram-type prior with piecewise constant realisations on bins forming a partition of the time interval. Specifically, these constants are realizations of independent inverse Gamma distributed randoma variables. We justify our approach by deriving the rate at which the corresponding posterior distribution asymptotically concentrates around the data-generating diffusion coefficient. This posterior contraction rate turns out to be optimal for estimation of a Hölder-continuous diffusion coefficient with smoothness parameter 0 < *λ* ≤ 1. Our approach is straightforward to implement, as the posterior distributions turn out to be inverse Gamma again, and leads to good practical results in a wide range of simulation examples. Finally, we apply our method on exchange rate data sets.\nIntroduction\n============\nProblem description\n-------------------\nStochastic differential equations (SDEs) have been widely used as models in numerous applications ranging from physics (see for example ) to engineering (see ) and to finance (see ). We assume observations from an SDE of the form\n \n$$\\label{sde}\n\\dd X\\_t=b\\_0(t,X\\_t)\\,\\dd t+s\\_0(t)\\,\\dd W\\_t, \\quad X\\_0=x, \\quad t\\in[0,T],$$ \nwith a drift coefficient *b*0, (deterministic) dispersion coefficient *s*0, and a deterministic initial condition *x*. Here *X* is real valued and *W* is a Brownian motion. We assume observations \n$$\\scr{X}\\_n =\\{X\\_{t\\_{0,n}}, \\ldots, X\\_{t\\_{n,n}}\\}$$ \n from the solution *X* to are available, where $t\\_{i,n}=\\tfrac{i}n T,$ *i* = 0, …, *n*. Our aim is to estimate *s*0 nonparametrically within the Bayesian setup.\nModel covers the case of linear SDEs, such as the popular Ornstein-Uhlenbeck process; see, e.g., Section 5.6 in. Among references that study as models for log returns of asset prices, we mention and, but the model has applications far beyond this context as well. A seemingly more general SDE\n \n$$\\label{sde2}\n\\dd X\\_t=\\widetilde{b}\\_0(t,X\\_t)\\,\\dd t+s\\_0(t)f\\_0(X\\_t)\\,\\dd W\\_t, \\quad X\\_0=x, \\quad t\\in[0,T],$$ \ncan be reduced to the form through a simple transformation of *X**t*, namely\n \n$$Y\\_t=x+\\int\\_0^{X\\_t}\\frac{1}{f\\_0(u)}\\dd u,$$ \nprovided *f*0 is known and sufficiently regular; see p. 186 in. Financial practitioners often are content with a model of the type as a simple and useful generalisation of the Black-Scholes model; see, e.g., pp. 7–14 in for Dupire, Derman and Kani’s pioneering work on local volatility. In particular, a discretely observed geometric Brownian motion with time-varying coefficients is also a special case of, once one passes to the corresponding log returns (see for additional information and applications). As the drift in is allowed to be non-linear, the distribution of *X**t* is in general not Gaussian and may well exhibit heavy tails, which is attractive from the point of view of financial applications. Finally, in some practical applications it is genuinely important to employ a time-dependent diffusion coefficient; a real data example is given in Section [section:realdata].\nRelated literature\n------------------\nStatistical inference for SDEs is a well studied and very active field of research, that is far from saturation. Relevant literature can be divided into two categories: works dealing with parametric and works dealing with nonparametric methods. Parametric approaches specify parametric forms for the drift and diffusion coefficients of SDEs. When these specifications use correct functional forms, such methods attain a higher statistical efficiency over the nonparametric ones. On the other hand, nonparametric approaches, where one only assumes qualitative features of the drift and diffusion coefficients, guard one against model misspecification, which may have dramatically negative consequences for valid inference; also, nonparametric techniques may suggest plausible parametric models in those situations where these models cannot be derived from the first principles; see, e.g.,. For parametric approaches to inference in SDE models, see, e.g., Chapter 2 in, Chapter 3 in, and references therein. Nonparametric statistical inference for SDEs of the type studied in the present work has been considered in, and within the frequentist setup, while and have explored the problem from the Bayesian perspective. Although the nonparametric methods these papers study are implementable in principle, these works are primarily of theoretical nature and practical performance of the corresponding approaches is not clear. Furthermore, except and, there is hardly any other work available on estimation of the dispersion coefficient (or diffusion coefficient) from the nonparametric Bayesian point of view, which constitutes the central topic of our paper. In this context we can mention only a theoretical contribution and a practically oriented paper, but the models considered there, as well as the sampling scheme, are different from ours, and the theory developed in does not cover the approach in. On a general level, apart from a philosophical appeal for Bayesians, advantages of a Bayesian approach include automatic quantification of uncertainty in parameter estimates through Bayesian credible sets, and the fact that it is a fundamentally likelihood-based method (see ). Furthermore, recent practical advances made in nonparametric Bayesian estimation of the drift coefficient, see, e.g.,, and, would suggest that comparable results can be obtained for estimation of the diffusion coefficient too. We note, however, that from an implementational point of view nonparametric Bayesian estimation of a dispersion coefficient is very different from drift coefficient estimation: the latter fundamentally relies on the equivalence of laws of continuously observed diffusion processes that have the same diffusion coefficient, which is not applicable when the diffusion coefficient itself is unknown and is a parameter to be estimated.\nApproach and results\n--------------------\nThe main practical challenges for Bayesian inference in SDE models from discrete observations are an *intractable likelihood* and absence of a closed form expression for the posterior distribution, which complicates considerably the inference; see, e.g.,,, and. We circumvent these difficulties by intentionally misspecifying the drift coefficient, and employing a (conjugate) histogram-type prior on the diffusion coefficient, that has piecewise constant realisations on bins forming a partition of [0, *T*] (this is different from and, where the drift *b*0 is in fact zero, and other priors are used). Due to this, our nonparametric Bayesian method to estimate the dispersion coefficient *s*0 in is easily implemented, fast and requires little fine-tuning from the user. We demonstrate its good practical performance on a wide range of simulated data examples and we apply it on real data from finance, yielding interesting conclusions.\nOn the theoretical side, we investigate the asymptotic performance of our Bayesian procedure from the frequentist point of view. Theoretical analysis of Bayesian procedures for inference in SDE models from discrete observations is in general challenging; see, e.g., the contributions, and for an impression, albeit in settings different from ours. We consider the ‘infill’ asymptotics with the time *T* horizon staying fixed and the number of observations *n* in the interval [0, *T*] increasing; this asymptotic regime is standard in the literature and can be thought of as reasonably satisfied in many financial applications. Complicating factors for a theoretical analysis in our setting are due to influence of the unknown drift coefficient *b*0, which we have intentionally misspecified. We address this through an argument based on Girsanov’s theorem. The main theoretical result we obtain tells us that the drift misspecification is asymptotically harmless and our procedure for estimating the diffusion coefficient is consistent at rate *n*− *β* in the *L*2-norm, with the precise value of *β* depending on the smoothness of the true dispersion coefficient. The corresponding posterior contraction rate is optimal for estimation of Hölder smooth dispersion coefficients of order 0 < *λ* ≤ 1.\nOrganisation of this paper\n--------------------------\nSection [section:model] contains the model specification and a detailed description of our nonparametric Bayesian approach. Section [section:asymp] contains the theoretical results. Our Bayesian method depends on a hyperparameter, the number of bins forming the partition of the interval [0, *T*], and Sections [sec:dic] and [sec:bayes:factor] discuss possible practical methods of its choice. In Section [section:simulation] we investigate practical performance of our method via simulations and provide illustrations of our theory from Section [section:asymp]. Examples with real data are studied in Section [section:realdata]. Proofs of the results from Section [section:asymp] can be found in Section [section:proofsgeneral]. In Appendix [appendix:asymp] we state and prove an additional theoretical result.\nFrequently used notation\n------------------------\nWe denote by ∥ ⋅ ∥2 the *L*2-norm with respect to the Lebesgue measure on the Borel sets of [0, 1]. We use the following notation to compare two sequences {*a**n*} and {*b**n*} of positive real numbers: $a\\_n\\lesssim b\\_n$ (or $b\\_n\\gtrsim a\\_n$) means that there exists a constant *C* > 0 that is independent of *n* and is such that *a**n* ≤ *C**b**n*. As a combination of the two we write *a**n* ≍ *b**n* if both $a\\_n\\lesssim b\\_n$ and $a\\_n\\gtrsim b\\_n$. We will also write *a**n* ≫ *b**n* to indicate that *a**n*/*b**n* → ∞ as *n* → ∞. By *a* ∨ *b* we denote the maximum of two numbers *a* and *b*.\nWe let $\\pp$ denote the law of the path (*X**t*: *t* ∈ [0, *T*]) from under the true parameter values (*b*0, *s*0). In particular the notation P0, *s*0 is used for such a law when the drift coefficient is equal to zero. We denote the prior distribution on the dispersion coefficient by Π*n* (with *n* the number of observations) and write the posterior as Π*n*( ⋅ ∣ X*n*). We denote the posterior expectation and variance by $\\expi(\\,\\cdot\\mid\\mathcal{X}\\_n)$ and $\\vpi(\\,\\cdot\\mid \\mathcal{X}\\_n),$ respectively.\nAssumptions and Bayesian setup\n==============================\nWe summarise the assumptions on our statistical model.\n[standing] Assume that\n1. the model is given with *x* = 0 and *T* = 1;\n2. the drift coefficient satisfies a linear growth condition and is Lipschitz in its second argument: for some *K* > 0 it holds that\n \n$$\\begin{aligned}\n|b\\_0(t,x)|^2 & \\leq K(1+|x|^2),\\quad \\forall x\\in{\\mathbb{R}},\\\\\n|b\\_0(t,x)-b\\_0(t,y)| &\\leq K|x-y|, \\quad \\forall t\\in[0,1], \\quad \\forall x,y\\in{\\mathbb{R}};\\end{aligned}$$\n3. the dispersion coefficient *s*0 is Hölder continuous on [0, 1] with Hölder constant *L* and Hölder exponent *λ* ∈ (0, 1], ∣*s*0(*u*) − *s*0(*v*)∣ ≤ *L*∣*u* − *v*∣*λ* for all *u*, *v* ∈ [0, 1], and is bounded away from zero and (by continuity also from infinity);\n4. a discrete time sample \n$$\\scr{X}\\_n =\\{X\\_{t\\_{0,n}}, \\ldots, X\\_{t\\_{n,n}}\\}$$ \n from the solution *X* to is available, where *t**i*, *n* = *i*/*n*, *i* = 0, …, *n*. For future reference, we also define *Y**i*, *n* = *X**t**i*, *n* − *X**t**i* − 1, *n*.\nUnder Assumption [standing], equation admits a unique strong solution, see, e.g., Theorems 2.5 and 2.9 of section 5.2 in. The minimal regularity conditions on the dispersion coefficient *s*0 in Assumption [standing] (c) are needed for our asymptotic statistical theory to work; see Section [section:asymp]. The sampling scheme in Assumption [standing] (d) is referred to as the high-frequency data setting and is a popular asymptotic setup for inference in SDE models, see, e.g.,,,,,, and. Financial data are often of a similar type, see. As remarked in, p. 1334, ``while the estimation of a drift coefficient function is theoretically of major interest in finance, for instance, for portfolio optimization, it can practically rarely be achieved. It is a fact that accurate estimation of drift coefficient functions requires considerably longer time series than typically available\". This example, then, provides an instance of a setting when the assumption *T* → ∞ cannot be thought to be reasonably satisfied. Other sampling schemes have been also considered in the literature on nonparametric inference for SDE models, e.g. when the time horizon *T* → ∞, but the distance Δ between observation times stays fixed (the so-called low frequency data setting; see, e.g., and ). Alternatively, one can assume observations are available at times Δ, 2Δ, …, and consider the case Δ → 0, *T* = *n*Δ → ∞, see, e.g., (this is again referred to as the high-frequency data setting). The question which of the asymptotic regimes reasonably applies to a given dataset in practice can be decided only on the case by case basis.\nOur first observation is that under the sampling scheme as in Assumption [standing] (d), consistent estimation of the drift coefficient *b*0 is impossible; see, e.g.,, p. 919. Furthermore, in many contexts, e.g. when pricing financial derivatives, knowledge of the drift coefficient is in fact of no interest, whereas the dispersion coefficient is of paramount importance; see. This motivates us to completely ignore the drift coefficient in our estimation procedure by intentionally misspecifying the model and acting as if the drift were equal to zero. We will justify this in Section [section:asymp]. Then the pseudo-likelihood associated with our observations is Gaussian and is given by\n \n$$\\label{likelih}\nL\\_n(s)=\\prod\\_{i=1}^{n} \\left\\{ \\frac{1}{\\sqrt{2\\pi\\int\\_{t\\_{i-1,n}}^{t\\_{i,n}}s^2(u)\\,\\dd u}}\\psi\\left( \\frac{Y\\_{i,n}}{\\sqrt{\\int\\_{t\\_{i-1,n}}^{t\\_{i,n}}s^2(u)\\,\\dd u}} \\right) \\right\\},$$ \nwhere *ψ*(*u*) = exp( − *u*2/2). Gaussian pseudo-likelihood is a widely used object in statistics, see, e.g., Section 5.2 in, and for some examples. Setting the drift to zero in our setting has a practical advantage of obtaining a simple and tractable expression for the pseudo-likelihood.\nWith Π*n* denoting a prior on the dispersion coefficient, provided all the involved quantities are suitably measurable, Bayes’ theorem gives that the posterior probability of any measurable subset *S* ⊂ S of dispersion coefficients is given by\n \n$$\\Pi\\_n(S\\mid X\\_{t\\_{0,n}}\\ldots,X\\_{t\\_{n,n}})=\\frac{\\int\\_{S} L\\_n(s) \\Pi\\_n(\\dd s)}{ \\int\\_{\\mathcal{S}} L\\_n(s) \\Pi\\_n(\\dd s) },$$ \nwhere S denotes a space on which the prior Π*n* is defined. From the above display various point estimates of *s*0 can be obtained, such as the posterior mean.\nIt follows from that the likelihood depends on the parameter of interest only through the integrals $\\int\\_{t\\_{i-1,n}}^{t\\_{i,n}}s^2(u)\\,\\dd u$ and is otherwise ‘blind’ to precise values the diffusion coefficient takes inside the intervals [*t**i* − 1, *n*, *t**i*, *n*]. Consequently, it appears natural to a priori model the diffusion coefficient as piecewise constant on intervals [*t**i* − 1, *n*, *t**i*, *n*]. However, some smoothing should also be performed, and this can be achieved by aggregated several neighbouring intervals [*t**i* − 1, *n*, *t**i*, *n*]. Thus, to construct the prior, we proceed as follows: Let *m* be an integer smaller than *n*. Then we can uniquely write *n* = *m**N* + *r* with 0 ≤ *r* < *m*, and in fact $N=\\lfloor \\frac{n}{m}\\rfloor$. Both *m* and *N* will depend on *n* (and we also write *m**n* and *N**n* to emphasize this when appropriate). With this assumption we have bins *B**k* = [*t**m*(*k* − 1), *n*, *t**m**k*, *n*), *k* = 1, …, *N* − 1 and *B**N* = [*t**m*(*N* − 1), *n*, 1]. Note that the length of *B**k* is then equal to *m*/*n* for *k* ≤ *N* − 1, whereas *B**N* has length $1-t\\_{m(N-1),n}=\\frac{r+m}{n}<\\frac{2m}{n}$. For notational convenience later on, we also write\n \n$$\\begin{aligned}\nB\\_k&=[a\\_{k-1},a\\_k), \\quad k=1,\\ldots,N-1,\\\\ \nB\\_{N}&=[a\\_{N-1},a\\_{N}]. \\end{aligned}$$ \nLet *s* = ∑*k* = 1*N**n**ξ**k***1***B**k*. The prior Π*n* on the dispersion coefficient *s* is defined by putting a prior on the coefficients *ξ**k*’s. Since\n \n*s*2 = ∑*k* = 1*N**n**ξ**k*2**1***B**k* = ∑*k* = 1*N**n**θ**k***1***B**k*, \nwhere we have put *θ**k* = *ξ**k*2, equivalently one can place the prior on the coefficients *θ**k*’s of the diffusion coefficient *s*2.\nWe call the prior Π*n* a histogram-type prior. Conceptually somewhat similar priors have already been employed in the nonparametric Bayesian density estimation context, as well as the Poisson intensity estimation context, see, e.g.,,,,,,, and, but our problem is rather different from density or Poisson intensity estimation and requires the use of many different ideas. In practice, e.g. in financial applications, one can hardly hope to estimate the diffusion coefficient to a very fine degree of detail, and in that respect using a prior with piecewise constant realisations does not appear to be unnatural. We note that a practical frequentist approach to nonparametric volatility estimation in likewise produces piecewise constant estimates. There is yet another practical advantage in using priors with piecewise constant realisations: financial time series often exhibit jumps, accurate detection of which is a delicate task. Due to their localised nature, our Bayesian estimates of the dispersion coefficient are likely to quickly recover from negative effects of a moderate number of undetected jumps.\nIf the prior coefficients *θ*1, …, *θ**N* are independent and have an inverse gamma IG(*α*, *β*) distribution with parameters *α*, *β* > 0, which will henceforth be our assumption, then the posterior is conjugate, as stated in the next lemma.\n[lem:post-ig] Assume *θ*1, …, *θ**N* are independent with the inverse gamma IG(*α*, *β*) distribution. Then *θ*1, …, *θ**N* are a posteriori independent and, for *k* = 1, …, *N* − 1, \n$${\\theta}\\_k \\mid \\scr{X}\\_n \\sim \\mathrm{IG}\\left(\\alpha+m/2, \\beta+ n Z\\_k/2\\right).$$ \n with \n*Z**k* = ∑*i* = (*k* − 1)*m* + 1*k**m**Y**i*, *n*2, \n whereas \n$${\\theta}\\_N \\mid \\scr{X}\\_n \\sim \\mathrm{IG}\\left(\\alpha+(m+r)/2, \\beta+ n Z\\_N/2\\right),$$ \n with \n*Z**N* = ∑*i* = (*N* − 1)*m* + 1*n**Y**i*, *n*2. \n[section:proofsig]\nWrite *θ* = (*θ*1, …, *θ**N*). The likelihood, considered as a function of *θ*, is given by\n \n$$\\begin{gathered}\n L\\_n({\\theta}) =\\prod\\_{i=1}^n \\phi\\left(0; Y\\_{i,n}, \\int\\_{t\\_{i-1}}^{t\\_i} s^2(u) d u \\right)\\\\ \\propto \\theta\\_N^{-(m+r)/2} \\exp\\left(-\\frac{n Z\\_N}{2{\\theta}\\_N} \\right) \\prod\\_{k=1}^{N-1} {\\theta}\\_k^{-m/2} \\exp\\left(-\\frac{n Z\\_k}{2{\\theta}\\_k} \\right). \\end{gathered}$$ \nHere *ϕ*(*x*; *μ*, *σ*2) denotes the density of the normal distribution with mean *μ* and variance *σ*2 evaluated at point *x*. As the prior satisfies *p*(*θ*1, …, *θ**N*) ∝ ∏*k* = 1*N**θ**k*− (*α* + 1)*e*− *β*/*θ**k*, the result easily follows.\nPosterior computations thus turn out to be elementary with our approach. For instance, the posterior mean of *s*2 can be obtained from the posterior means of *θ**k*’s. Recall that the IG(*α*, *β*) distribution has mean *β*/(*α* − 1) and variance *β*2/(*α* − 1)2(*α* − 2) (if finite, to which end one must have *α* > 2). Then, e.g., for *k* < *N* and *m* ≥ 2, the posterior mean of *θ**k* is equal to\n \n$$\\label{postgamma}\n\\ee\\_{\\Pi\\_n}(\\theta\\_k\\mid\\mathcal{X}\\_n)=\\frac{\\beta+ n Z\\_k/2}{\\alpha+m/2-1}.$$ \nConceptually the posterior mean of *s*2 in this context is similar to a regressogram; see, e.g., Examples 4.5 and 5.24 in. However, the Bayesian approach deals with the *entire posterior distribution* and does not reduce to a point estimate, such as the posterior mean.\nInstead of a histogram-type representation in, one could have tried to base the prior on some other series representation for *s*2. At first sight, e.g. splines are a sensible choice to that end. However, enforcing spline basis functions to have disjoint supports with endpoints at *t**n*, *i*’s does not appear to be a natural procedure, whereas other choices would not have lead to simple posterior computations as in Lemma [lem:post-ig].\nAsymptotic theory\n=================\nGeneralities\n------------\nA highly desirable property of a Bayesian procedure, in particular from the frequentist point of view, is that the posterior asymptotically concentrates around the true parameter value. In fact, studying the rate at which the posterior contracts around the true parameter is similar to studying convergence rates of frequentist estimators. Some of by now classical references, where general conditions for derivation of posterior contraction rates are given, include, and. However, we will follow a rather different and more direct path of the proof.\nFor *ɛ* > 0, we denote by\n \n*U**s*0, *ɛ* = {*s* ∈ S*n*: ∥*s* − *s*0∥2 < *ɛ*} \nthe *L*2-neighbourhood of *s*0 of radius *ɛ*.\nIn the next proposition, we show that without loss of generality one may assume *b*0 = 0 in the proofs. This proposition also explains why ignoring the drift in our estimation procedure by intentionally setting it to zero still leads to consistent Bayesian estimation of *s*0. The corresponding theoretical argument relies on an application of Girsanov’s theorem (see, e.g., Section 3.5 in ). We would like to stress the fact that given the simplicity of our prior and intentional misspecification of the likelihood, the possibility of consistent estimation of *s*0 with our approach is not obvious and requires a thorough investigation.\n[prop1] Let Assumption [standing] hold and assume that for *ɛ**n* → 0\n \n$$\\ee\\_{0,s\\_0}[\\Pi\\_n(U\\_{s\\_0,\\varepsilon\\_n}^c\\mid\\mathcal{X}\\_n) ] \\rightarrow 0$$ \nas *n* → ∞. Then also\n \n$$\\ee\\_{b,s\\_0}[\\Pi\\_n(U\\_{s\\_0,\\varepsilon\\_n}^c\\mid\\mathcal{X}\\_n)] \\rightarrow 0.$$ \nPosterior contraction rates\n---------------------------\nAs we consider the asymptotics *n* → ∞, we take the number of bins *N* to depend on the sample size *n*, and indicate this in our notation by writing *N**n*. Then also *m* depends on *n*, and we write *m**n* to emphasise this dependence.\nAssume that Assumption [standing] holds for the remainder of this subsection. The following theorem shows that posterior contracts at rate *n*− *λ*/(2*λ* + 1) in *L*2.\n[thm:rate-posterior-ig][cor:rate:b] Assume *N**n* ≍ *n*1/(2*λ* + 1). If we let $ \\eps\\_n\\asymp n^{-\\lambda/(2\\lambda+1)}$, then for any sequence *h**n* tending to infinity (as *n* → ∞) we have \n$$\\exzb \\left[ \\Pi\\_n(\\|s^2-s\\_0^2\\|\\_2 \\ge h\\_n \\eps\\_n \\mid \\scr{X}\\_n) \\right] \\to 0$$ \n as *n* → ∞.\nIn the next theorem we give the posterior contraction rate for the sup-norm.\n[cor:rate:b:sup] Assume *N**n* ≍ *n*1/(2*λ* + 1). If we let $ \\tilde\\eps\\_n\\asymp n^{-\\lambda/(2\\lambda+2)}$, then for any sequence *h**n* tending to infinity (as *n* → ∞) we have \n$$\\exzb \\left[ \\Pi\\_n\\Big(\\sup\\_{x\\in [0,1]}|s^2(x)-s\\_0^2(x)| \\ge h\\_n \\tilde\\eps\\_n \\mid \\scr{X}\\_n\\Big) \\right] \\to 0$$ \n as *n* → ∞.\nDiscussion\n----------\nNow we provide some discussion on the obtained theoretical results.\nEstablishing posterior contraction in the *L*2-metric is rather natural, as $\\int\\_0^1 s\\_0^2(t)\\dd t$ is the quadratic variation of the process (*X**t* : *t* ∈ [0, 1]) over the interval [0, 1]. It is also the variance of *X*1 when the drift coefficient *b*0 is a deterministic function depending only on time.\nThe inequality\n \n∥*s*2 − *s*02∥2 ≥ *κ*∥*s* − *s*0∥2, \nvalid for *s*0 satisfying Assumption [standing] (here *κ* > 0 is a lower bound of *s*0), implies that the rate of Theorem [thm:rate-posterior-ig] is also valid with ∥*s* − *s*0∥2. A similar remark applies to the posterior contraction in Theorem [cor:rate:b:sup].\nA comparison with the frequentist minimax convergence rate in shows that the posterior for the diffusion coefficient contracts at the optimal rate in the *L*2-metric (strictly speaking, the results in the latter paper are given for *B**p*, *q**s*-Besov smooth diffusion coefficients with *s* > 1, but general arguments for derivation of lower bounds in our statistical setup are classical and work also in the Hölder setting). With histogram-type priors considered in this work no further improvement in the posterior contraction rate, when *λ* = 1, is possible beyond *n*− 1/3, even if the function *s*0 is smoother than a Lipschitz function. An intuitive reason for this is that realisations of our histogram-type priors are too rough for this; cf. p. 629 in.\nThere exists an excellent and deep reference on Bayesian inference in misspecified infinite-dimensional models, namely. That paper provides some additional intuition why our approach is still successful despite the model misspecification. Summarised somewhat simplistically, the results from say that the posterior in misspecified statistical models asymptotically concentrates around that value from the parameter space that is closest to the ‘true’ parameter value in the sense of the minimal Kullback-Leibler distance between respective probability distributions. Asymptotically the observation scheme as in Assumption [standing] (d) is almost as good as observing the process *X* continuously over the interval [0, 1]. On the other hand, the laws corresponding to paths (*X**t* : *t* ∈ [0, 1]) with two different diffusion coefficients are mutually singular, see Theorem 3.24 in §III.3d,, with a consequence that the corresponding Kullback-Leibler divergence is infinite. Hence, irrespective of the prior assumptions on the drift coefficient, the posterior for the dispersion coefficient (equivalently, diffusion coefficient) should concentrate around the ‘true’ dispersion coefficient *s*0 (equivalently, the true diffusion coefficient *s*02), for it is precisely this parameter value that yields finite Kullback-Leibler divergence between the laws of the ‘true’ and various misspecified models.\nThe result in Theorem [cor:rate:b:sup] has to be compared to similar results on estimation of the volatility (not necessarily a deterministic function of time, as in our paper) in the *L*∞-norm. Such results have been obtained in,, and. As observed in, p. 273, ‘near to nothing is known on this topic’. establish consistency of their Fourier-based estimator of volatility, without deriving its convergence rate. The asymptotics considered in are different from the one in the present work, which makes a direct comparison difficult. Finally, Theorem 3.5 in gives a convergence rate of a kernel estimator of volatility; when translated to our setting, the corresponding optimal convergence rate, up to a log factor, is *n*− *λ*/(2*λ* + 1). This is a faster rate than the rate obtained in our Theorem [cor:rate:b:sup]. However, the suboptimal rate in Theorem [cor:rate:b:sup] is likely to be an artefact of our proof, specifically a somewhat rough bound employed in the second inequality of. Unlike our method, the kernel estimator in suffers from the boundary bias problem, which necessitates studying its asymptotic properties on a time interval strictly contained in [0, *T*].\nIn, the following frequentist estimator of *s*02 is introduced,\n \n*ŝ*2(*t*) = ∑*i* = 1*n**K**h*(*t*, *t**i*)*Y**i*, *n*2, \nwhere *K* is a kernel function, a constant *h* > 0 is a bandwidth, and $K\\_h(s,t)=\\frac{1}{h}K\\left(\\frac{t-s}{h}\\right)$ is a rescaled kernel. Suppose now *K* is a boxcar kernel, *K*(*u*) = (1/2)1∣*u*∣ ≤ 1, see p. 55 in. Then\n \n$$\\hat{s}^2(t)=\\frac{1}{2h} \\sum\\_{i\\colon|t-t\\_{i,n}|\\leq h} Y\\_{i,n}^2.$$ \nA brief reflection shows that for *k* < *N*, *n* large, and *h* = *m*/(2*n*) (half the bin length), *ŝ*2 is quite similar to, the difference being that in that formula averaging occurs over individual bins, while here one averages locally over observations in a neighbourhood of each time point *t*. We note, however, that the asymptotic theory in does not cover the asymptotics of the posterior mean as in, and also that the regularity conditions of that paper are different from ours. On the other hand, practical computation of the kernel estimator *ŝ*2 would typically require from a user some form of data binning, cf. Appendix D.2 in, so that from this point of view the posterior mean and the estimator *ŝ*2 are closely related.\nWe have already pointed out in the introduction that our setup covers more general SDE models than those with deterministic dispersion coefficients, see equation ; this follows from an application of Itô’s formula. Under regularity conditions, a further generalisation of our results is possible to the case when the dispersion coefficient *s*0 is in fact a stochastic process independent of the driving Wiener process in (we cite an interesting example in this context: a widely known stochastic volatility model arising as a diffusion limit of a GARCH(1, 1) process, see ). Namely, in this case one can simply follow the Bayesian methodology we described in Section [section:model] with no further changes, acting as if the dispersion coefficient were a deterministic function. Despite (yet another) purposeful misspecification, the resulting Bayesian procedure is consistent. This can be established by combining arguments in the present paper with the ones similar to those in, that deal with a kernel volatility estimator. Indeed, careful examination shows that one of the main steps in our proofs is what can be termed a Bayesian bias-variance decomposition, see the proof of Theorem [thm:rate-posterior-ig]. Both the ‘bias’ and ‘variance’ terms there are analysed using techniques similar to those employed in e.g. kernel regression or density estimation problems, whence a possibility for further generalisations. Space considerations preclude us from studying this interesting question in detail in the present work.\nBin number selection via DIC\n============================\nIn this section we describe a method of choosing *N* that is based on the Deviance Information Criterion (DIC) of ; see also and. Further discussion is given in Section [section:simulation].\nBy *M* we denote the posterior mean of *s*2; log*L**n*(*M*) is our notation for the log-likelihood evaluated at the posterior mean. Introduce the DIC measure of predictive accuracy, \n$$\\widehat{\\operatorname{elpd}}\\_{\\textrm{DIC}}=\\log L\\_n(M)-\\nu\\_{\\textrm{DIC}},$$ \n where “elpd” is an abbreviation for “expected log predictive density” and \n$$\\nu\\_{\\textrm{DIC}}=2 \\left\\{ \\log L\\_n(M) - \\expi ( \\log L\\_n(s) | \\mathcal{X}\\_n ) \\right\\}$$ \n is the effective number of parameters. Straightforward but tedious calculations employing Lemma [lem:post-ig] and properties of the (inverse) gamma distribution give that\n \n$$\\begin{aligned}\n\\log L\\_n(M)&=-\\frac{n}{2}\\log(2\\pi)-\\frac{n}{2}\\log\\left(\\frac{T}{n}\\right)\\\\\n& \\quad -\\frac{1}{2}\\sum\\_{k=1}^N m\\_k \\log\\left( \\frac{\\beta+nZ\\_k/2}{\\alpha+m\\_k/2-1} \\right)-\\frac{n}{2T} \\sum\\_{k=1}^N Z\\_k \\frac{\\alpha+m\\_k/2-1}{\\beta+nZ\\_k/2},\\end{aligned}$$ \nwhere *m**k* denotes the number of observations in the bin *B**k* (with a harmless abuse of notation) and *Z**k* is as in Lemma [lem:post-ig]. By similar calculations, \n$$\\nu\\_{\\textrm{DIC}}=\\frac{n}{T}\\sum\\_{k=1}^N \\frac{Z\\_k}{\\beta+nZ\\_k/2}-\\sum\\_{k=1}^N m\\_k \\left\\{ \\Psi\\left(\\alpha+\\frac{m\\_k}{2}\\right) - \\log\\left( \\alpha+\\frac{m\\_k}{2}-1 \\right) \\right\\},$$ \n where Ψ is the digamma function. The formulae simplify even further, when *m**k* = *m*, *k* = 1, …, *N*.\nNow the idea consists in evaluating $\\widehat{\\operatorname{elpd}}\\_{\\textrm{DIC}}$ for a range of values of *N*, and choosing the one that maximises $\\widehat{\\operatorname{elpd}}\\_{\\textrm{DIC}}$. This aims at optimising the predictive behaviour of the model. We note that using predictive performance criteria for Bayesian model selection is a well-established practice in the case of finite-dimensional, parametric models (see, e.g., ), but seems to be a new idea in the non-parametric Bayesian setting. DIC in some sense constitutes a Bayesian analogue of Akaike’s AIC, and conceptually our proposal is similar to employing information criteria for smoothing parameter selection in the frequentist literature; see, e.g., the widely cited work. On the computational side, our DIC-based method is very simple to implement and does not require heavy computations. We test its practical performance in Section [section:simulation].\nBin number selection via marginal likelihood\n============================================\nIn this section we describe an alternative method of the bin number selection to the one we discussed in Section [sec:dic]. This is based on maximising the marginal likelihood $\n\\int\\_{\\mathcal{S}} L\\_n(s) \\Pi\\_n(\\dd s)\n$ as a function of *N*, which can be viewed as model evidence given the data. Model selection based on the marginal likelihood (or Bayes factors) is well-established in Bayesian statistics. See, e.g., for an application to smoothing parameter selection in the context of smoothing spline regression, which is conceptually related to choosing the bin number *N* in our problem. On a more general level, this is nothing else but an instance of a well-known empirical Bayes method (cf. , Section 5.1).\nSince we identify a piecewise constant diffusion coefficient *s*2 with its coefficients *θ*1, …, *θ**N*, using a priori independence of *θ**k*’s and Fubini’s theorem, the marginal likelihood in our setting can be evaluated as (here *m**k* was defined in Section [sec:dic] and is the number of observations in bin *B**k*)\n \n$$\\begin{gathered}\n\\operatorname{ML}\\_N(\\mathcal{X}\\_n)=\\prod\\_{k=1}^N \\left\\{ \\int\\_{[0,\\infty)} \\left(\\frac{2\\pi}{n}\\right)^{-m\\_k/2} \\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)} \\theta\\_k^{-(\\alpha+m\\_k/2)-1} \\exp\\left(- \\frac{1}{\\theta\\_k}\\left( \\beta+\\frac{nZ\\_k}{2} \\right) \\right) \\dd \\theta\\_k \\right\\} \\\\\n\\propto \\frac{\\beta^{\\alpha N}}{\\Gamma(\\alpha)^N} \\prod\\_{k=1}^N \\frac{\\Gamma(\\alpha+m\\_k/2)}{(\\beta+nZ\\_k/2)^{\\alpha + m\\_k/2}}.\\end{gathered}$$ \nFrom a numerical point of view, rather than using analytic tools for optimisation, we recommend plotting the values of ML*N* versus its argument *N*, and performing graphical maximisation. This results in a computationally simple model selection procedure and we apply it in practice in Section [section:simulation].\nSimulated data examples\n=======================\nIn this section we use simulations of diffusion processes with known drift and diffusion coefficient to gain insight into the numerical performance of our method. We are particularly interested in both the practical consequence of using a pseudo-likelihood ignoring the drift and the empirical rate of posterior contraction attainable in examples.\nIn the first subsection we simulate realisations from the model for different dispersion and drift coefficients. Given subsamples of those realisations sampled at different rates we compute the posterior distribution of the dispersion function using the piecewise constant prior (histogram-type prior) with varying number of bins. As an illustration, plots of marginal posterior bands are compared with the true dispersion function. The marginal posterior bands are obtained by computing 1 − *α* central posterior intervals (see, p. 33) separately for the coefficients *θ**k*’s using Lemma [lem:post-ig].\nBy Proposition [prop1], assuming that there is no drift still leads to consistent Bayesian estimation of the dispersion coefficient, even if the data are from a diffusion process with nonzero drift. This is illustrated by our simulation results.\nIn the second subsection we use Monte Carlo methods to determine the distribution of the distance between samples from the posterior distributions and the true dispersion coefficient. In Theorem [thm:rate-posterior-ig] we showed that the posterior contraction rate in the *L*2-norm is optimal for estimation of Hölder smooth dispersion coefficients of order 0 < *λ* ≤ 1 for the prior based on the inverse gamma distribution. The results of the Monte Carlo simulation agree with this. Furthermore, we numerically determine the rate of posterior convergence in the supremum norm for two examples. The simulation results in this case are less conclusive, but suggest that the posterior contraction rate in the *L*∞-norm in Theorem [cor:rate:b:sup] is possibly suboptimal.\nOur analyses were done employing the programming language Julia, see.\nInfluence of the drift\n----------------------\nFor this numerical experiment we simulated sample paths of the diffusion (*X**t* : *t* ∈ [0, 1]) where the true dispersion coefficient is given by one of\n \n$$\\begin{aligned}\ns\\_1(t) &= 3/2 + \\sin(2(4t-2)) + 2\\exp(-16(4t-2)^2),\\\\\ns\\_2(t) &= W\\_t(\\omega\\_0) + 1,\n\\intertext{\nand the drift is given by one of }\nb\\_0(x) &= 0, \\\\\nb\\_1(x) &= -10x+20.\\end{aligned}$$ \nThe function *s*1 is a benchmark function used in in the context of nonparametric regression, up to a vertical shift to ensure positivity. To define *s*2(*t*), we took a fixed realisation of a Wiener path starting in 1, with *W**t*(*ω*0) > − 1 for *t* ∈ [0, 1] (sampled on an equidistant grid with 800001 points in [0, 1]). The function *s*1 is Lipschitz continuous, while *s*2 is Hölder continuous with coefficient essentially $\\frac12$.\nSpecifically, we used the Euler scheme on a grid with 800001 equidistant points in the interval [0, 1] to obtain a single diffusion path for each combination of drift and dispersion coefficients given above, which then was subsampled to obtain *n* = 4 000 ⋅ 2*j* + 1, *j* ∈ {1, 2, 3} observations each. As the prior on the coefficients on the individual bins we used independent IG(0.1,0.1) distributions.\nFigures [fig:s2] and [fig:s4] show marginal 98 % posterior bands for different combinations of bin number and observation regime for both dispersion coefficients when the drift is zero. Figure [fig:s2drift] shows the marginal posterior bands for *s*1 which are obtained if an affine drift term *b*1(*x*) = − 10*x* + 20 is present, but neglected in the estimation procedure. Comparison with Figure [fig:s2] shows that presence of a strong nonzero drift hardly affects the obtained credible bands. Note that credible bands successfully recover the overall shape of the functions *s**i*, although the recovery is not too refined; however, it would be misleading to visually compare the results obtained in the SDE setting to e.g. those obtainable in nonparametric regression, as the latter constitutes a much easier inferential problem. The functions *s**i* do not always pass through all the credible intervals, which is not surprising given the fact that these intervals are marginal. In general, construction of uniform confidence bands in nonparametric statistics is a long-studied and difficult problem, see Section 5.7 in ; for a general perspective on nonparametric uniform confidence bands see. Less is known about frequentist performance of nonparametric Bayesian confidence sets, although some interesting results have already been obtained in recent years, see, e.g.,, and. We do not address this issue in detail in this paper, but note that posterior contraction at an optimal rate does not automatically imply ‘good’ frequentist coverage properties of Bayesian credible sets. Following pp. 130–131 in in a similar study in the case of histograms employed as nonparametric probability density estimators, in our case it is arguably more natural to consider performance of Bayesian credible bands at the resolution of the histogram-type priors, i.e. for a histogramised version of a dispersion coefficient *s*, obtained as \n*s̄* = ∑*k* = 1*N**n**s̄**k***1***B**k* \n for \n$$\\bar{s}\\_{k} = \\frac{1}{\\Delta\\_k} \\int\\_{ B\\_k } s(t)\\dd t, \\quad k=1,\\ldots,N\\_n,$$ \n where Δ*k* denotes the length of the bin *B**k*. In a frequentist approach to nonparametric inference this constitutes an alternative to attempting to get rid of the bias of a nonparametric estimator for confidence band construction purposes via an artificial device like undersmoothing. Figure [fig:s2m] gives a detail of Figure [fig:s2] (panels for *N* = 40 and *N* = 160, *n* = 16 001, zoomed-in to show *t* ∈ [0.2, 0.5], posterior credible bands only) with a histogramised *s*1 superimposed. In comparison to Figure [fig:s2], the results appear to be visually even more pleasing, with the Bayesian credible band covering the curve *s*1 in its entirety in the left panel. This suggests to give the number of bins *N**n* an additional interpretation of a resolution at which one is interested in learning properties of the function *s*. Obviously, this resolution cannot be made arbitrarily fine, as this would distort the frequentist consistency property of our nonparametric Bayesian procedure as expressed in our theoretical results from Section [section:asymp]. We close this brief discussion on confidence bands by mentioning the fact that a number of authors have argued in favour of the so-called average coverage as a more natural concept of coverage of confidence bands than the uniform confidence bands, see Section 5.8 in and references therein.\nNote that the recovery is somewhat less accurate for function *s*2, see Figure [fig:s4], than for function *s*1, see Figure [fig:s2]. This is in perfect agreement with our theoretical results from Section [section:asymp], that give a slower posterior contraction rate for less smooth functions.\nOur posterior contraction theorems only specify that the optimal number of bins *N**n* is proportional to *n*− *β*, where the exponent *β* depends on the smoothness *λ* of a function to be estimated. This does not give a directly applicable recipe on how to choose the proportionality constant. In practice we recommend to use our theoretical results as guidance and to try out several choices of the number of bins, cf. Figures [fig:s2], [fig:s2drift] and [fig:s4]. This is not unlike the scale-space smoothing approach in the frequentist literature, see, e.g., Section 5.11 in. Furthermore, a useful point of reference in our setup is the number of non-overlapping neighbouring marginal posterior intervals. Figure [fig:s2m] shows that if *N* is too small to capture adequately the curvature of a dispersion coefficient, neighbouring marginal posterior credible intervals tend to be disjoint. On the other hand, choosing too many bins leads to undersmoothing and erratic appearance of marginal credible intervals.\nNumerical experiments similar to the above were also performed for other benchmark functions given in. As the results were similar, they are not reported here.\nWe also tested the performance of the procedure for choosing the number of bins as discussed in Section [sec:dic]. We considered the case with dispersion coefficient *s*1, drift *b*1 and 8001 observations. This corresponds to the leftmost column of Figure [fig:s2drift]. We computed $\\widehat{\\operatorname{elpd}}\\_{\\textrm{DIC}}$ for *N* ∈ {5, 10, 20, 40, 80, 160, 320}. The results are in Figure [fig:elpd], from which it is seen that the criterion is maximised for *N* = 40. This corresponds to the bottomleft panel of Figure [fig:s2drift]. In Figure [fig:ml] we plot the results obtained with the procedure for choosing the number of bins as discussed in Section [sec:bayes:factor]. Also this procedure suggests *N* = 40 as an optimal number of bins.\nEmpirical contraction rates\n---------------------------\nOf particular interest is the empirical size of the *L*2- and *L*∞-balls containing most of the posterior mass. To assess this, we approximate the distribution of the *L*2- or *L*∞-distance between posterior samples and the truth by sampling from the posterior. We do this for four different realisations of the model denoted by *X*(*ω*1), …, *X*(*ω*4). Note that *q* being the 90 %-quantile of this distribution entails that the *L*2-ball respective *L*∞-ball of size *q* contains 90 % of the posterior mass. To be specific we employ the following steps four times for each *s**i*, *i* = 1, 2 and both norms.\n1. Simulate the diffusion (*X**t*(*ω*))*t* ∈ [0, 1] with dispersion coefficient *s**i* (without drift) on a grid with 800001 points in the interval [0, 1].\n2. Subsample to obtain *n* = 2500 ⋅ 2*j* + 1, *j* ∈ {1, 2, 3, 4, 5} observations each.\n3. [step:bins] Draw *k* = 1, …, 2000 samples *S**k**i*, *n* from the posterior using $\\log(N\\_n) = \\log 5 + \\frac{1}{2\\lambda\\_i+1}\\log(n)$ bins and determine the distance ∥*S**i*, *n* − *s**i*∥2 (respective $\\log(N\\_n) = \\log 5 + \\frac{1}{2\\lambda\\_i+1}\\log(n/\\log(n))$ bins for ∥*S**i*, *n* − *s**i*∥∞).\n4. Determine the 90% quantile *q**i*, *n*(*ω*) of the distance samples and plot as function of *n*.\nFigure [fig:contraction-l2] shows the 90% quantile *q**i*(*n*) on a log-log scale for the *L*2-norm for *i* = 1, 2. The empirical findings for function *s*1 agree very well with the exponent $\\frac13 = \\frac{\\lambda}{ 2\\lambda+1}$ for *λ* = 1 obtained in Theorem [thm:rate-posterior-ig]. The function *s*2 is *λ*-Hölder smooth for any $\\lambda < \\frac12$. The empirically determined exponent is in excellent agreement with the exponent $\\frac14 = \\frac{\\lambda}{2\\lambda+1}$ for $\\lambda = \\frac12$ obtained in Theorem [thm:rate-posterior-ig].\nFigure [fig:contraction-sup] shows the 90% quantile *q**i*(*n*) for the *L*∞-norm. The number of bins *N**n* was chosen in analogy to nonparametric kernel regression, see Theorem 1.8 in, as $\\log(N\\_n) = \\log 5 + \\frac{1}{2\\lambda\\_i+1}\\log(n/\\log(n))$. Here the results are less conclusive than in the case of the *L*2-norm. The empirical findings suggest the rate $(n/\\log n)^\\frac13$ or similar for *s*1, and the rate $(n/\\log n)^\\frac14$ or similar for *s*2.\n[fig:s2]\n[fig:s2drift]\n[fig:s4]\nccc *N* = 40 & *N* = 160 \n& \n[fig:s2m]\n![A log plot of the \\widehat{\\operatorname{elpd}}_{\\textrm{DIC}}-values for estimating the dispersion coefficient s_1 versus different bin numbers N, n fixed at 8001, with drift b_1. ](\\loc elpd.png \"fig:\") [fig:elpd]\n![A log plot of suitably scaled and shifted \\operatorname{ML}-values for estimating the dispersion coefficient s_1 versus different bin numbers N, n fixed at 8001, with drift b_1. ](\\loc mlscaled.png \"fig:\") [fig:ml]\n![Estimate of the contraction rate of the size of L_2-balls around s covering 90 \\% of posterior probability mass by sampling (blue); compared with rate (purple). Top: Lipschitz continuous case, example s_1. Bottom: Hölder coefficient essentially \\frac12, example s_2.](\\loc contractions4.png \"fig:\") [fig:contraction-l2]\n![Estimate of the contraction rate of the size of L_\\infty-balls around s covering 90 \\% of posterior probability mass by sampling (blue); compared with likely rate (purple) and rate of Theorem [cor:rate:b:sup] (green). Top: Lipschitz continuous case, example s_1. Bottom: Hölder coefficient essentially \\frac12, example s_2.](\\loc contractions4sup.png \"fig:\") [fig:contraction-sup]\nReal data examples\n==================\nIn this section we apply our Bayesian method on two real data examples and study its implications. The daily exchange rates (noon buying rates in New York City for cable transfers payable in foreign currencies) JPY/USD and USD/GBP from January 1, 1999, to March 20, 2010 are available as data sets DEXJPUS and DEXUSUK from. We visualise the data in Figure [fig:exchangedata] (time in this and subsequent figures corresponds to the physical time, with unit being a year).\nThese exchange rate time series were considered e.g. in. Based on discrete time observations assumed to have arisen from the solution of an SDE\n \n$$\\label{ham}\n\\dd X\\_t=b(X\\_t)\\,\\dd t+\\sigma(X\\_t)\\,\\dd W\\_t, \\quad X\\_0=x, \\quad t\\in[0,T],$$ \nwith space-dependent dispersion coefficient *σ*, proposed a maximum penalised quasi-likelihood method to estimate nonparametrically the diffusion coefficient *σ*2.\nPlots of both series appear to indicate that the data are nonstationary, and this is confirmed also by the outcomes of the augmented Dickey-Fuller test we performed using urca package in **R**, see and. The test constitutes a standard unit root test in time series analysis, see, e.g., Chapter 17 in and Section 5.3 in for additional information. A similar conclusion on nonstationarity is reached in. As stationarity is no prerequisite for application of our nonparametric Bayesian method, we do not pursue this question any further in the paper. We retrieved the estimates $\\hat \\sigma$ given in from the figures published in the digital version of the publication using WebPlotDigitizer, see, and next used them to calculate the induced estimates $t \\mapsto \\hat s(t) = \\hat\\sigma(X\\_t)$ of the historical volatility at time *t*. In Figures [fig:JPUS] and [fig:USUK] we contrast those induced estimates $\\hat s$ with 90% marginal posterior bands for the deterministic dispersion coefficient *s*0 that were obtained through our Bayesian procedure. Since nominal exchange rates vary widely with the denomination used, we employed a non-informative IG(0.001,0.001) prior for coefficients *θ**k*’s from Lemma [lem:post-ig]. Our estimation results in Figures [fig:JPUS] and [fig:USUK] show that the volatility was high in the final years of the decade 2000–2010, coinciding with the sub-prime mortgage crisis and the following recession. On the other hand, the model from does not appear to capture this fact. It is reassuring to see that our method recovers this relevant event from the data. Based on this observation, we believe that including time-dependence of the volatility into the model is more appropriate in this example.\n \n![Daily exchange rates JPY/USD (top) and USD/GBP (bottom) from January 1, 1999 to March 20, 2010. Change-point estimates are indicated with dashed vertical lines. ](\\loc USUK.png \"fig:\") [fig:exchangedata]\n![90\\% marginal posterior band for the volatility of DEXJPUS. Plot of t \\mapsto \\hat \\sigma(X_t) as induced by the estimates in. ](\\loc JPUS16.png \"fig:\") [fig:JPUS]\n![90\\% marginal posterior band for volatility of DEXUSUK. Plot of t \\mapsto \\hat \\sigma(X_t). ](\\loc USUK16.png \"fig:\") [fig:USUK]\nA further confirmation of our findings comes from the change-point analysis of the data. For both the DEXJPUS and DEXUSUK series, a simple estimator for detection of a change-point in diffusivity of an SDE, see and Section 4.3.1 in, that is implemented in **R** in the sde package (see ), yields a change-point in the volatility level before and after 2007, with volatility prior to 2007 being lower. This is in excellent agreement with findings using our Bayesian method. See Figure [fig:exchangedata] below, where we plot the exchange rate data together with change point estimates, and compare to Figures [fig:JPUS] and [fig:USUK]. We note, however, that our Bayesian approach yields more, in that the method from assumes that the volatility is constant before and after the change-point, which is not what our Bayesian marginal credible sets suggest.\nProofs\n======\n[Proof of Proposition [prop1].] By Theorem 6.10 in and our Assumption [standing] (b)–(c), the laws $\\pp$ and P0, *s*0 of the path (*X**t* : *t* ∈ [0, 1]) are equivalent, the result that ultimately relies upon Girsanov’s theorem. Let $Z=\\frac{\\dd\\pp}{\\dd\\mathbb{P}\\_{0,s\\_0}}$, and let $Z^n=\\frac{\\dd\\mathbb{P}^n\\_{b\\_0,s\\_0}}{\\dd\\mathbb{P}^n\\_{0,s\\_0}}$ be the density of the respective laws of X*n*. By Theorem 2 on p. 245 in and Corollary 2 on p. 246 there, *Z**n* = E0, *s*0(*Z* ∣ X*n*). Then convergence of Π*n*(*U**s*0, *ɛ**n**c* ∣ X*n*) in P0, *s*0-probability to zero implies that it also converges to zero in $\\pp$-probability. Indeed, fix *η* > 0, let *A**n* = {Π*n*(*U**s*0, *ɛ**n**c* ∣ X*n*) > *η*} and let $\\eps>0$. Then \n$$\\pp(A\\_n)=\\mathbb{E}\\_{{0,s\\_0}}[Z^n\\mathbf{1}\\_{A^n}]=\\mathbb{E}\\_{{0,s\\_0}}[\\mathbb{E}\\_{{0,s\\_0}}(Z\\mathbf{1}\\_{A^n}\\mid \\mathcal{X}\\_n)]=\\mathbb{E}\\_{{0,s\\_0}}[Z\\mathbf{1}\\_{A^n}].$$ \n Choose *δ* > 0 such that P0, *s*0(*A*) < *δ* for any event *A* implies $\\pp(A)=\\mathbb{E}\\_{{0,s\\_0}}[Z\\mathbf{1}\\_{A}]<\\eps$, possible in view of Lemma 13.1 in. As eventually P0, *s*0(*A**n*) < *δ*, we have $\\pp(A\\_n)<\\eps$.\n[thm:rate] Define \n$$M(x):=\\expi (s^2(x) \\mid \\scr{X}\\_n),$$ \n the posterior mean of *s*2(*x*). Assume *m**n* ≍ *n*2*λ*/(2*λ* + 1) (equivalently, *N**n* ≍ *n*1/(2*λ* + 1)). If we let $ \\eps\\_n\\asymp m\\_n^{-1/2}$, i.e. $\\eps\\_n\\asymp n^{-\\lambda/(2\\lambda+1)}$, then for any sequence *h**n* tending to infinity we have for any fixed *x* ∈ [0, 1] \n$$\\ppz\\left(|M(x)-s\\_0^2(x)| \\ge \\eps\\_n h\\_n\\right) \\to 0$$ \n as *n* → ∞.\nAssume *x* ∈ *B**k* for some *k* < *N* (note that then $k=\\lfloor \\frac{nx}{m}\\rfloor+1$). The case *x* ∈ *B**N* follows later on. We compute\n \n$$\\begin{aligned}\n M(x) &=\\expi({\\theta}\\_k \\mid \\scr{X}\\_n) = \\frac{\\beta+\\frac{n}{2}\\sum\\_{i=(k-1)m+1}^{km}Y\\_{i,n}^2}{\\alpha+m/2-1}\\\\ &=\\frac{2\\beta}{2(\\alpha-1)+m}+\\frac{n\\sum\\_{i=(k-1)m+1}^{km}Y\\_{i,n}^2}{2(\\alpha-1)+m}.\\end{aligned}$$ \nNote that \n$$\\exz \\left[ \\sum\\_{i=(k-1)m+1}^{km}Y\\_{i,n}^2\\right]=\\int\\_{t\\_{m(k-1)}}^{t\\_{mk}}s\\_0^2(u)\\,\\dd u.$$ \n Hence, \n$$b(x):=\\exz [M(x)]-s\\_0^2(x)=\\frac{2\\beta}{2(\\alpha-1)+m}+\\frac{n\\int\\_{t\\_{m(k-1)}}^{t\\_{mk}}s\\_0^2(u)\\,\\dd u}{2(\\alpha-1)+m}-s\\_0^2(x).$$ \n We consider\n \n$$\\begin{aligned}\n\\frac{n\\int\\_{t\\_{m(k-1)}}^{t\\_{mk}}s\\_0^2(u)\\,\\dd u}{2(\\alpha-1)+m}-s\\_0^2(x) \n& =\\frac{n\\int\\_{t\\_{m(k-1)}}^{t\\_{mk}}s\\_0^2(u)\\,\\dd u}{2(\\alpha-1)+m}-\\frac{n}{m}\\int\\_{t\\_{m(k-1)}}^{t\\_{mk}}s\\_0^2(x)\\,\\dd u \\end{aligned}$$ \n \n$$= \\frac{n}{m+2(\\alpha-1)}\\int\\_{t\\_{m(k-1)}}^{t\\_{mk}}(s\\_0^2(u)-s\\_0^2(x))\\,\\dd u -\\frac{2(\\alpha-1)s\\_0^2(x)}{m+2(\\alpha-1)}.$$ \n As *s*0 is bounded from above by some constant K > 0, the last term is of order $\\frac{1}{m}$. We continue with the integral expression. By Hölder continuity of *s*0, it follows that ∣*s*02(*u*) − *s*02(*v*)∣ ≤ 2K*L*∣*u* − *v*∣*λ*. Using the sharp bound (attained at *x* = *t**m*(*k* − 1) and *x* = *t**m**k*) \n$$\\int\\_{t\\_{m(k-1)}}^{t\\_{mk}}|u-x|^\\lambda\\,\\dd u \\leq \\frac{1}{\\lambda+1}\\left(\\frac{m}{n}\\right)^{\\lambda+1},$$ \n we get the order bound (uniformly in *x* ∈ *B**k*) \n$$|b(x)|\\leq O\\left(\\frac{1}{m}\\right)+O\\left(\\frac{n}{m}\\left(\\frac{m}{n}\\right)^{\\lambda+1}\\right)=O\\left(\\frac{1}{m}\\right)+O\\left(\\frac{m}{n}\\right)^{\\lambda}.$$ \n For the variance we obtain (using that *Y**i*, *n* and *Y**j*, *n* are independent for *i* ≠ *j*)\n \n$$\\begin{aligned}\n \\varz [M(x)] &= \\left(\\frac{n}{2(\\alpha-1)+m}\\right)^2 \\sum\\_{i=(k-1)m+1}^{km} \\varz [Y\\_{i,n}^2] \\\\ & =\n\\left(\\frac{n}{2(\\alpha-1)+m}\\right)^2 2\\sum\\_{i=(k-1)m+1}^{km}\\left(\\int\\_{t\\_{i-1}}^{t\\_i} s\\_0^2(u) d u \\right)^2 \\\\&\\le C\\_\\alpha \\scr{K}^4\\frac{1}{m} = O\\left(\\frac1{m}\\right),\\end{aligned}$$ \nfor a positive constant *C**α*, uniformly for *x* ∈ *B**k*. Balancing bias and standard deviation yields the order equality \n$$O\\left(\\frac{1}{m}\\right)+O\\left(\\frac{m}{n}\\right)^{\\lambda}=O\\left(\\frac{1}{\\sqrt{m}}\\right),$$ \n or \n$$O(1)+O\\left(\\frac{m^{\\lambda+1}}{n^{\\lambda}}\\right)=O(\\sqrt{m}),$$ \n which gives *m**λ* + 1/2 ≍ *n**λ*, so that\n \n$$\\label{eq:N-m}\nm\\asymp n^{\\frac{2\\lambda}{2\\lambda+1}}.$$ \nA similar analysis holds for *x* ∈ *B**N*. We highlight the main steps for this case. For instance, we now get \n$$M(x) =\\expi({\\theta}\\_N | \\scr{X}\\_n) =\\frac{2\\beta}{2(\\alpha-1)+m+r}+\\frac{nZ\\_N}{2(\\alpha-1)+m+r},$$ \n and \n$$\\exz [Z\\_N]=\\int\\_{t\\_{m(N-1)}}^1s\\_0^2(u)\\,\\dd u.$$ \n This results in the bias \n$$b(x)=O\\left(\\frac{1}{m}\\right) + O\\left(\\frac{m+r}{n}\\right)^{\\lambda}=O\\left(\\frac{1}{m}\\right) + O\\left(\\frac{m}{n}\\right)^{\\lambda},$$ \n as 0 ≤ *r* < *m*. Similarly, we get \n$$\\varz [M(x)] = O\\left(\\frac1{m+r}\\right),$$ \n which is, using again 0 ≤ *r* < *m*, of order $O\\left(\\frac1{m}\\right)$. Hence balancing gives the same order relation $m\\asymp n^{\\frac{2\\lambda}{2\\lambda+1}}$ as for the case *k* < *N* in.\nWith the above choice for *m* we obtain, for any *x* ∈ [0, 1], for the mean squared error that \n$$\\exz [\\left(M(x) -s\\_0^2(x)\\right)^2] \\asymp n^{-\\frac{2{\\lambda}}{2{\\lambda}+1}}.$$ \n Hence, upon taking $\n\\eps\\_n\\asymp n^{-\\frac{\\lambda}{2\\lambda+1}}, \n$ the result follows from Chebyshev’s inequality: \n$$\\ppz\\left(|M(x)-s\\_0^2(x))| \\ge \\eps\\_n h\\_n\\right) \\le \\frac{ \\exz [\\left(M(x) -s\\_0^2(x)\\right)^2] }{\\eps\\_n^2 h\\_n^2} \\lesssim \\frac1{h\\_n^2} \\to 0$$ \n as *n* → ∞.\n[cor:rate] Under the assumptions of Lemma [thm:rate] we have \n$$\\exz [ \\|M-s\\_0^2\\|\\_2^2] \\asymp \\eps\\_n^2,$$ \n where ∥ ⋅ ∥2 is the *L*2-norm on [0, 1].\nSimply note that\n \n$$\\begin{aligned}\n {\\exz [\\|M-s\\_0\\|^2\\_2]}\n& = {\\int\\_0^1 \\exz [|M(x)-s\\_0(x)|^2]\\,\\dd x}\\\\ & = \\sum\\_{k=1}^{N\\_n} \\int\\_{B\\_k} \\exz [|M(x)-s\\_0(x)|^2]\\, \\dd x,\\end{aligned}$$ \nand apply the bounds derived in the proof of Lemma [thm:rate] on each bin *B**k* separately, 1 ≤ *k* ≤ *N**n*.\n[Proof of Theorem [thm:rate-posterior-ig]] First assume *b*0 ≡ 0. By Chebyshev’s inequality, one has\n \n$$\\label{eq:bv0}\n\\exz [\\Pi\\_n(\\|s^2-s\\_0^2\\|\\_2 \\ge h\\_n \\eps\\_n \\mid \\scr{X}\\_n)] \\le (h\\_n \\eps\\_n)^{-2}\\exz\\left[\\expi \\left(\\|s^2-s\\_0^2\\|\\_2^2 \\mid \\mathcal{X}\\_n\\right)\\right].$$ \nThe bias-variance decomposition gives\n \n$$\\label{eq:bv1}\n\\exz\\left[\\expi (\\|s^2-s\\_0^2\\|\\_2^2|\\mathcal{X}\\_n)\\right] =\\exz\\left [ \\|M-s\\_0^2\\|\\_2^2\\right] +\\sum\\_{k=1}^{N\\_n}\\int\\_{B\\_k}\\exz[\\vpi (s^2(x) \\mid \\mathcal{X}\\_n)]\\dd x.$$ \nThe behaviour of the first term on the righthand side was derived in the proof of Lemma [thm:rate]. For the second term, we consider *x* ∈ *B**k* for *k* < *N*. As in the proof of Lemma [thm:rate], the case *x* ∈ *B**N* is similar. We obtain\n \n$$\\label{eq:bv2}\n\\begin{split}\n\\vpi (s^2(x)|\\mathcal{X}\\_n)&=\\vpi ({\\theta}\\_k|\\mathcal{X}\\_n) = \\frac{(\\beta+nZ\\_k/2)^2}{(\\alpha+m/2-1)^2(\\alpha+m/2-2)}\\\\\n&= O\\left(\\frac1{m^3}\\right) + O\\left(\\frac{n^2}{m^3}\\right) Z\\_k^2.\n\\end{split}$$ \nBy Assumption [standing] (c), *s*0 is bounded by a positive constant $\\scr{K}$. We have\n \n$$\\begin{aligned}\n \\exz [Z\\_k^2] &= \\varz [Z\\_k] + \\left( \\exz [Z\\_k]\\right)^2=\\sum\\_{i=(k-1)m+1}^{km} \\varz [Y\\_{i,n}^2] + \\left(\\int\\_{B\\_k} s\\_0^2(u)\\,\\dd u \\right)^2\\\\ &\n \\le 2 \\sum\\_{i=(k-1)m+1}^{km} \\left(\\int\\_{t\\_{i-1}}^{t\\_i} s^2(u)\\,\\dd u \\right)^2 + \\left(\\frac{m}{n}\\right)^2 \\scr{K}^4 \\\\ &\n \\le 2\\sum\\_{i=(k-1)m+1}^{km} \\scr{K}^4 n^{-2} + \\left(\\frac{m}{n}\\right)^2 \\scr{K}^4 =O\\left(\\frac{m}{n^2}\\right)+O\\left(\\frac{m}{n}\\right)^2=O\\left(\\frac{m}{n}\\right)^2.\\end{aligned}$$ \nThis implies\n \n$$\\label{eq:bv3}\n\\exz\\left[\\vpi (s^2(x)|\\mathcal{X}\\_n)\\right]=O\\left(\\frac1{m^3}\\right) +O\\left(\\frac{n^2 }{m^3} \\frac{m^2}{n^2}\\right)=O\\left(\\frac1{m}\\right).$$ \nCombining the above order bounds then yields \n$$\\exz \\left[\\expi (\\|s^2-s\\_0^2\\|^2|\\mathcal{X}\\_n) \\right] = \\left[ O\\left(\\frac{1}{m}\\right)+O\\left(\\frac{m}{n}\\right)^{\\lambda}\\right]^2 + O\\left(\\frac1{m}\\right).$$ \n Balancing these terms was already done in the proof of Lemma [thm:rate], and gives the value for *m* depending on *n* as in display. This implies the stated result for *b*0 ≡ 0. Proposition [prop1] implies the result is not only true for *b*0 ≡ 0, but in general.\n[Proof of Theorem [cor:rate:b:sup]] We first prove\n \n$$\\label{eq:statement1}\n \\exz \\left[ \\sup\\_{x\\in [0,1]}|M(x)-s\\_0^2(x)| \\right] \\asymp \\tilde\\eps\\_n.$$ \nFor that, we first consider a selected bin *B**k* for *k* < *N* (the analysis on *B**N* would yield the same order estimates) and sup*x* ∈ *B**k*∣*M*(*x*) − *s*02(*x*)∣. Note that for *x* ∈ *B**k* one has that *M*(*x*) does not explicitly depend on *x*, being equal to $\\exz(\\theta\\_k|\\mathcal{X}\\_n).$ We thus occasionally write *M*(*x*) = *M**k*, whenever this is convenient. For any *x* ∈ *B**k* one has, recalling $b(x)=\\exz [M(x)]-s\\_0^2(x)$,\n \n$$\\begin{aligned}\n|M(x)-s\\_0^2(x)|^2 & \\leq 2|M(x)-\\exz [M(x)]|^2 + 2|\\exz [M(x)]-s\\_0^2(x)|^2 \\\\\n& = 2|M\\_k-\\exz [M\\_k]|^2 + 2|b(x)|^2, \\end{aligned}$$ \nand hence\n \n$$\\begin{aligned}\n\\sup\\_{x\\in B\\_k}|M(x)-s\\_0^2(x)|^2 & \\leq 2|M\\_k-\\exz [M\\_k]|^2 + 2\\sup\\_{x\\in B\\_k}|b(x)|^2. \\end{aligned}$$ \nTurning to sup*x* ∈ [0, 1]∣*M*(*x*) − *s*02(*x*)∣, we obtain from the above that\n \n$$\\label{rate:loss}\n\\begin{split}\n\\sup\\_{x\\in [0,1]}|M(x)-s\\_0^2(x)|^2 & = \\sup\\_k \\sup\\_{x\\in B\\_k}|M(x)-s\\_0^2(x)|^2 \\\\\n& \\leq 2\\sup\\_k\\, |M\\_k-\\exz [M\\_k]|^2 + 2\\sup\\_k\\sup\\_{x\\in B\\_k}|b(x)|^2 \\\\\n& \\leq 2\\sum\\_k\\, |M\\_k-\\exz [M\\_k]|^2 +2\\sup\\_k\\sup\\_{x\\in B\\_k}|b(x)|^2,\n\\end{split}$$ \nand therefore \n$$\\exz\\left[\\sup\\_{x\\in [0,1]}|M(x)-s\\_0^2(x)|^2\\right] \\leq 2\\sum\\_k\\varz [M\\_k] + 2\\sup\\_k\\sup\\_{x\\in B\\_k}|b(x)|^2.$$ \n Note that in the proof of Lemma [thm:rate] we obtained the uniform order bounds, not depending on *x* and *k*, $b(x)=O\\left(\\frac{1}{m}\\right) + O\\left(\\frac{m}{n}\\right)^{\\lambda}$ and $\\varz [M(x)] = O\\left(\\frac1{m}\\right)$. It follows that\n \n$$\\begin{aligned}\n\\exz \\left[\\sup\\_{x\\in [0,1]}|M(x)-s\\_0^2(x)|^2\\right]\n& \\leq N\\_n O\\left(\\frac{1}{m}\\right) + \\left(O\\left(\\frac{1}{m}\\right)+O\\left(\\frac{m}{n}\\right)^{\\lambda}\\right)^2 \\\\\n& \\leq O\\left(\\frac{n}{m^2}\\right) + \\left(O\\left(\\frac{1}{m}\\right)+O\\left(\\frac{m}{n}\\right)^{\\lambda}\\right)^2.\\end{aligned}$$ \nBalancing of the two summands, using *m* = *O*(*n**α*), is obtained for $\\alpha=\\frac{2\\lambda+1}{2\\lambda+2}$. This results in \n$$\\exz\\left[\\sup\\_{x\\in [0,1]}|M(x)-s\\_0^2(x)|^2\\right] = O(n^{-\\frac{\\lambda}{\\lambda+1}}),$$ \n so that \n$$\\exz\\left[\\sup\\_{x\\in [0,1]}|M(x)-s\\_0^2(x)| \\right]= O(n^{-\\frac{\\lambda}{2\\lambda+2}}),$$ \n which completes the proof of. The proof of the second statement of the theorem follows upon combining elements from the proofs of Theorems [thm:rate-posterior-ig] and of this result.\nProposition [prop1] implies the result is not only true for *b*0 ≡ 0, but in general.\nIn this appendix we present an alternative asymptotic frequentist analysis of our Bayesian procedure. A principal difference with the one in the main text is that our prior on the coefficients *θ**k*’s is not bound to be inverse gamma. On the downside, the *L*2-posterior contraction rate we obtain is slower than that in Theorem [thm:rate-posterior-ig]. This is due to the fact that in our arguments we cannot rely on the conjugacy of the inverse gamma prior anymore.\n[classX] Let S*n* denote a set of dispersion coefficients *s*: [0, 1] → [*κ*, K], that are piecewise constant on the bins *B**k*:\n \n*s* = ∑*k* = 1*N**n**ξ**k***1***B**k*. \nThe prior Π*n* on the dispersion coefficient *s* is defined by putting a prior on the coefficients *ξ**k*’s.\n \n*s*2 = ∑*k* = 1*N**n**ξ**k*2**1***B**k* = ∑*k* = 1*N**n**θ**k***1***B**k*, \nwhere we have set *θ**k* = *ξ**k*2.\nGeneral contraction rate\n------------------------\n[mainthm] Let Assumption [standing] hold with bounds 0 < *κ* ≤ *s*0(*t*) ≤ K < ∞ for all *t* ∈ [0, 1]. Assume the prior Π*n* is defined as the law of a random function *s* from, where the random variables *κ* ≤ *ξ**k* ≤ K, *k* = 1, …, *N**n*, are independent and identically distributed with a density that is bounded away from zero on the interval [*κ*, K]. Then for any sequence *m**n* ≍ *n*1 − *α*, equivalently *N**n* ≍ *n**α*, with $\\alpha \\in [\\frac{1}{2},1-\\frac{\\lambda}{2}]$, there exists a constant $\\widetilde{M}>0$, such that for $\\varepsilon\\_n= \\widetilde{M}n^{-\\beta}\\log^\\gamma n$ with $\\beta=\\frac{\\lambda}{4}$ and arbitrary *γ* > 1,\n \n$$\\label{eq:post}\n\\ee\\_{b\\_0,s\\_0}[\\Pi\\_n(U\\_{s\\_0,\\varepsilon\\_n}^c|\\scr{X}\\_n)]\\rightarrow 0$$ \nas *n* → ∞.\nThe essential term determining the posterior contraction rate is *n*− *β*. Here *β* depends on *λ* in an increasing way: a smoother function *s*0 allows for a faster contraction rate. The optimality of the choice of *α*, the exponent *β* and the condition *γ* > 1, at least within the context of our proof, is elaborated in Remark [rem:optimal] on page. The best possible rate obtainable from Theorem [mainthm] is achieved for *λ* = 1 and is (essentially) *n*− 1/4. The possibility that the posterior contraction rate for arbitrary priors is in fact faster than the one given in Theorem [mainthm] cannot be discarded: although our proofs for this general result use intricate technical arguments, they still might be not sharp enough.\nThe statement of Theorem [mainthm] is in some respect similar to that of Theorem 1 in. Significant differences are that choose a different prior and assume that the function *s*0 is differentiable; cf. our remarks in the introduction to the paper. Not only does this amount to difference in statements, but also proofs in the present paper are more involved than the ones in.\n[rem:l2] The inequality\n \n$$\\|s-s\\_0\\|\\_2 \\geq \\frac{1}{2\\mathcal{K}} \\|s^2-s\\_0^2\\|\\_2,$$ \nvalid for *s* ∈ S*n* and *s*0 satisfying Assumption [standing], together with Theorem [mainthm] implies that also the posterior for the diffusion coefficient *s*02 contracts around the truth with the rate *ɛ**n* given in that theorem.\nProof of Theorem [mainthm]\n--------------------------\nSome of by now classical references, where general conditions for derivation of posterior contraction rates are given, include, and. The proof of Theorem [mainthm] follows the same roadmap as these papers, however without appealing directly to their results, as our statistical setup is somewhat different from the ones covered by those papers. In particular, note that the distribution of the observations *X**t**i*, *n* ’s depends on the index *n*, which is not covered by the results in the above-mentioned references.\nWe first introduce some notation and definitions: *p**i*, *n*, *s* and *p**i*, *n*, 0 will be the densities of increments *Y**i*, *n* = *X**t**i*, *n* − *X**t**i* − 1, *n* under the parameter values *s* and *s*0, with drift equal to *b*0 = 0 in both cases. The notation *Z**i*, *n*, *s*(*Y**i*, *n*) = log(*p**i*, *n*, *s*(*Y**i*, *n*)/*p**i*, *n*, 0(*Y**i*, *n*)) will stand for the log-likelihood ratio corresponding to a single observation *Y**i*, *n*. *R**n*(*s*) = *L**n*(*s*)/*L**n*(*s*0) will denote the likelihood ratio. Furthermore, in line with the notation in, we let\n \n$$z\\_i=t\\_{i-1,n},\\quad\\mathcal{W}\\_i=1-\\frac{Y\\_{i,n}^2}{\\int\\_{t\\_{i-1,n}}^{t\\_{i,n}}s\\_0^2(u)\\,\\dd u}, \\quad f\\_s(z)=\\frac{\\int\\_{z}^{z+1/n}[s\\_0^2(u)-s^2(u)]\\,\\dd u}{\\int\\_{z}^{z+1/n}s^2(u)\\,\\dd u}.$$ \nUnder the parameter pair (0, *s*0), the random variables W*i*’s are i.i.d. with zero mean and variance equal to two. Since their distributions do not depend on *n*, we take the liberty to omit an extra index *n* in our notation. For notational simplicity, we also omit an extra index *n* in *z**i*’s and *f**s*, though, strictly speaking, it is still required there.\n[Proof of Theorem [mainthm]] As in the proofs of results from the main body of the paper, we may assume *b*0 = 0: the statement for a general *b*0 follows from this particular case, see Proposition [prop1]. The general structure of the proof is similar to the one of Theorem 1 in and ultimately and, but many details differ, as evidenced in particular by the proofs of the lemmas in Appendix [section:proofsremaining].\nWrite\n \n$$\\Pi\\_n({U}\\_{s\\_0,\\varepsilon\\_n}^c\\mid X\\_{t\\_{0,n}}\\ldots,X\\_{t\\_{n,n}})\n=\\frac{\\int\\_{{U}\\_{s\\_0,\\varepsilon\\_n}^c} R\\_n(s) \\Pi(\\dd s)}{ \\int\\_{\\mathcal{S}} R\\_n(s) \\Pi\\_n(\\dd s) }=\\frac{\\mathcal{N}\\_n}{\\mathcal{D}\\_n}.$$ \nLet *ε* > 0. For any events *E**n* and *F**n* we have\n \n$$\\label{eq:ab}\n\\mathbb{P}\\_{0,s\\_0}\\left(\\frac{\\mathcal{N}\\_n}{\\mathcal{D}\\_n}>\\epsilon\\right)\\leq \\mathbb{P}\\_{0,s\\_0}\\left(\\left\\{\\frac{\\mathcal{N}\\_n}{\\mathcal{D}\\_n}>\\epsilon\\right\\}\\cap E\\_n\\cap F\\_n\\right)+\\mathbb{P}\\_{0,s\\_0}(E\\_n^c)+\\mathbb{P}\\_{0,s\\_0}(F\\_n^c),$$ \nwhich we shall use for suitably chosen *E**n* and *F**n* having the property P0, *s*0(*E**n**c*) → 0 and P0, *s*0(*F**n**c*) → 0.\nDenote *S**n*(*s*) = *n*− 1log*R**n*(*s*) and note that $\n\\mathcal{D}\\_n=\\int\\_{\\mathcal{S}\\_n} \\exp( n S\\_n(s)) \\Pi\\_n(\\dd s).\n$ As in, we write\n \n$$S\\_n(s)=\\frac{1}{2}\\frac{1}{n}\\sum\\_{i=1}^{n} \\mathcal{W}\\_i f\\_s(z\\_{i})+\\frac{1}{2}\\frac{1}{n}\\sum\\_{i=1}^{n}\\left[ \\log\\left( 1+f\\_s(z\\_{i}) \\right) - f\\_s(z\\_{i}) \\right].$$ \nBelow and in Appendix [section:proofsremaining] we need the neighbourhoods *V**s*0, *ɛ* = {*s* ∈ S : ∥*s* − *s*0∥∞ < *ɛ*} for arbitrary $\\eps>0$, where ∥ ⋅ ∥∞ is the usual *L*∞-norm. We will use these for $\\eps=\\widetilde{\\eps}\\_n\\asymp n^{-\\beta}\\log^\\gamma n$. We have the following lower bound on D*n*, \n$$\\mathcal{D}\\_n\\geq \\int\\_{V\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}} R\\_n(s)\\Pi\\_n(\\dd s) \\geq \\inf\\_{s\\in V\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}} R\\_n(s)\\times\\Pi\\_n(V\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}).$$ \nLet\n \n$$E\\_n=\\left\\{\\sup\\_{s\\in V\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}}\\left|\\frac{1}{n}\\sum\\_{i=1}^{n} \\mathcal{W}\\_i f\\_s(z\\_{i})\\right|\\leq \\delta\\_n\\right\\},$$ \nwith $\\delta\\_n=\\widetilde{\\varepsilon}\\_n^2$. Lemmas [lemma1] and [lemma0] from Appendix [section:proofsremaining] yield that on *E**n* one has\n \n$$\\begin{aligned}\n\\inf\\_{s\\in V\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}}R\\_n(s)\n& \\geq \\exp\\left(- \\frac{2\\mathcal{K}^2}{\\kappa^4} n\\widetilde{\\varepsilon}\\_n^2-\\frac{n}{2}\\sup\\_{s\\in V\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}}\\left|\\frac{1}{n}\\sum\\_{i=1}^{n} \\mathcal{W}\\_i f\\_s(z\\_{i})\\right|\\right) \\\\\n& \\geq \\exp\\left(- \\frac{2\\mathcal{K}^2}{\\kappa^4} n\\widetilde{\\varepsilon}\\_n^2-\\frac{n\\widetilde{\\varepsilon}\\_n^2}{2}\\right),\\end{aligned}$$ \nwhereas P0, *s*0(*E**n**c*) → 0. Hence on *E**n* we have, using the prior mass result of Lemma [lemmaprior], \n$$\\mathcal{D}\\_n\\geq \\exp\\left(\n- \\left(\\frac{2\\mathcal{K}^2}{\\kappa^4} +\\frac{1}{2}+\\overline{C}\\right)n\\widetilde{\\varepsilon}\\_n^2\n\\right).$$ \n Next we consider N*n*, for which we have the trivial upper bound \nN*n* ≤ sup*U**s*0, *ɛ**n**c**R**n*(*s*). \n Let \n*F**n* = {sup*s* ∈ *U**s*0, *ɛ**n**c**R**n*(*s*) ≤ exp( − *c*1*n**ɛ**n*2)}, \n with $\\eps\\_n\\asymp n^{-\\beta}\\log^\\gamma n$ and the constant *c*1 > 0 as in Lemma [lemma2]. According to the latter lemma, we have P0, *s*0(*F**n**c*) → 0. Putting the above results together, we obtain on *E**n* ∩ *F**n* the inequality \n$$\\frac{\\mathcal{N}\\_n}{\\mathcal{D}\\_n}\\leq \\exp\\left(\\left(\n\\frac{2\\mathcal{K}^2}{\\kappa^4} +\\frac{1}{2}+\\overline{C}\\right)n\\widetilde{\\varepsilon}\\_n^2-c\\_1n{\\varepsilon}\\_n^2\n\\right).$$ \n Taking $\\varepsilon\\_n = \\widetilde{M} \\widetilde{\\varepsilon}\\_n,$ with $\\widetilde{M}>0$ large enough, gives a positive constant *M* for which on *E**n* ∩ *F**n* the inequality \n$$\\frac{\\mathcal{N}\\_n}{\\mathcal{D}\\_n}\\leq \\exp\\left(-Mn\\widetilde{\\varepsilon}\\_n^2\n\\right)$$ \n holds. It follows that \n$$\\mathbb{P}\\_{0,s\\_0}\\left(\\left\\{\\frac{\\mathcal{N}\\_n}{\\mathcal{D}\\_n}>\\epsilon\\right\\}\\cap E\\_n\\cap F\\_n\\right)=0$$ \n for all large *n*. We then obtain from that \n$$\\Pi\\_n(U\\_{s\\_0,\\varepsilon\\_n}^c|\\scr{X}\\_n)\\convp 0.$$ \n The assertion of the theorem now follows from this and the dominated convergence theorem. The optimal choice for *β* in $\\eps\\_n\\asymp n^{-\\beta}\\log^\\gamma n,$ as well as of *α* in *N**n* ≍ *n**α*, is a consequence of a discussion in Remark [rem:optimal].\nProofs of the remaining technical results\n-----------------------------------------\nWe use the following notation: *M**ɛ* will denote the smallest positive integer, such that 2*M**ɛ**ɛ*2 ≥ 4K2. Note that this definition implies 2*M**ɛ**ɛ*2 ≤ 8K2, so that *M**ɛ* ≍ log2(1/*ɛ*) for *ɛ* → 0. We also define sets *A**j*, *ɛ* = {*s* ∈ S : 2*j**ɛ*2 ≤ ∥*s* − *s*0∥22 < 2*j* + 1*ɛ*2} and *B**j*, *ɛ* = {*s* ∈ S : ∥*s* − *s*0∥22 < 2*j* + 1*ɛ*2} for *j* = 0, 1, …, *M**ɛ*. The measure *Q**n* will be the uniform discrete probability measure on points *z**i*’s, while ∥ ⋅ ∥*Q**n* will denote the *L*2(*Q**n*)-norm. In this appendix we use these sets for $\\eps=\\widetilde{\\eps}\\_n\\asymp n^{-\\beta}\\log^\\gamma n$.\nThe next lemma verifies the prior mass condition in the proof of our main theorem. This corresponds, roughly speaking, to e.g. condition (2.4) in Theorem 2.1 of. This prior mass condition is crucial in establishing posterior contraction rates and we refer to for an additional discussion on it. Note that in this condition is verified for another, somewhat artificial, prior.\n[lemmaprior] Under the conditions of Theorem [mainthm], the prior Π*n* satisfies\n \n$$\\label{priorC}\n\\Pi\\_n( V\\_{s\\_0,\\widetilde{\\varepsilon}\\_n} ) \\gtrsim e^{- \\overline{C} n \\widetilde{\\varepsilon}\\_n^2}$$ \nfor some constant $\\overline{C}>0.$\nSince *ξ**k*’s are independent, we have\n \n$$\\begin{aligned}\n\\Pi\\_n( V\\_{s\\_0,\\widetilde{\\varepsilon}\\_n} )&=\\Pi\\_n\\left( \\bigcap\\_{k=1}^{N\\_n} \\left\\{ \\sup\\_{x\\in B\\_k} |s(x)-s\\_0(x)|<\\widetilde{\\varepsilon}\\_n \\right\\} \\right)\\\\\n&=\\prod\\_{k=1}^{N\\_n}\\Pi\\_n\\left( \\sup\\_{x\\in B\\_k} | s(x) -s\\_0(x)|<\\widetilde{\\varepsilon}\\_n \\right) =\\prod\\_{k=1}^{N\\_n}\\Pi\\_n\\left( \\sup\\_{x\\in B\\_k} |\\xi\\_k-s\\_0(x)|<\\widetilde{\\varepsilon}\\_n \\right).\\end{aligned}$$ \nNow, since *s*0 is Hölder continuous and *B**k* is of length at most 2*m**n*/*n*, we have\n \n$$s\\_0(x)=s\\_0(a\\_{k-1})+O\\left(\\frac{m\\_n}{n}\\right)^\\lambda, \\quad x\\in B\\_k,$$ \nwhere the order term is uniform in *x* ∈ [0, 1]. Remember also that by our conditions\n \n$$\\label{eq:m}\n\\left(\\frac{m\\_n}{n}\\right)^\\lambda \\ll \\widetilde{\\varepsilon}\\_n.$$ \nThen for all *n* large enough and all *k*, \n \n$$\\begin{aligned}\n\\Pi\\_n\\left( \\sup\\_{x\\in B\\_k} |\\xi\\_k-s\\_0(x)|<\\widetilde{\\varepsilon}\\_n \\right) & \\geq \\Pi\\_n\\left( |\\xi\\_k-s\\_0(a\\_{k-1})|<\\frac{\\widetilde{\\varepsilon}\\_n}{2} \\right)\\\\\n&\\geq\\textrm{const} \\cdot \\widetilde{\\varepsilon}\\_n,\\end{aligned}$$ \nwhere const > 0 is\nsome constant independent of *k* and *n*, and the last inequality comes from the fact that *ξ**k* has a density bounded away from zero on [*κ*, K]. It then follows that\n \n$$\\begin{aligned}\n\\Pi\\_n( V\\_{s\\_0,\\widetilde{\\varepsilon}\\_n} )&\\geq (\\textrm{const} \\cdot \\widetilde{\\varepsilon}\\_n)^{N\\_n}\\\\\n&=e^{N\\_n \\log(\\textrm{const} \\cdot \\widetilde{\\varepsilon}\\_n)}.\\end{aligned}$$ \nWe want to show existence of a constant $\\overline{C}>0,$ such that for all *n* large enough,\n \n$$N\\_n \\log(\\textrm{const} \\cdot \\widetilde{\\varepsilon}\\_n) \\geq - \\overline{C} n \\widetilde{\\varepsilon}\\_n^2.$$ \nBut this is immediate from our conditions on *N**n* (equivalently, *m**n*) and $\\widetilde{\\varepsilon}\\_n,$ e.g. for $\\overline{C}=1.$ The proof of the lemma is completed.\nThe following lemma is an analogue of Lemma A.2 in. The statement is slightly different, and so is the proof.\n[lemma1] Let the conditions of Theorem [mainthm] hold and let $s\\in V\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}.$ Then, uniformly in *s*, \n \n$$\\frac{1}{2}\\frac{1}{n}\\sum\\_{i=1}^n \\left\\{ \\log(1+f\\_s(z\\_i)) - f\\_s(z\\_i) \\right\\}\n\\geq - \\frac{2\\mathcal{K}^2}{\\kappa^4} \\widetilde{\\varepsilon}\\_n^2.$$ \nAs in the proof in, one derives\n \n$$\\label{eq:lbound1}\n\\frac{1}{2}\\frac{1}{n}\\sum\\_{i=1}^n \\left\\{ \\log(1+f\\_s(z\\_i)) - f\\_s(z\\_i) \\right\\}\n\\geq \n-\\frac{1}{2n}\\sum\\_{i=1}^n f\\_s^2(z\\_i).$$ \nUsing the Cauchy-Schwarz inequality, the bounds on *s* and the fact that $s\\in V\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}$, one gets\n \n$$\\begin{aligned}\nf\\_s^2(z\\_i) & = \\left( \\frac{\\int\\_{z}^{z+1/n}[s\\_0^2(u)-s^2(u)]\\,\\dd u}{\\int\\_{z}^{z+1/n}s^2(u)\\,\\dd u}\\right)^2 \\leq \\frac{\\frac{1}{n}\\int\\_{z}^{z+1/n}[s\\_0^2(u)-s^2(u)]^2\\,\\dd u}{\\kappa^4/n^2} \\\\\n& \\leq \\frac{\\frac{4\\mathcal{K}^2}{n}\\int\\_{z}^{z+1/n}[s\\_0(u)-s(u)]^2\\,\\dd u}{\\kappa^4/n^2} \\leq \\frac{4\\mathcal{K}^2\\widetilde{\\varepsilon}\\_n^2}{\\kappa^4}.\\end{aligned}$$ \nThe result follows by inserting the latter upper bound into.\nLemma [lemma0] below is a preciser version of Lemma A.1 in. Its proof is similar in general terms, but differs substantially in the entropy estimates used, as well as some other arguments. In its proof and that of Lemma [lemma3] we will need the following lemma.\n[lemma:lipschitz] Let *g* : [0, ∞) → R be Hölder continuous of order *λ* > 0 and Hölder constant *L* > 0. Then \n$$\\left|\\int\\_z^{z+h}g(u)\\,\\dd u - hg(z)\\right|\\leq \\frac{Lh^{1+\\lambda}}{1+\\lambda}.$$ \n Furthermore, if $z\\_i=\\frac{i}{n}$, for *i* = 1, …, *n*, then \n$$\\left|\\int\\_0^{1}g(u)\\,\\dd u - \\frac{1}{n}\\sum\\_{i=1}^ng(z\\_i)\\right|\\leq \\frac{L}{(1+\\lambda)n^\\lambda}.$$ \nFor the first statement we consider\n \n$$\\begin{aligned}\n\\left|\\int\\_z^{z+h}g(u)\\,\\dd u - hg(z)\\right| & = \\left|\\int\\_z^{z+h}(g(u)-g(z))\\,\\dd u\\right| \\\\\n& \\leq \\int\\_z^{z+h}|g(u)-g(z)|\\,\\dd u \\leq \\int\\_z^{z+h}L(u-z)^{\\lambda}\\,\\dd u = \\frac{Lh^{1+\\lambda}}{1+\\lambda}.\\end{aligned}$$ \nFor the second one we have, using the first part,\n \n$$\\begin{gathered}\n\\left|\\int\\_0^{1}g(u)\\,\\dd u - \\frac{1}{n}\\sum\\_{i=1}^ng(z\\_i)\\right| =\\left|\\sum\\_{i=1}^n\\int\\_{\\frac{i-1}{n}}^{\\frac{i}{n}}(g(u) - g(z\\_i))\\,\\dd u\\right| \\\\\n \\leq \\sum\\_{i=1}^n\\left|\\int\\_{\\frac{i-1}{n}}^{\\frac{i}{n}}(g(u) - g(z\\_i))\\,\\dd u\\right| \\leq \\frac{L}{(1+\\lambda)n^\\lambda}.\\end{gathered}$$ \nThe proof is completed.\n[lemma0] Let the conditions of Theorem [mainthm] hold and assume *b*0 = 0. Then\n \n$$\\mathbb{P}\\_{0,s\\_0}\\left(\\sup\\_{f\\_s\\in\\mathcal{F}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}}\\left| \\frac{1}{n} \\sum\\_{i=1}^n \\mathcal{W}\\_i f\\_s(z\\_{i}) \\right|\\geq\\delta\\_n\\right) \\lesssim \\frac{1}{n^{\\lambda}{\\widetilde{\\varepsilon}\\_n}^2},$$ \nwhere $\\mathcal{F}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}=\\{ f\\_s:\\|s-s\\_0\\|\\_{\\infty}<\\widetilde{\\varepsilon}\\_n \\}=\\{f\\_s: s \\in V\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}\\},$ and *δ**n* is an arbitrary sequence of positive numbers, such that $\\delta\\_n \\asymp {\\widetilde{\\varepsilon}\\_n}^2.$ In particular, as *n* → ∞, the probability on the lefthand side of the above display converges to zero.\nIntroduce\n \n$$g\\_s(z)=\\frac{s\\_0^2(z)-s^2(z)}{s^2(z)}, \\quad \\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}=\\{g\\_s: \\|s-s\\_0\\|\\_{\\infty}<\\widetilde{\\varepsilon}\\_n \\}.$$ \nThe function *g**s* approximates *f**s* in the following sense:\n \n$$\\label{approx\\*}\n|f\\_s(z\\_i)-g\\_s(z\\_i)|\\leq \\frac{2\\mathcal{K}L}{\\kappa^2n^\\lambda}, \\quad i=1,\\ldots,n,$$ \nwhich can be seen as follows. Every interval [*z**i*, *z**i* + 1/*n*) is contained in some bin *B**k*, so that *s*2 is constant on [*z**i*, *z**i* + 1/*n*). Hence one obtains $f\\_s(z\\_i)=n\\int\\_{z\\_i}^{z\\_i+1/n}g\\_s(u)\\,\\dd u$. It follows from Definition [classX] and boundedness and Hölder continuity of *s*0 in Assumption [standing] that \n$$|g\\_s(u)-g\\_s(v)|\\leq \\frac{2\\mathcal{K}L}{\\kappa^2}|u-v|^\\lambda.$$ \n Hence equation follows from Lemma [lemma:lipschitz]. Note that the righthand side of is uniform in *s*.\nLet \n$$\\rho\\_n=\\frac{2\\mathcal{K}L}{\\kappa^2n^{1+\\lambda}}\\sum\\_{i=1}^n|\\mathcal{W}\\_i|.$$ \n Then\n \n$$\\sup\\_{f\\_s\\in\\mathcal{F}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}}\\left| \\frac{1}{n} \\sum\\_{i=1}^n \\mathcal{W}\\_i f\\_s(z\\_i) \\right| \\leq \\sup\\_{g\\_s\\in\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}}\\left| \\frac{1}{n} \\sum\\_{i=1}^n \\mathcal{W}\\_i g\\_s(z\\_i) \\right| + \\rho\\_n.$$ \nHence,\n \n$$\\begin{aligned}\n\\lefteqn{\\mathbb{P}\\_{0,s\\_0}\\left(\\sup\\_{f\\_s\\in\\mathcal{F}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}}\\left| \\frac{1}{n} \\sum\\_{i=1}^n \\mathcal{W}\\_i f\\_s(z\\_{i}) \\right|\\geq\\delta\\_n\\right)}\\\\\n& \\qquad \\leq \\mathbb{P}\\_{0,s\\_0}\\left(\\sup\\_{g\\_s\\in\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}}\\left| \\frac{1}{n} \\sum\\_{i=1}^n \\mathcal{W}\\_i g\\_s(z\\_{i}) \\right|+\\rho\\_n\\geq\\delta\\_n\\right) \\\\\n& \\qquad \\leq \\mathbb{P}\\_{0,s\\_0}\\left(\\sup\\_{g\\_s\\in\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}}\\left| \\frac{1}{n} \\sum\\_{i=1}^n \\mathcal{W}\\_i g\\_s(z\\_{i}) \\right|\\geq\\delta\\_n/2\\right) + \\mathbb{P}\\_{0,s\\_0}(\\rho\\_n\\geq \\delta\\_n/2).\\end{aligned}$$ \nThe Markov inequality gives \n$$\\mathbb{P}\\_{0,s\\_0}(\\rho\\_n\\geq\\delta\\_n/2)\\lesssim \\frac{1}{n^{\\lambda}\\delta\\_n}= \\frac{1}{n^{\\lambda}{\\widetilde{\\varepsilon}\\_n}^2}.$$ \n Therefore, in order to prove the lemma, it is enough to show that\n \n$$\\label{eq:G}\n\\mathbb{P}\\_{0,s\\_0}\\left(\\sup\\_{g\\_s\\in\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}}\\left| \\frac{1}{n} \\sum\\_{i=1}^n \\mathcal{W}\\_i g\\_s(z\\_{i}) \\right|\\geq\\delta\\_n\\right)\\lesssim \\frac{1}{n^{\\lambda}{\\widetilde{\\varepsilon}\\_n}^2}.$$ \nWe will apply Corollary 8.8 from to show that holds true, since the assertion of that corollary, inequality (8.30), implies that\n \n$$\\label{eq:830}\n\\mathbb{P}\\_{0,s\\_0}\\left(\\sup\\_{g\\_s\\in\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}}\\left| \\frac{1}{n} \\sum\\_{i=1}^n \\mathcal{W}\\_i g\\_s(z\\_{i}) \\right|\\geq\\delta\\_n\\right) \\leq \\tilde c\\_1\\exp(-\\tilde c\\_2\\,n\\widetilde{\\varepsilon}\\_n^2),$$ \nfor some positive constants $\\tilde c\\_1,\\tilde c\\_2$. The righthand side of the above display is asymptotically much smaller than $1/(n^{\\lambda}{\\widetilde{\\varepsilon}\\_n}^2).$\nApplication of the mentioned corollary amounts to verification of formulae (8.23)–(8.29) in. Exactly as in, conditions (8.23)–(8.27), (8.29) can be satisfied by choosing $R\\_n=2\\mathcal{K}\\widetilde{\\varepsilon}\\_n/\\kappa^2,$ *K*1 = 3, $\\sigma\\_0^2=2 \\ee\\_{0,s\\_0}\\left[ \\mathcal{W}\\_i^2 e^{|\\mathcal{W}\\_i|/3} \\right],$ $K\\_2=2\\mathcal{K}\\widetilde{\\varepsilon}\\_n/\\kappa^2,$ *C*1 = 3, *K* = 4*K*1*K*2, and *C*0 = 2*C*, with *C* a universal constant as in Corollary 8.8 in. Finally, we need to verify formula (8.28) in,\n \n$$\\label{8.28}\n\\sqrt{n}\\delta\\_n \\geq C\\_0 \\left( \\int\\_{\\delta\\_n/2^6}^{\\sqrt{2}R\\_n\\sigma\\_0} H\\_B^{1/2}\\left( \\frac{u}{\\sqrt{2}\\sigma\\_0},\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n},Q\\_n\\right) \\,\\dd u \\vee \\sqrt{2} R\\_n \\sigma\\_0 \\right).$$ \nHere $H\\_B\\left( \\delta,\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n},Q\\_n\\right)$ is the *δ*-entropy with bracketing of $\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}$ for the *L*2(*Q**n*)-metric (see Definition 2.2 in ).\nFirst of all, note that\n \n$$\\sqrt{n}\\delta\\_n \\asymp \\sqrt{n} \\widetilde{\\varepsilon}\\_n^2 \\gg R\\_n \\asymp \\widetilde{\\varepsilon}\\_n,$$ \nholds, since we have $n\\widetilde{\\varepsilon}\\_n^2\\rightarrow\\infty.$ Thus it suffices to show\n \n$$\\sqrt{n}\\delta\\_n \\geq C\\_0 \\int\\_{\\delta\\_n/2^6}^{\\sqrt{2}R\\_n\\sigma\\_0} H\\_B^{1/2}\\left( \\frac{u}{\\sqrt{2}\\sigma\\_0},\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n},Q\\_n\\right) \\,\\dd u.$$ \nIn order to do this, we will upper bound the righthand side by first upper bounding the integrand with a manageable and simple expression, and thereby we obtain a bound on the integral itself. We will use *H*∞, the entropy for the supremum norm (see Definition 2.3 in ) and the inequality *H**B*(*δ*, G, *Q*) ≤ *H*∞(*δ*/2, G), valid for any collection of functions G and a probability measure *Q*, see Lemma 2.1 in, to obtain\n \n$$\\begin{aligned}\n\\lefteqn{\\int\\_{\\delta\\_n/2^6}^{\\sqrt{2}R\\_n\\sigma\\_0} H\\_B^{1/2}\\left( \\frac{u}{\\sqrt{2}\\sigma\\_0},\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n},Q\\_n\\right) \\,\\dd u} \\\\\n & \\qquad\\leq \\int\\_{\\delta\\_n/2^6}^{\\sqrt{2}R\\_n\\sigma\\_0} H\\_{\\infty}^{1/2}\\left( \\frac{u}{2\\sqrt{2}\\sigma\\_0},\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}\\right) \\,\\dd u\\\\\n&\\qquad = 2\\sqrt{2}\\sigma\\_0 \\int\\_{0}^{R\\_n/2-\\delta\\_n/(2^7\\sqrt{2}\\sigma\\_0)} H\\_{\\infty}^{1/2}\\left( u + \\frac{\\delta\\_n}{2^7\\sqrt{2}\\sigma\\_0},\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}\\right) \\,\\dd u\\\\\n&\\qquad\\leq 2\\sqrt{2}\\sigma\\_0 \\int\\_{0}^{R\\_n} H\\_{\\infty}^{1/2}\\left( u + \\frac{\\delta\\_n}{2^7\\sqrt{2}\\sigma\\_0},\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}\\right) \\,\\dd u.\\end{aligned}$$ \nWe will now estimate the entropy in the last integral in the above display. Suppose *u* > 0 is fixed. For every $g\\in\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}$ construct an approximating function\n \n$$\\widetilde{g}=\\sum\\_{k=1}^{N\\_n} u \\left\\lfloor \\frac{g(a\\_{k-1})}{u} \\right\\rfloor 1\\_{B\\_k}.$$ \nThe quality of approximation can be assessed as follows: we have\n \n$$\\begin{aligned}\n\\|g-\\widetilde{g}\\|\\_{\\infty}&=\\max\\_k \\|(g-\\widetilde{g})1\\_{B\\_k}\\|\\_{\\infty}\\\\\n&=\\max\\_k\\sup\\_{x\\in B\\_k} \\left|g(x)-g(a\\_{k-1})+u\\frac{g(a\\_{k-1})}{u}-u \\left\\lfloor \\frac{g(a\\_{k-1})}{u} \\right\\rfloor\\right|\\\\\n&\\leq\\max\\_k\\sup\\_{x\\in B\\_k} |g(x)-g(a\\_{k-1})|+u.\\end{aligned}$$ \nSince *s*2 is constant on *B**k* and *s*02 is Hölder, while *B**k* is of length at most 2*m**n*/*n*, we have\n \n$$\\begin{aligned}\n\\sup\\_{x\\in B\\_k} |g(x)-g(a\\_{k-1})|&=\\sup\\_{x\\in B\\_k}\\left| \\frac{ s\\_0^2(x) - s\\_0^2(a\\_{k-1}) }{s^2(a\\_{k-1})} \\right|\\\\\n&\\leq \\frac{2\\mathcal{K}L}{\\kappa^2} \\left( \\frac{2m\\_n}{n} \\right)^\\lambda,\\end{aligned}$$ \nso that\n \n$$\\label{8\\*}\n\\|g-\\widetilde{g}\\|\\_{\\infty} \\leq u + \\frac{2\\mathcal{K}L}{\\kappa^2} \\left( \\frac{2m\\_n}{n} \\right)^\\lambda.$$ \nAccording to our assumptions,\n \n$$\\label{bound\\*}\n\\left( \\frac{m\\_n}{n} \\right)^\\lambda \\ll \\delta\\_n,$$ \nand hence for any pair of positive constants *A* and *B* one eventually has\n \n$$A\\left( \\frac{2m\\_n}{n} \\right)^\\lambda \\leq B\\delta\\_n.$$ \nThis implies that for all *n* large enough,\n \n$$H^{1/2}\\_{\\infty}\\left( u + \\frac{\\delta\\_n}{2^7\\sqrt{2}\\sigma\\_0},\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}\\right) \\leq H^{1/2}\\_{\\infty}\\left( u + \\frac{2\\mathcal{K}L}{\\kappa^2} \\left( \\frac{2m\\_n}{n} \\right)^\\lambda,\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}\\right).$$ \nThe entropy on the righthand side of the above display can be estimated by bounding the corresponding covering number of $\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}$, which can be achieved by counting the number of different $\\widetilde{g}$’s. In fact, as we shall see below, although $\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}$ is an infinite set, the number of different $\\widetilde{g}$’s is finite and can by easily estimated from above.\nFirstly, recall that $\\|g\\|\\_{\\infty}\\leq (2\\mathcal{K}/\\kappa^2)\\widetilde{\\varepsilon}\\_n$ for $g\\in \\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}$. This implies that there are at most\n \n$$\\left\\lfloor \\frac{4\\mathcal{K}}{\\kappa^2} \\frac{\\widetilde{\\varepsilon}\\_n}{u} \\right\\rfloor+1$$ \npossible values for $\\widetilde{g}(a\\_0).$ Next, by the triangle inequality and,\n \n$$\\begin{aligned}\n|\\widetilde{g}(a\\_k)-\\widetilde{g}(a\\_{k-1})|&\\leq |\\widetilde{g}(a\\_k)-g(a\\_k)|+|g(a\\_k)-g(a\\_{k-1})|+|\\widetilde{g}(a\\_{k-1})-g(a\\_{k-1})|\\\\\n&\\leq 2u+ \\frac{4\\mathcal{K}}{\\kappa^2} L \\left( \\frac{2m\\_n}{n} \\right)^\\lambda+2\\|g\\|\\_{\\infty}.\\end{aligned}$$ \nThus, once $\\widetilde{g}(a\\_{k-1})$ has been determined, $\\widetilde{g}(a\\_k)$ can take at most\n \n$$\\left\\lfloor \\frac{1}{u} \\left(4u+ \\frac{8\\mathcal{K}}{\\kappa^2} L \\left( \\frac{2m\\_n}{n} \\right)^\\lambda+4\\|g\\|\\_{\\infty} \\right) \\right\\rfloor+1$$ \nvalues. Therefore, in total there can be at most\n \n$$\\left( \\left\\lfloor \\frac{4\\mathcal{K}}{\\kappa^2} \\frac{\\widetilde{\\varepsilon}\\_n}{u} \\right\\rfloor +1 \\right) \\left( \\left\\lfloor 4+ \\frac{8\\mathcal{K}}{\\kappa^2} \\frac{L}{u} \\left( \\frac{2m\\_n}{n} \\right)^\\lambda+\\frac{8\\mathcal{K}}{\\kappa^2} \\frac{\\widetilde{\\varepsilon}\\_n}{u} \\right\\rfloor+1 \\right)^{N\\_n-1}$$ \ndifferent $\\widetilde{g}$’s, which yields an upper bound on the covering number of the set $\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}.$ Using, for large *n*, the logarithm of the above display is bounded by\n \n$$N\\_n \\log\\left( \\textrm{const}+ \\textrm{const} \\frac{\\widetilde{\\varepsilon}\\_n}{u} \\right)$$ \nfor some constant const > 0 independent of *n* and *u*. This expression gives an upper bound on the entropy of the set $\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}.$ Inserting this upper bound into the entropy integral, we get that for all *n* large enough,\n \n$$\\begin{aligned}\n\\int\\_{0}^{R\\_n} H\\_{\\infty}^{1/2}\\left( u + \\frac{\\delta\\_n}{2^7\\sqrt{2}\\sigma\\_0},\\mathcal{G}\\_{s\\_0,\\widetilde{\\varepsilon}\\_n}\\right) \\,\\dd u & \\leq \\sqrt{N\\_n} \\int\\_{0}^{R\\_n} \\log^{1/2}\\left( \\textrm{const}+ \\textrm{const} \\frac{\\widetilde{\\varepsilon}\\_n}{u} \\right) \\,\\dd u\\\\\n&=\\widetilde{\\varepsilon}\\_n \\sqrt{N\\_n} \\int\\_{0}^{2\\mathcal{K}/\\kappa^2} \\log^{1/2}\\left( \\textrm{const}+ \\textrm{const} \\frac{1}{u} \\right) \\,\\dd u \\\\\n&\\asymp \\widetilde{\\varepsilon}\\_n \\sqrt{N\\_n},\\end{aligned}$$ \nsince the integral in the second line is convergent, because the integrand is bounded by a constant times *u*− 1/2, and the latter function is integrable in the neighbourhood of zero.\nSummarising the above intermediate calculations, we obtain that in order to prove (8.28) in, we need\n \n$$\\label{eq:delta}\n\\sqrt{n}\\delta\\_n \\gg \\widetilde{\\varepsilon}\\_n \\sqrt{N\\_n}$$ \nand to hold. Both are satisfied with our choice of $\\widetilde{\\varepsilon}\\_n$ and *N**n* (equivalently, *m**n*). Hence all conditions of Corollary 8.8 in are satisfied and follows. This completes the proof.\nThe next lemma, to be used in the proof of Lemma [lemma2], is an analogue of Lemma A.4 in. Its proof is also similar in structure, but differs in details.\n[lemma3] Let the conditions of Theorem [mainthm] hold and assume *b*0 = 0.\nThere exist two positive constants $\\widetilde{c}\\_0$ and $\\widetilde{C}\\_0$, depending on *κ*, K and *L* only, such that for all *n* large enough and all *s* ∈ *A**j*, *ɛ**n*, *j* = 0, 1, …, *M**ɛ**n*, we have\n \n$$\\sum\\_{i=1}^n \\exz[Z\\_{i,n,s}(Y\\_{i,n})] \\leq - \\frac{\\widetilde{c}\\_0\\kappa^2}{\\mathcal{K}^4}2^j\\varepsilon\\_n^2 n +\\widetilde{C}\\_0n^{1-\\lambda}.$$ \nBelow we use the following elementary inequality: for any fixed constant *C* > 0 there exists another constant $\\widetilde{c}\\_0>0$, such that for *x* ∈ ( − 1, *C*), $\\log(1+x)-x\\leq - \\widetilde{c}\\_0 x^2$ holds.\nIt follows from Assumption [standing] that \n$$\\frac{ \\int\\_{[z\\_i,z\\_{i+1)}} [s\\_0^2(u)-s^2(u)]\\,\\dd u }{ \\int\\_{[z\\_i,z\\_{i+1)}} s^2(u)\\,\\dd u }\\leq \\frac{\\mathcal{K}^2}{\\kappa^2}-1=:C>0.$$ \n Hence, for some $\\widetilde{c}\\_0>0$,\n \n$$\\begin{aligned}\n\\exz[Z\\_{i,n,s}(Y\\_{i,n})] &=\\frac{1}{2}\\log\\left( 1 + \\frac{ \\int\\_{[z\\_i,z\\_{i+1)}} [s\\_0^2(u)-s^2(u)]\\,\\dd u }{ \\int\\_{[z\\_i,z\\_{i+1)}} s^2(u)\\,\\dd u } \\right)\\\\\n&-\\frac{1}{2}\\frac{ \\int\\_{[z\\_i,z\\_{i+1)}} [s\\_0^2(u)-s^2(u)]\\,\\dd u }{ \\int\\_{[z\\_i,z\\_{i+1)}} s^2(u)\\,\\dd u } \\\\\n&\\leq - \\frac{ \\widetilde{c}\\_0 }{2} \\left\\{ \\frac{ \\int\\_{[z\\_i,z\\_{i+1)}} [s\\_0^2(u)-s^2(u)]\\,\\dd u }{ \\int\\_{[z\\_i,z\\_{i+1)}} s^2(u)\\,\\dd u } \\right\\}^2.\\end{aligned}$$ \nWe now focus on the term in braces. Let $g(u)=\\frac{ (s^2(u)-s^2\\_0(u))^2 }{s^4(u)}$. On bins, and hence on intervals [*z**i*, *z**i* + 1), the function *s* is constant and positive, and by Hölder continuity and boundedness of *s*0 also *g* is Hölder continuous. Application of Lemma [lemma:lipschitz] gives\n \n$$\\label{eq:l1}\n\\int\\_{[z\\_i,z\\_{i+1)}} \\frac{ (s^2(u)-s^2\\_0(u))^2 }{s^4(u)}\\,\\dd u =\\frac{1}{n}\\frac{1}{s^4(z\\_i)}(s\\_0^2(z\\_i)-s^2(z\\_i))^2+O\\left(\\frac{1}{n^{1+\\lambda}}\\right).$$ \nNext, by a similar argument,\n \n$$\\label{eq:l2}\n\\frac{ \\int\\_{ [z\\_i,z\\_{i+1)}} [s^2(u)-s^2\\_0(u)] \\,\\dd u}{ \\int\\_{ [z\\_i,z\\_{i+1)}} s^2(u)\\,\\dd u } = \\frac{s^2(z\\_i)-s\\_0^2(z\\_i)}{s^2(z\\_i)}+O\\left(\\frac{1}{n^\\lambda}\\right).$$ \nCombination of and and the bounds in Assumption [standing] yields \n$$\\left\\{ \\frac{ \\int\\_{ [z\\_i,z\\_{i+1)}} [s^2(u)-s^2\\_0(u)] \\,\\dd u}{ \\int\\_{ [z\\_i,z\\_{i+1)}} s^2(u)\\,\\dd u } \\right\\}^2 - n\\int\\_{[z\\_i,z\\_{i+1)}} \\frac{ (s^2(u)-s^2\\_0(u))^2 }{s^4(u)}\\,\\dd u\n=O\\left(\\frac{1}{n^\\lambda}\\right).$$ \n Consequently, by summation,\n \n$$\\label{two\\*}\n\\frac{1}{n}\\sum\\_{i=1}^n\\left\\{ \\frac{ \\int\\_{ [z\\_i,z\\_{i+1)}} [s^2(u)-s^2\\_0(u)] \\,\\dd u}{ \\int\\_{ [z\\_i,z\\_{i+1)}} s^2(u)\\,\\dd u } \\right\\}^2 - \\int\\_0^1 \\frac{ (s^2(u)-s^2\\_0(u))^2 }{s^4(u)}\\,\\dd u\n=O\\left(\\frac{1}{n^\\lambda}\\right).$$ \nHence, for all *n* large enough,\n \n$$\\begin{aligned}\n\\sum\\_{i=1}^n \\exz[Z\\_{i,n,s}(Y\\_{i,n})] &\\leq \n-\\frac{ \\widetilde{c}\\_0 n }{2} \\int\\_0^1 \\frac{ (s^2(u)-s^2\\_0(u))^2 }{s^4(u)}\\,\\dd u+O(n^{1-\\lambda})\\\\\n&\\leq - \\frac{\\widetilde{c}\\_0\\kappa^2}{\\mathcal{K}^4}2^j\\varepsilon\\_n^2 n+\\widetilde{C}\\_0n^{1-\\lambda}.\\end{aligned}$$ \nHere the first inequality follows from and the last inequality from the assumption *s* ∈ *A**j*, *ɛ**n*. The proof is completed.\nThe following lemma is an analogue of Lemma A.3 in. The proof also shares similar general arguments, but differs in details.\n[lemma2]\nLet Assumption [standing] be satisfied and assume also *b*0 = 0. For a fixed and small enough constant *c*1 > 0, for any sequence *ɛ**n* ≍ *n*− *β*log*γ**n* of strictly positive numbers with *γ* > 0, and any sequence *m**n* ≍ *n*1 − *α* (equivalently, *N**n* ≍ *n**α*) as in Theorem [mainthm], there exist a constant *c*0 > 0 (depending on *κ*, K and *L*) and a universal constant *C* > 0 for which the inequality\n \n$$\\label{one\\*}\n\\ppz \\left( \\sup\\_{s\\in U\\_{s\\_0,\\varepsilon\\_n}^c} \\prod\\_{i=1}^n \\frac{p\\_{i,n,s}(Y\\_{i,n})}{ p\\_{i,n,0}(Y\\_{i,n}) } \\geq \\exp\\left( -c\\_1 n\\varepsilon\\_n^2 \\right)\\right)\n\\leq C M\\_{\\varepsilon\\_n} \\exp\\left( - c\\_0 n\\varepsilon\\_n^2 \\right)$$ \nholds for all *n* large enough.\nAs in the proof of Lemma A.3 in, one has\n \n$$\\begin{gathered}\n\\label{zero\\*}\n\\ppz \\left( \\sup\\_{s\\in U\\_{s\\_0,\\varepsilon\\_n}^c} \\prod\\_{i=1}^n \\frac{p\\_{i,n,s}(Y\\_{i,n})}{ p\\_{i,n,0}(Y\\_{i,n}) } \\geq \\exp\\left( -c\\_1 n\\varepsilon\\_n^2 \\right)\\right)\\\\\n=\\sum\\_{j=0}^{M\\_{\\varepsilon\\_n}} \\ppz\\left( \\sup\\_{s\\in A\\_{j,\\varepsilon\\_n}} \\prod\\_{i=1}^n \\frac{p\\_{i,n,s}(Y\\_{i,n})}{ p\\_{i,n,0}(Y\\_{i,n}) } \\geq \\exp\\left( -c\\_1 n\\varepsilon\\_n^2 \\right) \\right).\\end{gathered}$$ \nFurthermore, using Lemma [lemma3] above, by arguments identical to those in the proof of Lemma A.3 in, we have for all *n* large enough\n \n$$\\begin{gathered}\n\\label{pbjn}\n\\ppz\\left( \\sup\\_{s\\in A\\_{j,\\varepsilon\\_n}} \\prod\\_{i=1}^n \\frac{p\\_{i,n,s}(Y\\_{i,n})}{ p\\_{i,n,0}(Y\\_{i,n}) } \\geq \\exp\\left( -c\\_1 n\\varepsilon\\_n^2 \\right) \\right)\\\\\n\\leq\\ppz\\Biggl( \\sup\\_{s\\in B\\_{j,\\varepsilon\\_n}} \\left| \\frac{1}{"},"data_id":{"kind":"string","value":"arxiv_0183477"}}},{"rowIdx":45896,"cells":{"content":{"kind":"string","value":"Does Financial Literacy Impact Investment Participation and Retirement Planning in Japan?[1](#fn1)\n==================================================================================================\n**Abstract**\n**Keywords:** financial literacy, investment participation, retirement planning, causal discovery, Fast Causal Inference\nIntroduction\n============\nIn 2019, Japan’s Financial Services Agency (FSA) published a panel report titled “Asset Formation and Management in an Aging Society.” The report highlighted that an average older couple would need approximately 20 million yen for a 30-year post-retirement life, leading to a monthly deficit of around 50,000 yen not covered by public pension programs. This revelation garnered widespread media attention and sparked significant public concern regarding the financial asset formation of households in Japan. Three years later, in 2022, the Council of New Form of Capitalism Realization, established in the Cabinet Secretariat of Japan, formally adopted the “Doubling of Asset-based Incomes Plan.” This initiative aims to double the number of Nippon Individual Savings Account (NISA) holders and cumulative purchase amounts within five years. This plan is designed to encourage Japanese households to diversify their savings into risky assets, including stocks and mutual funds, which is viewed as crucial for boosting overall economic growth and enhancing personal financial well-being.\nThe FSA has implemented various measures to promote the transition from savings to investment over the past 20 years. For example, from 2003 to 2013, preferential tax treatment for securities investment allowed that the tax rate on dividends and capital gains from stock or investment trusts was reduced from 20% to 10%. In 2014, the NISA was initiated as an investment tax-break scheme. In 2018, an installment-type NISA was launched, drawing inspiration from the investment strategies advocated by Burton Malkiel and Charles Ellis, renowned economists in the asset management field. Despite these initiatives, the impact appears to be less dramatic than anticipated. As illustrated in the “Flow of Funds for the Fourth Quarter of 2023” report by the Bank of Japan (BOJ), the proportion of each type of asset has remained relatively stable over the last two decades. At the end of December 2023, the share of “currency and deposits” in the total outstanding financial assets held by households was still as high as 52.6%.\nDue to the prolonged ultra-low interest rate environment for Japanese Yen deposits, portfolios with a significant allocation to “currency and deposits” have experienced a notable disparity in total financial asset growth. According to the data published by FSA in 2019, in the 20-year period starting from 1998, the total amount of household financial assets in the United States (U.S.) and the United Kingdom (U.K.) expanded to 2.7 and 2.3 times their original values, while Japan’s growth was limited to only 1.4 times during the same period. This discrepancy is attributed significantly to the differences in investment returns among these countries.\nIn Japan, characterized by a rapidly aging population and a pay-as-you-go public pension system, the emphasis on financial asset accumulation and proactive retirement planning is becoming increasingly critical. Various academic studies and governmental surveys have been undertaken to elucidate the reasons behind the reluctance of Japanese households to invest in risky assets. With this context, the role of financial literacy, defined as individuals’ capacity to process information and make judicious decisions about financial planning, asset accumulation, and pension-related considerations, has emerged as a focal point. The Central Council for Financial Services Information, an organization of the BOJ responsible for conducting financial services information activities, has carried out nationwide large-scale questionnaire surveys, “Financial Literacy Survey,” in 2016, 2019, and 2022, aiming to assess the current state of financial literacy of individuals. Common selection questions were included in the questionnaire, and the percentage of correct answers on financial knowledge in Japan was lower compared to the U.S. and some other Organization for Economic Co-operation and Development (OECD) countries.\n developed three questions as a benchmark for measuring financial literacy, based on economic models of saving and portfolio choice with the focus on three key economic concepts vital for financial decision-making: the understanding of interest compounding, inflation, and risk diversification. The design of the questions was guided by four fundamental principles: simplicity, relevance, brevity, and capacity to differentiate. This guideline was applied in the Health and Retirement Study conducted in the U.S. in 2004. The same set of questions was also incorporated into the “Financial Literacy Survey” executed by the BOJ.\nPrior studies utilizing diverse survey datasets have explored the relationship between financial literacy and various financial activities. These investigations have elucidated that enhanced financial literacy contributes to more effective retirement planning and encourages increased investment in complex financial assets. Based on these research findings and the suboptimal financial literacy level among Japanese households, as revealed by the survey results, Japan revised its national school curriculum guidelines in 2022. This update mandates a heightened focus on personal finance education for high school students, with the objective of raising financial literacy within this young demographic.\nWhile existing research on financial literacy has predominately identified positive correlational relationships with financial activities, employing causal discovery methods offers the potential to draw conclusions about causality. However, no prior research has employed causal discovery methods to work on this topic, either within Japan or internationally. Therefore, this study adopts a causal discovery approach to investigate the causal connections between financial literacy and financial activities, including retirement planning and investment participation in risky financial assets in Japan. By doing so, this research not only sheds light on understanding the impact of Japan’s recent implementation of finance education in high schools but also provides a framework that can be applied to similar studies. Although our study primarily focuses on analyzing and interpreting data from Japan, the implications of applying the causal discovery approach also extend to other countries.\nLiterature Review\n=================\nOver the last two decades, financial literacy has emerged as a key area of study, with a focus on the relationship with an individual’s capability to make informed financial decisions. Previous studies posit that financial literacy is essential in enabling individuals to engage in financial investment and aiding in effective retirement planning.\n demonstrate that financial literacy significantly influences financial decision-making. Their analysis indicates that individuals with higher levels of financial literacy are more likely to participate in stock investment. This conclusion was derived from a regression analysis using the Ordinary Least Squares (OLS) method on data from the 2005 De Nederlandsche Banks Household Survey. applied OLS and probit models to a composite dataset from nine European countries: Austria, Belgium, Denmark, Germany, Italy, France, Switzerland, Sweden, and the Netherlands. Their study evaluated the influence of factors on the likelihood of stock market participation in 2010, affirming that financial literacy positively impacts this likelihood. The same conclusion is also applicable to Japan. conducted an online survey in 2019 on wealth building, securities, investment, and financial literacy in Japan. The analysis of this survey data, which included a straightforward regression model and an Instrumental Variables (IV) estimation, indicated that elevated financial literacy is generally linked to increased participation in the stock market.\nOther previous studies have focused on the association between financial literacy and retirement planning. analyzed data from the National Financial Capability Survey in the U.S., which is a comprehensive survey aimed at establishing benchmarks for key indicators of financial capability and associating these indicators with demographic, behavioral, attitudinal, and financial literacy characteristics. Utilizing OLS and an IV approach, the study suggests that financial literacy significantly influences retirement planning. This conclusion remains robust even when accounting for endogeneity and possible inaccuracies in measuring of financial literacy. leveraged the OLS and an IV estimation to analyze the January – February 2010 wave of the Survey of Living Preferences and Satisfaction. This research highlights the link between financial literacy and retirement planning, revealing that enhanced financial literacy increases the likelihood of holding a retirement savings plan.\nMost previous studies have consistently identified correlational associations related to financial literacy through regression analysis. Some of these studies have attempted to address specific types of endogeneity, such as reverse causality or selection bias, by adopting an IV approach . However, IV methods primarily concentrate on estimating specific causal relationships between selected sets of variables without consideration of latent confounding variables, which may not be sufficiently comprehensive to uncover the overarching causal mechanisms.\nData\n====\nThe “Financial Literacy Survey” is an extensive questionnaire survey to evaluate the current level of financial literacy among Japanese individuals. This online survey engaged 30,000 respondents aged between 18 and 79, selected to correspond closely with the demographic distribution in Japan.[2](#fn2) For this study, we obtained individual-level data from the 2022 survey from the Central Council for Financial Services Information (Secretariat: Public Relations Department of the BOJ). This survey is the third installment following the conduction in 2016 and 2019.\nWe used 13 variables from the survey data to represent the demographics, socioeconomic status, behavioral biases, financial activities, and the level of financial literacy of respondents. The list of variables is provided in [apd:third].\nFirst, we identified five exogenous variables from the survey data for demographics and socioeconomic status. \n(1) *Male*: Dummy Variable. Assigned a value of 1 if the response to “Q42. What is your gender?” is “Male.” \n(2) *Fin\\_Edu*: Dummy Variable. Set to 1 if the answer to “Q39. Was financial education offered by a school or college you attended, or a workplace where you were employed?” is “Yes, and I did participate in the financial education.” \n(3) *Fin\\_Edu\\_Home*: Dummy Variable. Assigned a value of 1 if the response to “Q40. Did your parents or guardians teach you how to manage your finances?” is “Yes.” \n(4) *Age*: Calculated as the mean of the age range indicated by the response to “Q43. What is your age?” For example, a response is “1,” denoting the age range of 18 – 19, results in an assigned value of “18.5.” The age ranges corresponding to each answer option are as follows: “2” represents 20 – 24, “3” for 25 – 29, up to “13” for 75 – 79. \n(5) *Education*: Determined by the total years required to complete the educational level indicated in “Q44. What is your educational background?” For instance, if “Primary and secondary schools only” is selected, then *Education* is set as “9,” and if “University” is selected, then *Education* is set as “16.” A missing value is assigned if the answer “Other” is selected.\nMoreover, we included two more variables to capture additional information about socioeconomic status. We incorporated a variable *Income* from the answer to “Q51. Which of these categories do your (your household) income for last year fall into?” We calculated the mean of the range for each category: a value of “0” for “Don’t have any financial assets,” “125” for “Less than 2.5 million yen,” “375” for “At least 2.5 million but less than 5 million yen,” “625” for “At least 5 million but less than 7.5 million yen,” “875” for “At least 7.5 million but less than 10 million yen,” and “1250” for “At least 10 million but less than 15 million yen.” For the answer “At least 15 million yen,” we assigned the lower limit value, “1500,” for *Income*. A missing value is set if the response is “Don’t know/Prefer not to say.” Similarly, we determined the variable *Asset\\_Amt* based on the answer to “Q52. Which of these categories do your (your household’s) financial assets (deposits, stocks, etc.) currently fall into?” The assigned values are “0” for “Don’t have any financial assets,” “125” for “Less than 2.5 million yen,” “375” for “At least 2.5 million but less than 5 million yen,” “625” for “At least 5 million but less than 7.5 million yen,” “875” for “At least 7.5 million but less than 10 million yen,” and “1500” for “At least 10 million but less than 20 million yen.” For “At least 20 million yen,” we assigned “2000” as the value of *Asset\\_Amt*. A missing value is set for the answer “Don’t know/Prefer not to say.” Variables for income and asset amount are considered crucial because they have been demonstrated to be related to the level of financial literacy .\nIn the 2011 report “National Financial Literacy Strategy,” the Australian Securities & Investments Commission identified six primary behavioral biases that may influence individuals’ decisions, including those related to complex financial matters. These biases are disengagement, overconfidence, loss aversion, myopic decision-making, mental accounting, and herd behavior. For myopic decision making, which involves prioritizing short-term gains without considering future consequences, we adopted the answer to “Q1\\_10. If I had the choice of (1) receiving 100,000 yen now or (2) receiving 110,000 yen in 1 year, I would choose (1), provided that I can definitely receive the money” as the indicator variable *Myopic\\_Bias*. Moreover, we identified herd behavior through *Herding\\_Bias*, based on the response to “Q1\\_3. When there are several similar products, I tend to buy what is recommended as the best-selling product, rather than what I actually think is a good product.” The answer options for Q1\\_10 and Q1\\_3, which range from “1” (Agree) to “5” (Disagree), employ the Likert scale, a psychometric scale commonly used in questionnaires. This scale, named after the social scientist Rensis Likert, allows respondents to indicate their level of agreement or disagreement with a statement by choosing from a specified range of options. In addition, we utilized the answer to “Q17. How would you rate your overall knowledge about financial matters compared with other people?” as a measure of respondents’ confidence in their financial literacy level (*Confidence*). The choices for Q17, ranging from “1” (Very high) to “5” (Very low), with “6” representing “Don’t know,” also follow the Likert scale format. A missing value is assigned to *Confidence* if option “6” is selected. The combination of objective financial literacy and subjective confidence can be used to measure the individuals’ overconfidence, and is reported to be positively correlated with stock market participation. Unfortunately, for the remaining three biases, we could not identify suitable corresponding questions within the survey.\nA variable *Invest* was constructed to qualify the number of risky asset types an individual has invested in, based on the responses to “Q34. Have you ever purchased any of the following financial products?” Risky assets are defined as stocks, investment trusts, and foreign currency deposits/money market funds. Another variable, *Planning*, was created to assess whether the individual is well prepared for retirement, according to the answers to a series of questions: “Q7. What expenses do you think you will have to cover in the future?,” “Q8\\_1. Are you aware of the amounts that will be required for living expenses for retirement in your case?,” “Q9\\_1. Do you have a financial plan for the expenses you think you will have to cover for living expenses for retirement?” and “Q10\\_1. Have you set aside funds for the expenses you think you will have to cover for living expenses for retirement?” If the option “1. Living expenses for retirement” is chosen in Q7, 1 is added to *Planning*. Additionally, for each “Yes” response to Q8\\_1, Q9\\_1, and Q10\\_1, *Planning* is incremented by 1. Therefore, *Planning* can range from “0” to “4,” where a higher value indicates more comprehensive planning and preparation for retirement.\nFinally, and of significant importance, *Fin\\_Literacy* represents the number of correct answers to the 25 questions designed to evaluate the individual’s financial literacy level, covering eight categories of the Financial Literacy Map. See [apd:second] for all the question numbers and corresponding categories. Although the BOJ does not publicly disclose the correct answer to each question, we deduced them from our background knowledge in finance. Moreover, the “Financial Literacy Survey: 2022 Results” include each option’s response ratio and the correct answer rate for every question. We compared the response ratios of the options deemed correct with the correct answer rates to ascertain the accuracy of our answers.\nApart from the three dummy variables – *Male*, *Fin\\_Edu*, *Fin\\_Edu\\_Home* – all other variables are treated as continuous. These continuous variables have been standardized before the subsequent analysis. We eliminated data entries with missing values for any of the variables, resulting in a reduction of the sample size from 30,000 to 19,333. This exclusion process could potentially introduce bias, particularly concerning the data entries for income and financial assets. The selection of the “Don’t know/Prefer not to say” option to these questions may not occur randomly. Instead, it is conceivable that individuals with either particularly low or high income (or financial assets) might tend to choose “Prefer not to say.”\nMethod\n======\nTo elucidate causal relationships, the traditional methods involve interventions or randomized controlled trials. However, these approaches are often too expensive, time-intensive, or impractical to implement for specific fields of study. For instance, it is hardly permissible to engage in randomized A/B tests concerning financial education and financial literacy, particularly from the perspective of social ethics and fairness. Considering these challenges, causal discovery methods are gaining increasing interest across various disciplines. These methods facilitate the derivation of causal structures from observational data, offering a practical alternative to more traditional, intervention-based approaches.\nAs the application of causal discovery models requires establishing specific foundational assumptions, our initial focus will be on discussing the assumptions to select the most appropriate model. As noted in Section 3, our inability to locate corresponding questions within the survey that align with concepts of disengagement, loss aversion, and mental accounting – all of which have been suggested to influence an individual’s financial activities – leads us to conclude that the assumption of “***causal sufficiency***” is likely violated. The causal sufficiency assumption necessitates the absence of unobserved (or latent) variables in the analysis. While the linear, non-Gaussian, and acyclic model (**LiNGAM**), as one of the structural equation models, is capable of identifying the unique directed acyclic graph (DAG), it requires causal sufficiency, which is not satisfied in the context of this study. Among the constraint-based methods, the **PC** algorithm, named after its inventors Peter Spirtes and Clark Glymour, necessitates the assumptions of acyclicity, causal faithfulness, and causal sufficiency. A first extension of the PC algorithm, the **FCI**, relaxes the assumptions by foregoing the necessity of causal sufficiency and permitting the presence of hidden variables. Therefore, we leverage the FCI algorithm to explore the mechanism behind financial literacy and financial activities using the FCI method in Causal-learn package in Python.[3](#fn3)\nGiven the mixed nature of our dataset, which comprises three discrete dummy variables and ten continuous variables, we initially considered the usage of FCI with a kernel-based conditional independence test, as this method is suited for handling the complexity of mixed data types. However, kernel methods’ computational complexity scales cubically with the sample size. Due to the constraints of computational resources and considering our substantial sample size of nearly 20,000, it is impractical to implement the FCI model with the kernel-based conditional independence test in this study. As an alternative approach, we divide our data into eight groups based on the combination of values of the three dummy variables: *Male*, *Fin\\_Edu*, *Fin\\_Edu\\_Home*. For each of these groups, we apply the FCI model with Fisher’s Z conditional independent test, which is suitable for continuous variables. A significance level of 0.05 is selected for the individual partial correlation test in this study.\nFurthermore, as previous research demonstrates that incorporating background knowledge into causal discovery enhances the accuracy of exploring causal relations and lends more realism to the interpretation of the results, we define *Age* and *Education* as exogenous variables when applying the FCI algorithm to the pre-segmented data of the eight groups.\nAdditionally, bootstrapping is used to confirm the statistical stability of the results from the causal discovery approach before we draw conclusions. The bootstrap method involves estimating the distribution of an estimator or a test statistic through the process of resampling from the original dataset. By resampling the data multiple times and regenerating the partial ancestral graphs (PAG), the outputs estimated from the FCI algorithm for each sample, we can estimate the bootstrap probability of the existence of each edge. This process enables the evaluation of the robustness of the PAG against variations in the data sample. In this study, we specify the number of bootstrap samplings as 100 and proceed to estimate the bootstrap probability. We exclude illustrations of edges with notably low probabilities, specifically those lower than or equal to 0.2, due to their unreliability. Instead, our focus was on edges with higher bootstrap probabilities.\nResults\n=======\nAs mentioned in Section 4, we divided the original survey data into eight groups based on three dummy variables: *Male*, *Fin\\_Edu*, and *Fin\\_Edu\\_Home*. The sample sizes for each group, along with their corresponding conditions, are summarized in Table [datasummary]. More than 75% of the 19,333 respondents in the research sample indicated that they did not receive financial education either at school or at home, which are segmented into Groups 4 and 8. These two groups are comparatively less financially literate, as evidenced by the number of correct answers to questions assessing financial knowledge (*Fin\\_Literacy)* illustrated in Figure [boxplot].\n[htbp] [datasummary]\n| **Group** | ***Male*** | ***Fin\\_Edu*** | ***Fin\\_Edu\\_Home*** | **Sample Size** |\n| --- | --- | --- | --- | --- |\n| Group 1 | 1 | 1 | 1 | 443 |\n| Group 2 | 1 | 0 | 1 | 1,482 |\n| Group 3 | 1 | 1 | 0 | 657 |\n| Group 4 | 1 | 0 | 0 | 7,961 |\n| Group 5 | 0 | 1 | 1 | 259 |\n| Group 6 | 0 | 0 | 1 | 1,659 |\n| Group 7 | 0 | 1 | 0 | 300 |\n| Group 8 | 0 | 0 | 0 | 6,572 |\nIn addition, Figure [boxplot] demonstrates that female, represented by Groups 5 to 8, exhibits a lower level of financial literacy compared to male categorized by Groups 1 to 4, which aligns with the conclusion drawn from previous research .\n![Boxplots of Fin_Literacy for Eight Groups](boxplot.png \"fig:\") [boxplot]\nThe PAGs derived from the FCI model, utilizing the processed survey data, are illustrated in Figure [14] for Groups 1 to 4 and Figure [58] for Groups 5 to 8. A PAG is a graphical representation used to depict the causal relationships among a set of variables, particularly when dealing with datasets that may have latent or hidden variables. In PAGs, the vertices and edges represent variables and the relationships between them. Unlike a DAG, a PAG can have four types of edges : → , ∘ → , ∘ – ∘ , and ↔ . *i* → *j* is drawn when *i* is an ancestor of *j* and *j* is not an ancestor of *i*. An edge *i* ↔ *j* is drawn when *i* is not an ancestor of *j* and *j* is not an ancestor of *i*, which indicates the influence of an unobserved confounder. The open dot ∘ is used in *i* ∘ → *j* when *i* → *j* is present in some maximal ancestral graphs (MAG), and *i* ↔ *j* exists in some other MAGs.\nTo ascertain the reliability of the generated PAGs, we employed the bootstrap method to assess the statistical robustness of the FCI model’s outputs. The comprehensive lists of edges generated from 100 bootstrap samplings, along with their corresponding bootstrap probabilities for each group, are included in [apd:fourth]. Interpretations of the findings presented in this section, as well as discussions and conclusions in subsequent sections, are based on edges with bootstrap probability larger than 0.2 in [apd:fourth].\n![Output PAGs of FCI for Groups 1 to 4](1_4.png \"fig:\") [14]\n![Output PAGs of FCI for Groups 5 to 8](5_8.png \"fig:\") [58]\nWe could not find a direct causal flow from *Fin\\_Literacy* to *Invest* or *Planning* in the PAGs of any of the eight groups, which demonstrates that the level of financial literacy is not a direct cause of participation in risky financial assets investment or retirement planning in Japan. Meanwhile, in the PAGs of Groups 1, 3, 4, 7, and 8, “ ↔ ” exists between *Fin\\_Literacy* and *Invest*, which might indicate the existence of latent confounders. Likewise, the presence of “ ↔ ” between *Fin\\_Literacy* and *Planning* in the PAGs of Groups 3, 4, 5, and 7 implies the possibility of unobserved variables. Moreover, “ ∘ → ” extends from *Invest* to *Fin\\_Literacy* in the PAGs of Groups 4, 6, and 8. This edge suggests the potential causal relationship where participation in financial investment might enhance financial literacy levels. Similarly, “ ∘ → ” can be found in the PAGs of Groups 4 and 6, and “ → ” can be found in the PAGs of Groups 6, 7, and 8 from *Planning* to *Fin\\_Literacy*, revealing the possibility of a causal impact of retirement planning on the improvement of financial literacy.\nConcerning the causal relationship about confidence in financial literacy, divergent outcomes were observed across different bootstrap samplings and groups. Specifically, for Groups 1, 2, 3, 4, 6, and 8, the PAGs of some bootstrap samples depict a possible causal flow from confidence levels to investment in risky financial assets, as represented by the “ → ” or “ ∘ → ” edges from *Confidence* to *Invest*. In contrast, the level of confidence can be a result of participation in financial investment, illustrated by the “ → ” from *Invest* to *Confidence* in some bootstrapping results of Groups 1, 2, 5, 6, and 8. Unobserved confounders can exist between *Invest* and *Confidence* represented by “ ↔ ” in PAGs generated from specific bootstrap samples in Groups 3 and 8. Regarding the causal link between confidence and retirement planning, “ → ” from *Confidence* to *Planning* and a reverse causal flow “ → ” from *Planning* to *Confidence* can be both observed in the PAGs emerged from different bootstrap samples of Groups 1, 2, 6, and 8. For Group 4, “ → ” and “ ↔ ” are both possible between *Confidence* and *Planning* in the results, indicating the possible causal flow from confidence to retirement planning and the potential latent confounders between them. “ ↔ ” between *Confidence* and *Planning* can also be found in Groups 2, 6, and 8 as a signal of latent variables.\nFurthermore, beyond the scope of our initial research question related to financial literacy, the results uncovered the causal influence of age on retirement planning, which is reflected by the indirect “ → ” from *Age* to *Planning* in the PAG of Groups 1, 4, 7, and 8, and direct “ → ” in Groups 2, 3, 4, 5, and 7. Additionally, direct causal influences from age (*Age*) to participation in risky asset investment (*Invest*) are confirmed in the results of female Groups 5, 6, and 8.\nDiscussion\n==========\nAlthough the previous research introduced in Section 2 demonstrates that the financial literacy level is highly associated with financial activities, which could be a causal relationship, we did not confirm the direct causal impact of financial literacy on either investment in risky financial assets or retirement planning in the results of the FCI model. Therefore, enhancing the financial literacy level of individuals in Japan through mandatory financial education in high school may not inevitably improve participation in financial investment and retirement planning in Japan. Meanwhile, we found the possibility of latent confounders between financial literacy and financial activities, either participation in financial investment or retirement planning, for Groups 1, 3, 4, 5, 7, and 8. Moreover, in Groups 4, 6, 7, and 8, financial activities can have a causal impact on financial literacy levels. The existence of latent confounders and reverse causation may explain the high correlation between the variables verified in previous research.\nWhen it comes to confidence in financial literacy, although results differed across the bootstrap samples and groups, it was a cause of investment participation in Groups 1, 2, 3, 4, 6, and 8 and of retirement planning in Groups 1, 2, 4, 6, and 8. As depicted in the “Financial Literacy Survey: 2022 Results,” a mere 12.1% of respondents expressed confidence in their financial knowledge by rating their overall knowledge about financial matters as “Very high” or “Quite high” compared with others. This figure is significantly lower than the 76% reported in the U.S. Subsequently, interventions, such as providing opportunities for simulated financial experiences and creating environments conducive to discussions about finances, to increase confidence in financial literacy among Japanese households, may potentially boost investment participation and retirement planning.\nWhile the primary focus of this study remains on the impact of financial literacy, the results indicate a direct or indirect causal effect of age on financial activities. has noted that women often enhance their financial knowledge when it becomes relevant, typically around the time of their spouses’ passing. In a parallel manner, the influence of age on retirement planning and investment participation might stem from a sense of “relevancy.” As individuals age, they may perceive an escalating need for financial planning. Women may start investing in riskier assets to pursue higher returns, as their income growth does not typically match the pace experienced by men in Japan. To extend and improve this study further, variables representing disengagement, loss aversion, and mental accounting can be included in the dataset, as these factors may affect individuals’ decision-making. Given that the PAGs imply the presence of latent confounders between financial literacy and financial activities, potential variables can be identified by delving into financial and economic theories. Subsequently, redesigning the survey questions and conducting a subsequent survey are necessary for acquiring the data for further research.\nMoreover, broadening the definition of financial literacy may provide deeper insights into its causal influence on financial activities. Traditionally, financial literacy has been primarily quantified in terms of financial knowledge. The practical application of this knowledge in decision-making processes was not a major focus until the OECD redefined financial literacy. This redefinition encompasses not only financial knowledge but also financial attitude and behavior. Therefore, financial behavior and attitude could be considered integral components of financial literacy. suggest that financial behavior, reflecting individuals’ actions in financial matters, can be accessed by the extent of agreement with the statements below from the “Financial Literacy Survey.” \nQ1\\_1. Before I buy something I carefully consider whether I can afford it. \nQ1\\_2. I pay my bills on time. \nQ1\\_4. I set long-term financial goals and strive to achieve them. \nQ1\\_7. I keep a close personal watch on my financial affairs. \nIn a similar fashion, financial attitude, indicating one’s perspective on financial issues, can be measurable through responses to the following statements. \nQ1\\_5. I find it more satisfying to spend money than to save it for the long term. \nQ1\\_6. I tend to live for today and let tomorrow take care of itself.\nAdditionally, future studies can apply the same method to the data from the 2019 iteration of the “Financial Literacy Survey.” As highlighted in Section 1, the “20-million-yen problem” garnered significant attention in 2019, intensifying public focus on financial asset formation in Japan. The onset of the COVID-19 pandemic from 2020 onwards likely influenced individuals’ lifestyles, consumption habits, and financial activities. Comparing the 2019 survey results with this study could elucidate any shifts in the underlying mechanism of households’ financial activities in Japan.\nConclusion\n==========\nIn this research, we applied the FCI model with Fisher’s Z conditional independent test to the individual-level data from the 2022 “Financial Literacy Survey” conducted by the BOJ. This approach was employed to investigate the causal relationships between financial literacy and financial activities, specifically focusing on retirement planning and investment in risky financial assets in Japan. The original dataset was divided into eight groups based on three exogenous discrete variables to facilitate the use of Fisher’s Z conditional independent test, which is appropriate for continuous variables. Additionally, two other variables were designated as exogenous, and this information was incorporated as background knowledge about the edges. The robustness of the edges in PAGs derived from the FCI model was verified by estimating their probabilities using the bootstrap method.\nWhile financial education at school and home may correlate with higher financial literacy, higher financial literacy does not necessarily lead to increased participation in financial investment and retirement planning in Japan. Our results did not identify direct causal links from financial literacy to financial activities. Consequently, the Japanese government may need to consider supplementary or alternative measures to achieve its objective of “a shift from savings to investment.” As confidence in financial literacy could support the adoption of desirable financial activities, initiatives can be undertaken to enhance confidence levels. The impact of age on financial activities could be attributed to perceived relevancy. Although age itself is an exogenous factor and cannot be intervened, measures could be implemented to strengthen the sense of relevancy.\nWe hope this study encourages further discussions on applying causal discovery algorithms to financial literacy and financial activity survey data. Equally important, we hope that the insights derived from this study will help governmental policymakers in their efforts to promote the financial well-being of households in Japan.\nQuestion Numbers for Category of Financial Literacy Map\n=======================================================\n|p6.5cm|p6.2cm| **Category of Financial Literacy Map** & **Question Numbers** \nFamily budget management & Q4, Q5 \nLife planning & Q12, Q13 \nFinancial knowledge & \nBasics of financial transactions & Q14, Q15, Q16 \nBasics of finance and economy & Q18, Q19, Q20, Q21\\_1, Q22, Q23 \nInsurance & Q25, Q26, Q28 \nLoans/credit & Q21\\_2, Q30, Q31 \nWealth building & Q21\\_3, Q21\\_4, Q33 \nUse of outside expertise & Q36, Q37, Q38 \nList of Variables\n=================\n**1. *Male*:** Dummy Variable. Assigned a value of 1 if the response to “Q42. What is our gender?” is “Male.” \n**2. *Fin\\_Edu*:** Dummy Variable. Set to 1 if the answer to “Q39. Was financial education offered by a school or college you attended, or a workplace where you were employed?” is “Yes, and I did participate in the financial education.” \n**3. *Fin\\_Edu\\_Home*:** Dummy Variable. Assigned a value of 1 if the response to “Q40. Did your parents or guardians teach you how to manage your finances?” is “Yes.” \n**4. *Age*:** Continuous Variable. Calculated as the mean of the age range indicated by the response to “Q43. What is your age?” \n**5. *Education*:** Continuous Variable. Determined by the total years required to complete the educational level indicated in “Q44. What is your educational background?” \n**6. *Income*:** Continuous Variable. The answer to “Q51. Which of these categories do your (your household) income for last year fall into?” \n**7. *Asset\\_Amt*:** Continuous Variable. The answer to “Q52. Which of these categories do your (your household’s) financial assets (deposits, stocks, etc.) currently fall into?” \n**8. *Myopic\\_Bias*:** Continuous Variable. The answer to “Q1 10. If I had the choice of (1) receiving 100,000 yen now or (2) receiving 110,000 yen in 1 year, I would choose (1), provided that I can definitely receive the money.” \n**9. *Herding\\_Bias*:** Continuous Variable. The response to “Q1 3. When there are several similar products, I tend to buy what is recommended as the best-selling product, rather than what I actually think is a good product.” \n**10. *Confidence*:** Continuous Variable. The answer to “Q17. How would you rate your overall knowledge about financial matters compared with other people?” \n**11. *Invest*:** Continuous Variable. The number of risky asset types an individual has invested in, based on the responses to “Q34. Have you ever purchased any of the following financial products?” \n**12. *Planning*:** Continuous Variable. Assessed whether the individual is well prepared for retirement, according to the answers to a series of questions: Q7, Q8\\_1, Q9\\_1, and Q10\\_1. \n**13. *Fin\\_Literacy*:** Continuous Variable. The number of correct answers to the 25 questions designed to evaluate the individual’s financial literacy level.\nEdges with Bootstrap Probabilities\n==================================\n|p5cm|p1cm| Age → Asset\\_Amt & 1 \nAge → Fin\\_Literacy & 1 \nAge → Herding\\_Bias & 0.05 \nAge → Income & 0.22 \nAge → Planning & 0.02 \nAsset\\_Amt → Confidence & 0.11 \nAsset\\_Amt → Fin\\_Literacy & 0.12 \nAsset\\_Amt ∘ → Fin\\_Literacy & 0.07 \nAsset\\_Amt → Income & 0.34 \nAsset\\_Amt ∘ → Income & 0.01 \nAsset\\_Amt → Invest & 0.05 \nAsset\\_Amt ↔ Invest & 0.03 \nAsset\\_Amt → Planning & 0.58 \nAsset\\_Amt ↔ Planning & 0.16 \nConfidence → Asset\\_Amt & 0.12 \nConfidence ↔ Asset\\_Amt & 0.08 \nConfidence ∘ → Asset\\_Amt & 0.06 \nConfidence ∘ → Herding\\_Bias & 0.01 \nConfidence → Income & 0.05 \nConfidence ↔ Income & 0.02 \nConfidence ∘ – ∘ Income & 0.01 \nConfidence → Invest & 0.5 \nConfidence ∘ → Invest & 0.16 \nConfidence ↔ Invest & 0.06 \nConfidence ∘ – ∘ Invest & 0.01 \nConfidence → Planning & 0.37 \nConfidence ↔ Planning & 0.16 \nConfidence ∘ → Planning & 0.11 \nConfidence ∘ – ∘ Planning & 0.01 \nEducation → Asset\\_Amt & 0.18 \nEducation → Confidence & 0.33 \nEducation → Fin\\_Literacy & 0.69 \nEducation → Income & 0.64 \nEducation → Invest & 0.35 \nEducation → Myopic\\_Bias & 0.07 \nEducation → Planning & 0.01 \nFin\\_Literacy ↔ Asset\\_Amt & 0.23 \nFin\\_Literacy → Asset\\_Amt & 0.1 \nFin\\_Literacy ∘ → Asset\\_Amt & 0.01 \nFin\\_Literacy → Herding\\_Bias & 0.21 \nFin\\_Literacy ∘ → Herding\\_Bias & 0.01 \nFin\\_Literacy ↔ Invest & 0.35 \nFin\\_Literacy → Myopic\\_Bias & 0.12 \nFin\\_Literacy ↔ Planning & 0.05 \nHerding\\_Bias ↔ Confidence & 0.03 \nHerding\\_Bias ∘ → Confidence & 0.01 \nHerding\\_Bias ∘ → Fin\\_Literacy & 0.41 \nHerding\\_Bias → Fin\\_Literacy & 0.23 \nHerding\\_Bias ↔ Fin\\_Literacy & 0.14 \nHerding\\_Bias ↔ Invest & 0.31 \nHerding\\_Bias ∘ → Invest & 0.09 \nHerding\\_Bias → Invest & 0.03 \nHerding\\_Bias ∘ – ∘ Myopic\\_Bias & 0.22 \nHerding\\_Bias ↔ Myopic\\_Bias & 0.21 \nHerding\\_Bias ∘ → Myopic\\_Bias & 0.11 \nHerding\\_Bias → Myopic\\_Bias & 0.04 \nHerding\\_Bias ↔ Planning & 0.01 \nIncome → Asset\\_Amt & 0.59 \nIncome ∘ – ∘ Asset\\_Amt & 0.03 \nIncome ∘ → Asset\\_Amt & 0.03 \nIncome → Confidence & 0.09 \nIncome → Invest & 0.54 \nIncome ↔ Invest & 0.11 \nIncome ∘ → Invest & 0.09 \nIncome ∘ – ∘ Invest & 0.02 \nIncome → Planning & 0.24 \nIncome ↔ Planning & 0.18 \nIncome ∘ → Planning & 0.04 \nInvest → Asset\\_Amt & 0.07 \nInvest → Confidence & 0.27 \nInvest → Fin\\_Literacy & 0.01 \nInvest → Herding\\_Bias & 0.01 \nInvest → Income & 0.04 \nInvest ∘ → Income & 0.01 \nInvest ∘ → Planning & 0.01 \nMyopic\\_Bias ↔ Asset\\_Amt & 0.03 \nMyopic\\_Bias → Asset\\_Amt & 0.01 \nMyopic\\_Bias ∘ → Fin\\_Literacy & 0.47 \nMyopic\\_Bias → Fin\\_Literacy & 0.23 \nMyopic\\_Bias ↔ Fin\\_Literacy & 0.15 \nMyopic\\_Bias ∘ → Herding\\_Bias & 0.23 \nMyopic\\_Bias → Herding\\_Bias & 0.13 \nMyopic\\_Bias ↔ Income & 0.02 \nMyopic\\_Bias ∘ → Invest & 0.01 \nMyopic\\_Bias ↔ Planning & 0.19 \nMyopic\\_Bias → Planning & 0.02 \nMyopic\\_Bias ∘ → Planning & 0.01 \nPlanning → Asset\\_Amt & 0.06 \nPlanning ∘ → Asset\\_Amt & 0.03 \nPlanning → Confidence & 0.22 \nPlanning ∘ → Confidence & 0.04 \nPlanning → Fin\\_Literacy & 0.01 \nPlanning → Income & 0.06 \nPlanning ∘ → Income & 0.02 \nPlanning ↔ Invest & 0.15 \nPlanning → Invest & 0.09 \nPlanning ∘ → Invest & 0.03 \n|p5cm|p1cm| Age → Asset\\_Amt & 1 \nAge → Fin\\_Literacy & 1 \nAge → Herding\\_Bias & 0.98 \nAge → Income & 0.06 \nAge → Myopic\\_Bias & 0.43 \nAge → Planning & 0.96 \nAsset\\_Amt → Confidence & 0.3 \nAsset\\_Amt → Fin\\_Literacy & 0.22 \nAsset\\_Amt → Income & 0.22 \nAsset\\_Amt ↔ Invest & 0.59 \nAsset\\_Amt → Invest & 0.4 \nAsset\\_Amt → Myopic\\_Bias & 0.02 \nAsset\\_Amt ↔ Planning & 0.21 \nAsset\\_Amt → Planning & 0.16 \nAsset\\_Amt ∘ – ∘ Planning & 0.01 \nConfidence ↔ Asset\\_Amt & 0.52 \nConfidence → Asset\\_Amt & 0.14 \nConfidence ∘ → Asset\\_Amt & 0.04 \nConfidence ↔ Fin\\_Literacy & 0.09 \nConfidence ↔ Income & 0.17 \nConfidence → Income & 0.06 \nConfidence ∘ – ∘ Income & 0.01 \nConfidence ∘ → Income & 0.01 \nConfidence → Invest & 0.35 \nConfidence ↔ Invest & 0.11 \nConfidence ∘ → Invest & 0.05 \nConfidence ∘ – ∘ Invest & 0.01 \nConfidence ↔ Planning & 0.41 \nConfidence → Planning & 0.31 \nConfidence ∘ → Planning & 0.05 \nEducation → Asset\\_Amt & 0.45 \nEducation → Confidence & 0.18 \nEducation → Fin\\_Literacy & 1 \nEducation → Herding\\_Bias & 0.05 \nEducation → Income & 0.99 \nEducation → Invest & 0.95 \nEducation → Myopic\\_Bias & 0.16 \nEducation → Planning & 0.04 \nFin\\_Literacy → Asset\\_Amt & 0.64 \nFin\\_Literacy ∘ → Asset\\_Amt & 0.05 \nFin\\_Literacy ↔ Asset\\_Amt & 0.05 \nFin\\_Literacy → Herding\\_Bias & 0.62 \nFin\\_Literacy → Myopic\\_Bias & 0.3 \nHerding\\_Bias ↔ Confidence & 0.27 \nHerding\\_Bias ↔ Fin\\_Literacy & 0.37 \nHerding\\_Bias ∘ → Fin\\_Literacy & 0.01 \nHerding\\_Bias ↔ Invest & 0.99 \nHerding\\_Bias ↔ Myopic\\_Bias & 0.12 \nHerding\\_Bias ↔ Planning & 0.01 \nIncome → Asset\\_Amt & 0.72 \nIncome ∘ → Asset\\_Amt & 0.05 \nIncome ↔ Asset\\_Amt & 0.01 \nIncome → Confidence & 0.26 \nIncome ↔ Invest & 0.22 \nIncome → Invest & 0.15 \nIncome ∘ → Invest & 0.08 \nIncome → Myopic\\_Bias & 0.01 \nIncome → Planning & 0.44 \nIncome ↔ Planning & 0.39 \nIncome ∘ → Planning & 0.01 \nInvest → Asset\\_Amt & 0.01 \nInvest → Confidence & 0.48 \nInvest → Planning & 0.04 \nMyopic\\_Bias ↔ Asset\\_Amt & 0.25 \nMyopic\\_Bias ∘ → Asset\\_Amt & 0.14 \nMyopic\\_Bias → Asset\\_Amt & 0.13 \nMyopic\\_Bias ↔ Fin\\_Literacy & 0.35 \nMyopic\\_Bias ∘ → Fin\\_Literacy & 0.3 \nMyopic\\_Bias → Fin\\_Literacy & 0.03 \nMyopic\\_Bias ∘ – ∘ Fin\\_Literacy & 0.02 \nMyopic\\_Bias → Herding\\_Bias & 0.05 \nMyopic\\_Bias ∘ → Herding\\_Bias & 0.02 \nMyopic\\_Bias ↔ Income & 0.35 \nMyopic\\_Bias ∘ → Income & 0.04 \nMyopic\\_Bias → Income & 0.02 \nPlanning ∘ → Asset\\_Amt & 0.06 \nPlanning → Asset\\_Amt & 0.02 \nPlanning → Confidence & 0.22 \nPlanning ∘ → Confidence & 0.01 \nPlanning → Income & 0.12 \nPlanning ∘ → Income & 0.02 \nPlanning ↔ Invest & 0.37 \nPlanning → Invest & 0.03 \n|p5cm|p1cm| Age → Asset\\_Amt & 1 \nAge → Fin\\_Literacy & 1 \nAge → Herding\\_Bias & 0.45 \nAge → Myopic\\_Bias & 0.09 \nAge → Planning & 0.98 \nAsset\\_Amt → Confidence & 0.06 \nAsset\\_Amt → Fin\\_Literacy & 0.62 \nAsset\\_Amt → Herding\\_Bias & 0.02 \nAsset\\_Amt → Income & 0.03 \nAsset\\_Amt ↔ Invest & 0.1 \nAsset\\_Amt → Planning & 0.13 \nAsset\\_Amt ↔ Planning & 0.13 \nConfidence ∘ → Asset\\_Amt & 0.31 \nConfidence ↔ Asset\\_Amt & 0.2 \nConfidence → Asset\\_Amt & 0.13 \nConfidence ↔ Fin\\_Literacy & 0.08 \nConfidence ∘ → Fin\\_Literacy & 0.04 \nConfidence → Fin\\_Literacy & 0.02 \nConfidence ∘ → Income & 0.03 \nConfidence ∘ – ∘ Income & 0.02 \nConfidence ↔ Invest & 0.37 \nConfidence ∘ → Invest & 0.36 \nConfidence → Invest & 0.19 \nConfidence ∘ – ∘ Invest & 0.02 \nConfidence ↔ Planning & 0.07 \nConfidence → Planning & 0.02 \nConfidence ∘ → Planning & 0.01 \nEducation → Asset\\_Amt & 0.03 \nEducation → Confidence & 0.41 \nEducation → Fin\\_Literacy & 0.75 \nEducation → Income & 0.89 \nEducation → Invest & 0.05 \nEducation → Myopic\\_Bias & 0.05 \nEducation → Planning & 0.04 \nFin\\_Literacy → Asset\\_Amt & 0.24 \nFin\\_Literacy ↔ Asset\\_Amt & 0.06 \nFin\\_Literacy → Confidence & 0.03 \nFin\\_Literacy → Herding\\_Bias & 0.27 \nFin\\_Literacy ↔ Invest & 0.72 \nFin\\_Literacy → Invest & 0.01 \nFin\\_Literacy ↔ Planning & 0.5 \nFin\\_Literacy → Planning & 0.07 \nHerding\\_Bias ↔ Asset\\_Amt & 0.11 \nHerding\\_Bias ∘ → Asset\\_Amt & 0.07 \nHerding\\_Bias ∘ → Fin\\_Literacy & 0.46 \nHerding\\_Bias → Fin\\_Literacy & 0.2 \nHerding\\_Bias ↔ Fin\\_Literacy & 0.07 \nHerding\\_Bias ↔ Invest & 0.36 \nHerding\\_Bias ∘ → Invest & 0.11 \nHerding\\_Bias → Invest & 0.1 \nHerding\\_Bias → Myopic\\_Bias & 0.03 \nHerding\\_Bias ↔ Myopic\\_Bias & 0.02 \nHerding\\_Bias ∘ → Myopic\\_Bias & 0.02 \nHerding\\_Bias ∘ – ∘ Myopic\\_Bias & 0.02 \nIncome → Asset\\_Amt & 0.91 \nIncome ↔ Asset\\_Amt & 0.05 \nIncome ∘ → Asset\\_Amt & 0.01 \nIncome → Confidence & 0.08 \nIncome ↔ Invest & 0.68 \nIncome → Invest & 0.12 \nIncome ∘ → Invest & 0.02 \nIncome ↔ Planning & 0.67 \nIncome → Planning & 0.16 \nInvest → Asset\\_Amt & 0.06 \nInvest → Confidence & 0.06 \nInvest → Fin\\_Literacy & 0.01 \nInvest ∘ → Income & 0.02 \nMyopic\\_Bias ∘ → Asset\\_Amt & 0.37 \nMyopic\\_Bias ↔ Asset\\_Amt & 0.16 \nMyopic\\_Bias → Asset\\_Amt & 0.08 \nMyopic\\_Bias ↔ Confidence & 0.04 \nMyopic\\_Bias ∘ → Confidence & 0.01 \nMyopic\\_Bias ∘ → Fin\\_Literacy & 0.29 \nMyopic\\_Bias ↔ Fin\\_Literacy & 0.1 \nMyopic\\_Bias → Fin\\_Literacy & 0.07 \nMyopic\\_Bias ∘ → Herding\\_Bias & 0.04 \nMyopic\\_Bias ↔ Income & 0.01 \nMyopic\\_Bias ↔ Invest & 0.02 \nMyopic\\_Bias ∘ → Invest & 0.01 \nMyopic\\_Bias ∘ → Planning & 0.04 \nMyopic\\_Bias ↔ Planning & 0.02 \nPlanning → Asset\\_Amt & 0.2 \nPlanning ∘ → Asset\\_Amt & 0.02 \nPlanning → Confidence & 0.01 \nPlanning → Fin\\_Literacy & 0.12 \nPlanning ∘ → Fin\\_Literacy & 0.07 \nPlanning → Income & 0.02 \nPlanning ↔ Invest & 0.01 \nPlanning → Myopic\\_Bias & 0.01 \n|p5cm|p1cm| Age → Asset\\_Amt & 1 \nAge → Fin\\_Literacy & 1 \nAge → Herding\\_Bias & 1 \nAge → Myopic\\_Bias & 1 \nAge → Planning & 1 \nAsset\\_Amt → Confidence & 0.23 \nAsset\\_Amt ∘ → Fin\\_Literacy & 0.18 \nAsset\\_Amt → Fin\\_Literacy & 0.15 \nAsset\\_Amt ↔ Invest & 0.58 \nAsset\\_Amt → Invest & 0.04 \nAsset\\_Amt → Myopic\\_Bias & 0.66 \nAsset\\_Amt → Planning & 0.43 \nAsset\\_Amt ∘ → Planning & 0.02 \nAsset\\_Amt ↔ Planning & 0.01 \nConfidence ↔ Asset\\_Amt & 0.46 \nConfidence → Asset\\_Amt & 0.31 \nConfidence ∘ → Fin\\_Literacy & 0.35 \nConfidence ↔ Fin\\_Literacy & 0.24 \nConfidence → Fin\\_Literacy & 0.16 \nConfidence ∘ – ∘ Income & 0.18 \nConfidence ↔ Income & 0.18 \nConfidence → Income & 0.14 \nConfidence ∘ → Income & 0.01 \nConfidence → Invest & 0.42 \nConfidence ∘ – ∘ Invest & 0.14 \nConfidence ∘ → Invest & 0.1 \nConfidence ↔ Invest & 0.09 \nConfidence → Myopic\\_Bias & 0.05 \nConfidence ∘ → Myopic\\_Bias & 0.01 \nConfidence ↔ Planning & 0.48 \nConfidence → Planning & 0.44 \nConfidence ∘ → Planning & 0.01 \nEducation → Asset\\_Amt & 0.99 \nEducation → Confidence & 1 \nEducation → Fin\\_Literacy & 1 \nEducation → Income & 1 \nEducation → Invest & 1 \nEducation → Myopic\\_Bias & 1 \nEducation → Planning & 0.11 \nFin\\_Literacy → Asset\\_Amt & 0.59 \nFin\\_Literacy ↔ Asset\\_Amt & 0.08 \nFin\\_Literacy → Confidence & 0.25 \nFin\\_Literacy ↔ Income & 0.34 \nFin\\_Literacy ↔ Invest & 0.4 \nFin\\_Literacy → Invest & 0.07 \nFin\\_Literacy → Myopic\\_Bias & 0.06 \nFin\\_Literacy ↔ Planning & 0.42 \nFin\\_Literacy → Planning & 0.09 \nHerding\\_Bias ↔ Confidence & 0.36 \nHerding\\_Bias → Confidence & 0.06 \nHerding\\_Bias ∘ → Fin\\_Literacy & 0.65 \nHerding\\_Bias ↔ Fin\\_Literacy & 0.31 \nHerding\\_Bias → Fin\\_Literacy & 0.04 \nHerding\\_Bias ↔ Income & 0.19 \nHerding\\_Bias ↔ Invest & 0.08 \nIncome → Asset\\_Amt & 0.45 \nIncome ↔ Asset\\_Amt & 0.38 \nIncome ∘ → Asset\\_Amt & 0.17 \nIncome → Confidence & 0.34 \nIncome ∘ → Confidence & 0.15 \nIncome ∘ → Fin\\_Literacy & 0.5 \nIncome → Fin\\_Literacy & 0.11 \nIncome ↔ Invest & 0.15 \nIncome → Invest & 0.15 \nIncome ∘ – ∘ Invest & 0.12 \nIncome ∘ → Invest & 0.08 \nIncome ∘ → Myopic\\_Bias & 0.01 \nIncome → Planning & 0.55 \nIncome ↔ Planning & 0.34 \nIncome ∘ → Planning & 0.03 \nInvest → Asset\\_Amt & 0.28 \nInvest ∘ → Asset\\_Amt & 0.1 \nInvest → Confidence & 0.18 \nInvest ∘ → Confidence & 0.07 \nInvest ∘ → Fin\\_Literacy & 0.41 \nInvest → Fin\\_Literacy & 0.12 \nInvest ∘ → Income & 0.07 \nInvest → Income & 0.04 \nInvest ∘ → Myopic\\_Bias & 0.01 \nInvest → Planning & 0.31 \nInvest ∘ → Planning & 0.01 \nMyopic\\_Bias → Asset\\_Amt & 0.34 \nMyopic\\_Bias ↔ Confidence & 0.25 \nMyopic\\_Bias → Confidence & 0.06 \nMyopic\\_Bias ∘ → Fin\\_Literacy & 0.4 \nMyopic\\_Bias ↔ Fin\\_Literacy & 0.09 \nMyopic\\_Bias ↔ Income & 0.11 \nMyopic\\_Bias ↔ Invest & 0.01 \nMyopic\\_Bias ↔ Planning & 0.09 \nMyopic\\_Bias → Planning & 0.04 \nPlanning → Asset\\_Amt & 0.51 \nPlanning ∘ → Asset\\_Amt & 0.03 \nPlanning → Confidence & 0.07 \nPlanning ∘ → Fin\\_Literacy & 0.35 \nPlanning → Fin\\_Literacy & 0.14 \nPlanning → Income & 0.08 \nPlanning ↔ Invest & 0.54 \nPlanning → Invest & 0.03 \nPlanning → Myopic\\_Bias & 0.03 \n|p5cm|p1cm| Age → Asset\\_Amt & 1 \nAge → Confidence & 0.01 \nAge → Fin\\_Literacy & 0.19 \nAge → Income & 0.02 \nAge → Invest & 0.33 \nAge → Myopic\\_Bias & 0.14 \nAge → Planning & 0.49 \nAsset\\_Amt → Confidence & 0.08 \nAsset\\_Amt → Fin\\_Literacy & 0.19 \nAsset\\_Amt ∘ → Fin\\_Literacy & 0.01 \nAsset\\_Amt → Income & 0.62 \nAsset\\_Amt → Invest & 0.63 \nAsset\\_Amt ↔ Invest & 0.13 \nAsset\\_Amt ∘ → Invest & 0.08 \nAsset\\_Amt ∘ – ∘ Invest & 0.05 \nConfidence → Asset\\_Amt & 0.02 \nConfidence ↔ Asset\\_Amt & 0.02 \nConfidence → Fin\\_Literacy & 0.13 \nConfidence ↔ Fin\\_Literacy & 0.08 \nConfidence ∘ → Fin\\_Literacy & 0.07 \nConfidence → Herding\\_Bias & 0.06 \nConfidence ∘ → Herding\\_Bias & 0.03 \nConfidence ↔ Income & 0.01 \nConfidence ∘ → Invest & 0.16 \nConfidence ↔ Invest & 0.07 \nConfidence → Invest & 0.07 \nConfidence ∘ – ∘ Invest & 0.01 \nConfidence → Planning & 0.07 \nConfidence ↔ Planning & 0.07 \nConfidence ∘ → Planning & 0.01 \nEducation → Confidence & 0.04 \nEducation → Fin\\_Literacy & 0.04 \nEducation → Herding\\_Bias & 0.03 \nEducation → Income & 0.63 \nEducation → Myopic\\_Bias & 0.12 \nFin\\_Literacy ↔ Asset\\_Amt & 0.3 \nFin\\_Literacy → Asset\\_Amt & 0.01 \nFin\\_Literacy ∘ → Asset\\_Amt & 0.01 \nFin\\_Literacy → Confidence & 0.02 \nFin\\_Literacy → Myopic\\_Bias & 0.01 \nFin\\_Literacy ↔ Planning & 0.29 \nHerding\\_Bias ↔ Confidence & 0.18 \nHerding\\_Bias → Confidence & 0.01 \nHerding\\_Bias ↔ Fin\\_Literacy & 0.82 \nHerding\\_Bias ∘ → Fin\\_Literacy & 0.13 \nHerding\\_Bias ↔ Income & 0.05 \nHerding\\_Bias ↔ Invest & 0.67 \nHerding\\_Bias ∘ → Invest & 0.04 \nIncome ↔ Asset\\_Amt & 0.25 \nIncome → Asset\\_Amt & 0.07 \nIncome ∘ → Asset\\_Amt & 0.06 \nIncome ↔ Invest & 0.02 \nIncome → Invest & 0.02 \nIncome → Myopic\\_Bias & 0.02 \nIncome ↔ Planning & 0.3 \nIncome → Planning & 0.21 \nIncome ∘ → Planning & 0.03 \nInvest → Asset\\_Amt & 0.08 \nInvest ∘ → Asset\\_Amt & 0.01 \nInvest → Confidence & 0.68 \nInvest ∘ → Confidence & 0.01 \nInvest → Fin\\_Literacy & 0.02 \nInvest → Herding\\_Bias & 0.02 \nInvest ∘ → Income & 0.01 \nInvest → Income & 0.01 \nMyopic\\_Bias ↔ Confidence & 0.02 \nMyopic\\_Bias ∘ → Confidence & 0.01 \nMyopic\\_Bias ∘ → Fin\\_Literacy & 0.63 \nMyopic\\_Bias ↔ Fin\\_Literacy & 0.32 \nMyopic\\_Bias → Fin\\_Literacy & 0.02 \nMyopic\\_Bias ∘ – ∘ Fin\\_Literacy & 0.01 \nMyopic\\_Bias ∘ → Herding\\_Bias & 0.01 \nMyopic\\_Bias ↔ Income & 0.09 \nMyopic\\_Bias ∘ → Income & 0.03 \nMyopic\\_Bias ↔ Invest & 0.03 \nMyopic\\_Bias ∘ → Invest & 0.02 \nMyopic\\_Bias ↔ Planning & 0.02 \nMyopic\\_Bias ∘ → Planning & 0.01 \nPlanning ∘ → Confidence & 0.02 \nPlanning → Confidence & 0.01 \nPlanning → Fin\\_Literacy & 0.1 \nPlanning ∘ → Fin\\_Literacy & 0.07 \nPlanning ∘ → Income & 0.17 \n|p5cm|p1cm| Age → Asset\\_Amt & 1 \nAge → Confidence & 0.01 \nAge → Fin\\_Literacy & 0.97 \nAge → Herding\\_Bias & 0.1 \nAge → Income & 0.02 \nAge → Invest & 0.95 \nAge → Myopic\\_Bias & 0.97 \nAsset\\_Amt → Confidence & 0.52 \nAsset\\_Amt → Fin\\_Literacy & 0.46 \nAsset\\_Amt ∘ → Fin\\_Literacy & 0.26 \nAsset\\_Amt → Invest & 0.72 \nAsset\\_Amt ↔ Invest & 0.04 \nAsset\\_Amt → Myopic\\_Bias & 0.08 \nAsset\\_Amt → Planning & 0.05 \nConfidence → Asset\\_Amt & 0.22 \nConfidence ↔ Asset\\_Amt & 0.19 \nConfidence ∘ → Asset\\_Amt & 0.05 \nConfidence ∘ → Fin\\_Literacy & 0.43 \nConfidence ↔ Fin\\_Literacy & 0.3 \nConfidence → Fin\\_Literacy & 0.16 \nConfidence ↔ Income & 0.08 \nConfidence → Invest & 0.4 \nConfidence ↔ Invest & 0.14 \nConfidence ∘ → Invest & 0.09 \nConfidence ∘ – ∘ Invest & 0.01 \nConfidence → Planning & 0.38 \nConfidence ↔ Planning & 0.26 \nConfidence ∘ – ∘ Planning & 0.02 \nConfidence ∘ → Planning & 0.01 \nEducation → Confidence & 0.15 \nEducation → Fin\\_Literacy & 0.99 \nEducation → Herding\\_Bias & 0.06 \nEducation → Income & 0.96 \nEducation → Invest & 0.59 \nEducation → Myopic\\_Bias & 0.16 \nEducation → Planning & 0.13 \nFin\\_Literacy → Asset\\_Amt & 0.25 \nFin\\_Literacy → Confidence & 0.01 \nFin\\_Literacy ↔ Income & 0.02 \nFin\\_Literacy ↔ Invest & 0.17 \nFin\\_Literacy → Invest & 0.02 \nFin\\_Literacy → Myopic\\_Bias & 0.13 \nFin\\_Literacy → Planning & 0.15 \nFin\\_Literacy ↔ Planning & 0.09 \nHerding\\_Bias ∘ → Fin\\_Literacy & 0.94 \nHerding\\_Bias ↔ Fin\\_Literacy & 0.06 \nHerding\\_Bias ↔ Income & 0.02 \nHerding\\_Bias ↔ Invest & 0.03 \nHerding\\_Bias ∘ → Invest & 0.02 \nHerding\\_Bias → Invest & 0.01 \nHerding\\_Bias ∘ → Myopic\\_Bias & 0.06 \nHerding\\_Bias ↔ Myopic\\_Bias & 0.02 \nHerding\\_Bias → Myopic\\_Bias & 0.01 \nHerding\\_Bias ↔ Planning & 0.01 \nIncome → Asset\\_Amt & 0.91 \nIncome ↔ Asset\\_Amt & 0.08 \nIncome ∘ → Asset\\_Amt & 0.01 \nIncome → Confidence & 0.09 \nIncome ∘ → Fin\\_Literacy & 0.4 \nIncome → Fin\\_Literacy & 0.03 \nIncome → Invest & 0.02 \nIncome ∘ → Myopic\\_Bias & 0.02 \nIncome → Myopic\\_Bias & 0.01 \nIncome → Planning & 0.69 \nIncome ↔ Planning & 0.09 \nIncome ∘ – ∘ Planning & 0.05 \nIncome ∘ → Planning & 0.05 \nInvest → Asset\\_Amt & 0.24 \nInvest → Confidence & 0.35 \nInvest ∘ → Confidence & 0.01 \nInvest ∘ → Fin\\_Literacy & 0.66 \nInvest → Fin\\_Literacy & 0.12 \nInvest → Planning & 0.09 \nMyopic\\_Bias ∘ → Asset\\_Amt & 0.09 \nMyopic\\_Bias ↔ Asset\\_Amt & 0.08 \nMyopic\\_Bias → Asset\\_Amt & 0.08 \nMyopic\\_Bias ∘ → Fin\\_Literacy & 0.6 \nMyopic\\_Bias ↔ Fin\\_Literacy & 0.16 \nMyopic\\_Bias → Fin\\_Literacy & 0.11 \nMyopic\\_Bias ↔ Income & 0.12 \nMyopic\\_Bias ↔ Planning & 0.12 \nPlanning → Confidence & 0.28 \nPlanning ∘ → Confidence & 0.02 \nPlanning ∘ → Fin\\_Literacy & 0.38 \nPlanning → Fin\\_Literacy & 0.37 \nPlanning → Income & 0.08 \nPlanning ∘ → Income & 0.04 \nPlanning ↔ Invest & 0.05 \nPlanning → Invest & 0.02 \nPlanning → Myopic\\_Bias & 0.01 \n|p5cm|p1cm| Age → Asset\\_Amt & 1 \nAge → Confidence & 0.17 \nAge → Fin\\_Literacy & 0.24 \nAge → Herding\\_Bias & 0.08 \nAge → Invest & 0.12 \nAge → Planning & 0.33 \nAsset\\_Amt → Confidence & 0.09 \nAsset\\_Amt → Fin\\_Literacy & 0.19 \nAsset\\_Amt → Income & 0.34 \nAsset\\_Amt → Invest & 0.24 \nAsset\\_Amt ↔ Invest & 0.08 \nAsset\\_Amt → Planning & 0.33 \nAsset\\_Amt ∘ → Planning & 0.02 \nAsset\\_Amt ∘ – ∘ Planning & 0.01 \nAsset\\_Amt ↔ Planning & 0.01 \nConfidence ∘ → Asset\\_Amt & 0.03 \nConfidence → Asset\\_Amt & 0.03 \nConfidence ↔ Fin\\_Literacy & 0.3 \nConfidence → Fin\\_Literacy & 0.01 \nConfidence → Herding\\_Bias & 0.01 \nConfidence ↔ Income & 0.1 \nConfidence ∘ → Income & 0.04 \nConfidence → Income & 0.01 \nConfidence ∘ – ∘ Invest & 0.2 \nConfidence → Invest & 0.16 \nConfidence ↔ Invest & 0.15 \nConfidence ∘ → Invest & 0.11 \nConfidence ∘ → Planning & 0.01 \nEducation → Asset\\_Amt & 0.01 \nEducation → Confidence & 0.01 \nEducation → Fin\\_Literacy & 0.2 \nEducation → Herding\\_Bias & 0.06 \nEducation → Income & 0.78 \nEducation → Myopic\\_Bias & 0.64 \nEducation → Planning & 0.12 \nFin\\_Literacy ↔ Asset\\_Amt & 0.3 \nFin\\_Literacy → Asset\\_Amt & 0.1 \nFin\\_Literacy ∘ → Asset\\_Amt & 0.02 \nFin\\_Literacy ↔ Invest & 0.23 \nFin\\_Literacy → Myopic\\_Bias & 0.04 \nFin\\_Literacy ∘ → Myopic\\_Bias & 0.01 \nFin\\_Literacy ↔ Planning & 0.25 \nFin\\_Literacy → Planning & 0.13 \nFin\\_Literacy ∘ → Planning & 0.03 \nHerding\\_Bias ↔ Confidence & 0.03 \nHerding\\_Bias ∘ → Confidence & 0.01 \nHerding\\_Bias ∘ → Fin\\_Literacy & 0.49 \nHerding\\_Bias ↔ Fin\\_Literacy & 0.13 \nHerding\\_Bias → Fin\\_Literacy & 0.05 \nHerding\\_Bias ∘ – ∘ Fin\\_Literacy & 0.02 \nHerding\\_Bias ↔ Income & 0.01 \nHerding\\_Bias ↔ Invest & 0.04 \nHerding\\_Bias ∘ → Invest & 0.01 \nHerding\\_Bias ↔ Myopic\\_Bias & 0.02 \nHerding\\_Bias ∘ → Myopic\\_Bias & 0.01 \nHerding\\_Bias ∘ – ∘ Planning & 0.01 \nIncome → Asset\\_Amt & 0.6 \nIncome ↔ Asset\\_Amt & 0.03 \nIncome ∘ → Asset\\_Amt & 0.03 \nIncome → Confidence & 0.01 \nIncome ↔ Invest & 0.02 \nIncome → Myopic\\_Bias & 0.2 \nIncome ∘ → Myopic\\_Bias & 0.05 \nIncome → Planning & 0.26 \nIncome ↔ Planning & 0.07 \nIncome ∘ → Planning & 0.01 \nInvest ∘ → Asset\\_Amt & 0.28 \nInvest → Asset\\_Amt & 0.09 \nInvest → Confidence & 0.2 \nInvest ∘ → Confidence & 0.18 \nInvest ∘ → Herding\\_Bias & 0.01 \nInvest ∘ → Income & 0.03 \nInvest → Income & 0.01 \nInvest ∘ → Myopic\\_Bias & 0.01 \nInvest → Planning & 0.02 \nMyopic\\_Bias ↔ Asset\\_Amt & 0.03 \nMyopic\\_Bias ∘ → Asset\\_Amt & 0.02 \nMyopic\\_Bias → Confidence & 0.01 \nMyopic\\_Bias ↔ Confidence & 0.01 \nMyopic\\_Bias → Fin\\_Literacy & 0.36 \nMyopic\\_Bias ↔ Fin\\_Literacy & 0.32 \nMyopic\\_Bias ∘ → Fin\\_Literacy & 0.23 \nMyopic\\_Bias ∘ – ∘ Fin\\_Literacy & 0.02 \nMyopic\\_Bias ↔ Income & 0.11 \nMyopic\\_Bias ∘ → Income & 0.1 \nMyopic\\_Bias ∘ – ∘ Income & 0.06 \nMyopic\\_Bias → Income & 0.05 \nMyopic\\_Bias → Planning & 0.03 \nMyopic\\_Bias ↔ Planning & 0.01 \nPlanning ∘ → Asset\\_Amt & 0.1 \nPlanning → Asset\\_Amt & 0.08 \nPlanning → Confidence & 0.01 \nPlanning → Fin\\_Literacy & 0.27 \nPlanning ∘ → Fin\\_Literacy & 0.19 \nPlanning ∘ → Income & 0.06 \nPlanning → Income & 0.01 \nPlanning → Myopic\\_Bias & 0.01 \nPlanning ∘ → Myopic\\_Bias & 0.01 \n|p5cm|p1cm| Age → Asset\\_Amt & 1 \nAge → Fin\\_Literacy & 1 \nAge → Herding\\_Bias & 0.89 \nAge → Invest & 0.88 \nAge → Myopic\\_Bias & 1 \nAsset\\_Amt → Confidence & 0.47 \nAsset\\_Amt → Fin\\_Literacy & 0.77 \nAsset\\_Amt → Invest & 0.7 \nAsset\\_Amt ↔ Invest & 0.05 \nAsset\\_Amt ∘ → Myopic\\_Bias & 0.28 \nAsset\\_Amt → Myopic\\_Bias & 0.25 \nAsset\\_Amt → Planning & 0.49 \nAsset\\_Amt ↔ Planning & 0.24 \nConfidence → Asset\\_Amt & 0.4 \nConfidence ↔ Asset\\_Amt & 0.13 \nConfidence → Fin\\_Literacy & 0.32 \nConfidence ∘ → Fin\\_Literacy & 0.31 \nConfidence ↔ Fin\\_Literacy & 0.29 \nConfidence ↔ Income & 0.22 \nConfidence → Income & 0.02 \nConfidence ∘ → Income & 0.01 \nConfidence → Invest & 0.4 \nConfidence ↔ Invest & 0.23 \nConfidence ∘ → Invest & 0.05 \nConfidence ∘ – ∘ Invest & 0.01 \nConfidence ↔ Planning & 0.39 \nConfidence → Planning & 0.3 \nEducation → Confidence & 0.99 \nEducation → Fin\\_Literacy & 1 \nEducation → Income & 1 \nEducation → Invest & 0.81 \nEducation → Myopic\\_Bias & 1 \nFin\\_Literacy → Asset\\_Amt & 0.23 \nFin\\_Literacy → Confidence & 0.08 \nFin\\_Literacy ↔ Invest & 0.31 \nFin\\_Literacy → Invest & 0.01 \nFin\\_Literacy → Myopic\\_Bias & 0.04 \nFin\\_Literacy ↔ Planning & 0.13 \nFin\\_Literacy → Planning & 0.01 \nHerding\\_Bias ↔ Confidence & 0.2 \nHerding\\_Bias → Confidence & 0.01 \nHerding\\_Bias ∘ → Confidence & 0.01 \nHerding\\_Bias ↔ Fin\\_Literacy & 0.54 \nHerding\\_Bias ∘ → Fin\\_Literacy & 0.46 \nHerding\\_Bias ↔ Income & 0.47 \nHerding\\_Bias → Invest & 0.01 \nIncome → Asset\\_Amt & 0.52 \nIncome ↔ Asset\\_Amt & 0.48 \nIncome ∘ → Confidence & 0.11 \nIncome → Confidence & 0.01 \nIncome ∘ → Fin\\_Literacy & 0.03 \nIncome ↔ Invest & 0.01 \nIncome ∘ → Myopic\\_Bias & 0.28 \nIncome → Myopic\\_Bias & 0.07 \nIncome → Planning & 0.52 \nIncome ↔ Planning & 0.34 \nInvest → Asset\\_Amt & 0.25 \nInvest → Confidence & 0.3 \nInvest ∘ → Confidence & 0.01 \nInvest ∘ → Fin\\_Literacy & 0.62 \nInvest → Fin\\_Literacy & 0.06 \nMyopic\\_Bias → Asset\\_Amt & 0.25 \nMyopic\\_Bias ↔ Asset\\_Amt & 0.21 \nMyopic\\_Bias ↔ Confidence & 0.01 \nMyopic\\_Bias → Fin\\_Literacy & 0.47 \nMyopic\\_Bias ↔ Fin\\_Literacy & 0.46 \nMyopic\\_Bias ∘ → Fin\\_Literacy & 0.03 \nMyopic\\_Bias ↔ Income & 0.52 \nMyopic\\_Bias → Income & 0.01 \nMyopic\\_Bias ↔ Planning & 0.63 \nPlanning → Asset\\_Amt & 0.21 \nPlanning ∘ → Asset\\_Amt & 0.06 \nPlanning → Confidence & 0.26 \nPlanning ∘ → Confidence & 0.05 \nPlanning → Fin\\_Literacy & 0.81 \nPlanning ∘ → Fin\\_Literacy & 0.05 \nPlanning → Income & 0.12 \nPlanning ∘ → Income & 0.02 \nPlanning ↔ Invest & 0.16 \nPlanning → Myopic\\_Bias & 0.23 \nPlanning ∘ → Myopic\\_Bias & 0.09 \n---\n1. Acknowledgements: This research utilized individual-level data from the “Financial Literacy Survey,” which was graciously provided by the Central Council for Financial Services Information (Secretariat: Public Relations Department, Bank of Japan). This work was partially supported by JSPS KAKENHI 20K11708.[↩](#fnref1)\n2. The questions and options can be found on page 38 – page 51 of <https://www.shiruporuto.jp/e/survey/kinyulite/pdf/22kinyulite.pdf>[↩](#fnref2)\n3. See <https://causal-learn.readthedocs.io/en/latest/>[↩](#fnref3)\n\nDoes Financial Literacy Impact Investment Participation and Retirement Planning in Japan?[1](#fn1)\n==================================================================================================\n**Abstract**\n**Keywords:** financial literacy, investment participation, retirement planning, causal discovery, Fast Causal Inference\nIntroduction\n============\nIn 2019, Japan’s Financial Services Agency (FSA) published a panel report titled “Asset Formation and Management in an Aging Society.” The report highlighted that an average older couple would need approximately 20 million yen for a 30-year post-retirement life, leading to a monthly deficit of around 50,000 yen not covered by public pension programs. This revelation garnered widespread media attention and sparked significant public concern regarding the financial asset formation of households in Japan. Three years later, in 2022, the Council of New Form of Capitalism Realization, established in the Cabinet Secretariat of Japan, formally adopted the “Doubling of Asset-based Incomes Plan.” This initiative aims to double the number of Nippon Individual Savings Account (NISA) holders and cumulative purchase amounts within five years. This plan is designed to encourage Japanese households to diversify their savings into risky assets, including stocks and mutual funds, which is viewed as crucial for boosting overall economic growth and enhancing personal financial well-being.\nThe FSA has implemented various measures to promote the transition from savings to investment over the past 20 years. For example, from 2003 to 2013, preferential tax treatment for securities investment allowed that the tax rate on dividends and capital gains from stock or investment trusts was reduced from 20% to 10%. In 2014, the NISA was initiated as an investment tax-break scheme. In 2018, an installment-type NISA was launched, drawing inspiration from the investment strategies advocated by Burton Malkiel and Charles Ellis, renowned economists in the asset management field. Despite these initiatives, the impact appears to be less dramatic than anticipated. As illustrated in the “Flow of Funds for the Fourth Quarter of 2023” report by the Bank of Japan (BOJ), the proportion of each type of asset has remained relatively stable over the last two decades. At the end of December 2023, the share of “currency and deposits” in the total outstanding financial assets held by households was still as high as 52.6%.\nDue to the prolonged ultra-low interest rate environment for Japanese Yen deposits, portfolios with a significant allocation to “currency and deposits” have experienced a notable disparity in total financial asset growth. According to the data published by FSA in 2019, in the 20-year period starting from 1998, the total amount of household financial assets in the United States (U.S.) and the United Kingdom (U.K.) expanded to 2.7 and 2.3 times their original values, while Japan’s growth was limited to only 1.4 times during the same period. This discrepancy is attributed significantly to the differences in investment returns among these countries.\nIn Japan, characterized by a rapidly aging population and a pay-as-you-go public pension system, the emphasis on financial asset accumulation and proactive retirement planning is becoming increasingly critical. Various academic studies and governmental surveys have been undertaken to elucidate the reasons behind the reluctance of Japanese households to invest in risky assets. With this context, the role of financial literacy, defined as individuals’ capacity to process information and make judicious decisions about financial planning, asset accumulation, and pension-related considerations, has emerged as a focal point. The Central Council for Financial Services Information, an organization of the BOJ responsible for conducting financial services information activities, has carried out nationwide large-scale questionnaire surveys, “Financial Literacy Survey,” in 2016, 2019, and 2022, aiming to assess the current state of financial literacy of individuals. Common selection questions were included in the questionnaire, and the percentage of correct answers on financial knowledge in Japan was lower compared to the U.S. and some other Organization for Economic Co-operation and Development (OECD) countries.\n developed three questions as a benchmark for measuring financial literacy, based on economic models of saving and portfolio choice with the focus on three key economic concepts vital for financial decision-making: the understanding of interest compounding, inflation, and risk diversification. The design of the questions was guided by four fundamental principles: simplicity, relevance, brevity, and capacity to differentiate. This guideline was applied in the Health and Retirement Study conducted in the U.S. in 2004. The same set of questions was also incorporated into the “Financial Literacy Survey” executed by the BOJ.\nPrior studies utilizing diverse survey datasets have explored the relationship between financial literacy and various financial activities. These investigations have elucidated that enhanced financial literacy contributes to more effective retirement planning and encourages increased investment in complex financial assets. Based on these research findings and the suboptimal financial literacy level among Japanese households, as revealed by the survey results, Japan revised its national school curriculum guidelines in 2022. This update mandates a heightened focus on personal finance education for high school students, with the objective of raising financial literacy within this young demographic.\nWhile existing research on financial literacy has predominately identified positive correlational relationships with financial activities, employing causal discovery methods offers the potential to draw conclusions about causality. However, no prior research has employed causal discovery methods to work on this topic, either within Japan or internationally. Therefore, this study adopts a causal discovery approach to investigate the causal connections between financial literacy and financial activities, including retirement planning and investment participation in risky financial assets in Japan. By doing so, this research not only sheds light on understanding the impact of Japan’s recent implementation of finance education in high schools but also provides a framework that can be applied to similar studies. Although our study primarily focuses on analyzing and interpreting data from Japan, the implications of applying the causal discovery approach also extend to other countries.\nLiterature Review\n=================\nOver the last two decades, financial literacy has emerged as a key area of study, with a focus on the relationship with an individual’s capability to make informed financial decisions. Previous studies posit that financial literacy is essential in enabling individuals to engage in financial investment and aiding in effective retirement planning.\n demonstrate that financial literacy significantly influences financial decision-making. Their analysis indicates that individuals with higher levels of financial literacy are more likely to participate in stock investment. This conclusion was derived from a regression analysis using the Ordinary Least Squares (OLS) method on data from the 2005 De Nederlandsche Banks Household Survey. applied OLS and probit models to a composite dataset from nine European countries: Austria, Belgium, Denmark, Germany, Italy, France, Switzerland, Sweden, and the Netherlands. Their study evaluated the influence of factors on the likelihood of stock market participation in 2010, affirming that financial literacy positively impacts this likelihood. The same conclusion is also applicable to Japan. conducted an online survey in 2019 on wealth building, securities, investment, and financial literacy in Japan. The analysis of this survey data, which included a straightforward regression model and an Instrumental Variables (IV) estimation, indicated that elevated financial literacy is generally linked to increased participation in the stock market.\nOther previous studies have focused on the association between financial literacy and retirement planning. analyzed data from the National Financial Capability Survey in the U.S., which is a comprehensive survey aimed at establishing benchmarks for key indicators of financial capability and associating these indicators with demographic, behavioral, attitudinal, and financial literacy characteristics. Utilizing OLS and an IV approach, the study suggests that financial literacy significantly influences retirement planning. This conclusion remains robust even when accounting for endogeneity and possible inaccuracies in measuring of financial literacy. leveraged the OLS and an IV estimation to analyze the January – February 2010 wave of the Survey of Living Preferences and Satisfaction. This research highlights the link between financial literacy and retirement planning, revealing that enhanced financial literacy increases the likelihood of holding a retirement savings plan.\nMost previous studies have consistently identified correlational associations related to financial literacy through regression analysis. Some of these studies have attempted to address specific types of endogeneity, such as reverse causality or selection bias, by adopting an IV approach . However, IV methods primarily concentrate on estimating specific causal relationships between selected sets of variables without consideration of latent confounding variables, which may not be sufficiently comprehensive to uncover the overarching causal mechanisms.\nData\n====\nThe “Financial Literacy Survey” is an extensive questionnaire survey to evaluate the current level of financial literacy among Japanese individuals. This online survey engaged 30,000 respondents aged between 18 and 79, selected to correspond closely with the demographic distribution in Japan.[2](#fn2) For this study, we obtained individual-level data from the 2022 survey from the Central Council for Financial Services Information (Secretariat: Public Relations Department of the BOJ). This survey is the third installment following the conduction in 2016 and 2019.\nWe used 13 variables from the survey data to represent the demographics, socioeconomic status, behavioral biases, financial activities, and the level of financial literacy of respondents. The list of variables is provided in [apd:third].\nFirst, we identified five exogenous variables from the survey data for demographics and socioeconomic status. \n(1) *Male*: Dummy Variable. Assigned a value of 1 if the response to “Q42. What is your gender?” is “Male.” \n(2) *Fin\\_Edu*: Dummy Variable. Set to 1 if the answer to “Q39. Was financial education offered by a school or college you attended, or a workplace where you were employed?” is “Yes, and I did participate in the financial education.” \n(3) *Fin\\_Edu\\_Home*: Dummy Variable. Assigned a value of 1 if the response to “Q40. Did your parents or guardians teach you how to manage your finances?” is “Yes.” \n(4) *Age*: Calculated as the mean of the age range indicated by the response to “Q43. What is your age?” For example, a response is “1,” denoting the age range of 18 – 19, results in an assigned value of “18.5.” The age ranges corresponding to each answer option are as follows: “2” represents 20 – 24, “3” for 25 – 29, up to “13” for 75 – 79. \n(5) *Education*: Determined by the total years required to complete the educational level indicated in “Q44. What is your educational background?” For instance, if “Primary and secondary schools only” is selected, then *Education* is set as “9,” and if “University” is selected, then *Education* is set as “16.” A missing value is assigned if the answer “Other” is selected.\nMoreover, we included two more variables to capture additional information about socioeconomic status. We incorporated a variable *Income* from the answer to “Q51. Which of these categories do your (your household) income for last year fall into?” We calculated the mean of the range for each category: a value of “0” for “Don’t have any financial assets,” “125” for “Less than 2.5 million yen,” “375” for “At least 2.5 million but less than 5 million yen,” “625” for “At least 5 million but less than 7.5 million yen,” “875” for “At least 7.5 million but less than 10 million yen,” and “1250” for “At least 10 million but less than 15 million yen.” For the answer “At least 15 million yen,” we assigned the lower limit value, “1500,” for *Income*. A missing value is set if the response is “Don’t know/Prefer not to say.” Similarly, we determined the variable *Asset\\_Amt* based on the answer to “Q52. Which of these categories do your (your household’s) financial assets (deposits, stocks, etc.) currently fall into?” The assigned values are “0” for “Don’t have any financial assets,” “125” for “Less than 2.5 million yen,” “375” for “At least 2.5 million but less than 5 million yen,” “625” for “At least 5 million but less than 7.5 million yen,” “875” for “At least 7.5 million but less than 10 million yen,” and “1500” for “At least 10 million but less than 20 million yen.” For “At least 20 million yen,” we assigned “2000” as the value of *Asset\\_Amt*. A missing value is set for the answer “Don’t know/Prefer not to say.” Variables for income and asset amount are considered crucial because they have been demonstrated to be related to the level of financial literacy .\nIn the 2011 report “National Financial Literacy Strategy,” the Australian Securities & Investments Commission identified six primary behavioral biases that may influence individuals’ decisions, including those related to complex financial matters. These biases are disengagement, overconfidence, loss aversion, myopic decision-making, mental accounting, and herd behavior. For myopic decision making, which involves prioritizing short-term gains without considering future consequences, we adopted the answer to “Q1\\_10. If I had the choice of (1) receiving 100,000 yen now or (2) receiving 110,000 yen in 1 year, I would choose (1), provided that I can definitely receive the money” as the indicator variable *Myopic\\_Bias*. Moreover, we identified herd behavior through *Herding\\_Bias*, based on the response to “Q1\\_3. When there are several similar products, I tend to buy what is recommended as the best-selling product, rather than what I actually think is a good product.” The answer options for Q1\\_10 and Q1\\_3, which range from “1” (Agree) to “5” (Disagree), employ the Likert scale, a psychometric scale commonly used in questionnaires. This scale, named after the social scientist Rensis Likert, allows respondents to indicate their level of agreement or disagreement with a statement by choosing from a specified range of options. In addition, we utilized the answer to “Q17. How would you rate your overall knowledge about financial matters compared with other people?” as a measure of respondents’ confidence in their financial literacy level (*Confidence*). The choices for Q17, ranging from “1” (Very high) to “5” (Very low), with “6” representing “Don’t know,” also follow the Likert scale format. A missing value is assigned to *Confidence* if option “6” is selected. The combination of objective financial literacy and subjective confidence can be used to measure the individuals’ overconfidence, and is reported to be positively correlated with stock market participation. Unfortunately, for the remaining three biases, we could not identify suitable corresponding questions within the survey.\nA variable *Invest* was constructed to qualify the number of risky asset types an individual has invested in, based on the responses to “Q34. Have you ever purchased any of the following financial products?” Risky assets are defined as stocks, investment trusts, and foreign currency deposits/money market funds. Another variable, *Planning*, was created to assess whether the individual is well prepared for retirement, according to the answers to a series of questions: “Q7. What expenses do you think you will have to cover in the future?,” “Q8\\_1. Are you aware of the amounts that will be required for living expenses for retirement in your case?,” “Q9\\_1. Do you have a financial plan for the expenses you think you will have to cover for living expenses for retirement?” and “Q10\\_1. Have you set aside funds for the expenses you think you will have to cover for living expenses for retirement?” If the option “1. Living expenses for retirement” is chosen in Q7, 1 is added to *Planning*. Additionally, for each “Yes” response to Q8\\_1, Q9\\_1, and Q10\\_1, *Planning* is incremented by 1. Therefore, *Planning* can range from “0” to “4,” where a higher value indicates more comprehensive planning and preparation for retirement.\nFinally, and of significant importance, *Fin\\_Literacy* represents the number of correct answers to the 25 questions designed to evaluate the individual’s financial literacy level, covering eight categories of the Financial Literacy Map. See [apd:second] for all the question numbers and corresponding categories. Although the BOJ does not publicly disclose the correct answer to each question, we deduced them from our background knowledge in finance. Moreover, the “Financial Literacy Survey: 2022 Results” include each option’s response ratio and the correct answer rate for every question. We compared the response ratios of the options deemed correct with the correct answer rates to ascertain the accuracy of our answers.\nApart from the three dummy variables – *Male*, *Fin\\_Edu*, *Fin\\_Edu\\_Home* – all other variables are treated as continuous. These continuous variables have been standardized before the subsequent analysis. We eliminated data entries with missing values for any of the variables, resulting in a reduction of the sample size from 30,000 to 19,333. This exclusion process could potentially introduce bias, particularly concerning the data entries for income and financial assets. The selection of the “Don’t know/Prefer not to say” option to these questions may not occur randomly. Instead, it is conceivable that individuals with either particularly low or high income (or financial assets) might tend to choose “Prefer not to say.”\nMethod\n======\nTo elucidate causal relationships, the traditional methods involve interventions or randomized controlled trials. However, these approaches are often too expensive, time-intensive, or impractical to implement for specific fields of study. For instance, it is hardly permissible to engage in randomized A/B tests concerning financial education and financial literacy, particularly from the perspective of social ethics and fairness. Considering these challenges, causal discovery methods are gaining increasing interest across various disciplines. These methods facilitate the derivation of causal structures from observational data, offering a practical alternative to more traditional, intervention-based approaches.\nAs the application of causal discovery models requires establishing specific foundational assumptions, our initial focus will be on discussing the assumptions to select the most appropriate model. As noted in Section 3, our inability to locate corresponding questions within the survey that align with concepts of disengagement, loss aversion, and mental accounting – all of which have been suggested to influence an individual’s financial activities – leads us to conclude that the assumption of “***causal sufficiency***” is likely violated. The causal sufficiency assumption necessitates the absence of unobserved (or latent) variables in the analysis. While the linear, non-Gaussian, and acyclic model (**LiNGAM**), as one of the structural equation models, is capable of identifying the unique directed acyclic graph (DAG), it requires causal sufficiency, which is not satisfied in the context of this study. Among the constraint-based methods, the **PC** algorithm, named after its inventors Peter Spirtes and Clark Glymour, necessitates the assumptions of acyclicity, causal faithfulness, and causal sufficiency. A first extension of the PC algorithm, the **FCI**, relaxes the assumptions by foregoing the necessity of causal sufficiency and permitting the presence of hidden variables. Therefore, we leverage the FCI algorithm to explore the mechanism behind financial literacy and financial activities using the FCI method in Causal-learn package in Python.[3](#fn3)\nGiven the mixed nature of our dataset, which comprises three discrete dummy variables and ten continuous variables, we initially considered the usage of FCI with a kernel-based conditional independence test, as this method is suited for handling the complexity of mixed data types. However, kernel methods’ computational complexity scales cubically with the sample size. Due to the constraints of computational resources and considering our substantial sample size of nearly 20,000, it is impractical to implement the FCI model with the kernel-based conditional independence test in this study. As an alternative approach, we divide our data into eight groups based on the combination of values of the three dummy variables: *Male*, *Fin\\_Edu*, *Fin\\_Edu\\_Home*. For each of these groups, we apply the FCI model with Fisher’s Z conditional independent test, which is suitable for continuous variables. A significance level of 0.05 is selected for the individual partial correlation test in this study.\nFurthermore, as previous research demonstrates that incorporating background knowledge into causal discovery enhances the accuracy of exploring causal relations and lends more realism to the interpretation of the results, we define *Age* and *Education* as exogenous variables when applying the FCI algorithm to the pre-segmented data of the eight groups.\nAdditionally, bootstrapping is used to confirm the statistical stability of the results from the causal discovery approach before we draw conclusions. The bootstrap method involves estimating the distribution of an estimator or a test statistic through the process of resampling from the original dataset. By resampling the data multiple times and regenerating the partial ancestral graphs (PAG), the outputs estimated from the FCI algorithm for each sample, we can estimate the bootstrap probability of the existence of each edge. This process enables the evaluation of the robustness of the PAG against variations in the data sample. In this study, we specify the number of bootstrap samplings as 100 and proceed to estimate the bootstrap probability. We exclude illustrations of edges with notably low probabilities, specifically those lower than or equal to 0.2, due to their unreliability. Instead, our focus was on edges with higher bootstrap probabilities.\nResults\n=======\nAs mentioned in Section 4, we divided the original survey data into eight groups based on three dummy variables: *Male*, *Fin\\_Edu*, and *Fin\\_Edu\\_Home*. The sample sizes for each group, along with their corresponding conditions, are summarized in Table [datasummary]. More than 75% of the 19,333 respondents in the research sample indicated that they did not receive financial education either at school or at home, which are segmented into Groups 4 and 8. These two groups are comparatively less financially literate, as evidenced by the number of correct answers to questions assessing financial knowledge (*Fin\\_Literacy)* illustrated in Figure [boxplot].\n[htbp] [datasummary]\n| **Group** | ***Male*** | ***Fin\\_Edu*** | ***Fin\\_Edu\\_Home*** | **Sample Size** |\n| --- | --- | --- | --- | --- |\n| Group 1 | 1 | 1 | 1 | 443 |\n| Group 2 | 1 | 0 | 1 | 1,482 |\n| Group 3 | 1 | 1 | 0 | 657 |\n| Group 4 | 1 | 0 | 0 | 7,961 |\n| Group 5 | 0 | 1 | 1 | 259 |\n| Group 6 | 0 | 0 | 1 | 1,659 |\n| Group 7 | 0 | 1 | 0 | 300 |\n| Group 8 | 0 | 0 | 0 | 6,572 |\nIn addition, Figure [boxplot] demonstrates that female, represented by Groups 5 to 8, exhibits a lower level of financial literacy compared to male categorized by Groups 1 to 4, which aligns with the conclusion drawn from previous research .\n![Boxplots of Fin_Literacy for Eight Groups](boxplot.png \"fig:\") [boxplot]\nThe PAGs derived from the FCI model, utilizing the processed survey data, are illustrated in Figure [14] for Groups 1 to 4 and Figure [58] for Groups 5 to 8. A PAG is a graphical representation used to depict the causal relationships among a set of variables, particularly when dealing with datasets that may have latent or hidden variables. In PAGs, the vertices and edges represent variables and the relationships between them. Unlike a DAG, a PAG can have four types of edges : → , ∘ → , ∘ – ∘ , and ↔ . *i* → *j* is drawn when *i* is an ancestor of *j* and *j* is not an ancestor of *i*. An edge *i* ↔ *j* is drawn when *i* is not an ancestor of *j* and *j* is not an ancestor of *i*, which indicates the influence of an unobserved confounder. The open dot ∘ is used in *i* ∘ → *j* when *i* → *j* is present in some maximal ancestral graphs (MAG), and *i* ↔ *j* exists in some other MAGs.\nTo ascertain the reliability of the generated PAGs, we employed the bootstrap method to assess the statistical robustness of the FCI model’s outputs. The comprehensive lists of edges generated from 100 bootstrap samplings, along with their corresponding bootstrap probabilities for each group, are included in [apd:fourth]. Interpretations of the findings presented in this section, as well as discussions and conclusions in subsequent sections, are based on edges with bootstrap probability larger than 0.2 in [apd:fourth].\n![Output PAGs of FCI for Groups 1 to 4](1_4.png \"fig:\") [14]\n![Output PAGs of FCI for Groups 5 to 8](5_8.png \"fig:\") [58]\nWe could not find a direct causal flow from *Fin\\_Literacy* to *Invest* or *Planning* in the PAGs of any of the eight groups, which demonstrates that the level of financial literacy is not a direct cause of participation in risky financial assets investment or retirement planning in Japan. Meanwhile, in the PAGs of Groups 1, 3, 4, 7, and 8, “ ↔ ” exists between *Fin\\_Literacy* and *Invest*, which might indicate the existence of latent confounders. Likewise, the presence of “ ↔ ” between *Fin\\_Literacy* and *Planning* in the PAGs of Groups 3, 4, 5, and 7 implies the possibility of unobserved variables. Moreover, “ ∘ → ” extends from *Invest* to *Fin\\_Literacy* in the PAGs of Groups 4, 6, and 8. This edge suggests the potential causal relationship where participation in financial investment might enhance financial literacy levels. Similarly, “ ∘ → ” can be found in the PAGs of Groups 4 and 6, and “ → ” can be found in the PAGs of Groups 6, 7, and 8 from *Planning* to *Fin\\_Literacy*, revealing the possibility of a causal impact of retirement planning on the improvement of financial literacy.\nConcerning the causal relationship about confidence in financial literacy, divergent outcomes were observed across different bootstrap samplings and groups. Specifically, for Groups 1, 2, 3, 4, 6, and 8, the PAGs of some bootstrap samples depict a possible causal flow from confidence levels to investment in risky financial assets, as represented by the “ → ” or “ ∘ → ” edges from *Confidence* to *Invest*. In contrast, the level of confidence can be a result of participation in financial investment, illustrated by the “ → ” from *Invest* to *Confidence* in some bootstrapping results of Groups 1, 2, 5, 6, and 8. Unobserved confounders can exist between *Invest* and *Confidence* represented by “ ↔ ” in PAGs generated from specific bootstrap samples in Groups 3 and 8. Regarding the causal link between confidence and retirement planning, “ → ” from *Confidence* to *Planning* and a reverse causal flow “ → ” from *Planning* to *Confidence* can be both observed in the PAGs emerged from different bootstrap samples of Groups 1, 2, 6, and 8. For Group 4, “ → ” and “ ↔ ” are both possible between *Confidence* and *Planning* in the results, indicating the possible causal flow from confidence to retirement planning and the potential latent confounders between them. “ ↔ ” between *Confidence* and *Planning* can also be found in Groups 2, 6, and 8 as a signal of latent variables.\nFurthermore, beyond the scope of our initial research question related to financial literacy, the results uncovered the causal influence of age on retirement planning, which is reflected by the indirect “ → ” from *Age* to *Planning* in the PAG of Groups 1, 4, 7, and 8, and direct “ → ” in Groups 2, 3, 4, 5, and 7. Additionally, direct causal influences from age (*Age*) to participation in risky asset investment (*Invest*) are confirmed in the results of female Groups 5, 6, and 8.\nDiscussion\n==========\nAlthough the previous research introduced in Section 2 demonstrates that the financial literacy level is highly associated with financial activities, which could be a causal relationship, we did not confirm the direct causal impact of financial literacy on either investment in risky financial assets or retirement planning in the results of the FCI model. Therefore, enhancing the financial literacy level of individuals in Japan through mandatory financial education in high school may not inevitably improve participation in financial investment and retirement planning in Japan. Meanwhile, we found the possibility of latent confounders between financial literacy and financial activities, either participation in financial investment or retirement planning, for Groups 1, 3, 4, 5, 7, and 8. Moreover, in Groups 4, 6, 7, and 8, financial activities can have a causal impact on financial literacy levels. The existence of latent confounders and reverse causation may explain the high correlation between the variables verified in previous research.\nWhen it comes to confidence in financial literacy, although results differed across the bootstrap samples and groups, it was a cause of investment participation in Groups 1, 2, 3, 4, 6, and 8 and of retirement planning in Groups 1, 2, 4, 6, and 8. As depicted in the “Financial Literacy Survey: 2022 Results,” a mere 12.1% of respondents expressed confidence in their financial knowledge by rating their overall knowledge about financial matters as “Very high” or “Quite high” compared with others. This figure is significantly lower than the 76% reported in the U.S. Subsequently, interventions, such as providing opportunities for simulated financial experiences and creating environments conducive to discussions about finances, to increase confidence in financial literacy among Japanese households, may potentially boost investment participation and retirement planning.\nWhile the primary focus of this study remains on the impact of financial literacy, the results indicate a direct or indirect causal effect of age on financial activities. has noted that women often enhance their financial knowledge when it becomes relevant, typically around the time of their spouses’ passing. In a parallel manner, the influence of age on retirement planning and investment participation might stem from a sense of “relevancy.” As individuals age, they may perceive an escalating need for financial planning. Women may start investing in riskier assets to pursue higher returns, as their income growth does not typically match the pace experienced by men in Japan. To extend and improve this study further, variables representing disengagement, loss aversion, and mental accounting can be included in the dataset, as these factors may affect individuals’ decision-making. Given that the PAGs imply the presence of latent confounders between financial literacy and financial activities, potential variables can be identified by delving into financial and economic theories. Subsequently, redesigning the survey questions and conducting a subsequent survey are necessary for acquiring the data for further research.\nMoreover, broadening the definition of financial literacy may provide deeper insights into its causal influence on financial activities. Traditionally, financial literacy has been primarily quantified in terms of financial knowledge. The practical application of this knowledge in decision-making processes was not a major focus until the OECD redefined financial literacy. This redefinition encompasses not only financial knowledge but also financial attitude and behavior. Therefore, financial behavior and attitude could be considered integral components of financial literacy. suggest that financial behavior, reflecting individuals’ actions in financial matters, can be accessed by the extent of agreement with the statements below from the “Financial Literacy Survey.” \nQ1\\_1. Before I buy something I carefully consider whether I can afford it. \nQ1\\_2. I pay my bills on time. \nQ1\\_4. I set long-term financial goals and strive to achieve them. \nQ1\\_7. I keep a close personal watch on my financial affairs. \nIn a similar fashion, financial attitude, indicating one’s perspective on financial issues, can be measurable through responses to the following statements. \nQ1\\_5. I find it more satisfying to spend money than to save it for the long term. \nQ1\\_6. I tend to live for today and let tomorrow take care of itself.\nAdditionally, future studies can apply the same method to the data from the 2019 iteration of the “Financial Literacy Survey.” As highlighted in Section 1, the “20-million-yen problem” garnered significant attention in 2019, intensifying public focus on financial asset formation in Japan. The onset of the COVID-19 pandemic from 2020 onwards likely influenced individuals’ lifestyles, consumption habits, and financial activities. Comparing the 2019 survey results with this study could elucidate any shifts in the underlying mechanism of households’ financial activities in Japan.\nConclusion\n==========\nIn this research, we applied the FCI model with Fisher’s Z conditional independent test to the individual-level data from the 2022 “Financial Literacy Survey” conducted by the BOJ. This approach was employed to investigate the causal relationships between financial literacy and financial activities, specifically focusing on retirement planning and investment in risky financial assets in Japan. The original dataset was divided into eight groups based on three exogenous discrete variables to facilitate the use of Fisher’s Z conditional independent test, which is appropriate for continuous variables. Additionally, two other variables were designated as exogenous, and this information was incorporated as background knowledge about the edges. The robustness of the edges in PAGs derived from the FCI model was verified by estimating their probabilities using the bootstrap method.\nWhile financial education at school and home may correlate with higher financial literacy, higher financial literacy does not necessarily lead to increased participation in financial investment and retirement planning in Japan. Our results did not identify direct causal links from financial literacy to financial activities. Consequently, the Japanese government may need to consider supplementary or alternative measures to achieve its objective of “a shift from savings to investment.” As confidence in financial literacy could support the adoption of desirable financial activities, initiatives can be undertaken to enhance confidence levels. The impact of age on financial activities could be attributed to perceived relevancy. Although age itself is an exogenous factor and cannot be intervened, measures could be implemented to strengthen the sense of relevancy.\nWe hope this study encourages further discussions on applying causal discovery algorithms to financial literacy and financial activity survey data. Equally important, we hope that the insights derived from this study will help governmental policymakers in their efforts to promote the financial well-being of households in Japan.\nQuestion Numbers for Category of Financial Literacy Map\n=======================================================\n|p6.5cm|p6.2cm| **Category of Financial Literacy Map** & **Question Numbers** \nFamily budget management & Q4, Q5 \nLife planning & Q12, Q13 \nFinancial knowledge & \nBasics of financial transactions & Q14, Q15, Q16 \nBasics of finance and economy & Q18, Q19, Q20, Q21\\_1, Q22, Q23 \nInsurance & Q25, Q26, Q28 \nLoans/credit & Q21\\_2, Q30, Q31 \nWealth building & Q21\\_3, Q21\\_4, Q33 \nUse of outside expertise & Q36, Q37, Q38 \nList of Variables\n=================\n**1. *Male*:** Dummy Variable. Assigned a value of 1 if the response to “Q42. What is our gender?” is “Male.” \n**2. *Fin\\_Edu*:** Dummy Variable. Set to 1 if the answer to “Q39. Was financial education offered by a school or college you attended, or a workplace where you were employed?” is “Yes, and I did participate in the financial education.” \n**3. *Fin\\_Edu\\_Home*:** Dummy Variable. Assigned a value of 1 if the response to “Q40. Did your parents or guardians teach you how to manage your finances?” is “Yes.” \n**4. *Age*:** Continuous Variable. Calculated as the mean of the age range indicated by the response to “Q43. What is your age?” \n**5. *Education*:** Continuous Variable. Determined by the total years required to complete the educational level indicated in “Q44. What is your educational background?” \n**6. *Income*:** Continuous Variable. The answer to “Q51. Which of these categories do your (your household) income for last year fall into?” \n**7. *Asset\\_Amt*:** Continuous Variable. The answer to “Q52. Which of these categories do your (your household’s) financial assets (deposits, stocks, etc.) currently fall into?” \n**8. *Myopic\\_Bias*:** Continuous Variable. The answer to “Q1 10. If I had the choice of (1) receiving 100,000 yen now or (2) receiving 110,000 yen in 1 year, I would choose (1), provided that I can definitely receive the money.” \n**9. *Herding\\_Bias*:** Continuous Variable. The response to “Q1 3. When there are several similar products, I tend to buy what is recommended as the best-selling product, rather than what I actually think is a good product.” \n**10. *Confidence*:** Continuous Variable. The answer to “Q17. How would you rate your overall knowledge about financial matters compared with other people?” \n**11. *Invest*:** Continuous Variable. The number of risky asset types an individual has invested in, based on the responses to “Q34. Have you ever purchased any of the following financial products?” \n**12. *Planning*:** Continuous Variable. Assessed whether the individual is well prepared for retirement, according to the answers to a series of questions: Q7, Q8\\_1, Q9\\_1, and Q10\\_1. \n**13. *Fin\\_Literacy*:** Continuous Variable. The number of correct answers to the 25 questions designed to evaluate the individual’s financial literacy level.\nEdges with Bootstrap Probabilities\n==================================\n|p5cm|p1cm| Age → Asset\\_Amt & 1 \nAge → Fin\\_Literacy & 1 \nAge → Herding\\_Bias & 0.05 \nAge → Income & 0.22 \nAge → Planning & 0.02 \nAsset\\_Amt → Confidence & 0.11 \nAsset\\_Amt → Fin\\_Literacy & 0.12 \nAsset\\_Amt ∘ → Fin\\_Literacy & 0.07 \nAsset\\_Amt → Income & 0.34 \nAsset\\_Amt ∘ → Income & 0.01 \nAsset\\_Amt → Invest & 0.05 \nAsset\\_Amt ↔ Invest & 0.03 \nAsset\\_Amt → Planning & 0.58 \nAsset\\_Amt ↔ Planning & 0.16 \nConfidence → Asset\\_Amt & 0.12 \nConfidence ↔ Asset\\_Amt & 0.08 \nConfidence ∘ → Asset\\_Amt & 0.06 \nConfidence ∘ → Herding\\_Bias & 0.01 \nConfidence → Income & 0.05 \nConfidence ↔ Income & 0.02 \nConfidence ∘ – ∘ Income & 0.01 \nConfidence → Invest & 0.5 \nConfidence ∘ → Invest & 0.16 \nConfidence ↔ Invest & 0.06 \nConfidence ∘ – ∘ Invest & 0.01 \nConfidence → Planning & 0.37 \nConfidence ↔ Planning & 0.16 \nConfidence ∘ → Planning & 0.11 \nConfidence ∘ – ∘ Planning & 0.01 \nEducation → Asset\\_Amt & 0.18 \nEducation → Confidence & 0.33 \nEducation → Fin\\_Literacy & 0.69 \nEducation → Income & 0.64 \nEducation → Invest & 0.35 \nEducation → Myopic\\_Bias & 0.07 \nEducation → Planning & 0.01 \nFin\\_Literacy ↔ Asset\\_Amt & 0.23 \nFin\\_Literacy → Asset\\_Amt & 0.1 \nFin\\_Literacy ∘ → Asset\\_Amt & 0.01 \nFin\\_Literacy → Herding\\_Bias & 0.21 \nFin\\_Literacy ∘ → Herding\\_Bias & 0.01 \nFin\\_Literacy ↔ Invest & 0.35 \nFin\\_Literacy → Myopic\\_Bias & 0.12 \nFin\\_Literacy ↔ Planning & 0.05 \nHerding\\_Bias ↔ Confidence & 0.03 \nHerding\\_Bias ∘ → Confidence & 0.01 \nHerding\\_Bias ∘ → Fin\\_Literacy & 0.41 \nHerding\\_Bias → Fin\\_Literacy & 0.23 \nHerding\\_Bias ↔ Fin\\_Literacy & 0.14 \nHerding\\_Bias ↔ Invest & 0.31 \nHerding\\_Bias ∘ → Invest & 0.09 \nHerding\\_Bias → Invest & 0.03 \nHerding\\_Bias ∘ – ∘ Myopic\\_Bias & 0.22 \nHerding\\_Bias ↔ Myopic\\_Bias & 0.21 \nHerding\\_Bias ∘ → Myopic\\_Bias & 0.11 \nHerding\\_Bias → Myopic\\_Bias & 0.04 \nHerding\\_Bias ↔ Planning & 0.01 \nIncome → Asset\\_Amt & 0.59 \nIncome ∘ – ∘ Asset\\_Amt & 0.03 \nIncome ∘ → Asset\\_Amt & 0.03 \nIncome → Confidence & 0.09 \nIncome → Invest & 0.54 \nIncome ↔ Invest & 0.11 \nIncome ∘ → Invest & 0.09 \nIncome ∘ – ∘ Invest & 0.02 \nIncome → Planning & 0.24 \nIncome ↔ Planning & 0.18 \nIncome ∘ → Planning & 0.04 \nInvest → Asset\\_Amt & 0.07 \nInvest → Confidence & 0.27 \nInvest → Fin\\_Literacy & 0.01 \nInvest → Herding\\_Bias & 0.01 \nInvest → Income & 0.04 \nInvest ∘ → Income & 0.01 \nInvest ∘ → Planning & 0.01 \nMyopic\\_Bias ↔ Asset\\_Amt & 0.03 \nMyopic\\_Bias → Asset\\_Amt & 0.01 \nMyopic\\_Bias ∘ → Fin\\_Literacy & 0.47 \nMyopic\\_Bias → Fin\\_Literacy & 0.23 \nMyopic\\_Bias ↔ Fin\\_Literacy & 0.15 \nMyopic\\_Bias ∘ → Herding\\_Bias & 0.23 \nMyopic\\_Bias → Herding\\_Bias & 0.13 \nMyopic\\_Bias ↔ Income & 0.02 \nMyopic\\_Bias ∘ → Invest & 0.01 \nMyopic\\_Bias ↔ Planning & 0.19 \nMyopic\\_Bias → Planning & 0.02 \nMyopic\\_Bias ∘ → Planning & 0.01 \nPlanning → Asset\\_Amt & 0.06 \nPlanning ∘ → Asset\\_Amt & 0.03 \nPlanning → Confidence & 0.22 \nPlanning ∘ → Confidence & 0.04 \nPlanning"},"data_id":{"kind":"string","value":"arxiv_0183481"}}},{"rowIdx":45897,"cells":{"content":{"kind":"string","value":"{I}\\_{4\\times 4}\\} \\\\\n\\mbox{Case 2: } & x\\_{100} \\sim \\mathcal{N}\\{0,0.25\\times\\mathbf{I}\\_{4\\times 4}\\} \\end{aligned}$$ \n[fig:Scenario]\nwhere the target state is position and velocity in two dimensions. Snapshots of one dimension of both cases are shown in Fig. [fig:Scenario]. It is well-known that JPDA and related algorithms suffer from *coalescence*, *i.e.*, after targets come in close proximity, mean estimates remain on the mid-point of the targets for a significant duration (*e.g.*, ). In case 1, targets are completely indistinguishable at the mid-point, hence coalescence-like effects are at their worst. Coalescence-like effects also occur in case 2, but to a lesser extent (mainly due to the discernible difference in velocity). In case 1, targets all exist throughout the simulation (tracks are not pre-initialised). In case 2, the targets are born at times {0, 10, …, 10(*n* − 1)} (any targets not existing prior to time *t* = 100 are born at that time; consequently, for case 2 with *n* = 20, ten targets are born at time *t* = 100). Targets follow a linear-Gaussian model with nominally constant velocity, *x**t* = **F***x**t* − 1 + *w**t*, where *w**t* ∼ N{0, **Q**}, \n$$\\mathbf{F} = \\left[\\begin{array}{cc}\n1 & T \\\\\n0 & 1\n\\end{array}\\right] \\otimes \\mathbf{I}\\_{2\\times 2}, \\quad\n\\mathbf{Q} = q \\left[\\begin{array}{cc}\nT^3/3 & T^2/2 \\\\\nT^2/2 & T\n\\end{array}\\right] \\otimes \\mathbf{I}\\_{2\\times 2}$$ \n and *q* = 0.01, *T* = 1. Target-originated measurements provide position corrupted by Gaussian noise, *i.e.*, *z**t* = **H***x**t* + *v**t*, where **H** = [ 1 0 ] ⊗ **I**2 × 2, and *v**t* ∼ N{0, **I**2 × 2}. The initial unknown target intensity for the proposed methods is assumed to be *λ*0∣0*u*(*x*) = 10N{*x*; 0, **P**}, where **P** = diag[1002, 1, 1002, 1] covers the position and velocity region of interest. The birth intensity uses the same covariance, *λ*b(*x*) = 0.05N{*x*; 0, **P**}. Cases are considered with the average number of false alarms per scan as *λ*fa ∈ {10, 20, 40, 60, 80}, and their distribution is uniform on [ − 100, 100]2. All filters assume *P*s = 0.999. We consider cases with *P*d ∈ {0.3, 0.5, 0.7, 0.98}, representing a range of SNR values.\nThe scenarios examined are exceptionally challenging due to the large number of targets in close proximity. While others have considered larger numbers of targets, in most cases these positioned uniformly in space and rarely come into close contact. Cases such as this can be effectively decoupled into series of single target tracking problems. In the present study, up to 20 targets have effectively the same position and velocity at the mid-point in time, and the dependency between targets is inescapable.\nWe compare the proposed methods to CPHD, CB-MeMBer, JITS and linear multi-target ITS (LMITS) (a variant of JITS which replaces the marginal association probabilities with a heuristic approximation). The CB-MeMBer approximates the initial distribution by 30 identical Bernoulli components with *r* = 1/3, and uses a single birth component with *r* = 0.05 at each time. JITS and LMITS truncate the unknown target intensity to use *ρ*target(*z*) = *P*d⟨*λ*b, *f*(*z*∣ ⋅ )⟩. The implementation of JITS initiates a new track on every measurement (via ([eq:InitialExistenceProbability])), but only updates new tracks with measurements that are not in the gate of any hypothesis of an existing track; this was necessary for tractability. Exact marginal association probabilities were calculated using the junction tree algorithm described in, which in turn uses the library for discrete approximate inference. The exact calculation is not possible for the simulations with *n* > 6 targets, or with *λ*fa > 10. Since the weights provided by LMITS do not provide a probability that a measurement is not used by any other track, the LMITS implementation initiates tracks only on measurements that are not within the gate of any hypothesis. Where applicable, all tracks with *r**t*∣*t**i* ≥ 10− 4 are maintained (the threshold needed to be increased to 10− 3 for JITS to be able to successfully execute). All methods utilise Gaussian mixture implementations, with hypothesis management performed using *n*-scan merging, with *n* = 2.[17](#fn17) All methods tested share the same filtering and hypothesis management code, so run times are somewhat comparable. All code was written in MATLAB except for the C++ junction tree routine used for JITS. Clustering is used only in the JITS marginal probability calculation. The TOMB/P, JITS and LMITS implementations output estimates[18](#fn18) for all tracks with existence probability ≥ 0.8, while the remaining methods find the mode of the cardinality distribution (excluding the Poisson component for the MOMB/P), and output estimates for the corresponding number of tracks with the highest probability of existence (or, in the case of CPHD, the Gaussian components with highest weight).\nTracking performance\n--------------------\nThe results are shown in Fig. [fig:Results]; the figure shows the mean optimal sub-pattern assignment (OSPA)[19](#fn19) with *p* = 1 and *c* = 20 (calculated over both position and velocity) as a function of time, averaged over 200 Monte Carlo trials. The following observations are made:\n[p] \n[fig:Results]\n* The effect of coalescence is clearly visible from *t* = 100 to the end of the simulation for JITS, TOMB/P and LMITS. The cause of this difficulty is illustrated in figure [fig:MarginalisationDiagram](i): after targets have been closely spaced, the posterior marginal distribution for each track may contain a mode representing each target. Thus the product of these marginal distributions contains “alias” peaks with multiple targets in the same region, and in the output of the tracker, multiple estimates may be placed on the same target. Interestingly, LMITS is affected less than TOMB/P, which is affected less than JITS. The difference in each of these is the approximation of the marginal association probabilities used—the approximation used by LMITS is shown in to be inferior to the LBP-based method used for the TOMB/P, whereas the JITS uses the exact values. Counter-intuitively, the poorer approximations perform better. This suggests that the approximations tend to be less equivocal between the competing hypotheses than the exact probabilities, e.g., giving higher probability to the most likely hypothesis, and lower probability to alternative hypotheses, thus aiding earlier resolution of coalescence. A similar difference between JIPDA and LMIPDA was observed in. Various methods exist for modifying association weights to resolve coalescence (*e.g.*, ), but all suffer from similar exponential complexity to the exact marginal association calculation. An extension of the current paper which provides a tractable approximation of the multi-Bernoulli distribution that minimises the RFS KL divergence from the full distribution is available in.\n* MOMB/P (as well as CPHD and CB-MeMBer) exhibits good robustness to coalescence effects. An example of this was illustrated in figure [fig:MarginalisationDiagram](j): when coalescence effects are present, the MOMB approximation of the association distribution ([eq:MOApprox]) yields a multi-Bernoulli RFS density which tends to be closer to the full RFS density. This is due to the fact that all hypotheses in the Bernoulli component *j* ∈ {1, …, *m**t*} have been updated with measurement *z**t**j*, hence the components tend to be spatially concentrated. Consequently, the product of the marginal distributions will tend not to contain “alias” peaks, as shown in figure [fig:MarginalisationDiagram](j).\n* CPHD, CB-MeMBer and MOMB/P also react more quickly to cardinality changes due to the use of the cardinality mode estimator rather than component-by-component thresholding. Where coalescence is not dominant, all of these methods exhibit reduced performance in comparison to the various track oriented approaches. The reduction in performance is larger in lower *P*d cases. The MOMB/P outperforms CPHD and CB-MeMBer throughout, especially in the low *P*d cases. The primary error in CPHD and CB-MeMBer in these cases is the so-called “spooky effect”, *i.e.*, in cases in which one or more targets are not detected, their positions are estimated to be close to other targets that are detected, regardless of target spacing. Thus, as described in, the global cardinality is estimated accurately but the local cardinality is inaccurate. Neither TOMB/P nor MOMB/P exhibit the same difficulty. One possible explanation of this is that, while CB-MeMBer retains a multi-Bernoulli distribution, the cardinality correction is based on the overall intensity, hence, like CPHD, some information on local cardinality is lost. The approximation used in the proposed methods does not utilise the global intensity in the same way.\n* With *P*d = 0.3, TOMB/P, MOMB/P and CB-MeMBer all maintain around 200 tracks on average, whereas JITS and LMITS maintain 20–30. This is the consequence of the track initiation logic necessitated by JITS and LMITS, and the higher track deletion threshold for JITS (which was necessary to successfully execute the algorithm); it directly leads to the degradation in performance of these methods compared to the others with this value of *P*d. Thus the apparent lower complexity of JITS in Fig. [fig:TimeResults] is the result of its solving a much smaller problem, which leads to its lower performance, due to slow initiation. Scenarios with *n* ∈ {10, 20} targets or *λ*fa > 10 could not be executed at all for JITS due to the large memory and computation requirements. The extension of the current paper in demonstrates how the PPP component can be used to represent tracks with a low probability of existence to avoid maintaining such a large number of tracks.\nComputational complexity\n------------------------\n[!t] \n[fig:TimeResults]\nThe computation time for the various algorithms on a subset of scenarios is examined in Fig. [fig:TimeResults]. The left column show the total processing time for each scan (*i.e.*, each time step of the simulation) using each algorithm. While the computation times are quite long, the vast majority of the time is spent in calculation of a large number of Gaussian mixture hypotheses. For example, in the scenario with *n* = 20 targets, the maximum number of 1000 single-target hypotheses will be maintained around the mid-point in time. Each of these hypotheses needs to be updated with each of the target-originated measurements in the next time step (since the targets are closely spaced), yielding around 20, 000 hypotheses which are subsequently simplified back to 1000. The times spent in this process differ slightly for the various methods, as the differences in hypothesis weights yield variations in the number of hypotheses and tracks that need to be maintained. More sophisticated hypothesis reduction methods have been successfully applied, but are not presented here as they cannot easily be applied to the comparison algorithms (CPHD and CB-MeMBer).\nThe results with *λ*fa = 80 show that the complexity of CPHD is strongly affected by the total number of measurements, whereas the other methods are more impacted by the number of targets in close proximity. Through implementation methods such as gating and clustering (which are widely used in practical systems, *e.g.*, ), complexity scales very well to very large problems involving well-spaced targets (or small groups of well-spaced targets) since the sub-problems in each cluster are independent.\nThe only operation which is performed by TOMB/P and MOMB/P but not the other methods is the LBP approximation of marginal association probabilities. The total time consumed in the iterative calculation for each scan of the simulations is shown in the right column of Fig. [fig:TimeResults] (note that time is in milli-seconds, and clustering is not applied). Even in the most challenging case (*n* = 20), average computation time peaks at 15 ms; the values in other cases are significantly smaller. As discussed previously, the algorithms as stated maintain a large number of tracks with low probability of existence. The extension of the current paper in demonstrates that these can be efficiently represented using the PPP component, resulting in a significant computational saving.\nConclusion and Extensions\n=========================\nIn this paper, we have presented a derivation of a conjugate prior for a full Bayes RFS filter under common assumptions, and shown that the resulting data structure is similar to that utilised in TOMHT. Subsequently, the TOMB and MOMB methods were derived based on approximations of the marginal association distribution, which can be efficiently approximated using the LBP algorithm. Experimental scenarios demonstrated promising performance on a challenging problem.\nBy comparing the resulting TOMB/P algorithm with JITS/JIPDA, we showed how variants of these popular methods can be derived using the RFS framework. The newly-proposed MOMB/P algorithm follows the philosophy of the MeMBer alongside the LBP data association approximation to provide a solution that is robust to coalescence, and considerably improves upon the performance of CPHD and CB-MeMBer, especially with lower *P*d values.\nTo date, this work has been extended in three directions:\n* In, we demonstrate the significant computational benefit of representing Bernoulli components with a small probability of existence through the PPP component of the distribution.\n* In, we derive a variational method for (approximately) finding the multi-Bernoulli distribution with the smallest KL divergence from the full RFS distribution ([eq:PreexistingTargets]) (rather than minimising the KL divergence of the discrete association distribution, as in ([eq:TOMKLOpt]) and ([eq:MOMKLOpt])). The result combines the superior performance of TOMB/P when targets are well-separated with the robustness against coalescence of MOMB/P.\n* In, the derivation in Section [sec:Derivation] is extended to single-cluster processes, and the resulting method is applied to simultaneous localisation and mapping.\nProof of prediction step\n========================\n[Proof of Theorem [th:Prediction]] The proof follows these steps:\n1. Substitute the p.g.fl form of the density ([eq:FullDensity]) into the Bayes RFS prediction formula ([eq:PGFLPrediction])\n2. Observe that the result is the product of two p.g.fls\n3. Show that the first component is a PPP, show that the intensity of the PPP is calculated using a standard PHD prediction step\n4. Show that the second component is a MBM, and that the prediction step applies to each hypothesis of each track separately\nSubstituting ([eq:FullDensity]) into ([eq:PGFLPrediction]):\nrCl Gt|t-1[h] && {,h} Gt-1|t-1[1-P+P ph] \n&=& {,h} Gt-1|t-1[1-P+P ph] Gt-1|t-1[1-P+P ph] [eq:PredictionPGFl]\nrCl Gt|t-1[h] && {,h} Gt-1|t-1[1-P+P ph] \n&=& {,h} Gt-1|t-1[1-P+P ph] \n&& Gt-1|t-1[1-P+P ph] [eq:PredictionPGFl]\nWe assign the first two factors in this product to *G**t*∣*t* − 1ppp[*h*] and the final to *G**t*∣*t* − 1mbm[*h*] to obtain *G**t*∣*t* − 1[*h*] = *G**t*∣*t* − 1ppp[*h*]*G**t*∣*t* − 1mbm[*h*], where\n \n$$\\begin{aligned}\nG^\\mathrm{ppp}\\_{t|t-1}[h] &\\propto \\exp\\{\\langle\\lambda^\\mathrm{b},h\\rangle\\} G^\\mathrm{ppp}\\_{t-1|t-1}[1-P^{\\mathrm{s}}+P^{\\mathrm{s}} p\\_h] \\\\\nG^\\mathrm{mbm}\\_{t|t-1}[h] &= G^\\mathrm{mbm}\\_{t-1|t-1}[1-P^{\\mathrm{s}}+P^{\\mathrm{s}} p\\_h]\\end{aligned}$$ \nWe will show that each of these retain their respective forms, *i.e.*, ([eq:UndetectedTargets]) and ([eq:PreexistingTargets]). *G**t*∣*t* − 1ppp[*h*] corresponds directly to the prediction step in the PHD :\n \n$$\\begin{aligned}\nG^\\mathrm{ppp}\\_{t|t-1}[h] \n&\\propto \\exp\\{\\langle\\lambda^\\mathrm{b},h\\rangle\\} G^\\mathrm{ppp}\\_{t-1|t-1}[1-P^{\\mathrm{s}}+P^{\\mathrm{s}} p\\_h] \\\\\n&\\propto \\exp\\{\\langle\\lambda^\\mathrm{b},h\\rangle + \\langle\\lambda^u\\_{t-1|t-1},1-P^{\\mathrm{s}}+P^{\\mathrm{s}} p\\_h\\rangle\\} \\\\\n&\\propto \\exp\\{\\langle\\lambda^\\mathrm{b},h\\rangle + \\langle\\lambda^u\\_{t-1|t-1},P^{\\mathrm{s}} p\\_h\\rangle\\} \\\\\n&= \\exp\\{\\langle\\lambda^u\\_{t|t-1},h\\rangle\\} \\end{aligned}$$ \nwhere $\\langle\\lambda^u\\_{t|t-1},h\\rangle \\triangleq \\langle\\lambda^\\mathrm{b},h\\rangle + \\langle\\lambda^u\\_{t-1|t-1},P^{\\mathrm{s}} p\\_h\\rangle$. Thus *G**t*∣*t* − 1ppp[*h*] is indeed a PPP, with intensity given by ([eq:UndetectedPropagation]).\nWe now consider *G**t*∣*t* − 1mbm[*h*]; since *G**t* − 1∣*t* − 1mbm[*h*] is a mixture of MeMBer distributions, this can be calculated via the MeMBer prediction equation[20](#fn20) :\n \n$$\\begin{aligned}\nG^\\mathrm{mbm}\\_{t|t-1}[h] &\\triangleq G^\\mathrm{mbm}\\_{t-1|t-1}[1-P^{\\mathrm{s}}+P^{\\mathrm{s}} p\\_h] \\notag\\\\\n&= \\sum\\_a w^a\\prod\\_{i\\in{\\cal T}\\_{t-1|t-1}}G\\_{t-1|t-1}^{i,a^i}[1-P^{\\mathrm{s}}+P^{\\mathrm{s}} p\\_h] \\\\\n&= \\sum\\_a w^a\\prod\\_{i\\in{\\cal T}\\_{t-1|t-1}}G\\_{t|t-1}^{i,a^i}[h] \\label{eq:PredictionPGFlDecompos}\\end{aligned}$$ \n \n$$\\begin{aligned}\n\\lefteqn{G^\\mathrm{mbm}\\_{t|t-1}[h] \\triangleq G^\\mathrm{mbm}\\_{t-1|t-1}[1-P^{\\mathrm{s}}+P^{\\mathrm{s}} p\\_h]} \\notag\\\\\n&= \\sum\\_a w^a\\prod\\_{i\\in{\\cal T}\\_{t-1|t-1}}G\\_{t-1|t-1}^{i,a^i}[1-P^{\\mathrm{s}}+P^{\\mathrm{s}} p\\_h] \\\\\n&= \\sum\\_a w^a\\prod\\_{i\\in{\\cal T}\\_{t-1|t-1}}G\\_{t|t-1}^{i,a^i}[h] \\label{eq:PredictionPGFlDecompos}\\end{aligned}$$ \nwhere\nrCl Gt|t-1i,ai[h] && Gt-1|t-1i,ai[1-P+P ph] \n&=& 1-rt-1|t-1i,ai + rt-1|t-1i,ai + \n&& + rt-1|t-1i,ai \n&=& 1-rt|t-1i,ai + rt|t-1i,ai [eq:BernoulliPropagation]\nrCl \n&=& 1-rt-1|t-1i,ai + rt-1|t-1i,ai + \n&& + rt-1|t-1i,ai \n&& \n&=& 1-rt|t-1i,ai + rt|t-1i,ai [eq:BernoulliPropagation]\nwhere the final equality results from the definitions of ([eq:ExistProbPropagation]) and ([eq:KinematicDistPropagation]). Recognising ([eq:BernoulliPropagation]) as being Bernoulli, we obtain the result that *G**t*∣*t* − 1mbm[*h*] is in the form ([eq:PreexistingTargets]), with *n**t*∣*t* − 1 = *n**t* − 1∣*t* − 1, *h**t*∣*t* − 1*i* = *h**t* − 1∣*t* − 1*i*, and *w**t*∣*t* − 1*i*, *a**i* = *w**t* − 1∣*t* − 1*i*, *a**i*.\nProof of update step\n====================\nThe structure of the proof of the Bayes update is as follows:\n1. Substitute the p.g.fl form of the density ([eq:FullDensity]) into the Bayes RFS update formula ([eq:PGFLUpdate]), where the joint p.g.fl of targets and measurements is given by ([eq:JointPGFL])\n2. Show that the result is a product of two p.g.fls, where the first is a PPP; this is proven in Lemma [lem:Structure]\n3. Calculate the elements of the update of the second component (which we refer to as the *remaining component*)\n\t1. Updating the detected portion of a PPP with PPP-distributed clutter (Lemma [lem:PoisUpdate])\n\t2. Updating a Bernoulli component (Lemma [lem:BernoulliUpdate])Both of these elements use the simple result in Lemma [lem:ScaledBernoulli]. The complete update of the remaining component can then be calculated from these elements using the product rule ([eq:ProductRule]).\n4. By linearity of the derivative operator, the update in the remaining component can be broken into a sum with a term for each global hypothesis *a* in the prior MBM *G**t*∣*t* − 1mbm[*h*] (the first step in Theorem [th:Update]), effectively computing the update separately for each prior association hypothesis\n5. Given a prior global hypothesis (*i.e.*, a global association history hypothesis involving measurements from previous time steps), the update can be calculated using the product rule ([eq:ProductRule]); the resulting structure is shown in Lemma [lem:ConvolutionEquiv] to be in a form similar to ([eq:PreexistingTargets]) with weights decomposing according to ([eq:GlobalHypWeight])\n6. Subsequently (in Theorem [th:Update]) the sum over prior association hypotheses and hypotheses in the new scan are combined into a single sum\n[lem:Structure] The updated distribution *G**t*∣*t*[*h*] has the form\n \n*G**t*∣*t*[*h*] = *G**t*∣*t*ppp[*h*]*G**t*∣*t*mbm[*h*] \nwhere\nrCl Gt|t[h] && {t|tu,h} [eq:UpdateUndetectedPGFl] \nGt|t[h] && { {,g+ ut|t-1,h P pg} Gt|t-1[h(1-P + P pg)] }|g=0 [eq:UpdateMBMPGFl]\nrCl Gt|t[h] && {t|tu,h} [eq:UpdateUndetectedPGFl] \nGt|t[h] && { {,g+ ut|t-1,h P pg} \n&& Gt|t-1[h(1-P + P pg)] }|g=0 [eq:UpdateMBMPGFl]\nand *λ**t*∣*t**u*(*x*) is given in ([eq:UndetectedUpdate]).\nSubstituting ([eq:JointPGFL]) and ([eq:FullDensity]) into ([eq:PGFLUpdate]),\nrCl Gt|t[h] && { {,g} Gt|t-1[h(1-P + P pg)]}|g=0 [eq:UpdateStructure1] \n&=& { {,g} Gt|t-1[h(1-P + P pg)] Gt|t-1[h(1-P + P pg)] }|g=0 [eq:UpdateStructure2] \n&& { {,g+ ut|t-1,h(1-P + P pg)} Gt|t-1[h(1-P + P pg)] }|g=0 [eq:UpdateStructure3] \n&=& {ut|t-1,h(1-P)} { {,g+ ut|t-1,h P pg} Gt|t-1[h(1-P + P pg)] }|g=0 [eq:UpdateStructure4] [eq:UpdateStructure]\nrCl \n&& { {,g} Gt|t-1[h(1-P + P pg)]}|g=0 [eq:UpdateStructure1] \n&=& { {,g} Gt|t-1[h(1-P + P pg)] \n&& Gt|t-1[h(1-P + P pg)] }|g=0 [eq:UpdateStructure2] \n&& { {,g+ ut|t-1,h(1-P + P pg)} \n&& Gt|t-1[h(1-P + P pg)] }|g=0 [eq:UpdateStructure3] \n&=& {ut|t-1,h(1-P)} \n&& { {,g+ ut|t-1,h P pg} \n&& Gt|t-1[h(1-P + P pg)] }|g=0 [eq:UpdateStructure4] [eq:UpdateStructure]\nwhere ([eq:UpdateStructure1]) substitutes ([eq:JointPGFL]) into ([eq:PGFLUpdate]), ([eq:UpdateStructure2]) substitutes ([eq:FullDensity]) for *G**t*∣*t* − 1[ ⋅ ], ([eq:UpdateStructure3]) substitutes ([eq:UndetectedTargets]) for *G**t*∣*t* − 1ppp[*h*], and ([eq:UpdateStructure4]) exploits the fact that\nrCl ut|t-1,h(1-P + P pg)&=& \n&=& + \n&=& ut|t-1,h(1-P)+ ut|t-1,h(P pg)\nrCl \n&=& \n&=& \n&& + \n&=& ut|t-1,h(1-P)+ ut|t-1,h(P pg)\nand that the factor exp{⟨*λ**t*∣*t* − 1*u*, *h*(1 − *P*d)⟩} is independent of the functional of differentiation *g*, and thus can be taken outside the derivative. Thus, by setting *G**t*∣*t*ppp[*h*] to be the first line in ([eq:UpdateStructure4]), and *G**t*∣*t*mbm[*h*] to be equal to the second and third lines, we arrive at the results in ([eq:UndetectedUpdate]), ([eq:UpdateRetainsStructure]), ([eq:UpdateUndetectedPGFl]) and ([eq:UpdateMBMPGFl]).\n[lem:ScaledBernoulli] Any p.g.fl of the following form[21](#fn21) is an unnormalised Bernoulli distribution:\n \n*a* + ⟨*b*, *h*⟩ = *w*(1 − *r* + *r* ⋅ ⟨*f*, *h*⟩) \nwhere *w* = *a* + ∫*b*(*x*)d*x* = *a* + ⟨*b*, 1⟩, $r = \\frac{\\langle b,1\\rangle}{a+\\langle b,1\\rangle}$ and $f(x) = \\frac{b(x)}{\\langle b,1\\rangle}$.\nThe proof of Lemma [lem:ScaledBernoulli] follows immediately from substituting *w*, *r* and *f*(*x*) into the RHS of ([eq:ScaledBernoulli]). Lemma [lem:PoisUpdate] calculates the unnormalised Bayes update for an unnormalised Poisson distribution in Poisson clutter (note that the undetected portion of the Poisson distribution has been removed as it is represented separately, in *G**t*∣*t*ppp[*h*]).\n[lem:PoisUpdate]\n \n$$\\frac{\\delta}{\\delta Z} \\exp\\{\\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle\\}\\bigg|\\_{g=0} \n= \\prod\\_{z\\in Z} w\\_{t|t}^z (1-r\\_{t|t}^z + r\\_{t|t}^z \\langle f\\_{t|t}^z,h\\rangle)$$ \n \n$$\\begin{gathered}\n\\frac{\\delta}{\\delta Z} \\exp\\{\\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle\\}\\bigg|\\_{g=0} \\\\\n= \\prod\\_{z\\in Z} w\\_{t|t}^z (1-r\\_{t|t}^z + r\\_{t|t}^z \\langle f\\_{t|t}^z,h\\rangle) \\end{gathered}$$ \nwhere *w**t*∣*t**z*, *r**t*∣*t**z* and *f**t*∣*t**z*(*x*) are given in ([eq:PoisUpdateW]), ([eq:PoisUpdatePex]) and ([eq:PoisUpdateKin]) respectively (replacing *z* with *z**t**j*).\nWe use the derivative of a linear functional, and the chain rule: \n \n$$\\begin{aligned}\n\\frac{\\delta}{\\delta z}\\langle f,g\\rangle =& f(z) \\label{eq:LinearFunctional} \\\\\n\\frac{\\delta}{\\delta z} f(F[g]) &= \n\\frac{\\delta}{\\delta z}F[g] \\cdot \\left.\\frac{\\mathrm{d}}{\\mathrm{d} y} f(y)\\right|\\_{y=F[g]} \\label{eq:ChainRule}\\end{aligned}$$ \nUsing the result for linear functionals ([eq:LinearFunctional]):\n \n$$\\frac{\\delta}{\\delta z} \\left( \\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle \\right) \n= \\lambda^\\mathrm{fa}(z) + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} f(z|\\cdot)\\rangle$$ \n \n$$\\begin{gathered}\n\\frac{\\delta}{\\delta z} \\left( \\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle \\right) \\\\\n= \\lambda^\\mathrm{fa}(z) + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} f(z|\\cdot)\\rangle\\end{gathered}$$ \nSubsequently applying the chain rule ([eq:ChainRule]):\n \n$$\\frac{\\delta}{\\delta z} \\exp\\{ \\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle \\} \n= \\exp\\{ \\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle \\} \n\\cdot ( \\lambda^\\mathrm{fa}(z) + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} f(z|\\cdot)\\rangle )$$ \n \n$$\\begin{gathered}\n\\frac{\\delta}{\\delta z} \\exp\\{ \\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle \\} \\\\\n= \\exp\\{ \\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle \\} \\\\\n\\times ( \\lambda^\\mathrm{fa}(z) + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} f(z|\\cdot)\\rangle )\\end{gathered}$$ \nIterating the derivative, we find\n \n$$\\frac{\\delta}{\\delta Z} \\exp\\{ \\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle \\} \n= \\exp\\{ \\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle \\} \n\\cdot \\prod\\_{z\\in Z}\\left( \\lambda^\\mathrm{fa}(z) + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} f(z|\\cdot)\\rangle \\right)$$ \n \n$$\\begin{gathered}\n\\frac{\\delta}{\\delta Z} \\exp\\{ \\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle \\} \\\\\n= \\exp\\{ \\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle \\} \\\\\n\\times \\prod\\_{z\\in Z}\\left( \\lambda^\\mathrm{fa}(z) + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} f(z|\\cdot)\\rangle \\right)\\end{gathered}$$ \nSetting *g* = 0, and applying Lemma [lem:ScaledBernoulli], we obtain the desired result.\nLemma [lem:BernoulliUpdate] calculates the p.g.fl form of the Bayes update for an unnormalised Bernoulli distribution.\n[lem:BernoulliUpdate]\n \n$$\\begin{aligned}\n\\frac{\\delta}{\\delta Z}& w\\_{t|t-1}^{i,\\tilde{a}^i} \\Big( 1-r\\_{t|t-1}^{i,\\tilde{a}^i} \n + r\\_{t|t-1}^{i,\\tilde{a}^i}\\langle f\\_{t|t-1}^{i,\\tilde{a}^i},h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)\\rangle\\Big)\\bigg|\\_{g=0}\\notag \\\\\n=& \\begin{cases}\nw^{i,\\tilde{a}^i}\\_{t|t}(1-r^{i,\\tilde{a}^i}\\_{t|t}+r^{i,\\tilde{a}^i}\\_{t|t}\\langle f^{i,\\tilde{a}^i}\\_{t|t},h\\rangle), & Z=\\emptyset \\\\\nw^{i,a^i}\\_{t|t}(1-r^{i,a^i}\\_{t|t}+r^{i,a^i}\\_{t|t}\\langle f^{i,a^i}\\_{t|t},h\\rangle), & Z=\\{z\\_t^j\\} \\\\\n0, & \\mbox{otherwise}\n\\end{cases}\\end{aligned}$$ \n \n$$\\begin{aligned}\n\\frac{\\delta}{\\delta Z}& w\\_{t|t-1}^{i,\\tilde{a}^i} \\Big( 1-r\\_{t|t-1}^{i,\\tilde{a}^i} + \\notag\\\\\n& + r\\_{t|t-1}^{i,\\tilde{a}^i}\\langle f\\_{t|t-1}^{i,\\tilde{a}^i},h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)\\rangle\\Big)\\bigg|\\_{g=0}\\notag \\\\\n=& \\begin{cases}\nw^{i,\\tilde{a}^i}\\_{t|t}(1-r^{i,\\tilde{a}^i}\\_{t|t}+r^{i,\\tilde{a}^i}\\_{t|t}\\langle f^{i,\\tilde{a}^i}\\_{t|t},h\\rangle), & Z=\\emptyset \\\\\nw^{i,a^i}\\_{t|t}(1-r^{i,a^i}\\_{t|t}+r^{i,a^i}\\_{t|t}\\langle f^{i,a^i}\\_{t|t},h\\rangle), & Z=\\{z\\_t^j\\} \\\\\n0, & \\mbox{otherwise}\n\\end{cases}\\end{aligned}$$ \nwhere *w**t*∣*t**i*, *ã**i*, *r**t*∣*t**i*, *ã**i* and *f**t*∣*t**i*, *ã**i*(*x*) are given in ([eq:MissUpdateW]), ([eq:MissUpdatePex]) and ([eq:MissUpdateKin]) respectively, and *w**t*∣*t**i*, *a**i*, *r**t*∣*t**i*, *a**i* and *f**t*∣*t**i*, *a**i*(*x*) are given in ([eq:DetUpdateW]), ([eq:DetUpdatePex]) and ([eq:DetUpdateKin]) respectively.\nThe case with *Z* = ∅ results simply from Lemma [lem:ScaledBernoulli]. When *Z* = {*z**t**j*}, we observe that:\n \n⟨*f**t*∣*t* − 1*i*, *ã**i*, *h*(1 − *P*d + *P*d*p**g*)⟩ = ⟨*f**t*∣*t* − 1*i*, *ã**i*, *h*(1 − *P*d)⟩ + ⟨*f**t*∣*t* − 1*i*, *ã**i*, *h**P*d*p**g*⟩ \n \n$$\\begin{gathered}\n\\langle f\\_{t|t-1}^{i,\\tilde{a}^i},h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)\\rangle \\\\\n= \\langle f\\_{t|t-1}^{i,\\tilde{a}^i},h(1-P^{\\mathrm{d}})\\rangle + \n\\langle f\\_{t|t-1}^{i,\\tilde{a}^i},hP^{\\mathrm{d}} p\\_g\\rangle\\end{gathered}$$ \nwhere\n \n$$\\begin{aligned}\n\\langle f\\_{t|t-1}^{i,\\tilde{a}^i},hP^{\\mathrm{d}} p\\_g\\rangle \n&= \\int f\\_{t|t-1}^{i,\\tilde{a}^i}(x)h(x)P^{\\mathrm{d}}(x)\\int f(z|x)g(z)\\mathrm{d} z\\mathrm{d} x \\\\\n&= \\int g(z)\\int h(x)f\\_{t|t-1}^{i,\\tilde{a}^i}(x)P^{\\mathrm{d}}(x)f(z|x)\\mathrm{d} x\\mathrm{d} z\\end{aligned}$$ \n \n$$\\begin{aligned}\n\\langle & f\\_{t|t-1}^{i,\\tilde{a}^i},hP^{\\mathrm{d}} p\\_g\\rangle \\notag\\\\\n&= \\int f\\_{t|t-1}^{i,\\tilde{a}^i}(x)h(x)P^{\\mathrm{d}}(x)\\int f(z|x)g(z)\\mathrm{d} z\\mathrm{d} x \\\\\n&= \\int g(z)\\int h(x)f\\_{t|t-1}^{i,\\tilde{a}^i}(x)P^{\\mathrm{d}}(x)f(z|x)\\mathrm{d} x\\mathrm{d} z\\end{aligned}$$ \nSubsequently, we apply ([eq:LinearFunctional]) and Lemma [lem:ScaledBernoulli].\nLemma [lem:ConvolutionEquiv] shows that the product rule ([eq:ProductRule]) can be expressed equivalently in a form similar to ([eq:PreexistingTargets]). This equivalence is the point at which data association arises in the derivation; the terms of this sum can be interpreted as global association hypotheses. When we employ Lemma [lem:ConvolutionEquiv], *F*0*Z*[*g*] represents measurements arising from false alarms or previously unknown targets, while *F**i**Z*[*g*], *i* > 0 represents measurements from pre-existing tracks.\n[lem:ConvolutionEquiv] Suppose we are given set-parameterised functionals $F\\_i^Z[g]=\\frac{\\delta F\\_i}{\\delta Z}[g]$, *i* ∈ {0, ..., *n*}, such that *F*0*Z*[*g*] = ∏*z* ∈ *Z**f*0*z*[*g*] and if *i* ≥ 1 and ∣*Z*∣ ≥ 2 then *F**i**Z*[*g*] = 0. If *Z* = {*z*1, …, *z**m*}, then\n \n$$\\sum\\_{W\\_0 \\uplus \\cdots \\uplus W\\_n=Z}\\prod\\_{i=0}^n F\\_i^{W\\_i}[g]\n= \\sum\\_{\\bar{a}\\in{\\cal A}\\_t}\\prod\\_{i=1}^{n+m} F\\_i^{Z^{\\bar{a}^i}}[g]$$ \nwhere\n \n$$\\begin{gathered}\n{\\cal A}\\_t = \\bigg\\{\n(\\bar{a}^1,\\dots,\\bar{a}^{n+m})\n\\bigg| \n\\bar{a}^i\\in\\{0,\\dots,m\\},\\mbox{ for } i\\in\\{1,\\dots,n\\}, \n\\bar{a}^{n+j}\\in\\{0,j\\},\\mbox{ for } j\\in\\{1,\\dots,m\\}, \\\\\n{\\textstyle\\bigcup\\_{i=1}^{n+m}}{\\cal M}\\_t(i,\\bar{a}^i)=\\{(t,1),\\dots,(t,m)\\},\n{\\cal M}^t(i,\\bar{a}^i)\\cap{\\cal M}^t(i',\\bar{a}^{i'})=\\emptyset \\;\\forall\\; i\\neq i' \n\\bigg\\}\\end{gathered}$$ \n \n$$\\begin{gathered}\n{\\cal A}\\_t = \\bigg\\{\n(\\bar{a}^1,\\dots,\\bar{a}^{n+m})\n\\bigg| \n\\bar{a}^i\\in\\{0,\\dots,m\\},\\mbox{ for } i\\in\\{1,\\dots,n\\}, \\\\\n\\bar{a}^{n+j}\\in\\{0,j\\},\\mbox{ for } j\\in\\{1,\\dots,m\\}, \\\\\n{\\textstyle\\bigcup\\_{i=1}^{n+m}}{\\cal M}\\_t(i,\\bar{a}^i)=\\{(t,1),\\dots,(t,m)\\},\n\\\\\n{\\cal M}^t(i,\\bar{a}^i)\\cap{\\cal M}^t(i',\\bar{a}^{i'})=\\emptyset \\;\\forall\\; i\\neq i' \n\\bigg\\}\\end{gathered}$$ \nwith ${\\cal M}\\_t(i,0)=\\emptyset$ and ${\\cal M}\\_t(i,j)=\\{(t,j)\\}$. Finally,\n \n$$Z^{\\bar{a}^i} \\triangleq \\begin{cases}\n\\emptyset, & \\bar{a}^i = 0 \\\\\nz^{\\bar{a}^i}, & \\bar{a}^i \\in\\{1,\\dots,m\\}\n\\end{cases}$$ \nand for *j* ∈ {1, …, *m*},\n \n$$F\\_{n+j}^Z[g] \\triangleq \\begin{cases}\n1, & Z=\\emptyset \\\\\nf\\_0^{z^j}[g], & Z=\\{z^j\\} \\\\\n0, & \\mbox{otherwise}\n\\end{cases}$$ \nThe result follows from the construction of ${\\cal A}\\_t$ and the other components. To observe the correspondence of LHS and RHS terms, take $\\bar{a}=(\\bar{a}^1,\\dots,\\bar{a}^{n+m})\\in{\\cal A}\\_t$. Set $W\\_0=\\bigcup\\_{i=n+1}^{n+m}{\\cal M}\\_t(i,\\bar{a}^i)$, and for *i* ∈ {1, …, *n*} set $W\\_i={\\cal M}\\_t(i,a^i)$. By inspection, the corresponding LHS and RHS terms are equivalent. Similarly, take any non-zero LHS term, and for *i* ∈ {1, …, *n*} set\n \n$$\\bar{a}^i = \\begin{cases}\n0, & W^i = \\emptyset \\\\\nj, & W^i = \\{z^j\\} \n\\end{cases}$$ \nSimilarly, for *j* ∈ {1, …, *m*}, set\n \n$$\\bar{a}^{n+j} = \\begin{cases}\n0, & z\\_j\\notin W\\_0 \\\\\nj, & z\\_j\\in W\\_0\n\\end{cases}$$ \nAgain, by inspection, these terms will be in correspondence.\nWith these preliminary results, we are now ready to prove the main theorem.\n[Proof of Theorem [th:Update]] Following Lemma [lem:Structure], the necessary result is that *G**t*∣*t*mbm[*h*] in ([eq:UpdateMBMPGFl]) is made equal (or proportional) to ([eq:PreexistingTargets]) through the equalities made in the statement of the theorem (*i.e.*, that *G**t*∣*t*mbm[*h*] is a mixture of multi-Bernoulli distributions). Expanding the form of *G**t*∣*t* − 1mbm[*h*] using ([eq:PreexistingTargets]) and ([eq:GlobalHypWeight]) and applying the linearity property of differentials, we can write:\n \n$$\\label{eq:MBMUpdate1}\nG^\\mathrm{mbm}\\_{t|t}[h] \\propto\n\\sum\\_{\\tilde{a}\\in{\\cal A}^{t|t-1}} \\frac{\\delta}{\\delta Z\\_t}\\bigg( \n\\exp\\{\\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle\\} \n\\cdot \\prod\\_{i\\in{\\cal T}\\_{t|t-1}}w^{i,\\tilde{a}^i}\\_{t|t-1}G^{i,\\tilde{a}^i}\\_{t|t-1}[h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)]\\bigg)\\Bigg|\\_{g=0}$$ \n \n$$\\begin{gathered}\n\\label{eq:MBMUpdate1}\nG^\\mathrm{mbm}\\_{t|t}[h] \\propto\n\\sum\\_{\\tilde{a}\\in{\\cal A}^{t|t-1}} \\frac{\\delta}{\\delta Z\\_t}\\bigg( \n\\exp\\{\\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle\\} \\\\\n\\times \\prod\\_{i\\in{\\cal T}\\_{t|t-1}}w^{i,\\tilde{a}^i}\\_{t|t-1}G^{i,\\tilde{a}^i}\\_{t|t-1}[h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)]\\bigg)\\Bigg|\\_{g=0}\\end{gathered}$$ \nConsidering a single term in this sum (*i.e.*, a particular choice of *ã*), we apply the product rule ([eq:ProductRule]), taking derivatives with respect to the functional *g*, setting *n* = *n**t*∣*t* − 1, and\n \n$$\\begin{aligned}\nF\\_{\\tilde{a},0}[g,h] &= \\exp\\{\\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle\\} \\\\\nF\\_{\\tilde{a},i}[g,h] &= w^{i,\\tilde{a}^i}\\_{t|t-1}G^{i,\\tilde{a}^i}\\_{t|t-1}[h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)], \\; i>0\\end{aligned}$$ \nto obtain\n \n$$\\begin{gathered}\n\\frac{\\delta}{\\delta Z\\_t}\\bigg(\\exp\\{\\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle\\} \n\\cdot \\prod\\_{i\\in{\\cal T}\\_{t|t-1}}w^{i,\\tilde{a}^i}\\_{t|t-1}G^{i,\\tilde{a}^i}\\_{t|t-1}[h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)]\\bigg) \\\\\n=\n\\sum\\_{W\\_0 \\uplus \\cdots \\uplus W\\_n = Z\\_t}\n\\frac{\\delta F\\_{\\tilde{a},0}}{\\delta W\\_0}[g,h] \\cdots\n\\frac{\\delta F\\_{\\tilde{a},n}}{\\delta W\\_n}[g,h]\\end{gathered}$$ \n \n$$\\begin{gathered}\n\\frac{\\delta}{\\delta Z\\_t}\\bigg(\\exp\\{\\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle\\} \\\\\n\\times \\prod\\_{i\\in{\\cal T}\\_{t|t-1}}w^{i,\\tilde{a}^i}\\_{t|t-1}G^{i,\\tilde{a}^i}\\_{t|t-1}[h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)]\\bigg) \\\\\n=\n\\sum\\_{W\\_0 \\uplus \\cdots \\uplus W\\_n = Z\\_t}\n\\frac{\\delta F\\_{\\tilde{a},0}}{\\delta W\\_0}[g,h] \\cdots\n\\frac{\\delta F\\_{\\tilde{a},n}}{\\delta W\\_n}[g,h]\\end{gathered}$$ \nSubsequently applying Lemma [lem:ConvolutionEquiv], we obtain:\n \n$$\\begin{gathered}\n\\frac{\\delta}{\\delta Z\\_t}\\bigg(\\exp\\{\\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle\\} \\cdot \\prod\\_{i\\in{\\cal T}\\_{t|t-1}}w^{i,\\tilde{a}^i}\\_{t|t-1}G^{i,\\tilde{a}^i}\\_{t|t-1}[h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)]\\bigg) \\\\\n= \\sum\\_{\\bar{a}\\in{\\cal A}\\_t}\\prod\\_{i=1}^{n\\_{t|t-1}+m\\_t} F\\_{\\tilde{a},i}^{Z^{\\bar{a}^i}}[g,h]\\end{gathered}$$ \n \n$$\\begin{gathered}\n\\frac{\\delta}{\\delta Z\\_t}\\bigg(\\exp\\{\\langle\\lambda^\\mathrm{fa},g\\rangle + \\langle\\lambda^u\\_{t|t-1},h P^{\\mathrm{d}} p\\_g\\rangle\\} \\\\\n\\times \\prod\\_{i\\in{\\cal T}\\_{t|t-1}}w^{i,\\tilde{a}^i}\\_{t|t-1}G^{i,\\tilde{a}^i}\\_{t|t-1}[h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)]\\bigg) \\\\\n= \\sum\\_{\\bar{a}\\in{\\cal A}\\_t}\\prod\\_{i=1}^{n\\_{t|t-1}+m\\_t} F\\_{\\tilde{a},i}^{Z^{\\bar{a}^i}}[g,h]\\end{gathered}$$ \nSubstituting this into ([eq:MBMUpdate1]), we obtain\n \n$$G^\\mathrm{mbm}\\_{t|t}[h] \\propto \\sum\\_{\\tilde{a}\\in{\\cal A}^{t|t-1},\\bar{a}\\in{\\cal A}\\_t}\\prod\\_{i=1}^{n\\_{t|t-1}+m\\_t} F\\_{\\tilde{a},i}^{Z^{\\bar{a}^i}}[g,h]\\bigg|\\_{g=0}$$ \nObserving that the set of events $a\\in{\\cal A}^{t|t}$ covers the same events as $(\\tilde{a}\\in{\\cal A}^{t|t-1},\\bar{a}\\in{\\cal A}\\_t)$, we arrive at our final result\n \n$$G^\\mathrm{mbm}\\_{t|t}[h] \\propto \\sum\\_{a\\in{\\cal A}^{t|t}} \\prod\\_{i\\in{\\cal T}\\_{t|t}}w^{i,a^i}\\_{t|t}G\\_{t|t}^{i,a^i}[h]$$ \nwhere, for the term $a\\in{\\cal A}^{t|t}$ corresponding to $\\tilde{a}\\in{\\cal A}^{t|t-1}$ and $\\bar{a}\\in{\\cal A}\\_t$, (matching terms are identified by ${\\cal M}^{t}(i,a^i)={\\cal M}^{t-1}(i,\\tilde{a}^i)\\cup{\\cal M}\\_t(i,\\bar{a}^i)$)\n \n$$w^{i,a^i}\\_{t|t}G\\_{t|t}^{i,a^i}[h] = F\\_{\\tilde{a},i}^{Z^{\\bar{a}^i}}[g,h]\\bigg|\\_{g=0} \n= w\\_{t|t}^{i,a^i}(1-r^{i,a^i}\\_{t|t} + r^{i,a^i}\\_{t|t}\\langle f^{i,a^i}\\_{t|t},h\\rangle)$$ \n \n$$\\begin{aligned}\nw^{i,a^i}\\_{t|t}G\\_{t|t}^{i,a^i}[h] &= F\\_{\\tilde{a},i}^{Z^{\\bar{a}^i}}[g,h]\\bigg|\\_{g=0} \\\\\n&= w\\_{t|t}^{i,a^i}(1-r^{i,a^i}\\_{t|t} + r^{i,a^i}\\_{t|t}\\langle f^{i,a^i}\\_{t|t},h\\rangle)\\end{aligned}$$ \nLemmas [lem:PoisUpdate] and [lem:BernoulliUpdate] confirm that the definitions in the statement of Theorem [th:Update] achieve this equality with the respective parameters given in ([eq:MissUpdateW])-([eq:MissUpdateKin]) (missed detections on existing tracks, *i.e.*, *i* ∈ {1, …, *n**t*∣*t* − 1}, *a**i* ∈ {1, …, *h**t*∣*t* − 1*i*}), ([eq:DetUpdateW])-([eq:DetUpdateKin]) (detections updating existing tracks, *i.e.*, *i* ∈ {1, …, *n**t*∣*t* − 1}, *a**i* ∈ {*h**t*∣*t* − 1*i* + 1, …, *h**t*∣*t**i*}), ([eq:NewTargetNonExistWQ]) (new tracks without measurements, *i.e.*, *i* ∈ {*n**t*∣*t* − 1 + 1, …, *n**t*∣*t* − 1 + *m**t*}, *a**i* = 1), and ([eq:PoisUpdateW])-([eq:PoisUpdateKin]) (updates of new tracks, *i.e.*, *i* ∈ {*n**t*∣*t* − 1 + 1, …, *n**t*∣*t* − 1 + *m**t*}, *a**i* = 2).\nImplementation pseudo-code\n==========================\nThe simplest implementation of the algorithm uses a Gaussian approximation of each track, and a Gaussian mixture approximation of the PPP intensity of unknown targets. Pseudo-code for this is provided in this section for completeness; an earlier variant appeared in along with experimental results. Note that the experiments in Section [sec:Results] utilised a Gaussian mixture implementation, which is complicated by hypothesis management, *etc*. A Matlab implementation of the simplified version described in this section can be found in the ancillary files to.\nAdditional assumptions include:\n* Probability of detection and survival are uniform, *i.e.*, *P*d(*x*) = *P*d, *P*s(*x*) = *P*s\n* Target-originated measurements follow a linear-Gaussian model, *z**t* = **H***x**t* + *v**t*, where *v**t* ∼ N{*v**t*; 0, **R**}\n* Target dynamics follow a linear-Gaussian model, *x**t* = **F***x**t* − 1 + *w**t*, where *w**t* ∼ N{*w**t*; 0, **Q**}\n* Target birth intensity is a Gaussian mixture: \n*λ*b(*x**t*) = ∑*k* = 1*n**b**λ**b*, *k*N{*x**t*; *x̄**b*, *k*, **P***b*, *k*}\nThe state maintained by the algorithm is:\n* A Gaussian mixture representation of the intensity of the PPP of unknown targets, \n*λ**t*∣*t*ʹ*u*(*x**t*) = ∑*k* = 1*n**t*∣*t*ʹ*u**λ**t*∣*t*ʹ*u*, *k*N{*x**t*; *x̄**t*∣*t*ʹ*u*, *k*, **P***t*∣*t*ʹ*u*, *k*}\n* *n**t*∣*t*ʹ tracks (Bernoulli components), each comprised of a probability of existence *r**t*∣*t*ʹ*i*, a mean *x̄**t*∣*t*ʹ*i* and covariance **P***t*∣*t*ʹ*i*\nThe algorithm operates by executing the prediction step in Fig. [fig:algPrediction], the update step in Fig. [fig:algUpdate], LBP calculation of marginal association probabilities Fig. [fig:algLBP], and then either the TOMB/P procedure for forming new tracks, Fig. [fig:algTOMMeMBerP], or the MOMB/P procedure for forming new tracks, Fig. [fig:algMOMMeMBerP]. Steps for eliminating tracks with a low probability of existence and components in *λ**t*∣*t*ʹ*u*(*x*) with a low weight are not shown. Steps for extracting estimates are also not shown (TOMB/P outputs the mean estimate of each track with probability of existence greater than a threshold, while MOMB/P calculates the MAP cardinality estimate of the multi-Bernoulli distribution *n̂*, and outputs the mean estimates of the *n̂* components with the highest probability of existence). Between update and re-forming of tracks, (*m**t* + 1) hypotheses exist for each pre-existing track *i*, parameterised by a hypothesis weight *w**t*∣*t**i*, *a*, and hypothesis-conditioned probability of existence *r**t*∣*t**i*, *a*, mean *x̄**t*∣*t*ʹ*i*, *a* and covariance **P***t*∣*t*ʹ*i*, *a*. A single hypothesis exists for the new tracks, denoted by *a* = 1 (the non-existence hypothesis in ([eq:NewTargetNonExistWQ]) is accounted for without explicitly storing its weight, probability of existence, mean and covariance).\n [fig:algPrediction]\n [fig:algUpdate]\n [fig:algLBP]\n [fig:algTOMMeMBerP]\n [fig:algMOMMeMBerP]\n[]Jason L. Williams (S’01–M’07) received degrees of BE(Electronics)/BInfTech from Queensland University of Technology in 1999, MSEE from the United States Air Force Institute of Technology in 2003, and PhD from Massachusetts Institute of Technology in 2007.\nHe worked for several years as an engineering officer in the Royal Australian Air Force, before joining Australia’s Defence Science and Technology Organisation in 2007. He is also an adjunct senior lecturer at the University of Adelaide. His research interests include target tracking, sensor resource management, Markov random fields and convex optimisation.\n---\n1. Manuscript received September 11, 2012; revised December 7, 2012; resubmitted August 15, 2013; revised May 22, 2014 and November 28, 2014; accepted December 1, 2014. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Ba-Ngu Vo.[↩](#fnref1)\n2. The author is with the National Security, Intelligence, Surveillance and Reconnaissance Division, Defence Science and Technology Organisation, Australia, and the School of Electrical and Electronic Engineering, University of Adelaide, Australia (e-mail: jason.williams@dsto.defence.gov.au).[↩](#fnref2)\n3. We use the convention of capital letters (*e.g.*, *X*) representing sets, and lower-case letters representing single members (*e.g.*, *x*).[↩](#fnref3)\n4. Note that, whereas the argument of a pgf *G*(*s*) is the scalar *s*, the argument of a p.g.fl *G*[*h*] is the function *h*(*x*).[↩](#fnref4)\n5. A measure theoretic exploration of this can be found in.[↩](#fnref5)\n6. For each track, there is also a hypothesis that the target *never* existed, and thus uses no measurements.[↩](#fnref6)\n7. The term *track* in refers to the single-target hypothesis of definition [def:SingleTargetHypothesis], while the *target tree*, or the set of *track hypotheses* in refers to the track of definition [def:Track].[↩](#fnref7)\n8. A hypothesis at the previous time with *r**t*∣*t* − 1*i*, *a**i* = 0 need not be updated since the posterior weight in ([eq:DetUpdateW]) would be zero. For simplicity, the hypothesis numbering does not account for this exclusion.[↩](#fnref8)\n9. We will see that *f**t*∣*t**i*(*x*) is a marginal distribution averaging over association hypotheses. The importance of there being a single association hypothesis is that constraints between hypotheses in different tracks are discarded.[↩](#fnref9)\n10. JPDA/JIPDA additionally approximates the posterior distribution of each target as being Gaussian.[fn:JPDAGaussian][↩](#fnref10)\n11. We also incorporate the kinematic distribution of targets into the initial update in ([eq:PoisUpdateKin]); this distribution is assumed to be uniform in.[↩](#fnref11)\n12. As opposed to definition [def:Track], which defines a track as a collection of single-target hypotheses which all *begin* with the same measurement, the Bernoulli components maintained by the MOMB/P consist of single-target hypotheses which all *end* with the same measurement, i.e., all hypothesise the same final detection.[↩](#fnref12)\n13. Since we assumed that the prior was multi-Bernoulli, our hypotheses only consider the latest time step. For clarity of explanation, we initially omit missed detections, new targets and false alarms.[↩](#fnref13)\n14. Again, practical implementations will require additional approximations in pruning unlikely tracks, reducing the number of components in the mixture-conditioned distributions, and in calculating the marginal distributions *P**t*∣*t**j*(*b**j*) themselves.[↩](#fnref14)\n15. See for references.[↩](#fnref15)\n16. As discussed in Section [sec:Derivation], hypotheses in our derivation are multi-Bernoulli, and thus they imply a distribution over cardinality; in contrast, MHT hypotheses uniquely determine cardinality. This difference permits us to represent many MHT hypotheses (*e.g.*, those proposing missed detection or death of the same targets) via a single multi-Bernoulli hypothesis.[↩](#fnref16)\n17. Histories are only considered to be identical if the last two measurement updates are the same—*i.e.*, missed detection events are not considered to be an element in the history. A maximum of 1000 single-target hypotheses are retained (in total), and hypotheses with *w**t*∣*t**i*, *a**i**r**t*∣*t**i*, *a**i* ≤ 10− 4 are deleted.[↩](#fnref17)\n18. The estimate is the mean of the Gaussian component with the highest weight.[fn:Estimate][↩](#fnref18)\n19. OSPA provides a single metric for measuring distance between two finite sets, thus summarising errors in both position and cardinality in a single value.[↩](#fnref19)\n20. With target survival/death but without target birth, as this is incorporated in the separate Poisson component *G**t*∣*t* − 1ppp[*h*].[↩](#fnref20)\n21. with *a* ≥ 0, *b*(*x*) ≥ 0, 0 ≤ ∫*b*(*x*)d*x* < ∞. If ∫*b*(*x*)d*x* = 0 then any choice of *f*(*x*) will satisfy ([eq:ScaledBernoulli]). If *a* = 0 and ∫*b*(*x*)d*x* = 0 then any choice of *r* and *f*(*x*) will suffice.[↩](#fnref21)\n\nMarginal multi-Bernoulli filters: RFS derivation of MHT, JIPDA and association-based MeMBer\n===========================================================================================\nRecent developments in random finite sets (RFSs) have yielded a variety of tracking methods that avoid data association. This paper derives a form of the full Bayes RFS filter and observes that data association is implicitly present, in a data structure similar to MHT. Subsequently, algorithms are obtained by approximating the distribution of associations. Two algorithms result: one nearly identical to JIPDA, and another related to the MeMBer filter. Both improve performance in challenging environments.\nrandom finite sets, conjugate prior, multiple hypothesis tracking, joint probabilistic data association, MeMBer, Poisson point process, multi-Bernoulli, loopy belief propagation\nIntroduction\n============\nRecently, much work has been devoted to random finite set (RFS)-based approximations such as the probability hypothesis density (PHD), cardinalised PHD (CPHD) and multiple target multi-Bernoulli (MeMBer) filters. A key characteristic of these methods has been their tractability, which is the consequence of clever approximations that avoid data association, *i.e.*, reasoning over the correspondence of measurements and targets. For example, the PHD approximates the posterior distribution as a Poisson point process (PPP), for which updates can be calculated without data association. The CPHD improves upon this by calculating the cardinality distribution, and using an independently, identically distributed (iid) cluster process approximation for which updates can be calculated efficiently; CPHD has been shown to improve performance significantly over PHD in challenging environments (*e.g.*, ). The MeMBer makes approximations in its derivation that avoid association and are reasoned to be of little consequence when the false alarm rate is low; these approximations introduced a significant cardinality bias that was corrected in the cardinality balanced MeMBer (CB-MeMBer).\nOutside of the RFS field, two major approaches are joint probabilistic data association (JPDA) and multiple hypothesis tracking (MHT) ; both explicitly formulate and reason over association hypotheses, *i.e.*, hypotheses for the correspondence of measurements and targets. JPDA seeks to marginalise over the random variables representing association in order to calculate the marginal distribution of each track; joint integrated PDA (JIPDA) and joint integrated track splitting (JITS) extend this by incorporating target existence as an additional random variable to be estimated. MHT seeks to calculate the most likely association hypothesis, returning a tracking estimate conditioned on the hypothesis. Track oriented MHT (TOMHT) provides a tractable approach for implementing MHT by maintaining a hypothesis tree for each track, and finding the most likely global hypothesis indirectly (*e.g.*, using Lagrangian relaxation ).\nContributions\n-------------\nThis paper presents a derivation of the full Bayes RFS filter under commonly invoked assumptions (point measurements, PPP birth and false alarms), and proposes approximations that yield tractable solutions of the tracking problem. We observe that, while the RFS framework avoids the need for *explicitly* modelling data association, it *implicitly* arises in the full Bayes RFS filter. Contributions include:\n* The derivation in Section [sec:Derivation] proves a conjugate prior form (*i.e.*, a form that is preserved by prediction and update) for the full Bayes RFS filter in the model of interest. Various aspects of the form are similar to TOMHT (maintenance of a hypothesis tree for each track) and (use of a PPP to model targets which so-far remain undetected). In effect, the result provides an RFS-based derivation of the data structure employed by TOMHT, where the posterior involves a summation over association hypotheses that are similar to those utilised in MHT.\n* Observing that data association implicitly arises in the derivation, we subsequently seek to approximate the discrete distribution of data association. The first approximation yields the track oriented marginal MeMBer/Poisson (TOMB/P) filter, which is very similar to JITS and JIPDA (and related to the RFS-based derivation in ); the differences relate to the inclusion of a Bayesian model of target birth in our RFS derivation. Thus, with minor changes, the popular JITS/JIPDA algorithms (and extensions thereof) can be derived within the RFS framework.\n* The second approximation yields the measurement oriented marginal MeMBer/Poisson (MOMB/P) filter, which collects all hypotheses using a given measurement into a single Bernoulli component in a manner similar to the MeMBer filter of, but weighting hypotheses using the marginal association weights. Both TOMB/P and MOMB/P can be implemented using the high quality, tractable approximation of the marginal measurement-to-track association probabilities described in.\n* Section [sec:Results] demonstrates the proposed methods in a challenging scenario, showing that improved performance is obtained in comparison to CPHD and CB-MeMBer (particularly in cases with a lower probability of detection) for a similar computational load.\nPreliminary versions of this paper were available in.\nBackground\n==========\nRFS-based methods have been developed in order to conduct statistical inference in problems in which the variables of interest and/or the observations form finite sets. The methods are particularly applicable to tracking since they address two of the major challenges involved:\n* The number of targets present in the scene is unknown\n* Measurements are unordered, and measurement-to-target correspondence is unknown\nProblems such as this can be conveniently formulated by modelling the system state as a set of target states[3](#fn3) *X**t* = {*x**t*1, …, *x**t**n**t*} (at time *t*), and incorporating set-valued measurements *Z**t* = {*z**t*1, …, *z**t**m**t*}. RFS densities encode both uncertainty in the number of targets, and their states. A RFS density can be constructed from a cardinality distribution *p*(*n*), *n* ≥ 0, and a series of cardinality-conditioned joint distributions *f**n*(*x*1, …, *x**n*), yielding\n \n*f*({*x*1, …, *x**n*}) = *p*(*n*)∑*π**f**n*(*x**π*(1), …, *x**π*(*n*)) \nwhere the sum over *π* spans the *n*! permutation functions, ensuring that *f*(*X*) is permutation invariant. Bayesian estimation can be performed in this framework (conceptually, at least) by interleaving prediction and update steps: \n \n$$\\begin{aligned}\nf\\_{t|t-1}(X) &= \\int{f\\_{t|t-1}(X|X')f\\_{t-1|t-1}(X')\\delta X'} \\label{eq:RFSPrediction} \\\\\nf\\_{t|t}(X) &= \\frac{f(Z\\_t|X) f\\_{t|t-1}(X)}{f\\_{t|t-1}(Z\\_t)} \\propto f(Z\\_t|X) f\\_{t|t-1}(X) \\label{eq:RFSUpdate}\\end{aligned}$$ \nwhere *f**t*∣*t*ʹ(*X*) is the RFS density at time *t* given measurements up to and including time *t*ʹ (we leave off conditioning on the measurement set history *Z**t*ʹ = (*Z*1, …, *Z**t*ʹ) throughout, as this is implicit in the second subscript *f**t*∣*t*ʹ), *f**t*∣*t* − 1(*X*∣*X*ʹ) is the RFS transition density, *f*(*Z*∣*X*) is the RFS measurement likelihood, and the set integral is defined as: \n \n$$\\int{g(X) \\delta X}\n\\triangleq g(\\emptyset) + \\sum\\_{n=1}^{\\infty}\\frac{1}{n!}\\int{g(\\{x\\_1,\\dots,x\\_n\\})\\mathrm{d} x\\_1 \\cdots \\mathrm{d} x\\_n} \\label{eq:SetIntegral}$$ \nProbability generating functions (pgfs) are widely used in queueing systems, *etc*, as they greatly simplify analysis of independent random variables (*e.g.*, ). Likewise, probability generating functionals[4](#fn4) (p.g.fls) simplify analysis of RFSs that possess independence relationships. The p.g.fl of a RFS density *f* is defined as: \n \n*G*[*h*] = ∫*h**X**f*(*X*)*δ**X* \nwhere $h^X \\triangleq \\prod\\_{x\\in X}h(x)$. Just as the Fourier, Laplace and *z* transforms are alternative representations of signals, a p.g.fl is an alternative representation of a RFS distribution. If *X* and *Y* are independent RFSs and *Z* = *X* ∪ *Y*, then \n \n$$\\label{eq:Convolution}\n\\begin{split}\nf\\_Z(Z) &= \\sum\\_{X\\subseteq Z}f\\_X(X) f\\_Y(Z-X) \\\\\nG\\_Z[h] &= G\\_X[h] G\\_Y[h] \n\\end{split}$$ \nThis convolution identity is analogous to the result that the pgf of the sum of two independent variables *z* = *x* + *y* is *G**z*(*s*) = *G**x*(*s*)*G**y*(*s*).\nThis work utilises two basic forms of RFS distributions: PPPs and Bernoulli processes. A PPP with intensity function *λ*(*x*) has RFS density and p.g.fl: \n \n$$\\label{eq:Poisson}\n\\begin{split}\nf(X) &= \\exp\\left\\{-\\int{\\lambda(x)\\mathrm{d} x}\\right\\} \\cdot \\prod\\_{x\\in X} \\lambda(x) \\\\\nG[h] &= \\exp\\{\\langle \\lambda,h-1\\rangle\\} \\propto \\exp\\{\\langle \\lambda,h\\rangle\\} \n\\end{split}$$ \nwhere ⟨*λ*, *h*⟩ denotes the inner product of *λ*( ⋅ ) and *h*( ⋅ ):\n \n$$\\langle \\lambda,h\\rangle \\triangleq \\int{\\lambda(x)h(x)\\mathrm{d} x}$$ \nand thus ⟨*λ*, *h* − 1⟩ = ∫*λ*(*x*)*h*(*x*)d*x* − ∫*λ*(*x*)d*x*.\nA Bernoulli process with probability of existence *r* and existence-conditioned probability density function (PDF) *f*(*x*) has RFS density and p.g.fl:\n \n$$\\label{eq:Bernoulli}\n\\begin{split}\nf(X) &= \\begin{cases}\n1-r, & X = \\emptyset \\\\\nr \\cdot f(x), & X = \\{x\\} \\\\\n0, & \\mbox{otherwise}\n\\end{cases} \\\\\nG[h] &= 1 - r + r \\cdot \\langle f,h\\rangle \n\\end{split}$$ \nBy ([eq:Bernoulli]) and ([eq:Convolution]), we can write the p.g.fl of a multi-Bernoulli process, *i.e.*, the process resulting from the union *Y* = ⋃*i* = 1*N**X**i* of independent Bernoulli processes *X**i*:\n \n*G**Y*[*h*] = ∏*i* = 1*N*(1 − *r**i* + *r**i*⟨*f**i*, *h*⟩) \nwhere *r**i* and *f**i*(*x*) are the existence probability and existence-conditioned PDF of the *i*-th Bernoulli process *X**i*. The RFS density of a multi-Bernoulli process is more difficult to deal with due to the sum over permutations. If *X* = {*x*1, …, *x**n*}, gives the multi-Bernoulli form as:\n \n$$f(X) = \\left[\\prod\\_{i=1}^N(1-r\\_i)\\right]\\cdot\\sum\\_{1\\leq i\\_1\\neq\\cdots\\neq i\\_n\\leq N} \\prod\\_{k=1}^n\\left[\\frac{r\\_{i\\_k}}{1-r\\_{i\\_k}}f\\_{i\\_k}(x\\_i) \\right]$$ \nAn alternative form for this expression is:\n \n$$\\label{eq:MultiBernoulli}\nf(X) = \\sum\\_{\\alpha\\in{\\cal P}\\_N^n}\n\\prod\\_{i=1}^N f\\_i(X\\_{\\alpha(i)})$$ \nwhere *f**i*(*X*) is the Bernoulli RFS density of the form ([eq:Bernoulli]) with parameters *r**i* and *f**i*(*x*), and ${\\cal P}\\_N^n$ is the set of all functions\n \n$$\\begin{gathered}\n\\label{eq:DefinitionP}\n{\\cal P}\\_N^n = \\big\\{\\alpha : \\{1,\\dots,N\\} \\rightarrow \\{0,\\dots,n\\} \\big| \n\\{1,\\dots,n\\}\\subseteq\\alpha(\\{1,\\dots,N\\}), \\\\\n\\mbox{ and if } \\alpha(i)>0,\\; i\\neq j \\mbox{ then } \\alpha(i)\\neq\\alpha(j)\n\\big\\}\\end{gathered}$$ \n \n$$\\begin{gathered}\n\\label{eq:DefinitionP}\n{\\cal P}\\_N^n = \\big\\{\\alpha : \\{1,\\dots,N\\} \\rightarrow \\{0,\\dots,n\\} \\big| \\\\\n\\{1,\\dots,n\\}\\subseteq\\alpha(\\{1,\\dots,N\\}), \\\\\n\\mbox{ and if } \\alpha(i)>0,\\; i\\neq j \\mbox{ then } \\alpha(i)\\neq\\alpha(j)\n\\big\\}\\end{gathered}$$ \nThis is the set of permutation-like functions that map exactly one Bernoulli component index onto each *k* ∈ {1, …, *n*}, and the remaining Bernoulli component indices onto 0. The sets *X**α*(*i*) are defined as:\n \n$$\\label{eq:BernoulliSet}\nX\\_{\\alpha(i)} \\triangleq \\begin{cases}\n\\emptyset, & \\alpha(i) = 0 \\\\\n\\{x\\_{\\alpha(i)}\\}, & \\alpha(i)>0\n\\end{cases}$$ \nDynamics model and prediction step\n----------------------------------\nAssumption [ass:Dynamics] describes the dynamics model that we utilise in this work.\nThe multiple target state evolves according to the following time dynamics process:\n* Targets arrive at each time according to a non-homogeneous PPP with birth intensity *λ*b(*x*), independent of existing targets\n* Targets depart according to iid Markovian processes; the survival probability in state *x* is *P*s(*x*)\n* Target motion follows iid Markovian processes; the single-target transition PDF is *f**t*∣*t* − 1(*x*∣*x*ʹ)\n[ass:Dynamics]\nAssumption [ass:Dynamics] describes a multiple target dynamics process which is the union of a PPP (describing arrival of new targets), and an independent Bernoulli process for each existing target conditioning on the prior existence and state, *G*[*h*∣*x*ʹ] = 1 − *P*s(*x*ʹ) + *P*s(*x*ʹ)*p**h*(*x*ʹ), where $p\\_h(x') \\triangleq \\int{h(x) f\\_{t|t-1}(x|x')\\mathrm{d} x}$. Thus, the multiple target transition p.g.fl is: \n \n$$\\begin{aligned}\nG[h|X'] &= \\int{h^X f\\_{t|t-1}(X|X') \\delta X} \\\\\n&= \\exp\\{\\langle\\lambda^\\mathrm{b},h-1\\rangle\\} \\prod\\_{x'\\in X'}\\{1 - P^{\\mathrm{s}}(x') + P^{\\mathrm{s}}(x') p\\_h(x')\\} \\label{eq:PGFLTransition}\\end{aligned}$$ \n \n$$\\begin{aligned}\n\\lefteqn{G[h|X'] = \\int{h^X f\\_{t|t-1}(X|X') \\delta X}} \\\\\n&= \\exp\\{\\langle\\lambda^\\mathrm{b},h-1\\rangle\\} \\prod\\_{x'\\in X'}\\{1 - P^{\\mathrm{s}}(x') + P^{\\mathrm{s}}(x') p\\_h(x')\\} \\label{eq:PGFLTransition}\\end{aligned}$$ \nTaking the p.g.fl of ([eq:RFSPrediction]), changing the order of integration and substituting ([eq:PGFLTransition]), we obtain the p.g.fl form of the prediction equation: \n \n$$\\begin{aligned}\nG\\_{t|t-1}[h] &\\triangleq \\int{h^X f\\_{t|t-1}(X) \\delta X} \\\\\n&= \\iint{h^X f\\_{t|t-1}(X|X')\\delta X f\\_{t-1|t-1}(X') \\delta X'} \\\\\n&= \\exp\\{\\langle\\lambda^\\mathrm{b},h-1\\rangle\\}\\cdot \\int{\\{1-P^{\\mathrm{s}} + P^{\\mathrm{s}} p\\_h\\}^{X'}f\\_{t-1|t-1}(X')\\delta X'} \\\\\n&= \\exp\\{\\langle\\lambda^\\mathrm{b},h-1\\rangle\\} \\cdot G\\_{t-1|t-1}[1-P^{\\mathrm{s}} + P^{\\mathrm{s}} p\\_h] \\label{eq:PGFLPrediction}\\end{aligned}$$ \n \n$$\\begin{aligned}\n\\lefteqn{G\\_{t|t-1}[h] \\triangleq \\int{h^X f\\_{t|t-1}(X) \\delta X}} \\\\\n&= \\iint{h^X f\\_{t|t-1}(X|X')\\delta X f\\_{t-1|t-1}(X') \\delta X'} \\\\\n&= \\exp\\{\\langle\\lambda^\\mathrm{b},h-1\\rangle\\}\\cdot \\int{\\{1-P^{\\mathrm{s}} + P^{\\mathrm{s}} p\\_h\\}^{X'}f\\_{t-1|t-1}(X')\\delta X'} \\\\\n&= \\exp\\{\\langle\\lambda^\\mathrm{b},h-1\\rangle\\} \\cdot G\\_{t-1|t-1}[1-P^{\\mathrm{s}} + P^{\\mathrm{s}} p\\_h] \\label{eq:PGFLPrediction}\\end{aligned}$$ \nMeasurement update step and measurement model\n---------------------------------------------\nIn analogy to the similar result for pgfs, the RFS distribution can be recovered from the p.g.fl via the iterated functional derivative: \n \n$$\\label{eq:PGFLInversion}\nf(X) = \\left.\\frac{\\delta}{\\delta X}G[h]\\right|\\_{h=0} = \\left.\\frac{\\delta^{|X|}}{\\prod\\_{x\\in X}\\delta x}G[h]\\right|\\_{h=0}$$ \nwhere $\\frac{\\delta}{\\delta x}G[h] \\triangleq \\lim\\_{\\epsilon\\downarrow 0}\\frac{G[h + \\epsilon\\delta\\_x] - G[h]}{\\epsilon}$, and *δ**x*ʹ(*x*) is a Dirac delta function concentrated at *x* = *x*ʹ.[5](#fn5) Thus we can recover the p.g.fl of the updated distribution ([eq:RFSUpdate]) via \n \n$$\\begin{aligned}\nG\\_{t|t}[h] &\\triangleq \\int{h^X f\\_{t|t}(X)\\delta X} \\\\\n&\\propto \\int{h^X f(Z\\_t|X)f\\_{t|t-1}(X)\\delta X} \\\\\n&= \\left.\\frac{\\delta}{\\delta Z\\_t}\\iint{h^X g^{Z\\_t} f(Z\\_t|X)f\\_{t|t-1}(X)\\delta X\\delta Z\\_t}\\right|\\_{g=0} \\\\\n&= \\left.\\frac{\\delta}{\\delta Z\\_t}\nF[g,h]\n\\right|\\_{g=0} \\label{eq:PGFLUpdate}\\end{aligned}$$ \n \n$$\\begin{aligned}\n\\lefteqn{G\\_{t|t}[h] \\triangleq \\int{h^X f\\_{t|t}(X)\\delta X}} \\\\\n&\\propto \\int{h^X f(Z\\_t|X)f\\_{t|t-1}(X)\\delta X} \\\\\n&= \\left.\\frac{\\delta}{\\delta Z\\_t}\\iint{h^X g^{Z\\_t} f(Z\\_t|X)f\\_{t|t-1}(X)\\delta X\\delta Z\\_t}\\right|\\_{g=0} \\\\\n&= \\left.\\frac{\\delta}{\\delta Z\\_t}\nF[g,h]\n\\right|\\_{g=0} \\label{eq:PGFLUpdate}\\end{aligned}$$ \nwhere the derivative with respect to *Z**t* operates on the corresponding functional *g*, and *F*[*g*, *h*] is the joint p.g.fl of measurements *Z**t* (functional *g*) and targets *X* (functional *h*). Assumption [ass:Measurement] describes the measurement model that we utilise in this work.\nThe multiple target measurement process is as follows:\n* Each target may give rise to at most one measurement; probability of detection in state *x* is *P*d(*x*)\n* Each measurement is the result of at most one target\n* False alarm measurements arrive according to a non-homogeneous PPP with intensity *λ*fa(*z*), independent of targets and target-related measurements\n* Each target-derived measurement is independent of all other targets and measurements conditioned on its corresponding target; the single target measurement likelihood is *f*(*z*∣*x*)\n[ass:Measurement]\nIn analogy with ([eq:PGFLTransition]), the p.g.fl of the measurement likelihood *f*(*Z*∣*X*) is \n \n$$\\begin{aligned}\nG[g|X] &= \\int{g^Z f(Z|X) \\delta Z} \\\\\n&= \\exp\\{\\langle\\lambda^\\mathrm{fa},g-1\\rangle\\}\\prod\\_{z\\in Z}\\{1-P^{\\mathrm{d}}(x) + P^{\\mathrm{d}}(x)p\\_g(x)\\} \\label{eq:PGFLMeasurement}\\end{aligned}$$ \n \n$$\\begin{aligned}\n\\lefteqn{G[g|X] = \\int{g^Z f(Z|X) \\delta Z}} \\\\\n&= \\exp\\{\\langle\\lambda^\\mathrm{fa},g-1\\rangle\\}\\prod\\_{x\\in X}\\{1-P^{\\mathrm{d}}(x) + P^{\\mathrm{d}}(x)p\\_g(x)\\} \\label{eq:PGFLMeasurement}\\end{aligned}$$ \nwhere $p\\_g(x) \\triangleq \\int{g(z)f(z|x)\\mathrm{d} z}$. Thus: \n \n$$\\begin{aligned}\nF[g,h] &= \\int{h^X G[g|X] f\\_{t|t-1}(X)\\delta X} \\\\\n&= \\exp\\{\\langle\\lambda^\\mathrm{fa},g-1\\rangle\\}\\int{\\{h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)\\}^X f\\_{t|t-1}(X)\\delta X} \\\\\n&= \\exp\\{\\langle\\lambda^\\mathrm{fa},g-1\\rangle\\} G\\_{t|t-1}[h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)] \\label{eq:JointPGFL}\\end{aligned}$$ \n \n$$\\begin{aligned}\n\\lefteqn{F[g,h] = \\int{h^X G[g|X] f\\_{t|t-1}(X)\\delta X}} \\\\\n&= \\exp\\{\\langle\\lambda^\\mathrm{fa},g-1\\rangle\\}\\int{\\{h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)\\}^X f\\_{t|t-1}(X)\\delta X} \\\\\n&= \\exp\\{\\langle\\lambda^\\mathrm{fa},g-1\\rangle\\} G\\_{t|t-1}[h(1-P^{\\mathrm{d}} + P^{\\mathrm{d}} p\\_g)] \\label{eq:JointPGFL}\\end{aligned}$$ \nThus, using ([eq:PGFLPrediction]), ([eq:PGFLUpdate]) and ([eq:JointPGFL]), a Bayes filter may be derived directly in p.g.fl form. A key component of the derivation is the product rule: \n \n$$\\frac{\\delta}{\\delta Z}(F\\_0[h]\\cdots F\\_n[h]) = \\sum\\_{W\\_0 \\uplus \\cdots \\uplus W\\_n = Z}\n\\frac{\\delta F\\_0}{\\delta W\\_0}[h] \\cdots\n\\frac{\\delta F\\_n}{\\delta W\\_n}[h]\n\\label{eq:ProductRule}$$ \nwhere the notation ⊎ denotes that the sum is over all disjoint sets *W*0, …, *W**n* such that *W*0 ∪ ⋯ ∪ *W**n* = *Z*. This permits calculation of the derivative of a product of several component p.g.fls (*e.g.*, a multi-Bernoulli p.g.fl).\nRandom set filter derivation\n============================\nThe model of interest was detailed in Assumptions [ass:Dynamics] and [ass:Measurement]. The derivation does not change if the birth density, survival probability, detection probability, false alarm density and measurement likelihood are time varying (*e.g.*, depending on a known, time-varying sensor state); we omit the time index from these parameters for notational simplicity. The multiple sensor case may be addressed by performing update steps for each sensor sequentially in between prediction steps.\nBefore we commence, we make several definitions that arise in the derivation, and define some of the elements that comprise the filter structure.\n[def:Unknown] An **unknown target** is a target that is hypothesised to exist but has never been detected.\nThis terminology follows. The explicit modelling of targets that have never been detected may appear unusual at first glance. For example, how can one distinguish targets that have never been detected from those that do not exist? Mathematically, it arises naturally: the RFS prediction model hypothesises birth of a PPP of targets (with intensity *λ*b(*x*)), and the detection model hypothesises that missed detections will occur on some proportion of these (1 − *P*d(*x*)); thus we are obliged to carry them over to the next time step as *unknown* targets. Practically, if *P*d ≈ 1, the density of unknown targets will be very low and can be safely neglected. The practical necessity for maintaining awareness of unknown targets arises in cases in which sensor characteristics are non-uniform and non-stationary. For example, suppose that the sensor observes a region at irregular intervals, achieving a different probability of detection with each observation (as commonly occurs in phased array radar systems). In this case, even if the birth intensity is uniform and stationary, with each look one would expect to discover a different number of new targets; the unknown target density provides a mechanism for modelling this effect.\nWhile measurement-to-target association is not explicitly introduced in the derivation, we will find that it arises implicitly, as a consequence of the sum over all decompositions of the measurements *Z**t* into the disjoint subsets *W**i* assigned to prior tracks (*i.e.*, the factors of the p.g.fl of the prior), which appears in the product rule of ([eq:ProductRule]). Thus we define:\n[def:GlobalHypothesis] A **global association history hypothesis** (or **global hypothesis** for short) is a partitioning of all measurements received so far in to subsets, where each subset is hypothesised to correspond to a particular potential target.\nNote that a consequence of assumption [ass:Measurement] is that the subset of measurements for each target can contain at most one measurement from each time. Global association history hypotheses are made up of single target association history hypotheses, as defined below.\n[def:SingleTargetHypothesis] A **single target association history hypothesis** (or **single target hypothesis** for short) is a subset of measurements that are hypothesised to correspond to the same potential target.\nThe term *potential* target is used in the definitions above because single-target hypotheses correspond to Bernoulli distributions (i.e., the form ([eq:Bernoulli]), incorporating a probability of existence and a PDF of target state). Consequently, a global hypothesis specifies a *distribution* over target cardinality, rather than a unique cardinality. This enables a significantly more compact set of hypotheses than in conventional MHT developments. For example, one single target hypothesis can incorporate a distribution over the following events:\n1. The target never existed\n2. The target did exist but death occurred at some time since the last detection\n3. The target continues to exists\nIncorporation of the first event arises from the derivation in theorem [th:Update], which shows that the update step naturally creates a new Bernoulli component for each measurement, where the probability of existence is the ratio of the PPP intensity of measurements arising from unknown targets to the sum of this intensity and the PPP intensity of false alarms, *λ*fa(*z*). Thus, one single target hypothesis represents both the event that the corresponding measurement was false alarm, and the event that it was the first detection of a new target, via a single Bernoulli distribution.\nA global hypothesis at time *t* may be represented as *a* = (*a*1, …, *a**n*), where *a**i* indexes the single target hypothesis utilised for the *i*-th target. We denote by ${\\cal M}^{t'}$ the set of all measurement indices up to and including time *t*ʹ, *i.e.*, the elements of ${\\cal M}^{t'}$ are of the form (*τ*, *j*), where *j* ∈ {1, …, *m**τ*} is an index of a measurement in scan *τ* ≤ *t*ʹ, where *t*ʹ denotes the last scan of measurements incorporated into the filter, *e.g.*, *t*ʹ = *t* − 1 following the prediction step, and *t*ʹ = *t* following the update step. Each single-target association history hypothesis incorporates the following information:\n* the history of measurement indices that are hypothesised to correspond to the target under the hypothesis, ${\\cal M}^{t'}(i,a^i)\\subseteq{\\cal M}^{t'}$ (*e.g.*, if ${\\cal M}^{t'}(i,a^i)=\\{(3,7),(5,8)\\}$ at time *t*ʹ = 6, then *a**i* hypothesises that the *i*-th hypothesised target was first detected as measurement index 7 at time 3, a missed detection occurred at time 4, it was detected as measurement 8 at time 5, and a missed detection occurred again at time 6)\n* the hypothesis weight *w**t*∣*t*ʹ*i*, *a**i* (utilised in calculation of the probability of the global hypotheses)\n* the hypothesis-conditioned Bernoulli distribution *f**t*∣*t*ʹ*i*, *a**i*(*X*) (with p.g.fl *G**t*∣*t*ʹ*i*, *a**i*[*h*]) of the form ([eq:Bernoulli]), parameterised by the probability of existence under the hypothesis, *r**t*∣*t*ʹ*i*, *a**i*, and the PDF under the hypothesis, *f**t*∣*t*ʹ*i*, *a**i*(*x*)\nSingle-target hypotheses are grouped together into *tracks*, where the point in common is that all hypothesise the same measurement corresponding to the first detection of the target.[6](#fn6)\n[def:Track] A **track** is the collection of information available about the target that was first detected in a particular measurement, consisting of a collection of single-target hypotheses (and the accompanying information) representing different possibilities of measurement sequences corresponding to the target.\nNote that this definition is more in line with the use of the term in JPDA and related methods than MHT.[7](#fn7) As a consequence of definition [def:Track], a new track is created for each measurement received (conceptually, at least). The structure of global hypotheses, single-target hypotheses and tracks is similar to that in the TOMHT ; it is illustrated in Fig. [fig:MBMStructure]. The advantage of the structure is the compactness of representation that it achieves, since the same single-target hypotheses are utilised in many global hypotheses.\nWe denote by ${\\cal T}\\_{t|t'}=\\{1,\\dots,n\\_{t|t'}\\}$ the tracks at time *t* conditioned on measurements up to and including time *t*ʹ, and by ${\\cal H}^i\\_{t|t'}=\\{1,\\dots,h^i\\_{t|t'}\\}$ the single-target hypotheses in track $i\\in{\\cal T}\\_{t|t'}$.\nOur derivation is based on an induction process, assuming a particular form for the full multiple target probability distribution, and showing that this form is maintained by prediction and update steps. The form of the full multi-target distribution at time *t*, conditioned on measurements up to and including time *t*ʹ (*i.e.*, if *t*ʹ = *t* − 1 then *f**t*∣*t*ʹ(*X**t*) = *f*(*X**t*∣*Z**t* − 1) is the predicted distribution, and if *t*ʹ = *t* then *f**t*∣*t*ʹ(*X**t*) = *f*(*X**t*∣*Z**t*) is the updated distribution), consists of two independent components:\n \n$$\\begin{split}\nf\\_{t|t'}(X) &= \\sum\\_{Y\\subseteq X}f^\\mathrm{ppp}\\_{t|t'}(Y) f^\\mathrm{mbm}\\_{t|t'}(X-Y) \\\\\nG\\_{t|t'}[h] &= G^\\mathrm{ppp}\\_{t|t'}[h] G^\\mathrm{mbm}\\_{t|t'}[h]\n\\end{split}\n\\label{eq:FullDensity}$$ \nwhere *f**t*∣*t*ʹppp(*X*) (and the corresponding p.g.fl *G**t*∣*t*ʹppp[*h*]) is a PPP representing unknown targets, with intensity *λ**t*∣*t*ʹ*u*(*x*):\n \n$$\\begin{split}\nf^\\mathrm{ppp}\\_{t|t'}(X) &= \\exp\\{-\\langle\\lambda^u\\_{t|t'},1\\rangle\\}\\prod\\_{x\\in X}\\lambda^u\\_{t|t'}(x) \\\\\nG^\\mathrm{ppp}\\_{t|t'}[h] &= \\exp\\{\\langle\\lambda^u\\_{t|t'},h-1\\rangle\\}\n\\end{split}\n\\label{eq:UndetectedTargets}$$ \nand *f**t*∣*t*ʹmbm(*X*) (and the corresponding p.g.fl *G**t*∣*t*ʹmbm[*h*]) is a multi-Bernoulli mixture (MBM), *i.e.*, a linear combination of multi-Bernoulli distributions:\n \n$$\\begin{aligned}\nf^\\mathrm{mbm}\\_{t|t'}(\\{x\\_1,\\dots,x\\_n\\}) &= \\sum\\_{\\alpha\\in{\\cal P}\\_{n\\_{t|t'}}^n,a\\in{\\cal A}^{t|t'}} w^a\n\\prod\\_{i\\in{\\cal T}\\_{t|t'}}f\\_{t|t'}^{i,a^i}(X\\_{\\alpha(i)}) \\notag\\\\\nG^\\mathrm{mbm}\\_{t|t'}[h] &= \\sum\\_{a\\in{\\cal A}^{t|t'}} w^a\\prod\\_{i\\in{\\cal T}\\_{t|t'}}G\\_{t|t'}^{i,a^i}[h]\n\\label{eq:PreexistingTargets}\\end{aligned}$$ \nwhere ${\\cal P}\\_{n\\_{t|t'}}^n$ and *X**α*(*i*) were defined in ([eq:DefinitionP]) and ([eq:BernoulliSet]), and ${\\cal A}^{t|t'}$ is the set of global association history hypotheses. Each global hypothesis must incorporate a single-target hypothesis for each track, and each global hypothesis must explain the origin of each measurement (up to and including time *t*ʹ), thus\n \n$$\\label{eq:AssocAlphabet}\n{\\cal A}^{t|t'} = \\bigg\\{\n(a^1,\\dots,a^{n\\_{t|t'}})\n\\bigg| \na^i\\in{\\cal H}^i\\_{t|t'}, \n\\bigcup\\_{i\\in{\\cal T}\\_{t|t'}}{\\cal M}^{t'}(i,a^i)={\\cal M}^{t'},\n{\\cal M}^{t'}(i,a^i)\\cap{\\cal M}^{t'}(j,a^j)=\\emptyset \\;\\forall\\; i\\neq j \n\\bigg\\}$$ \n \n$$\\begin{gathered}\n\\label{eq:AssocAlphabet}\n{\\cal A}^{t|t'} = \\bigg\\{\n(a^1,\\dots,a^{n\\_{t|t'}})\n\\bigg| \na^i\\in{\\cal H}^i\\_{t|t'}, \n\\bigcup\\_{i\\in{\\cal T}\\_{t|t'}}{\\cal M}^{t'}(i,a^i)={\\cal M}^{t'},\n\\\\\n{\\cal M}^{t'}(i,a^i)\\cap{\\cal M}^{t'}(j,a^j)=\\emptyset \\;\\forall\\; i\\neq j \n\\bigg\\}\\end{gathered}$$ \nThe global hypothesis weights *w**a* are related to the single-target hypothesis weights via the expression\n \n$$\\label{eq:GlobalHypWeight}\nw^a \\propto \\prod\\_{i\\in{\\cal T}\\_{t|t'}}w\\_{t|t'}^{i,a^i}$$ \nwhere the proportionality denotes that normalisation is required to ensure that $\\sum\\_{a\\in{\\cal A}^{t|t'}}w^a=1$.\nPrediction step\n---------------\nTheorem [th:Prediction] derives the prediction step by substituting ([eq:FullDensity]) into ([eq:PGFLPrediction]). The theorem confirms what one would intuitively expect: that the prediction of the PPP component ([eq:UndetectedPropagation]) follows the standard PHD prediction step of, and that the multi-Bernoulli tracks are predicted independently in an equivalent manner to the MeMBer (excluding birth of new targets, since we utilise a PPP birth model as opposed to a multi-Bernoulli birth model).\n[th:Prediction] Assume that the distribution from the previous time step *G**t* − 1∣*t* − 1[*h*] is of the form given in ([eq:FullDensity])-([eq:GlobalHypWeight]). Then the predicted distribution for the next step *G**t*∣*t* − 1[*h*] is of the same form, with:\n \n$$\\begin{aligned}\n\\lambda^u\\_{t|t-1}(x) &= \\lambda^\\mathrm{b}(x) + \\int{f\\_{t|t-1}(x|x')P^{\\mathrm{s}}(x')\\lambda^u\\_{t-1|t-1}(x')\\mathrm{d} x'}\n\\label{eq:UndetectedPropagation} \\\\\nn\\_{t|t-1} &= n\\_{t-1|t-1}; \\quad h^i\\_{t|t-1} = h^i\\_{t-1|t-1} \\; \\forall \\; i \\\\\nw\\_{t|t-1}^{i,a^i} &= w\\_{t-1|t-1}^{i,a^i} \\; \\forall \\; i, a^i \\\\\nr\\_{t|t-1}^{i,a^i} &= r\\_{t-1|t-1}^{i,a^i}\\langle f\\_{t-1|t-1}^{i,a^i},P^{\\mathrm{s}}\\rangle \\; \\forall \\; i, a^i \\label{eq:ExistProbPropagation} \\\\\nf\\_{t|t-1}^{i,a^i}(x) &= \\frac{\\int{f\\_{t|t-1}(x|x')P^{\\mathrm{s}}(x')f\\_{t-1|t-1}^{i,a^i}(x')\\mathrm{d} x'}}{\\langle f\\_{t-1|t-1}^{i,a^i},P^{\\mathrm{s}}\\rangle}\\; \\forall \\; i, a^i \\label{eq:KinematicDistPropagation}\\end{aligned}$$ \nThe proof of the theorem is in Appendix [app:PredictionProof].\nMeasurement update step\n-----------------------\nTheorem [th:Update] derives the measurement update step by substituting ([eq:JointPGFL]) and ([eq:FullDensity]) into ([eq:PGFLUpdate]). The PPP intensity update equation is ([eq:UndetectedUpdate]); this is identical to a PHD update with no measurements. The MBM update is illustrated and described in Fig. [fig:MBMStructure].\n[fig:MBMStructure]\n[th:Update] Assume that the predicted distribution *G**t*∣*t* − 1[*h*] is of the form given in ([eq:FullDensity])-([eq:GlobalHypWeight]). Then the updated distribution *G**t*∣*t*[*h*] (updated with the measurement set *Z**t* = {*z**t*1, …, *z**t**m**t*}) is of the same form, with *n**t*∣*t* = *n**t*∣*t* − 1 + *m**t*,\n \n$$\\begin{aligned}\n\\lambda^u\\_{t|t}(x) &= \\{1-P^{\\mathrm{d}}(x)\\}\\lambda^u\\_{t|t-1}(x) \\label{eq:UndetectedUpdate} \\\\\n{\\cal M}^t &= {\\cal M}^{t-1} \\cup \\big\\{(t,j)|j\\in\\{1,\\dots,m\\_t\\}\\big\\} \\end{aligned}$$ \nFor tracks continuing from previous time steps (*i* ∈ {1, …, *n**t*∣*t* − 1}), a hypothesis is included for each combination of a hypothesis from a previous time, and either a missed detection, or an update using one of the *m**t* new measurements, such that the number of hypotheses becomes *h**t*∣*t**i* = *h**t*∣*t* − 1*i*(1 + *m**t*). For missed detection hypotheses (*i* ∈ {1, …, *n**t*∣*t* − 1}, *a**i* ∈ {1, …, *h**t*∣*t* − 1}):\n \n$$\\begin{aligned}\n{\\cal M}^t(i,a^i) &= {\\cal M}^{t-1}(i,a^i) \\label{eq:MissUpdateMeasSet} \\\\\nw\\_{t|t}^{i,a^i} &= w^{i,a^i}\\_{t|t-1}(1-r\\_{t|t-1}^{i,a^i}+r\\_{t|t-1}^{i,a^i}\\langle f\\_{t|t-1}^{i,a^i},1-P^{\\mathrm{d}}\\rangle) \\label{eq:MissUpdateW} \\\\\nr^{i,a^i}\\_{t|t} &= \\frac{r\\_{t|t-1}^{i,a^i}\\langle f\\_{t|t-1}^{i,a^i},1-P^{\\mathrm{d}}\\rangle}{1-r\\_{t|t-1}^{i,a^i}+r\\_{t|t-1}^{i,a^i}\\langle f\\_{t|t-1}^{i,a^i},1-P^{\\mathrm{d}}\\rangle} \\label{eq:MissUpdatePex}\\\\\nf^{i,a^i}\\_{t|t}(x) &= \\frac{\\{1-P^{\\mathrm{d}}(x)\\}f\\_{t|t-1}^{i,a^i}(x)}{\\langle f\\_{t|t-1}^{i,a^i},1-P^{\\mathrm{d}}\\rangle} \\label{eq:MissUpdateKin}\\end{aligned}$$ \nFor hypotheses updating existing tracks (*i* ∈ {1, …, *n**t*∣*t* − 1}, *a**i* = *ã**i* + *h**t*∣*t* − 1*i**j*, *ã**i* ∈ {1, …, *h**t*∣*t* − 1*i*}, *j* ∈ {1, …, *m**t*}, *i.e.*, the previous hypothesis *ã**i*, updated with measurement *z**t**j*):[8](#fn8)\n \n$$\\begin{aligned}\n{\\cal M}^t(i,a^i) &= {\\cal M}^{t-1}(i,\\tilde{a}^i) \\cup \\{(t,j)\\} \\label{eq:DetUpdateMeasSet} \\\\\nw\\_{t|t}^{i,a^i} &= w^{i,\\tilde{a}^i}\\_{t|t-1}r\\_{t|t-1}^{i,\\tilde{a}^i}\\langle f\\_{t|t-1}^{i,\\tilde{a}^i},f(z\\_t^j|\\cdot)P^{\\mathrm{d}}\\rangle \\label{eq:DetUpdateW}\\\\\nr^{i,a^i}\\_{t|t} &= 1 \\label{eq:DetUpdatePex}\\\\\nf^{i,a^i}\\_{t|t}(x) &= \\frac{f(z\\_t^j|x)P^{\\mathrm{d}}(x)f\\_{t|t-1}^{i,\\tilde{a}^i}(x)}{\\langle f\\_{t|t-1}^{i,\\tilde{a}^i},f(z\\_t^j|\\cdot)P^{\\mathrm{d}}\\rangle} \\label{eq:DetUpdateKin}\\end{aligned}$$ \nFinally, for new tracks, *i* ∈ {*n**t*∣*t* − 1 + *j*}, *j* ∈ {1, …, *m**t*} (*i.e.*, the new track commencing on measurement *z**t**j*),\n \n$$\\begin{aligned}\nh^i\\_{t|t} &= 2 \\\\\n{\\cal M}^t(i,1) &= \\emptyset, \\quad\nw^{i,1}\\_{t|t} = 1, \\quad r^{i,1}\\_{t|t} = 0 \\label{eq:NewTargetNonExistWQ} \\\\\n{\\cal M}^t(i,2) &= \\{(t,j)\\} \\label{eq:PoisUpdateMeasSet} \\\\\nw^{i,2}\\_{t|t} &= \\lambda^\\mathrm{fa}(z\\_t^j) + \\langle\\lambda^u\\_{t|t-1},f(z\\_t^j|\\cdot)P^{\\mathrm{d}}\\rangle \\label{eq:PoisUpdateW} \\\\\nr\\_{t|t}^{i,2} &= \\frac{\\langle\\lambda^u\\_{t|t-1},f(z\\_t^j|\\cdot)P^{\\mathrm{d}}\\rangle}{\\lambda^\\mathrm{fa}(z\\_t^j) + \\langle\\lambda^u\\_{t|t-1},f(z\\_t^j|\\cdot)P^{\\mathrm{d}}\\rangle} \\label{eq:PoisUpdatePex}\\\\\nf\\_{t|t}^{i,2}(x) &= \\frac{f(z\\_t^j|x)P^{\\mathrm{d}}(x)\\lambda^u\\_{t|t-1}(x)}{\\langle\\lambda^u\\_{t|t-1},f(z\\_t^j|\\cdot)P^{\\mathrm{d}}\\rangle} \\label{eq:PoisUpdateKin} \\end{aligned}$$ \nNote that from the last section of the theorem, a new track is created for each measurement. These tracks contain two single target hypotheses (*h**t*∣*t**i* = 2), the first of which covers the case that the measurement is associated with another track, hence the new track has probability of existence equal to zero (the PDF *f**t*∣*t**i*, 1(*x*) has no effect and has not been specified); and the second of which covers the case that the measurement is not associated with any previous track, hence it is either a false alarm or a new target (the probability of existence *r**t*∣*t**i*, 2 models the relative likelihood of these two events).\nThe proof of Theorem [th:Update] is in Appendix [app:UpdateProof]. The first part of the proof (Lemma [lem:Structure]) shows that, in the updated distribution, the portion of the component *G**t*∣*t* − 1ppp[*h*] that is not detected is independent (in the posterior) of the remainder of the distribution (which comprises the portion of *G**t*∣*t* − 1ppp[*h*] that is detected, and both detected and undetected portions of the component *G**t*∣*t* − 1mbm[*h*]). Note that this is *not* claiming that the *entire* undetected portion is independent of the detected portion.\nThe updates in ([eq:MissUpdateKin]), ([eq:DetUpdateKin]) and ([eq:PoisUpdateKin]) are standard single target measurement updates, and thus can be calculated with well-known methods (*e.g.*, ). The derivation incorporates hypotheses updating every prior hypothesis with every measurement; however, in practical implementations, gating can be used to reduce the computational burden by excluding hypotheses with negligible weights. Other standard approximations such as clustering, mixture reduction, *etc*are also required in practice.\nInitialisation\n--------------\nOne significant benefit of the inclusion of a Poisson component is in initialisation of the tracker. Generally, when a surveillance system is first activated, the fact that prior measurements are not available does not imply that no targets are present. The Poisson distribution provides a convenient mechanism for specifying a prior distribution on the number and position of targets when little information is available. By initialising *n*0∣0 = 0 (*i.e.*, no MBM components), and setting *λ*0∣0*u*(*x*) = *λ*0*u**f*0(*x*) where *λ*0*u* is the expected number of targets present, and *f*0(*x*) is the (presumably diffuse) prior state distribution, the Bayes prediction and update steps incorporate this prior knowledge, adjusting the accrual of confidence in the existence probabilities *r**t*∣*t**i*, *a**i* accordingly. In, it was demonstrated that this approach can considerably improve the speed of track initiation in low signal-to-noise ratio (SNR) environments over methods that do not utilise such prior knowledge.\nMarginal association filters\n============================\nThe previous section derived a conjugate prior form for the full Bayes RFS filter. Not surprisingly, the filter is intractable, and approximations must be made. Noting its similarity to the PHD, the PPP component of the posterior is tractable (*e.g.*, see ). The difficulty is the number of global association history hypotheses, each of which gives rise to a term in the sum in the MBM ([eq:PreexistingTargets]). To motivate our approximation, consider the definition of a probability of joint association history:\n \n$$\\label{eq:JointAssocProb}\nP\\_{t|t}(a) = w^a \\propto \\prod\\_{i\\in{\\cal T}\\_{t|t}}w\\_{t|t}^{i,a^i}$$ \nfollowing which ([eq:PreexistingTargets]) can be equivalently viewed as a total probability expansion over this random variable\n \n$$f^\\mathrm{mbm}\\_{t|t}(X) = \\sum\\_{\\alpha\\in{\\cal P}\\_{n\\_{t|t}}^{|X|},a\\in{\\cal A}^{t|t}} P\\_{t|t}(a) \\prod\\_{i\\in{\\cal T}\\_{t|t}}f\\_{t|t}^{i,a^i}(X\\_{\\alpha(i)}) \n\\label{eq:LCMBIndepApprox}$$ \nThe methods in this section approximate the posterior distribution *f**t*∣*t*mbm(*X*) by directly approximating the discrete probability distribution of global association history hypotheses *P**t*∣*t*(*a*). Herein we will assume that the prior distribution *f**t* − 1∣*t* − 1mbm(*X*) is multi-Bernoulli, such that the predicted distribution *f**t*∣*t* − 1mbm(*X*) is multi-Bernoulli, and our task is to gain an approximation of *f**t*∣*t*mbm(*X*) that is multi-Bernoulli (since the true distribution is not). The predicted distribution in the form ([eq:PreexistingTargets]) will be multi-Bernoulli if there is a single association hypothesis, *e.g.*,\n \n$$\\label{eq:MultiBernoulliAssumption}\n\\renewcommand{\\arraycolsep}{1pt}\n\\begin{array}{rlrl}\n{\\cal M}^{t-1} &= \\emptyset, & h\\_{t|t-1}^i &= 1\\;\\forall\\;i, \\\\\n{\\cal M}^{t-1}(i,a^i) &= \\emptyset\\;\\forall\\;i,a^i,\\quad & w\\_{t|t-1}^{i,a^i} &= 1\\;\\forall\\;i,a^i \n\\end{array}$$ \nIn this case, we refer to the weight, existence probability and PDF for track *i* as *w**t*∣*t**i* = 1, *r**t*∣*t**i* and *f**t*∣*t**i*(*x*) respectively.[9](#fn9)\nTrack-oriented marginal MeMBer-Poisson filter\n---------------------------------------------\nSuppose we seek an approximate representation of the distribution *P**t*∣*t*(*a*) in which the hypotheses for each track *a**i* comprising the global hypothesis *a* = (*a*1, …, *a**n**t*∣*t*) are forced to be independent, *i.e.*, a distribution of the form:\n \n$$\\label{eq:MargTrackFilterWeights}\nP\\_{t|t}(a) \\approx \\prod\\_{i\\in{\\cal T}\\_{t|t}} q^i(a^i)$$ \nwhere we effectively expand the alphabet of the discrete random variable *a* from ${\\cal A}^{t|t}$ defined in ([eq:AssocAlphabet]) to the Cartesian product $\\tilde{{\\cal A}}^{t|t}\\triangleq\\prod\\_i{\\cal H}^i\\_{t|t}$, defining *P**t*∣*t*(*a*) = 0 for infeasible association events $a\\notin{\\cal A}^{t|t}$. Then with no further approximation, the joint multi-target distribution in ([eq:LCMBIndepApprox]) becomes:\n \n$$\\begin{aligned}\nf^\\mathrm{mbm}\\_{t|t}(X) &\\approx \\sum\\_{\\alpha\\in{\\cal P}\\_{n\\_{t|t}}^{|X|},a\\in\\tilde{{\\cal A}}^{t|t}} \\left[\\prod\\_{i\\in{\\cal T}\\_{t|t}} q^i(a^i)\\right] \\cdot \\prod\\_{i\\in{\\cal T}\\_{t|t}}f\\_{t|t}^{i,a^i}(X\\_{\\alpha(i)}) \\\\\n&= \\sum\\_{\\alpha\\in{\\cal P}\\_{n\\_{t|t}}^{|X|},a\\in\\tilde{{\\cal A}}^{t|t}} \\prod\\_{i\\in{\\cal T}\\_{t|t}}q^i(a^i)f\\_{t|t}^{i,a^i}(X\\_{\\alpha(i)}) \\\\\n&= \\sum\\_{\\alpha\\in{\\cal P}\\_{n\\_{t|t}}^{|X|}}\\prod\\_{i\\in{\\cal T}\\_{t|t}}f\\_{t|t}^i(X\\_{\\alpha(i)}) \\label{eq:MargTrackFilter}\\\\\nf\\_{t|t}^i(X) &= \\sum\\_{a^i\\in{\\cal H}\\_{t|t}^i}q^i(a^i)f\\_{t|t}^{i,a^i}(X)\\end{aligned}$$ \nwhere ([eq:MargTrackFilter]) exploits separability of the sum over $a\\in\\tilde{{\\cal A}}\\_{t|t}$, e.g., that $\\sum\\_{i\\in{\\cal I},j\\in{\\cal J}} f(i)g(j)=\\big(\\sum\\_{i\\in{\\cal I}}f(i)\\big)\\big(\\sum\\_{j\\in{\\cal J}}g(j)\\big)$. Comparing ([eq:MargTrackFilter]) with ([eq:MultiBernoulli]), we see that the resulting distribution is multi-Bernoulli.\nThe problem of finding the choice of *q**i*(*a**i*) which best fits the assumed form of the distribution can be solved by minimising the Kullback-Leibler (KL) divergence:\n \n$$\\label{eq:TOMKLOpt}\n\\left[q^{i\\*}(a^i)\\right]\\_{i\\in{\\cal T}\\_{t|t}} = \\operatorname\\*{arg~min}\\_{q^i(a^i),\\,i\\in{\\cal T}\\_{t|t}}D\\bigg(P\\_{t|t}(a)\\bigg|\\bigg|\\prod\\_{i\\in{\\cal T}\\_{t|t}}q^i(a^i) \\bigg)$$ \nwhere $D(p||q) = \\sum\\_a p(a)\\log\\frac{p(a)}{q(a)}$. It is shown in that *q**i* \\*(*a**i*) = *P**t*∣*t**i*(*a**i*) = ∑*ã*∣*ã**i* = *a**i**P**t*∣*t*(*ã*), *i.e.*, the marginal association distribution for track *i*.\nAccordingly, the data carried forward to the next time for the MBM component are simply the single hypothesis Bernoulli distributions *f**t*∣*t**i*(*X*) (parameterised by an existence probability, and a position distribution consisting of a weighted mixture of the old hypothesis-conditioned distributions). We refer to this as the track oriented marginal MeMBer/Poisson filter (TOMB/P), as it forms tracks out of the marginal track-to-measurement association distributions. The approximation made in the TOMB/P is exactly that made in JPDA: approximating the joint association distribution by the product of its marginals.[10](#fn10) Whereas JPDA assumes target existence, applying the marginal approximation within the RFS framework yields an algorithm that naturally captures target existence. In what follows, we show that the TOMB/P is equivalent to a particular variant of JITS/JIPDA with two minor modifications. The specific variant is JITS/JIPDA with a parametric clutter model, and using the one-point track initialisation method incorporating a new object spatial density. We assume that *P*d(*x*) = *P*d and *P*s(*x*) = *P*s (*i.e.*, that both of these parameters are spatially constant). From ([eq:JointAssocProb]), ([eq:MissUpdateW]), ([eq:DetUpdateW]), ([eq:PoisUpdateW]) and ([eq:MultiBernoulliAssumption]), the probability of a global association hypothesis *a* is:\n \n$$\\begin{aligned}\nP\\_{t|t}(a) \\propto& \\prod\\_{i=1}^{n\\_{t|t}} w^{i,a^i}\\_{t|t} \\notag \\\\\n=& \\prod\\_{i\\in\\{1,\\dots,n\\_{t|t-1}\\}|a^i=0} \\left\\{ 1- r^i\\_{t|t-1}P^{\\mathrm{d}} \\right\\} \n\\cdot \\prod\\_{i\\in\\{1,\\dots,n\\_{t|t-1}\\}|a^i>0} \\left\\{r^i\\_{t|t-1}P^{\\mathrm{d}} \\langle f^i\\_{t|t-1},f(z\\_t^{a^i}|\\cdot)\\rangle \\right\\} \\notag\\\\\n& \\times \\prod\\_{j\\in\\{1,\\dots,m\\_t\\}|a^{n\\_{t|t-1}+j}=2}\\left\\{\\lambda^\\mathrm{fa}(z\\_t^j)+P^{\\mathrm{d}}\\langle\\lambda^u\\_{t|t-1},f(z\\_t^j|\\cdot)\\rangle\\right\\} \\label{eq:JointEventProb1}\\end{aligned}$$ \n \n$$\\begin{aligned}\nP\\_{t|t}&(a) \\propto \\prod\\_{i=1}^{n\\_{t|t}} w^{i,a^i}\\_{t|t} \\notag \\\\\n=& \\prod\\_{i\\in\\{1,\\dots,n\\_{t|t-1}\\}|a^i=0} \\left\\{ 1- r^i\\_{t|t-1}P^{\\mathrm{d}} \\right\\} \\notag\\\\\n& \\times \\prod\\_{i\\in\\{1,\\dots,n\\_{t|t-1}\\}|a^i>0} \\left\\{r^i\\_{t|t-1}P^{\\mathrm{d}} \\langle f^i\\_{t|t-1},f(z\\_t^{a^i}|\\cdot)\\rangle \\right\\} \\notag\\\\\n& \\times \\prod\\_{j\\in\\{1,\\dots,m\\_t\\}|a^{n\\_{t|t-1}+j}=2}\\left\\{\\lambda^\\mathrm{fa}(z\\_t^j)+P^{\\mathrm{d}}\\langle\\lambda^u\\_{t|t-1},f(z\\_t^j|\\cdot)\\rangle\\right\\} \\label{eq:JointEventProb1}\\end{aligned}$$ \nSince, under any hypothesis, a measurement will either be explained as belonging to an existing track (∃ *i*s.t.\\ *a**i* = *j*), or a new track (*a**n**t*∣*t* − 1 + *j* = 2) (which incorporates the possibility of it being a false alarm), we can divide ([eq:JointEventProb1]) by the constant ∏*j* = 1*m**t*{*λ*fa(*z**t**j*) + *P*d⟨*λ**t*∣*t* − 1*u*, *f*(*z**t**j*∣ ⋅ )⟩} to obtain:\n \n$$\\begin{aligned}\nP\\_{t|t}(a) \\propto& \\prod\\_{i\\in\\{1,\\dots,n\\_{t|t-1}\\}|a^i=0} \\left\\{ 1- r^i\\_{t|t-1}P^{\\mathrm{d}} \\right\\} \n\\cdot \\prod\\_{i\\in\\{1,\\dots,n\\_{t|t-1}\\}|a^i>0} \\left\\{\\frac{\nr^i\\_{t|t-1}P^{\\mathrm{d}} \\langle f^i\\_{t|t-1},f(z\\_t^{a^i}|\\cdot)\\rangle\n}{\n\\lambda^\\mathrm{fa}(z\\_t^{a^i})+P^{\\mathrm{d}}\\langle\\lambda^u\\_{t|t-1},f(z\\_t^{a^i}|\\cdot)\\rangle\n} \\right\\} \\label{eq:JointEventProb2}\\end{aligned}$$ \n \n$$\\begin{aligned}\nP\\_{t|t}&(a) \\propto \\prod\\_{i\\in\\{1,\\dots,n\\_{t|t-1}\\}|a^i=0} \\left\\{ 1- r^i\\_{t|t-1}P^{\\mathrm{d}} \\right\\} \\notag\\\\\n& \\times \\prod\\_{i\\in\\{1,\\dots,n\\_{t|t-1}\\}|a^i>0} \\left\\{\\frac{\nr^i\\_{t|t-1}P^{\\mathrm{d}} \\langle f^i\\_{t|t-1},f(z\\_t^{a^i}|\\cdot)\\rangle\n}{\n\\lambda^\\mathrm{fa}(z\\_t^{a^i})+P^{\\mathrm{d}}\\langle\\lambda^u\\_{t|t-1},f(z\\_t^{a^i}|\\cdot)\\rangle\n} \\right\\} \\label{eq:JointEventProb2}\\end{aligned}$$ \nwhich differs from only in the inclusion of the term *P*d⟨*λ**t*∣*t* − 1*u*, *f*(*z**t**a**i*∣ ⋅ )⟩ in the denominator. Thus it is seen that JITS/JIPDA neglects the influence of unknown targets in the calculation of probabilities of association events. From ([eq:NewTargetNonExistWQ]), ([eq:PoisUpdatePex]) and ([eq:MargTrackFilter]), the posterior probability of existence of a track started on a measurement *z**t**j* is (*i* = *n**t*∣*t* − 1 + *j*, *j* ∈ {1, …, *m**t*})\n \n$$\\label{eq:InitialExistenceProbability}\nr^i\\_{t|t} = \\frac{P\\_{t|t}^i(2)P^{\\mathrm{d}}\\langle\\lambda^u\\_{t|t-1},f(z\\_t^j|\\cdot)\\rangle}{\\lambda^\\mathrm{fa}(z\\_t^j) + P^{\\mathrm{d}}\\langle\\lambda^u\\_{t|t-1},f(z\\_t^j|\\cdot)\\rangle}$$ \nThis is exactly, with *ρ*target(*z*) = *P*d⟨*λ**t*∣*t* − 1*u*, *f*(*z*∣ ⋅ )⟩. Thus, track initiation in the variant of JITS/JIPDA modelling a birth intensity *ρ*target(*z*) can be seen to be equivalent to TOMB/P assuming a stationary intensity of unknown targets.[11](#fn11) The benefit of dynamically estimating this time-varying quantity was demonstrated in. Accordingly, we have shown that after two minor modifications and the approximation of ([eq:MargTrackFilter]), JITS/JIPDA may be derived (and extended) using the RFS formalism. Similar algorithms have also been proposed within the RFS community; for example, updates existing tracks in a similar manner, but uses a two-point differencing approach to calculate initial states, weighting the new target hypotheses using a constant density of new targets (motivated as a truncated Poisson); and considers a bank of joint target-detection and tracking (JoTT) filters interacting through modified weights.\nThe approximation made in the TOMB/P can be easily seen to preserve the first moment of the distribution. The first moment of the form ([eq:LCMBIndepApprox]) can be written as:\n \n$$\\begin{aligned}\nD\\_{t|t}(x) &= \\sum\\_{a\\in{\\cal A}^{t|t}} P\\_{t|t}(a) \\sum\\_{i\\in{\\cal T}\\_{t|t}}r\\_{t|t}^{i,a^i}f\\_{t|t}^{i,a^i}(x) \\\\\n& = \\sum\\_{i\\in{\\cal T}\\_{t|t}} \\left(\\sum\\_{\\tilde{a}\\in{\\cal A}^{t|t}|\\tilde{a}^i=a^i} P\\_{t|t}(a)\\right) r\\_{t|t}^{i,a^i}f\\_{t|t}^{i,a^i}(x) \\\\\n&= \\sum\\_{i\\in{\\cal T}\\_{t|t}} \\sum\\_{a^i\\in{\\cal H}^i\\_{t|t}} P^i\\_{t|t}(a^i) r\\_{t|t}^{i,a^i}f\\_{t|t}^{i,a^i}(x) \n\\label{eq:TOMBFirstMoment}\\end{aligned}$$ \nwhich can be seen to be the first moment of the TOMB/P approximation ([eq:MargTrackFilter]).\nWhile our derivation has studied the unlabelled case, TOMB/P maintains tracks that are constructed from hypotheses which all begin with the same first detection of the target. Thus track continuity is implicitly maintained in the same way as in JPDA and related methods. This can be made explicit by incorporating a label element into the underlying state space, as recently proposed in ; it can easily be shown that each track maintained by TOMB/P would retain a unique label.\nThe only approximation made in the TOMB/P derivation was the approximation of the distribution *P**t*"},"data_id":{"kind":"string","value":"arxiv_0183482"}}},{"rowIdx":45898,"cells":{"content":{"kind":"string","value":"( \\pm 0.007)$, with *z* ∈ [0.1, 0.7]. At the cluster distance *D**L* = 1670 ± 17 Mpc, the corresponding scale is ∼ 4.6 Kpc/ʺ, and at the cluster range 0.28 < *z* < 0.34 the scale spans values from ∼ 4.3Kpc/ʺ to ∼ 4.9Kpc/ʺ.\nAC114 has a nonspherical shape, being elongated in the southeast-northwest direction (, ). The dynamical study of P15 shows also that AC114 is a very elongated radial filament, spanning $12\\ 000\\ \\mathrm{km\\ s^{-1}}$ with a broad velocity dispersion of $\\sim 1893\\_{-82}^{73} \\ \\mathrm{km \\ s^{-1}}$. These dimensions correspond to ∼ 330 Mpc in the radial direction and ∼ 9 Mpc in the perpendicular component, the latter being a lower limit, as the calculations are restricted to the FoV of the VIMOS observations.\nThe three-dimensional representation of the cluster members is shown in Figure [fig12] for SF and passive galaxies with blue and red dots, respectively. Figure [fig12] shows that the center of the cluster (at 0,0,0 in X,Y,Z) is populated mostly by passive galaxies; SF galaxies tend to be found at the outskirts.\nThe mass and metallicity profiles are presented in Figure [fig13]. The top panel shows the mass profile of AC114. Just as in Figure [fig13], we observe a high concentration of galaxies toward the cluster center, which is dominated by passive galaxies. A negative radial mass gradient is detected. Black diamonds are the mean stellar mass of the galaxies in bins of ∼ 23 Mpc. The slope of the mass profile of − 0.005 ± 0.002 (Spearman rank is − 0.57, p-value = 0.005) shows a weak but statistically significant trend to find massive galaxies in the inner regions of the cluster. This is consistent with AC114 being in an early stage of radial relaxation, as is also suggested by the large velocity dispersion estimated in P15. Furthermore, passive galaxies present a slightly flatter slope (-0.004 ± 0.003, red linear fit) compared with SF galaxies (-0.007 ± 0.003, blue linear fit), which is indicative of ongoing dynamical activity in the galaxy cluster. The bottom panel of Figure [fig13] presents a weak, non-statistically significant (Spearman rank of − 0.38 with a p-value = 0.11) metallicity gradient of the SF galaxies, with a slope of − 0.002 ± 0.002.\n![Three-dimensional projection of the AC114 members, expressed in Mpc with respect to the cluster center. The x, y, and z axes correspond to the RA,Dec coordinates and to the redshift, respectively. Blue and red points are SF and passive galaxies, respectively. Colored crosses are the projected distribution of the cluster sample in the xz plane. The black cross indicates the center of the galaxy cluster.](plots/3d_cluster.png \"fig:\") [fig12]\n![Radial profiles of AC114. Top panel: Stellar masses of galaxies as a function of the clustercentric radius. Blue and red crosses are SF and passive galaxies. Black diamonds represent mean mass bins of \\sim23 \\ \\mathrm{h^{-1}\\ Mpc}. Black, blue, and red lines are the linear fits that show a negative mass-gradient as a function of the clustercentric radius for the complete sample, SF, and passive galaxies, respectively. Bottom panel: Metallicity profile of the SF galaxies with available mean metallicity estimations. Black diamonds are mean metallicities in bins of \\sim33 \\ \\mathrm{h^{-1}\\ Mpc}, and the black line is the linear fit that shows a weak negative gradient.](plots/met_profile.png \"fig:\") [fig13]\nWe applied a kernel density estimator (KDE) to the redshift distribution of the galaxy cluster to search for possible substructures and construct the MZR of the sub-components. We made use of the machine-learning *KernelDensity* algorithm from the Scikit-learn package[6](#fn6) and used a linear kernel model. Three substructures were identified in the redshift distribution with this procedure. The red lines in the top panel of Figure [fig14] show the redshift limits of the substructures, namely SS1 (0.2850 ≤ *z* < 0.3029) being the front part of the galaxy cluster with three SF galaxies, SS2 (0.3029 ≤ *z* < 0.3243) for the main component of AC114 with ten SF galaxies, and SS3 (0.3243 ≤ *z* < 0.3380) being the back part of the galaxy cluster with six SF galaxies.\nOnce the substructures were identified, we went ahead to construct the respective MZRs. The bottom panel of Figure [fig14] shows the SF galaxies of the substructures in the mass-metallicity plane. Sky blue, black, and red are colors for SS1, SS2, and SS3, respectively. From SS1, only three SF galaxies were identified, grouped in the same region of the mass-metallicity plane. A statistically significant positive correlation is detected for SS3 (slope of 0.26 ± 0.14, Spearman rank *r* = 0.860, and p-value = 0.019, black dots). Furthermore, most of the galaxies in the sub-sample lie below the cMZR, suggesting that outskirt galaxies of AC114 are slowly evolving due to external processes that are suppressing the chemical evolution. On the other hand, the main component of AC114 (SS2) shows more scatter (0.28 and 0.33 for SS3 and SS2, respectively) in the mass-metallicity plane (red dots). The slope of the fit of SS2 is 0.19 ± 0.13, and no statistically significant trend is seen (Spearman rank of = 0.54, p-value = 0.13). The central region of AC114 consists of ten SF galaxies and 42 passive galaxies. Therefore, SF galaxies closer to the central region of AC114 are suffering not only from interaction with the ICM, but also more likely to be suffering from interaction with their neighbors.\nIn the previous subsection we were able to estimate clustercentric radial distances, so we repeated the same KDE procedure but applying it to the clustercentric radial distribution of AC114 galaxies. We divided the galaxies according to the following radial ranges (blue lines in the top panel of Figure [fig15]): the central region at *r* ≤ 32.5 *h*− 1Mpc with four SF galaxies, the intermediate region between 32.5 ≤ *r* < 57.4 Mpc with seven SF galaxies, and the outskirts of AC114 in *r* > 57.4 Mpc with eight SF galaxies.\nThe bottom panel of Figure [fig15] shows the SF galaxies of AC114 in the mass-metallicity plane colored with green, black, and blue for the central, intermediate, and external regions, respectively. Despite the central region only having four galaxies, two of them are the metal-richer ones of the galaxy cluster. On the other hand, the intermediate and outskirt regions of AC114 show an increase in oxygen abundances with the increase in the stellar mass, with a slope of 0.35 ± 0.08, Spearman rank *r* = 0.86, and *p* − value = 0.014, and a slope of 0.162 ± 0.09, Spearman rank *r* = 0.40, and *p* − value = 0.319, respectively.\n![Top panel: Kernel density estimator in the redshift distribution of the galaxy cluster region: 0.28\\leq z \\leq 0.34. Red lines show the redshift cuts of the three substructures identified: SS1 in 0.2850\\leq z< 0.3029, SS2 with 0.3029\\leq z< 0.3243, and SS3 covering 0.3243\\leq z< 0.3380. Vertical black segments are the clustercentric radial distance of the galaxies. Bottom panel: Galaxies of the three substructures in the mass-metallicity plane. Sky-blue triangles, black dots, and red dots represent SS1, SS2, and SS3, respectively. The black line is the cMZR. ](plots/KDE_MZR.png \"fig:\") [fig14]\n![Same as Figure [fig14], but in this instance the KDE was applied in the radial distribution. Top panel: Kernel density estimator in the radial distribution of the galaxy cluster. Blue lines show radial cuts at r= 32.5, r=57.4, and r=140\\ Mpc. The vertical black segments are the clustercentric radial distances of the galaxies. Bottom panel: Galaxies in the mass-metallicity plane of the three radial regions. Green triangles, black dots, and blue dots represent the central, intermediate, and outskirt regions, respectively.](plots/KDE_MZR_r.png \"fig:\") [fig15]\nDiscussion\n==========\nP15 and S23 studied the dynamical state of AC114 and the stellar populations of the cluster members. The radial profiles in Figure [fig13] show a negative mass gradient as a function of the clustercentric radius, consistent with the cluster being in an early stage of radial relaxation, as is also suggested by P15. The steeper mass gradient up to *r* ∼ 130 Mpc of SF members suggests that these galaxies are more sensitive to the ongoing dynamical processes of AC114 because of their lower masses with respect to the passive ones. Both the cMZR and the bMZR shown in Figure [fig11] are downshifted with respect to the local Universe MZR, showing that AC114 SF members are more metal-poor than the field galaxies at the same redshift, and similar in terms of metallicity to background galaxies at a given mass.\nS23, studying stellar populations of SF and passive galaxies, has shown that the stellar MZR of AC114 is steeper than for galaxies in the local Universe. However, the gas-phase cMZR in Figure [fig11] has a shallower slope than galaxies in the local Universe. This could be due to galaxies quenching by (i) strangulation (, ), where the gas is consumed faster when there is no fresh supply, and (ii) the ICM is boosting the metal-poor inflows. By adding our analysis to that reported in S23, stellar metallicities are lower than gas-phase metallicities because those metals were produced in the past, while the gas-phase metallicities are present-day ones. Therefore, the SF galaxies should be in an early stage of the quenching.\nWhen cluster members were separated into substructures, important imprints of environmental effects were revealed. Figures [fig14] and [fig15] shows that galaxies located outside the central region set the slope of the cMZR (red and black dots in Figure [fig14], and blue and black dots in Figure [fig15]). Star-forming galaxies within the main component of AC114 and the central region do not show a correlation between mass and gas-phase metallicity, with a large scatter. \n \nThe MZRs of SF galaxies in massive clusters ( > 1014*M*⊙) appear to follow those of field galaxies at their corresponding redshift. For example,, studying four CLASH clusters, namely MACS2129, MACS1311, MACS1931, and MS2131 (*M*200 ∼ 1014 − 1015 *M*⊙; ), in the redshift range *z* ∼ 0.2 − 0.6, indicate that in these clusters the MZRs do not show any changes with respect to that constructed with field SF galaxies. Additionally, and, by using SF members of the Fornax cluster (*M**v**i**r* ∼ 7 × 1013*M*⊙; ) and the RX J1532 + 30 (*z* ∼ 0.35, *M**v**i**r* ∼ 6.4 × 1014*M*⊙) galaxy cluster, respectively, are also consistent, with no variations with respect to the field MZR. However, these works report flatter galactic metallicity gradients, negative cluster metallicity gradients, and a negative trend in the residuals of the MZRs as a function of the clustercentric radius. Several physical processes are discussed, but the main physical phenomena that drive those trends are strangulation and RPS. It is well known that RPS is the physical result of the galaxy-ICM interaction that affects the outskirts of galaxy disks: in the presence of negative metallicity gradients, the stripped-out gas is metal-poorer than their nuclei, which leads to increased oxygen abundances.\nStrangulation is produced when the pristine gas inflow ends and the gas reservoir is consumed faster, increasing the SFR until the quenching begins. Therefore, the H*α* disks become smaller than the stellar disks, and oxygen abundances are more metal-rich than galaxies that do not experience galaxy-ICM interactions at the same mass.\nOn the other hand, galaxy clusters such as Coma, Hercules (A2151), MACSJ1115+01, and those in the SDSS DR4 survey, show enhanced metallicities with respect to the field MZR, up to ∼ 0.05 dex in galaxies affected by RPS and/or galaxy-galaxy interactions inside the virial radius of their host dark matter halos (,,, ). This indicates that extremely dense environments, such as galaxy clusters with *M**v**i**r* > 1015*M*⊙, may show variations in the cluster MZRs with respect to the field ones.\nThe MZR of AC114 also shows this feature for at least two cluster members when it is compared with the field MZR at *z* ∼ 0.3, as is indicated in Figure [figcomparison]. In this case, we used the Maiolino parameterization at redshift 0.3 to compare with the AC114-MZR. However, the two galaxies are more metal-rich, ∼ 0.10 dex, which is more than that reported by. Additionally, the other galaxies (below the linear fit of the cMZR with the red line) drive the shallower slope and have lower metallicities than the field SF galaxies at the same redshift up to ∼ 0.22 dex. The striking feature is that these galaxies are located outside the central region of AC114 (see Figure [fig15]). Several reasons can explain this feature: (i) the AGN feedback, which is important at low *z*, can expel the metal-rich gas component in the central region of SF members with negative metallicity gradients (, ); (ii) the presence of strong metal-poor inflows (, ), where galaxies accreting metal-poor gas decrease their average metallicities; and (iii) mergers and tidal interactions, where the interaction with closer companions can dilute the gas-phase metallicities.\nIn order to understand the main physical phenomena behind the metal-poor content of the SF galaxies of AC114 with respect to the field MZR at the same redshift, the AGN feedback can be discarded because our cluster sample has just two composite galaxies. One of them is over the local Universe MZR, and the other shows the largest deviation below the MZR in the cluster sample. According to, the location of these two composite galaxies in the mass-metallicity plane can be explained as the former being a LINER-composite galaxy and the latter a Seyfert-composite galaxy. Additionally, Seyfert galaxies have stronger [OIII]*λ*5007 emission lines than SF galaxies. The O3N2 index anticorrelates linearly with the oxygen abundances, so it may explain the location of this galaxy in the mass-metallicity plane. The same applies to the LINER galaxy in our cluster sample, but in this case, LINERS have strong [NII]*λ*6584 emission lines. According to observation and simulations from and, despite the AGN feedback being important at low redshift, for satellite galaxies (non-cDs in galaxy clusters) this effect is negligible.\nOn the other hand, a strong metal-poor inflow and dynamical events can explain the observed feature., simulating mergers and flybys between disk galaxies, showed that galaxy-galaxy interactions can dilute gas-phase metallicities and also increase the SFR, due to the accretion of metal-poor gas from galaxies with negative metallicity gradients. The estimated dilution agrees with, which reports that galaxy pair interaction dilutes gas-phase metallicities up to ∼ 0.2 − 0.3 dex. Those values are also in line with the downshifts of the outlier foreground galaxies with respect to the fMZR, which are ∼ 0.18 on average. Additionally, also explain that galaxy-galaxy interaction increases the gas-phase metallicities after the dilution due to the increase in the SFRs. Therefore, this mechanism increases the scatter of the MZR, as is also seen in.\nThis scenario is also supported by ongoing dynamical activity (P15) and by the fact that this galaxy cluster has an extended soft X-ray tail, together with the hardness ratio map, which suggests that this cluster has an ongoing merger activity with a smaller structure ().\nSummary and conclusions\n=======================\nWe explored the role of galaxy clusters in the MZR through SF galaxies of the AC114 galaxy cluster. We analyzed optical, intermediate-resolution VIMOS/VLT spectra for a sample of 182 galaxies in the FOV of AC114.\nThe search in photometric catalogues allowed us to get optical and NIR luminosities for 76% and 66% of the galaxies of the sample from the DECaLS DR10 and VIKING DR4 ESO surveys, respectively. Synthetic photometry was performed using the VIMOS spectra when optical photometry was missing. Optical and NIR-based stellar masses of the galaxies were estimated via theoretical stellar mass-to-light ratios computed from the E-MILES models by considering the age dependence for both sdss − *i* and *K**s* band passes. A comparison between optical and NIR-based mass estimations shows agreement between them for SF galaxies, but there is a slight deviation in massive, passive galaxies of 0.23 dex. However, the offset is within the propagated uncertainties of *σ* = 0.29 mags, so it does not change our results.\nWe classified the galaxies into ELGs and passive galaxies, based on the presence or absence of [OII]\\lambda3727, [OIII]\\lambda\\lambda4959,5007, H*β*, and H*α* emission lines, respectively. Using the BPT diagram, together with the and demarcation limits, we found that the galaxy sample has seven composite galaxies, where two of them (a Seyfert composite and a LINER composite) are cluster members. The remaining galaxies in the sample are, in fact, pure SF galaxies.\nGas-phase metallicities were estimated from the O3N2, which are calibrated on *T**e* − based metallicities. \n \nOur main conclusions can be summarized as follows:\n* Foreground and background MZRs span masses from 108.7 to 1010.3*M*⊙ and 109.4 to 1010.7*M*⊙, respectively, with both the fMZR and the bMZR showing a scatter of ∼ 0.12 dex. The fMZR was constructed with less-massive galaxies than bMZR due to an observational bias. However, the downshifts reported show a coherent evolution in redshift of the MZR, where foreground galaxies are 0.09 dex metal-poorer, and background galaxies are 0.20 dex metal-poorer on average with respect to the local Universe galaxies.\n* The AC114-MZR presents different features with respect to field galaxies at similar redshifts. The cMZR span masses from 109.2 to 1010.8*M*⊙, with 0.17 dex of scatter. A downshift of 0.19 dex on average and a slightly shallower slope is seen with respect to both the fMZR and the bMZR. This shows that, in general, AC114 SF galaxies have lower metallicities at the same mass than field galaxies at the same redshift. The metallicity measurements of the SF cluster members are comparable to the background galaxies in our galaxy sample. Besides, star-forming galaxies within the main component of AC114 and the central region do not show a correlation between mass and gas-phase metallicity, with a large scatter. This gives important clues about the environmental effects that are suppressing the chemical evolution of these galaxies.\n* The mass and metallicity profiles as a function of the clustercentric radius show that AC114 started its radial relaxation phase, in agreement with. The negative mass gradient suggests that SF galaxies are the most sensitive to the dynamical state of the galaxy cluster. Additionally, the negative (although not statistically significant) metallicity gradient can be a sign that massive galaxies located near the central region of AC114 are slightly more metal-rich than those found in external regions.\n* The substructures identified with the KDE show that for SF galaxies in the outskirts of the galaxy cluster the metallicity is dependent on the stellar mass. Moreover, these galaxies are responsible for the shallower slope of the cMZR (compared with the field MZR at *z* ∼ 0.3), which means that these galaxies are also the more metal-poor ones.\n* When the cMZR is compared with the field MZR at *z* ∼ 0.3, two galaxies are located above the field MZR by ∼ 0.10 dex. This feature is consistent with the literature, which suggests that galaxies are suffering from RPS, strangulation, and/or galaxy-galaxy interactions.\n* Galaxies falling below the linear fit of the cMZR are the drivers of the shallower slope. These galaxies are also found outside the central region of AC114. This means that outskirt galaxies are evolving slowly in terms of their chemical content, showing metallicities up to 0.22 dex lower than the local Universe galaxies and on average 0.17 dex lower than the field MZR at *z* ∼ 0.3. Strong metal-poor inflows produced by dynamical events such as galaxy-galaxy interactions may explain the strikingly low metallicities of the SF members of AC114. This scenario is also supported by the dynamical state of the cluster and the ongoing merger event between the main component of AC114 and a smaller structure, observed in the X-rays.\n* We emphasize the role of the mass of a galaxy cluster. Environmental and dynamical effects drive the chemical evolution of members in a unique way with respect to field galaxies. However, only galaxy clusters with *M* > 1015*M*⊙ appear to show observable changes in the MZR.\nWe thank to the anonymous referee for their useful comments which improved the quality of the paper. A special thanks goes to the Dr. Piercarlo Bonifacio for the feedback presented in the early stage of this research. Additionally, we want to thanks Felipe Vivanco and Gustavo Worner, for the opportunity to discuss our results and for the valuable contributions to the development of the Python scripts used in this work. This publication is based on observations collected at the European Southern Observatory under ESO programme ID 083.A-0566. Also, this program has made use of data from the VIKING survey from VISTA at the ESO Paranal Observatory, programme ID 179.A-2004. Data processing has been contributed by the VISTA Data Flow System at CASU, Cambridge and WFAU, Edinburgh.\nAlonso, M. S., Lambas, D. G., Tissera, P., et al. 2007,, 375, 1017. doi:10.1111/j.1365-2966.2007.11367.x\nAller, L. 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I., Kudritzki, R.-P., et al. 2014,, 791, 130. doi:10.1088/0004-637X/791/2/130\nTables:\n=======\n[table:0]\n| Figure | Structure | Slope | *σ* | *r* (Spearman rank) | *p* − value |\n| --- | --- | --- | --- | --- | --- |\n| 9 | fMZR | 0.236 ± 0.003 | 0.118 | 0.97 | 2.6 × 10− 8 |\n| 9 | bMZR | 0.312 ± 0.009 | 0.120 | 0.90 | 1.5 × 10− 4 |\n| 9 | cMZR | 0.241 ± 0.008 | 0.170 | 0.65 | 0.009 |\n| 12 | Mass profile | -0.005 ± 0.002 | 0.540 | -0.57 | 0.005 |\n| 12 | SF mass profile | -0.007 ± 0.003 | 0.470 | -0.50 | 0.020 |\n| 12 | Pasive mass profile | -0.004 ± 0.003 | 0.520 | -0.24 | 0.090 |\n| 12 | Metallicity profile | -0.002 ± 0.002 | 0.190 | -0.38 | 0.110 |\n| 13 | SS2 (cMZR) | 0.192 ± 0.129 | 0.330 | 0.54 | 0.125 |\n| 13 | SS3 (cMZR) | 0.269 ± 0.135 | 0.280 | 0.86 | 0.019 |\n| 14 | Middle region (cMZR) | 0.352 ± 0.078 | 0.270 | 0.86 | 0.014 |\n| 14 | External region (cMZR) | 0.162 ± 0.096 | 0.350 | 0.40 | 0.319 |\n[table:1]\n| ID | RA | Dec | z | *i* | *g* | *K**s* | ELGs |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| Q1-3 | 344.889 | -34.711 | 0.057330 ± 0.000014 | | | | 1 |\n| Q1-A1 | 344.877 | -34.713 | 0.277520 ± 0.000054 | | | | 1 |\n| Q1-5 | 344.884 | -34.716 | 0.057850 ± 0.000012 | | | | 1 |\n| Q1-A2 | 344.815 | -34.721 | 0.289160 ± 0.000022 | | | | 0 |\n| Q1-8 | 344.902 | -34.724 | 0.379530 ± 0.000015 | | | | 1 |\n| Q1-9 | 344.858 | -34.726 | 0.416980 ± 0.000011 | 22.2 ± 0.6 | | 17.80 ± 0.16 | 0 |\n| Q1-10 | 344.827 | -34.729 | 0.219600 ± 0.000017 | 18.7 ± 0.5 | 20.3 ± 0.8 | 15.78 ± 0.03 | 1 |\n| Q1-11 | 344.823 | -34.733 | 0.282340 ± 0.000007 | 21.6 ± 0.6 | 23.7 ± 1.1 | 18.46 ± 0.24 | 0 |\n| Q1-12 | 344.833 | -34.736 | 0.220480 ± 0.000016 | 19.6 ± 0.4 | 20.4 ± 0.8 | 17.13 ± 0.15 | 1 |\n| Q1-A3 | 344.800 | -34.740 | 0.271680 ± 0.000007 | | | | 1 |\n| Q1-15 | 344.867 | -34.741 | 0.071500 ± 0.000013 | 21.5 ± 0.6 | 21.9 ± 1.0 | | 1 |\n| Q1-16 | 344.870 | -34.746 | 0.408880 ± 0.000021 | 21.7 ± 0.5 | 23.7 ± 1.0 | 18.16 ± 0.24 | 0 |\n| Q1-17 | 344.910 | -34.748 | 0.342210 ± 0.000007 | | | | 0 |\n| Q1-18 | 344.846 | -34.751 | 0.312460 ± 0.000020 | 21.2 ± 0.6 | 22.4 ± 1.1 | 18.31 ± 0.18 | 1 |\n| Q1-19 | 344.827 | -34.754 | 0.408190 ± 0.000009 | 21.4 ± 0.5 | 22.5 ± 1.0 | 18.97 ± 0.34 | 1 |\n| Q1-20 | 344.918 | -34.756 | 0.231980 ± 0.000016 | | | | 0 |\n| Q1-A4 | 344.897 | -34.757 | 0.357780 ± 0.000010 | | | | 0 |\n| Q1-22 | 344.853 | -34.760 | 0.409870 ± 0.000019 | 19.0 ± 0.4 | 21.5 ± 0.8 | 15.92 ± 0.04 | 1 |\n| Q1-24 | 344.869 | -34.764 | 0.333100 ± 0.000005 | 19.3 ± 0.4 | 20.8 ± 0.8 | 16.52 ± 0.06 | 0 |\n| Q1-25 | 344.896 | -34.766 | 0.414380 ± 0.000192 | 20.5 ± 0.5 | 22.1 ± 1.0 | 17.73 ± 0.17 | 1 |\n| Q1-26 | 344.883 | -34.770 | 0.417650 ± 0.000038 | 21.3 ± 0.5 | 22.9 ± 1.0 | 17.91 ± 0.14 | 0 |\n| Q1-27 | 344.884 | -34.773 | 0.390830 ± 0.000017 | 22.1 ± 0.5 | 23.5 ± 0.9 | | 0 |\n| Q1-28 | 344.859 | -34.776 | 0.343650 ± 0.000011 | 19.1 ± 0.3 | 19.8 ± 0.6 | 17.15 ± 0.12 | 0 |\n| Q1-30 | 344.900 | -34.779 | 0.318050 ± 0.000014 | 19.1 ± 0.4 | 21.1 ± 0.8 | 15.30 ± 0.02 | 1 |\n| Q1-31 | 344.890 | -34.780 | 0.331760 ± 0.000021 | 20.6 ± 0.6 | 22.6 ± 1.0 | 17.34 ± 0.12 | 0 |\n| Q1-34 | 344.855 | -34.785 | 0.301330 ± 0.000011 | 20.3 ± 0.6 | 22.4 ± 1.0 | 17.04 ± 0.08 | 0 |\n| Q1-36 | 344.892 | -34.789 | 0.498080 ± 0.000010 | 20.7 ± 0.5 | 22.2 ± 0.8 | 18.75 ± 0.20 | 0 |\n| Q1-37 | 344.849 | -34.791 | 0.315480 ± 0.000042 | 19.5 ± 0.6 | 21.2 ± 1.0 | 16.51 ± 0.05 | 1 |\n| Q1-A5 | 344.799 | -34.793 | 0.343630 ± 0.000018 | 21.0 ± 0.4 | 22.8 ± 0.7 | 16.96 ± 0.09 | 0 |\n| Q1-38 | 344.825 | -34.794 | 0.485360 ± 0.000018 | | | | 1 |\n| Q1-40 | 344.847 | -34.799 | 0.323390 ± 0.000019 | 19.1 ± 0.3 | | 16.29 ± 0.05 | 1 |\n| Q1-41 | 344.820 | -34.802 | 0.576610 ± 0.000008 | | | 17.09 ± 0.08 | 0 |\n| Q1-A6 | 344.871 | -34.808 | 0.289270 ± 0.000014 | | | | 0 |\n| Q1-44 | 344.816 | -34.812 | 0.576030 ± 0.000009 | 21.6 ± 0.4 | 23.1 ± 0.7 | | 1 |\n| ID | RA | Dec | z | *i* | *g* | *K**s* | ELGs |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| Q2-1 | 344.817 | -34.561 | 0.259580 ± 0.000022 | | | | 1 |\n| Q2-2 | 344.825 | -34.564 | 0.334680 ± 0.000010 | | | | 0 |\n| Q2-3 | 344.824 | -34.566 | 0.409690 ± 0.000012 | | | | 1 |\n| Q2-4 | 344.806 | -34.570 | 0.408470 ± 0.000019 | | | | 1 |\n| Q2-5 | 344.812 | -34.573 | 0.256670 ± 0.000559 | | | | 0 |\n| Q2-6 | 344.849 | -34.576 | 0.423330 ± 0.000025 | 18.9 ± 0.4 | 21.0 ± 0.7 | 15.80 ± 0.05 | 0 |\n| Q2-7 | 344.839 | -34.578 | 0.242100 ± 0.000610 | 22.1 ± 0.7 | 25.2 ± 1.1 | 19.45 ± 0.52 | 0 |\n| Q2-8 | 344.803 | -34.580 | 0.445090 ± 0.000009 | 21.6 ± 0.6 | 22.7 ± 1.0 | 18.88 ± 0.29 | 0 |\n| Q2-9 | 344.823 | -34.582 | 0.370950 ± 0.000048 | 22.1 ± 0.6 | 23.2 ± 1.1 | | 1 |\n| Q2-10 | 344.828 | -34.584 | 0.408990 ± 0.000012 | 21.3 ± 0.7 | 23.4 ± 1.1 | 17.04 ± 0.12 | 1 |\n| Q2-11 | 344.820 | -34.586 | 0.298630 ± 0.000016 | 22.1 ± 0.4 | 22.9 ± 0.8 | | 1 |\n| Q2-12 | 344.886 | -34.590 | 0.370700 ± 0.000016 | 21.1 ± 0.5 | 22.9 ± 0.9 | 17.69 ± 0.19 | 0 |\n| Q2-13 | 344.859 | -34.592 | 0.360210 ± 0.000047 | | | | 0 |\n| Q2-14 | 344.834 | -34.595 | 0.373620 ± 0.000044 | 20.7 ± 0.4 | 22.6 ± 0.9 | 17.73 ± 0.15 | 0 |\n| Q2-15 | 344.815 | -34.598 | 0.410010 ± 0.000133 | 21.7 ± 0.6 | 23.9 ± 1.1 | 18.37 ± 0.20 | 0 |\n| Q2-16 | 344.829 | -34.600 | 0.573910 ± 0.000024 | 21.3 ± 0.4 | 23.5 ± 0.9 | 18.48 ± 0.27 | 1 |\n| Q2-A1 | 344.833 | -34.605 | 0.267590 ± 0.000278 | | | | 0 |\n| Q2-18 | 344.800 | -34.608 | 0.345920 ± 0.000037 | 21.1 ± 0.5 | 23.2 ± 0.9 | 17.98 ± 0.17 | 0 |\n| Q2-19 | 344.830 | -34.611 | 0.377380 ± 0.000030 | 19.3 ± 0.4 | 21.5 ± 0.8 | 16.37 ± 0.05 | 0 |\n| Q2-A2 | 344.855 | -34.612 | 0.466710 ± 0.000057 | | | | 0 |\n| Q2-21 | 344.840 | -34.615 | 0.413680 ± 0.000105 | 20.5 ± 0.5 | 22.5 ± 0.8 | 17.45 ± 0.11 | 0 |\n| Q2-22 | 344.903 | -34.615 | 0.391390 ± 0.000022 | | | | 1 |\n| Q2-23 | 344.839 | -34.619 | 0.413930 ± 0.000038 | 19.5 ± 0.5 | 22.1 ± 0.9 | 16.31 ± 0.04 | 0 |\n| Q2-24 | 344.903 | -34.623 | 0.380780 ± 0.000055 | 22.1 ± 0.6 | 23.2 ± 1.0 | | 1 |\n| Q2-A3 | 344.834 | -34.625 | 0.429260 ± 0.000014 | | | | 0 |\n| Q2-26 | 344.801 | -34.627 | 0.487890 ± 0.000018 | 21.5 ± 0.5 | 23.2 ± 0.9 | 19.30 ± 0.48 | 1 |\n| Q2-27 | 344.837 | -34.629 | 0.497530 ± 0.000033 | 20.3 ± 0.5 | 22.1 ± 0.9 | 17.11 ± 0.11 | 1 |\n| Q2-28 | 344.814 | -34.633 | 0.613460 ± 0.000019 | 20.4 ± 0.3 | 21.7 ± 0.6 | 17.36 ± 0.17 | 1 |\n| Q2-29 | 344.823 | -34.634 | 0.523020 ± 0.000010 | 20.6 ± 0.2 | | 17.90 ± 0.21 | 1 |\n| Q2-31 | 344.852 | -34.638 | 0.497110 ± 0.000022 | | | | 0 |\n| Q2-A4 | 344.866 | -34.638 | 0.497370 ± 0.000008 | | | | 1 |\n| Q2-32 | 344.854 | -34.643 | 0.308930 ± 0.000018 | | | 18.13 ± 0.23 | 1 |\n| Q2-33 | 344.819 | -34.645 | 0.579620 ± 0.000006 | | | 17.52 ± 0.11 | 1 |\n| Q2-A5 | 344.926 | -34.648 | 0.345720 ± 0.000017 | | | | 0 |\n| Q2-35 | 344.919 | -34.651 | 0.313670 ± 0.000056 | 19.7 ± 0.6 | 21.9 ± 1.1 | 16.51 ± 0.04 | 0 |\n| Q2-36 | 344.857 | -34.653 | 0.465160 ± 0.000210 | 20.2 ± 0.6 | 22.2 ± 1.0 | 17.42 ± 0.11 | 1 |\n| Q2-37 | 344.808 | -34.659 | 0.139900 ± 0.000017 | 20.4 ± 0.4 | 21.2 ± 0.7 | | 1 |\n| Q2-38 | 344.846 | -34.663 | 0.298510 ± 0.000033 | 21.5 ± 0.6 | 22.7 ± 1.0 | 19.46 ± 0.46 | 1 |\n| Q2-39 | 344.912 | -34.665 | 0.371870 ± 0.000021 | 20.6 ± 0.6 | 21.7 ± 1.1 | 17.88 ± 0.16 | 1 |\n| Q2-40 | 344.901 | -34.666 | 0.420140 ± 0.000154 | 20.9 ± 0.6 | 23.0 ± 1.0 | 18.49 ± 0.31 | 0 |\n| Q2-41 | 344.932 | -34.669 | 0.540080 ± 0.000014 | 21.3 ± 0.6 | 23.3 ± 1.0 | 18.04 ± 0.22 | 0 |\n| ID | RA | Dec | z | *i* | *g* | *K**s* | ELGs |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| Q3-1 | 344.673 | -34.555 | 0.329830 ± 0.000060 | 21.4 ± 0.6 | 22.5 ± 1.1 | 19.04 ± 0.40 | 1 |\n| Q3-2 | 344.672 | -34.557 | 0.330890 ± 0.000003 | 18.8 ± 0.5 | 20.1 ± 0.8 | 15.78 ± 0.04 | 1 |\n| Q3-3 | 344.667 | -34.559 | 0.331850 ± 0.000079 | 19.1 ± 0.6 | 21.3 ± 1.0 | 15.96 ± 0.03 | 0 |\n| Q3-4 | 344.667 | -34.561 | 0.212750 ± 0.000021 | 20.9 ± 0.7 | 22.1 ± 1.1 | 18.49 ± 0.26 | 1 |\n| Q3-5 | 344.671 | -34.563 | 0.331820 ± 0.000056 | 21.0 ± 0.5 | 22.2 ± 1.0 | 18.49 ± 0.36 | 1 |\n| Q3-6 | 344.641 | -34.564 | 0.495360 ± 0.000019 | | | | 0 |\n| Q3-A1 | 344.695 | -34.566 | 0.315910 ± 0.000481 | | | | 0 |\n| Q3-8 | 344.678 | -34.573 | 0.400050 ± 0.000009 | | | | 1 |\n| Q3-10 | 344.594 | -34.576 | 0.308500 ± 0.000136 | 18.5 ± 0.3 | 19.4 ± 0.5 | 16.83 ± 0.13 | 0 |\n| Q3-11 | 344.650 | -34.579 | 0.212680 ± 0.000023 | 20.8 ± 0.7 | 21.5 ± 1.1 | 18.78 ± 0.34 | 1 |\n| Q3-12 | 344.654 | -34.582 | 0.310390 ± 0.000230 | 20.0 ± 0.5 | 21.2 ± 0.8 | 17.49 ± 0.21 | 1 |\n| Q3-A2 | 344.685 | -34.587 | 0.497010 ± 0.000009 | | | | 0 |\n| Q3-15 | 344.671 | -34.589 | 0.415820 ± 0.000014 | | | | 0 |\n| Q3-16 | 344.691 | -34.591 | 0.176550 ± 0.000028 | 21.9 ± 0.5 | 22.7 ± 1.1 | | 1 |\n| Q3-17 | 344.702 | -34.594 | 0.497500 ± 0.000019 | 22.6 ± 0.6 | 24.8 ± 1.1 | 19.42 ± 0.44 | 1 |\n| Q3-18 | 344.735 | -34.596 | 0.221070 ± 0.000019 | 19.5 ± 0.4 | 20.5 ± 0.8 | 17.24 ± 0.14 | 1 |\n| Q3-19 | 344.597 | -34.599 | 0.159050 ± 0.000011 | 18.6 ± 0.4 | 19.6 ± 0.6 | 15.88 ± 0.06 | 0 |\n| Q3-20 | 344.635 | -34.602 | 0.354770 ± 0.000036 | 23.1 ± 0.5 | 24.3 ± 1.0 | | 1 |\n| Q3-21 | 344.635 | -34.604 | 0.353350 ± 0.000020 | 19.3 ± 0.4 | 20.8 ± 0.7 | 16.48 ± 0.08 | 1 |\n| Q3-23 | 344.596 | -34.607 | 0.353750 ± 0.000046 | 19.9 ± 0.6 | 21.6 ± 1.0 | 16.89 ± 0.07 | 0 |\n| Q3-24 | 344.621 | -34.609 | 0.271760 ± 0.000016 | 22.4 ± 0.7 | 24.2 ± 1.1 | 18.57 ± 0.27 | 0 |\n| Q3-25 | 344.670 | -34.611 | 0.595430 ± 0.000085 | 20.5 ± 0.5 | 23.2 ± 0.8 | 17.07 ± 0.07 | 0 |\n| Q3-26 | 344.618 | -34.613 | 0.315900 ± 0.000024 | 20.2 ± 0.7 | 22.1 ± 1.1 | 17.36 ± 0.09 | 0 |\n| Q3-27 | 344.623 | -34.615 | 0.593510 ± 0.000068 | 22.0 ± 0.7 | 23.4 ± 1.1 | 19.74 ± 0.48 | 1 |\n| Q3-28 | 344.601 | -34.617 | 0.258540 ± 0.000026 | 18.1 ± 0.5 | 20.1 ± 0.8 | 15.05 ± 0.02 | 0 |\n| Q3-29 | 344.635 | -34.618 | 0.462490 ± 0.000004 | 20.2 ± 0.6 | 21.0 ± 0.9 | 18.27 ± 0.22 | 1 |\n| Q3-30 | 344.616 | -34.621 | 0.402360 ± 0.000025 | | | | 1 |\n| Q3-31 | 344.685 | -34.623 | 0.332340 ± 0.000228 | 20.8 ± 0.5 | 22.0 ± 0.9 | 17.89 ± 0.20 | 1 |\n| Q3-A3 | 344.643 | -34.626 | 0.319970 ± 0.000009 | 23.8 ± 0.7 | 24.7 ± 1.1 | 19.79 ± 0.67 | 0 |\n| Q3-33 | 344.663 | -34.631 | 0.112810 ± 0.000010 | 22.3 ± 0.4 | 22.9 ± 0.6 | | 1 |\n| Q3-34 | 344.665 | -34.634 | 0.351610 ± 0.000046 | | | 18.38 ± 0.18 | 1 |\n| Q3-35 | 344.640 | -34.638 | 0.060200 ± 0.000009 | | | | 1 |\n| Q3-36 | 344.631 | -34.640 | 0.202140 ± 0.000016 | | | 17.70 ± 0.20 | 1 |\n| Q3-A4 | 344.668 | -34.642 | 0.345010 ± 0.000078 | | | | 0 |\n| Q3-38 | 344.672 | -34.644 | 0.432870 ± 0.000022 | | | 18.63 ± 0.28 | 1 |\n| Q3-39 | 344.694 | -34.646 | 0.399430 ± 0.000059 | | | 18.19 ± 0.22 | 1 |\n| Q3-40 | 344.642 | -34.648 | 0.318120 ± 0.000081 | 19.5 ± 0.5 | 21.7 ± 0.8 | 16.26 ± 0.04 | 0 |\n| Q3-A5 | 344.610 | -34.649 | 0.479200 ± 0.000065 | | | | 0 |\n| Q3-A6 | 344.588 | -34.651 | 0.325310 ± 0.000167 | | | | 1 |\n| Q3-43 | 344.664 | -34.653 | 0.509810 ± 0.000015 | 21.1 ± 0.5 | 22.3 ± 0.9 | | 1 |\n| Q3-44 | 344.664 | -34.656 | 0.224270 ± 0.000168 | 19.6 ± 0.5 | 21.2 ± 0.8 | 16.40 ± 0.07 | 1 |\n| Q3-45 | 344.681 | -34.659 | 0.327200 ± 0.000073 | 21.3 ± 0.5 | 22.9 ± 0.9 | 18.64 ± 0.34 | 0 |\n| Q3-46 | 344.678 | -34.661 | 0.309010 ± 0.000041 | 21.0 ± 0.5 | 22.6 ± 0.9 | 18.70 ± 0.32 | 0 |\n| Q3-A7 | 344.585 | -34.662 | 0.345340 ± 0.000153 | 19.9 ± 0.5 | 21.3 ± 0.8 | 17.01 ± 0.11 | 0 |\n| Q3-A8 | 344.620 | -34.665 | 0.412960 ± 0.000773 | | | | 0 |\n| Q3-49 | 344.655 | -34.668 | 0.309000 ± 0.000033 | 18.0 ± 0.3 | 19.2 ± 0.5 | 15.62 ± 0.05 | 0 |\n| ID | RA | Dec | z | *i* | *g* | *K**s* | ELGs |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| Q4-A1 | 344.669 | -34.704 | 0.133610 ± 0.000008 | | | | 1 |\n| Q4-2 | 344.646 | -34.706 | 0.318380 ± 0.000216 | 20.0 ± 0.6 | 22.0 ± 1.0 | 16.91 ± 0.08 | 0 |\n| Q4-3 | 344.688 | -34.708 | 0.412180 ± 0.000024 | 20.7 ± 0.6 | 22.2 ± 1.1 | 18.05 ± 0.18 | 1 |\n| Q4-4 | 344.648 | -34.709 | 0.310170 ± 0.000273 | 21.3 ± 0.7 | 23.0 ± 1.0 | 18.57 ± 0.25 | 0 |\n| Q4-5 | 344.677 | -34.711 | 0.411570 ± 0.000089 | 21.2 ± 0.6 | 22.6 ± 1.0 | 18.47 ± 0.32 | 1 |\n| Q4-A2 | 344.660 | -34.712 | 0.268420 ± 0.000033 | | | | 0 |\n| Q4-7 | 344.635 | -34.715 | 0.169450 ± 0.000183 | 19.5 ± 0.5 | 20.1 ± 0.8 | 17.62 ± 0.20 | 1 |\n| Q4-8 | 344.636 | -34.717 | 0.169800 ± 0.000028 | 21.4 ± 0.7 | 22.2 ± 1.1 | 19.17 ± 0.32 | 1 |\n| Q4-9 | 344.632 | -34.721 | 0.256730 ± 0.000144 | 18.9 ± 0.6 | 20.8 ± 0.9 | 15.69 ± 0.02 | 0 |\n| Q4-10 | 344.605 | -34.723 | 0.311730 ± 0.000114 | | | | 0 |\n| Q4-11 | 344.635 | -34.725 | 0.291170 ± 0.000046 | 22.4 ± 0.7 | 24.0 ± 1.1 | | 0 |\n| Q4-13 | 344.611 | -34.728 | 0.316650 ± 0.000101 | 21.3 ± 0.6 | 23.2 ± 1.0 | 18.69 ± 0.30 | 0 |\n| Q4-14 | 344.673 | -34.729 | 0.304220 ± 0.000072 | 21.4 ± 0.7 | 22.1 ± 1.1 | 19.63 ± 0.65 | 1 |\n| Q4-15 | 344.608 | -34.731 | 0.306580 ± 0.000008 | 21.9 ± 0.7 | 22.7 ± 1.1 | | 1 |\n| Q4-16 | 344.623 | -34.732 | 0.318090 ± 0.000212 | 20.1 ± 0.7 | 22.1 ± 1.2 | 16.79 ± 0.06 | 0 |\n| Q4-A3 | 344.596 | -34.733 | 0.362090 ± 0.000022 | | | | 1 |\n| Q4-18 | 344.608 | -34.735 | 0.290200 ± 0.000282 | 21.4 ± 0.8 | 24.2 ± 1.2 | 18.08 ± 0.16 | 0 |\n| Q4-19 | 344.609 | -34.736 | 0.309670 ± 0.000160 | 21.0 ± 0.6 | 21.8 ± 1.0 | 17.97 ± 0.18 | 1 |\n| Q4-20 | 344.664 | -34.738 | 0.306440 ± 0.000191 | 19.9 ± 0.6 | 22.0 ± 1.1 | 16.67 ± 0.05 | 0 |\n| Q4-21 | 344.711 | -34.737 | 0.314790 ± 0.000241 | | | | 0 |\n| Q4-22 | 344.666 | -34.741 | 0.317210 ± 0.000205 | 21.4 ± 0.7 | 22.3 ± 1.2 | 19.57 ± 0.58 | 1 |\n| Q4-23 | 344.646 | -34.742 | 0.320540 ± 0.000446 | 21.0 ± 0.7 | 23.0 ± 1.1 | 18.05 ± 0.24 | 0 |\n| Q4-24 | 344.638 | -34.744 | 0.314080 ± 0.000204 | 20.7 ± 0.7 | 22.6 ± 1.2 | 18.02 ± 0.16 | 0 |\n| Q4-25 | 344.626 | -34.745 | 0.297410 ± 0.001417 | 22.5 ± 0.8 | 24.3 ± 1.2 | | 0 |\n| Q4-26 | 344.675 | -34.747 | 0.324030 ± 0.000127 | 19.7 ± 0.5 | 20.4 ± 0.9 | 17.66 ± 0.17 | 1 |\n| Q4-27 | 344.624 | -34.748 | 0.323200 ± 0.000040 | 21.9 ± 0.6 | 24.7 ± 1.0 | 19.01 ± 0.34 | 0 |\n| Q4-28 | 344.612 | -34.750 | 0.314550 ± 0.000265 | 20.8 ± 0.7 | 22.7 ± 1.2 | 17.83 ± 0.17 | 0 |\n| Q4-A4 | 344.600 | -34.751 | 0.358350 ± 0.000065 | 21.6 ± 0.5 | 22.8 ± 0.8 | 19.53 ± 0.45 | 1 |\n| Q4-A5 | 344.625 | -34.752 | 0.427510 ± 0.000011 | | | | 0 |\n| Q4-31 | 344.652 | -34.754 | 0.317740 ± 0.000258 | 19.5 ± 0.6 | 21.7 ± 1.0 | 16.39 ± 0.04 | 0 |\n| Q4-32 | 344.658 | -34.756 | 0.316410 ± 0.000275 | 16.7 ± 0.1 | 18.9 ± 0.3 | 14.08 ± 0.02 | 0 |\n| Q4-33 | 344.672 | -34.758 | 0.101960 ± 0.000016 | 19.8 ± 0.4 | 20.6 ± 0.7 | 18.33 ± 0.19 | 1 |\n| Q4-A6 | 344.599 | -34.759 | 0.295250 ± 0.000024 | | | | 1 |\n| Q4-35 | 344.700 | -34.762 | 0.170350 ± 0.000007 | 20.7 ± 0.5 | 21.4 ± 1.1 | 17.46 ± 0.15 | 1 |\n| Q4-36 | 344.650 | -34.763 | 0.312860 ± 0.000277 | 17.6 ± 0.3 | 19.4 ± 0.5 | 14.66 ± 0.01 | 1 |\n| Q4-37 | 344.624 | -34.765 | 0.273050 ± 0.000022 | 22.8 ± 0.5 | 24.5 ± 0.9 | | 1 |\n| Q4-38 | 344.618 | -34.767 | 0.310670 ± 0.000239 | 19.2 ± 0.5 | 21.2 ± 0.9 | 16.28 ± 0.04 | 0 |\n| Q4-39 | 344.661 | -34.769 | 0.312550 ± 0.001159 | | | | 0 |\n| Q4-40 | 344.595 | -34.771 | 0.316250 ± 0.000482 | | | | 1 |\n| Q4-41 | 344.616 | -34.773 | 0.317970 ± 0.000324 | 20.5 ± 0.7 | 22.5 ± 1.1 | 17.59 ± 0.13 | 0 |\n| Q4-42 | 344.624 | -34.775 | 0.497280 ± 0.000614 | 21.1 ± 0.6 | 22.7 ± 1.0 | 18.42 ± 0.34 | 0 |\n| Q4-43 | 344.620 | -34.777 | 0.317200 ± 0.000339 | 20.9 ± 0.6 | 22.6 ± 1.0 | 18.57 ± 0.32 | 0 |\n| Q4-44 | 344.671 | -34.779 | 0.315630 ± 0.000309 | 18.8 ± 0.5 | 21.0 ± 0.9 | 15.38 ± 0.02 | 0 |\n| Q4-45 | 344.652 | -34.780 | 0.314980 ± 0.000326 | 20.6 ± 0.6 | 22.5 ± 1.0 | 17.85 ± 0.21 | 0 |\n| Q4-46 | 344.712 | -34.783 | 0.328510 ± 0.000097 | 19.5 ± 0.3 | 20.5 ± 0.6 | 17.61 ± 0.13 | 0 |\n| Q4-47 | 344.645 | -34.785 | 0.316340 ± 0.000236 | 20.0 ± 0.7 | 21.9 ± 1.1 | 17.08 ± 0.09 | 0 |\n| Q4-48 | 344.621 | -34.786 | 0.314460 ± 0.000288 | 19.6 ± 0.5 | 21.5 ± 0.8 | 16.45 ± 0.07 | 0 |\n| Q4-49 | 344.665 | -34.788 | 0.317640 ± 0.000297 | 18.9 ± 0.5 | 21.1 ± 0.8 | 15.76 ± 0.04 | 0 |\n| Q4-50 | 344.700 | -34.789 | 0.330460 ± 0.000464 | 22.0 ± 0.7 | 23.4 ± 1.2 | | 0 |\n| Q4-51 | 344.693 | -34.792 | 0.309450 ± 0.000221 | 18.8 ± 0.4 | 20.9 ± 0.8 | 15.92 ± 0.03 | 0 |\n| Q4-52 | 344.609 | -34.793 | 0.313230 ± 0.000288 | 20.5 ± 0.7 | 22.5 ± 1.1 | 17.54 ± 0.12 | 0 |\n| Q4-54 | 344.641 | -34.796 | 0.350100 ± 0.000059 | 20.4 ± 0.5 | 21.9 ± 0.6 | 17.44 ± 0.17 | 0 |\n| Q4-55 | 344.642 | -34.798 | 0.318480 ± 0.000254 | 20.9 ± 0.4 | 22.4 ± 0.5 | 17.74 ± 0.19 | 0 |\n| Q4-56 | 344.684 | -34.799 | 0.324800 ± 0.000005 | | | | 1 |\n| Q4-57 | 344.707 | -34.800 | 0.313880 ± 0.000143 | | | 18.17 ± 0.18 | 0 |\n| Q4-58 | 344.674 | -34.801 | 0.311070 ± 0.000131 | | | 15.90 ± 0.03 | 0 |\n| Q4-59 | 344.690 | -34.804 | 0.316080 ± 0.000118 | | | | 0 |\n| Q4-60 | 344.685 | -34.805 | 0.312500 ± 0.000122 | | | 17.87 ± 0.13 | 0 |\n| Q4-61 | 344.659 | -34.807 | 0.717580 ± 0.000108 | | | 17.34 ± 0.13 | 0 |\n| Q4-62 | 344.678 | -34.809 | 0.309600 ± 0.000172 | | | 17.94 ± 0.21 | 0 |\n| Q4-63 | 344.719 | -34.810 | 0.318810 ± 0.000127 | | | 17.33 ± 0.13 | 0 |\n| Q4-64 | 344.631 | -34.814 | 0.314870 ± 0.000135 | 20.6 ± 0.7 | 22.6 ± 0.9 | 17.10 ± 0.11 | 0 |\n[table:2]\n| ID | *M**i* | *M**g* | *M**K**s* | log(M*i*\\*/M⊙) | log(M*K**s*\\*/M⊙) | ELGs |\n| --- | --- | --- | --- | --- | --- | --- |\n| Q1-1 | -22.3 ± 0.5 | -21.2 ± 0.5 | | 11.1 ± 0.2 | | 0 |\n| Q1-3 | -18.8 ± 0.5 | -18.8 ± 0.5 | | 9.2 ± 0.2 | | 1 |\n| Q1-A1 | -19.4 ± 0.5 | -18.7 ± 0.5 | | 9.5 ± 0.2 | | 1 |\n| Q1-5 | -17.5 ± 0.5 | -17.6 ± 0.5 | | 8.7 ± 0.2 | | 1 |\n| Q1-A2 | -19.4 ± 0.5 | -19.0 ± 0.5 | | 10.0 ± 0.2 | | 0 |\n| Q1-8 | -22.6 ± 0.5 | -21.3 ± 0.5 | | 10.7 ± 0.2 | | 1 |\n| Q1-9 | -19.6 ± 0.6 | -18.4 ± 0.5 | -24.01 ± 0.16 | 10.0 ± 0.2 | 10.84 ± 0.06 | 0 |\n| Q1-10 | -21.5 ± 0.5 | -20.0 ± 0.8 | -24.44 ± 0.03 | 10.3 ± 0.2 | 10.51 ± 0.01 | 1 |\n| Q1-11 | -19.2 ± 0.6 | -17.2 ± 1.1 | -22.38 ± 0.24 | 9.9 ± 0.3 | 10.18 ± 0.10 | 0 |\n| Q1-12 | -20.6 ± 0.4 | -19.8 ± 0.8 | -23.10 ± 0.15 | 10.0 ± 0.2 | 9.97 ± 0.06 | 1 |\n| Q1-A3 | -20.0 ± 0.5 | -19.5 ± 0.5 | | 9.7 ± 0.2 | | 1 |\n| Q1-15 | -16.0 ± 0.6 | -15.6 ± 1.0 | | 8.1 ± 0.2 | | 1 |\n| Q1-16 | -20.0 ± 0.5 | -18.1 ± 1.0 | -23.61 ± 0.24 | 10.2 ± 0.2 | 10.67 ± 0.10 | 0 |\n| Q1-17 | -21.0 ± 0.5 | -19.3 ± 0.5 | | 10.6 ± 0.2 | | 0 |\n| Q1-18 | -19.9 ± 0.6 | -18.7 ± 1.1 | -22.77 ± 0.18 | 9.7 ± 0.2 | 9.84 ± 0.07 | 1 |\n| Q1-19 | -20.3 ± 0.5 | -19.2 ± 1.0 | -22.79 ± 0.34 | 9.8 ± 0.2 | 9.85 ± 0.13 | 1 |\n| Q1-20 | -22.7 ± 0.5 | -21.6 ± 0.5 | | 11.3 ± 0.2 | | 0 |\n| Q1-A4 | -19.8 ± 0.5 | -19.2 ± 0.5 | | 10.1 ± 0.2 | | 0 |\n| Q1-22 | -22.8 ± 0.4 | -20.3 ± 0.8 | -25.85 ± 0.04 | 10.8 ± 0.2 | 11.07 ± 0.01 | 1 |\n| Q1-24 | -21.9 ± 0.4 | -20.4 ± 0.8 | -24.72 ± 0.06 | 11.0 ± 0.1 | 11.12 ± 0.03 | 0 |\n| Q1-25 | -21.3 ± 0.5 | -19.7 ± 1.0 | -24.07 ± 0.17 | 10.2 ± 0.2 | 10.36 ± 0.07 | 1 |\n| Q1-26 | -20.6 ± 0.5 | -18.9 ± 1.0 | -23.91 ± 0.14 | 10.4 ± 0.2 | 10.79 ± 0.05 | 0 |\n| Q1-27 | -19.6 ± 0.5 | -18.1 ± 0.9 | | 10.0 ± 0.2 | | 0 |\n| Q1-28 | -22.2 ± 0.3 | -21.5 ± 0.6 | -24.17 ± 0.12 | 11.1 ± 0.1 | 10.90 ± 0.05 | 0 |\n| Q1-30 | -22.1 ± 0.4 | -20.0 ± 0.8 | -25.83 ± 0.02 | 10.5 ± 0.2 | 11.06 ± 0.01 | 1 |\n| Q1-31 | -20.6 ± 0.6 | -18.6 ± 1.0 | -23.90 ± 0.12 | 10.4 ± 0.2 | 10.79 ± 0.05 | 0 |\n| Q1-34 | -20.7 ± 0.6 | -18.6 ± 1.0 | -23.96 ± 0.08 | 10.5 ± 0.2 | 10.81 ± 0.03 | 0 |\n| Q1-36 | -21.5 ± 0.5 | -20.1 ± 0.8 | -23.52 ± 0.20 | 10.8 ± 0.2 | 10.64 ± 0.08 | 0 |\n| Q1-37 | -21.6 ± 0.6 | -19.9 ± 1.0 | -24.60 ± 0.05 | 10.3 ± 0.2 | 10.57 ± 0.02 | 1 |\n| Q1-A5 | -20.3 ± 0.4 | -18.5 ± 0.7 | -24.36 ± 0.09 | 10.3 ± 0.2 | 10.97 ± 0.04 | 0 |\n| Q1-38 | -21.3 ± 0.5 | -20.8 ± 0.5 | | 10.2 ± 0.2 | | 1 |\n| Q1-40 | -22.8 ± 0.5 | -19.8 ± 0.5 | -24.89 ± 0.05 | 10.8 ± 0.2 | 10.68 ± 0.02 | 1 |\n| Q1-41 | -22.6 ± 0.5 | -21.3 ± 0.5 | -25.57 ± 0.08 | 11.2 ± 0.2 | 11.46 ± 0.03 | 0 |\n| Q1-A6 | -18.8 ± 0.5 | -18.8 ± 0.5 | | 9.7 ± 0.2 | | 0 |\n| Q1-44 | -21.1 ± 0.4 | -19.6 ± 0.7 | | 10.1 ± 0.1 | | 1 |\n| ID | *M**i* | *M**g* | *M**K**s* | log(M*i*\\*/M⊙) | log(M*K**s*\\*/M⊙) | ELGs |\n| --- | --- | --- | --- | --- | --- | --- |\n| Q2-1 | -20.7 ± 0.5 | -20.3 ± 0.5 | | 10.0 ± 0.2 | | 1 |\n| Q2-2 | -19.4 ± 0.5 | -19.1 ± 0.5 | | 10.0 ± 0.2 | | 0 |\n| Q2-3 | -20.2 ± 0.5 | -19.8 ± 0.5 | | 9.8 ± 0.2 | | 1 |\n| Q2-4 | -19.7 ± 0.5 | -19.5 ± 0.5 | | 9.6 ± 0.2 | | 1 |\n| Q2-5 | -19.2 ± 0.5 | -18.5 ± 0.5 | | 9.9 ± 0.2 | | 0 |\n| Q2-6 | -22.9 ± 0.4 | -20.9 ± 0.7 | -26.05 ± 0.05 | 11.4 ± 0.2 | 11.65 ± 0.02 | 0 |\n| Q2-7 | -18.3 ± 0.7 | -15.3 ± 1.1 | -21.01 ± 0.52 | 9.5 ± 0.3 | 9.63 ± 0.21 | 0 |\n| Q2-8 | -20.4 ± 0.6 | -19.3 ± 1.0 | -23.11 ± 0.29 | 10.4 ± 0.2 | 10.47 ± 0.12 | 0 |\n| Q2-9 | -19.4 ± 0.6 | -18.3 ± 1.1 | | 9.4 ± 0.2 | | 1 |\n| Q2-10 | -20.4 ± 0.7 | -18.4 ± 1.1 | -24.72 ± 0.12 | 9.9 ± 0.3 | 10.62 ± 0.05 | 1 |\n| Q2-11 | -18.8 ± 0.4 | -18.1 ± 0.8 | | 9.2 ± 0.2 | | 1 |\n| Q2-12 | -20.4 ± 0.5 | -18.6 ± 0.9 | -23.83 ± 0.19 | 10.4 ± 0.2 | 10.76 ± 0.07 | 0 |\n| Q2-13 | -19.2 ± 0.5 | -19.7 ± 0.5 | | 9.9 ± 0.2 | | 0 |\n| Q2-14 | -20.8 ± 0.4 | -18.9 ± 0.9 | -23.81 ± 0.15 | 10.5 ± 0.2 | 10.75 ± 0.06 | 0 |\n| Q2-15 | -20.0 ± 0.6 | -17.8 ± 1.1 | -23.40 ± 0.20 | 10.2 ± 0.2 | 10.59 ± 0.08 | 0 |\n| Q2-16 | -21.3 ± 0.4 | -19.1 ± 0.9 | -24.17 ± 0.27 | 10.2 ± 0.2 | 10.40 ± 0.11 | 1 |\n| Q2-A1 | -22.3 ± 0.5 | -20.0 ± 0.5 | | 11.1 ± 0.2 | | 0 |\n| Q2-18 | -20.3 ± 0.5 | -18.2 ± 0.9 | -23.36 ± 0.17 | 10.3 ± 0.2 | 10.58 ± 0.07 | 0 |\n| Q2-19 | -22.2 ± 0.4 | -20.0 ± 0.8 | -25.19 ± 0.05 | 11.1 ± 0.2 | 11.31 ± 0.02 | 0 |\n| Q2-A2 | -20.5 ± 0.5 | -20.0 ± 0.5 | | 10.4 ± 0.2 | | 0 |\n| Q2-21 | -21.3 ± 0.5 | -19.3 ± 0.8 | -24.34 ± 0.11 | 10.7 ± 0.2 | 10.97 ± 0.04 | 0 |\n| Q2-22 | -20.1 ± 0.5 | -20.0 ± 0.5 | | 9.7 ± 0.2 | | 1 |\n| Q2-23 | -22.3 ± 0.5 | -19.7 ± 0.9 | -25.49 ± 0.04 | 11.1 ± 0.2 | 11.42 ± 0.01 | 0 |\n| Q2-24 | -19.4 ± 0.6 | -18.4 ± 1.0 | | 9.5 ± 0.2 | | 1 |\n| Q2-A3 | -20.3 ± 0.5 | -20.2 ± 0.5 | | 10.3 ± 0.2 | | 0 |\n| Q2-26 | -20.7 ± 0.5 | -19.0 ± 0.9 | -22.92 ± 0.48 | 10.0 ± 0.2 | 9.90 ± 0.19 | 1 |\n| Q2-27 | -22.0 ± 0.5 | -20.1 ± 0.9 | -25.17 ± 0.11 | 10.5 ± 0.2 | 10.80 ± 0.04 | 1 |\n| Q2-28 | -22.5 ± 0.3 | -21.2 ± 0.6 | -25.47 ± 0.17 | 10.7 ± 0.1 | 10.92 ± 0.07 | 1 |\n| Q2-29 | -21.8 ± 0.2 | -20.9 ± 0.5 | -24.50 ± 0.21 | 10.4 ± 0.1 | 10.53 ± 0.08 | 1 |\n| Q2-31 | -20.4 ± 0.5 | -20.3 ± 0.5 | | 10.4 ± 0.2 | | 0 |\n| Q2-A4 | -22.6 ± 0.5 | -21.7 ± 0.5 | | 10.7 ± 0.2 | | 1 |\n| Q2-32 | -20.7 ± 0.5 | -20.5 ± 0.5 | -22.93 ± 0.23 | 10.0 ± 0.2 | 9.90 ± 0.09 | 1 |\n| Q2-33 | -22.6 ± 0.5 | -22.1 ± 0.5 | -25.15 ± 0.11 | 10.7 ± 0.2 | 10.79 ± 0.05 | 1 |\n| Q2-A5 | -19.9 ± 0.5 | -19.4 ± 0.5 | | 10.2 ± 0.2 | | 0 |\n| Q2-35 | -21.4 ± 0.6 | -19.2 ± 1.1 | -24.59 ± 0.04 | 10.8 ± 0.2 | 11.06 ± 0.02 | 0 |\n| Q2-36 | -21.8 ± 0.6 | -19.9 ± 1.0 | -24.67 ± 0.11 | 10.4 ± 0.2 | 10.60 ± 0.05 | 1 |\n| Q2-37 | -18.7 ± 0.4 | -17.9 ± 0.7 | | 9.2 ± 0.2 | | 1 |\n| Q2-38 | -19.4 ± 0.6 | -18.3 ± 1.0 | -21.51 ± 0.46 | 9.5 ± 0.2 | 9.34 ± 0.18 | 1 |\n| Q2-39 | -20.9 ± 0.6 | -19.8 ± 1.1 | -23.64 ± 0.16 | 10.0 ± 0.3 | 10.19 ± 0.06 | 1 |\n| Q2-40 | -21.0 ± 0.6 | -18.9 ± 1.0 | -23.34 ± 0.31 | 10.6 ± 0.2 | 10.57 ± 0.13 | 0 |\n| Q2-41 | -21.2 ± 0.6 | -19.1 ± 1.0 | -24.45 ± 0.22 | 10.7 ± 0.2 | 11.01 ± 0.09 | 0 |\nccccccc ID & *M**i* & *M**g* & *M**K**s* & log(M*i*\\*/M⊙) & log(M*K**s*\\*/M⊙) & ELGs \nQ3-1 & -19.8 ± 0.6 & -18.8 ± 1.1 & -22.18 ± 0.40 & 9.6 ± 0.2 & 9.60 ± 0.16 & 1 \nQ3-2 & -22.4 ± 0.5 & -21.1 ± 0.8 & -25.45 ± 0.04 & 10.7 ± 0.2 & 10.91 ± 0.01 & 1 \nQ3-3 & -22.1 ± 0.6 & -19.9 ± 1.0 & -25.28 ± 0.03 & 11.1 ± 0.2 & 11.34 ± 0.01 & 0 \nQ3-4 & -19.2 ± 0.7 & -18.1 ± 1.1 & -21.66 ± 0.26 & 9.4 ± 0.3 & 9.39 ± 0.11 & 1 \nQ3-5 & -20.3 ± 0.5 & -19.1 ± 1.0 & -22.75 ± 0.36 & 9.8 ± 0.2 & 9.83 ± 0.14 & 1 \nQ3-6 & -19.7 ± 0.5 & -19.8 ± 0.5 & & 10.1 ± 0.2 & & 0 \nQ3-A1 & -24.5 ± 0.5 & -24.0 ± 0.5 & & 12.0 ± 0.2 & & 0 \nQ3-8 & -21.2 ± 0.5 & -20.1 ± 0.5 & & 10.2 ± 0.2 & & 1 \nQ3-10 & -22.6 ± 0.3 & -21.7 ± 0.5 & -24.23 ± 0.13 & 11.2 ± 0.1 & 10.92 ± 0.05 & 0 \nQ3-11 & -19.4 ± 0.7 & -18.6 ± 1.1 & -21.36 ± 0.34 & 9.4 ± 0.3 & 9.28 ± 0.14 & 1 \nQ3-12 & -21.1 ± 0.5 & -19.9 ± 0.8 & -23.58 ± 0.21 & 10.1 ± 0.2 & 10.16 ± 0.08 & 1 \nQ3-A2 & -19.6 ± 0.5 & -19.8 ± 0.5 & & 10.0 ± 0.2 & & 0 \nQ3-15 & -19.0 ± 0.5 & -19.2 ± 0.5 & & 9.8 ± 0.2 & & 0 \nQ3-16 & -17.8 ± 0.5 & -17.0 ± 1.1 & & 8.8 ± 0.2 & & 1 \nQ3-17 & -19.7 ± 0.6 & -17.5 ± 1.1 & -22.85 ± 0.44 & 9.6 ± 0.2 & 9.87 ± 0.18 & 1 \nQ3-18 & -20.7 ± 0.4 & -19.8 ± 0.8 & -22.99 ± 0.14 & 10.0 ± 0.2 & 9.93 ± 0.06 & 1 \nQ3-19 & -20.9 ± 0.4 & -19.8 ± 0.6 & -23.56 ± 0.06 & 10.6 ± 0.1 & 10.66 ± 0.02 & 0 \nQ3-20 & -18.2 ± 0.5 & -17.1 ± 1.0 & & 9.0 ± 0.2 & & 1 \nQ3-21 & -22.1 ± 0.4 & -20.6 ± 0.7 & -24.91 ± 0.08 & 10.5 ± 0.2 & 10.70 ± 0.03 & 1 \nQ3-23 & -21.4 ± 0.6 & -19.8 ± 1.0 & -24.50 ± 0.07 & 10.8 ± 0.3 & 11.03 ± 0.03 & 0 \nQ3-24 & -18.4 ± 0.7 & -16.6 ± 1.1 & -22.17 ± 0.27 & 9.6 ± 0.3 & 10.10 ± 0.11 & 0 \nQ3-25 & -22.2 ± 0.5 & -19.5 ± 0.8 & -25.68 ± 0.07 & 11.1 ± 0.2 & 11.50 ± 0.03 & 0 \nQ3-26 & -20.9 ± 0.7 & -19.0 ± 1.1 & -23.76 ± 0.09 & 10.6 ± 0.3 & 10.73 ± 0.04 & 0 \nQ3-27 & -20.7 ± 0.7 & -19.3 ± 1.1 & -22.99 ± 0.48 & 10.0 ± 0.3 & 9.93 ± 0.19 & 1 \nQ3-28 & -22.5 ± 0.5 & -20.5 ± 0.8 & -25.57 ± 0.02 & 11.2 ± 0.2 & 11.46 ± 0.01 & 0 \nQ3-29 & -21.9 ± 0.6 & -21.1 ± 0.9 & -23.81 ± 0.22 & 10.4 ± 0.2 & 10.25 ± 0.09 & 1 \nQ3-30 & -19.7 ± 0.5 & -19.5 ± 0.5 & & 9.6 ± 0.2 & & 1 \nQ3-31 & -20.5 ± 0.5 & -19.2 ± 0.9 & -23.35 ± 0.20 & 9.9 ± 0.2 & 10.07 ± 0.08 & 1 \nQ3-A3 & -17.3 ± 0.7 & -16.5 ± 1.1 & -21.36 ± 0.67 & 9.1 ± 0.3 & 9.77 ± 0.27 & 0 \nQ3-33 & -16.3 ± 0.4 & -15.7 ± 0.6 & & 8.2 ± 0.2 & & 1 \nQ3-34 & -21.3 ± 0.5 & -18.4 ± 0.6 & -23.00 ± 0.18 & 10.2 ± 0.2 & 9.93 ± 0.07 & 1 \nQ3-35 & -16.6 ± 0.5 & -14.2 ± 0.6 & & 8.3 ± 0.2 & & 1 \nQ3-36 & -20.0 ± 0.5 & -17.1 ± 0.6 & -22.32 ± 0.20 & 9.7 ± 0.2 & 9.66 ± 0.08 & 1 \nQ3-A4 & -19.9 ± 0.5 & -19.2 ± 0.5 & & 10.2 ± 0.2 & & 0 \nQ3-38 & -20.9 ± 0.5 & -19.7 ± 0.5 & -23.28 ± 0.28 & 10.0 ± 0.2 & 10.04 ± 0.11 & 1 \nQ3-39 & -21.2 ± 0.5 & -19.5 ± 0.5 & -23.51 ± 0.22 & 10.2 ± 0.2 & 10.14 ± 0.09 & 1 \nQ3-40 & -21.7 ± 0.5 & -19.4 ± 0.8 & -24.87 ± 0.04 & 10.9 ± 0.2 & 11.18 ± 0.02 & 0 \nQ3-A5 & -20.1 ± 0.5 & -20.4 ± 0.5 & & 10.2 ± 0.2 & & 0 \nQ3-A6 & -19.5 ± 0.5 & -19.3 ± 0.5 & & 9.5 ± 0.2 & & 1 \nQ3-43 & -21.2 ± 0.5 & -20.0 ± 0.9 & & 10.2 ± 0.2 & & 1 \nQ3-44 & -20.6 ± 0.5 & -19.0 ± 0.8 & -23.87 ± 0.07 & 9.9 ± 0.2 & 10.28 ± 0.03 & 1 \nQ3-45 & -19.9 ± 0.5 & -18.3 ± 0.9 & -22.56 ± 0.34 & 10.2 ± 0.2 & 10.26 ± 0.14 & 0 \nQ3-46 & -20.0 ± 0.5 & -18.4 ± 0.9 & -22.36 ± 0.32 & 10.2 ± 0.2 & 10.17 ± 0.13 & 0 \nQ3-A7 & -21.4 ± 0.5 & -20.1 ± 0.8 & -24.33 ± 0.11 & 10.8 ± 0.2 & 10.96 ± 0.05 & 0 \nQ3-A8 & -20.9 ± 0.5 & -20.1 ± 0.5 & & 10.6 ± 0.2 & & 0 \nQ3-49 & -23.0 ± 0.3 & -21.9 ± 0.5 & -25.44 ± 0.05 & 11.4 ± 0.1 & 11.41 ± 0.02 & 0 \n| ID | *M**i* | *M**g* | *M**K**s* | log(M*i*\\*/M⊙) | log(M*K**s*\\*/M⊙) | ELGs |\n| --- | --- | --- | --- | --- | --- | --- |\n| Q4-A1 | -17.3 ± 0.5 | -16.9 ± 0.5 | | 8.6 ± 0.2 | | 1 |\n| Q4-2 | -21.1 ± 0.6 | -19.1 ± 1.0 | -24.22 ± 0.08 | 10.6 ± 0.3 | 10.92 ± 0.03 | 0 |\n| Q4-3 | -21.1 ± 0.6 | -19.6 ± 1.1 | -23.74 ± 0.18 | 10.1 ± 0.3 | 10.23 ± 0.07 | 1 |\n| Q4-4 | -19.8 ± 0.7 | -18.0 ± 1.0 | -22.50 ± 0.25 | 10.1 ± 0.3 | 10.23 ± 0.10 | 0 |\n| Q4-5 | -20.6 ± 0.6 | -19.2 ± 1.0 | -23.31 ± 0.32 | 9.9 ± 0.2 | 10.06 ± 0.13 | 1 |\n| Q4-A2 | -18.7 ± 0.5 | -18.4 ± 0.5 | | 9.7 ± 0.2 | | 0 |\n| Q4-7 | -20.1 ± 0.5 | -19.5 ± 0.8 | -21.98 ± 0.20 | 9.7 ± 0.2 | 9.52 ± 0.08 | 1 |\n| Q4-8 | -18.2 ± 0.7 | -17.3 ± 1.1 | -20.44 ± 0.32 | 9.0 ± 0.3 | 8.90 ± 0.13 | 1 |\n| Q4-9 | -21.7 ± 0.6 | -19.8 ± 0.9 | -24.91 ± 0.02 | 10.9 ± 0.2 | 11.2 ± 0.01 | 0 |\n| Q4-10 | -19.7 ± 0.5 | -18.6 ± 0.5 | | 10.1 ± 0.2 | | 0 |\n| Q4-11 | -18.5 ± 0.7 | -16.9 ± 1.1 | | 9.6 ± 0.3 | | 0 |\n| Q4-13 | -19.8 ± 0.6 | -18.0 ± 1.0 | -22.43 ± 0.30 | 10.1 ± 0.2 | 10.20 ± 0.12 | 0 |\n| Q4-14 | -19.7 ± 0.7 | -18.9 ± 1.1 | -21.39 ± 0.65 | 9.6 ± 0.3 | 9.29 ± 0.26 | 1 |\n| Q4-15 | -19.1 ± 0.7 | -18.4 ± 1.1 | | 9.3 ± 0.3 | | 1 |\n| Q4-16 | -21.1 ± 0.7 | -19.0 ± 1.2 | -24.34 ± 0.06 | 10.6 ± 0.3 | 10.97 ± 0.02 | 0 |\n| Q4-A3 | -19.3 ± 0.5 | -18.8 ± 0.5 | | 9.4 ± 0.2 | | 1 |\n| Q4-18 | -19.5 ± 0.8 | -16.7 ± 1.2 | -22.82 ± 0.16 | 10.0 ± 0.3 | 10.36 ± 0.07 | 0 |\n| Q4-19 | -20.1 ± 0.6 | -19.3 ± 1.0 | -23.09 ± 0.18 | 9.7 ± 0.3 | 9.97 ± 0.07 | 1 |\n| Q4-20 | -21.1 ± 0.6 | -19.0 ± 1.1 | -24.37 ± 0.05 | 10.7 ± 0.3 | 10.98 ± 0.02 | 0 |\n| Q4-21 | -21.2 ± 0.5 | -19.7 ± 0.5 | | 10.7 ± 0.2 | | 0 |\n| Q4-22 | -19.7 ± 0.7 | -18.8 ± 1.2 | -21.56 ± 0.58 | 9.6 ± 0.3 | 9.35 ± 0.23 | 1 |\n| Q4-23 | -20.2 ± 0.7 | -18.1 ± 1.1 | -23.10 ± 0.24 | 10.3 ± 0.3 | 10.47 ± 0.09 | 0 |\n| Q4-24 | -20.3 ± 0.7 | -18.4 ± 1.2 | -23.08 ± 0.16 | 10.3 ± 0.3 | 10.46 ± 0.06 | 0 |\n| Q4-25 | -18.4 ± 0.8 | -16.7 ± 1.2 | | 9.6 ± 0.3 | | 0 |\n| Q4-26 | -21.5 ± 0.5 | -20.8 ± 0.9 | -23.51 ± 0.17 | 10.3 ± 0.2 | 10.14 ± 0.07 | 1 |\n| Q4-27 | -19.3 ± 0.6 | -16.5 ± 1.0 | -22.16 ± 0.34 | 9.9 ± 0.3 | 10.10 ± 0.14 | 0 |\n| Q4-28 | -20.3 ± 0.7 | -18.4 ± 1.2 | -23.27 ± 0.17 | 10.3 ± 0.3 | 10.54 ± 0.07 | 0 |\n| Q4-A4 | -19.8 ± 0.5 | -18.6 ± 0.8 | -21.90 ± 0.45 | 9.6 ± 0.2 | 9.49 ± 0.18 | 1 |\n| Q4-A5 | -19.6 ± 0.5 | -19.5 ± 0.5 | | 10.0 ± 0.2 | | 0 |\n| Q4-31 | -21.6 ± 0.6 | -19.4 ± 1.0 | -24.74 ± 0.04 | 10.9 ± 0.3 | 11.13 ± 0.02 | 0 |\n| Q4-32 | -24.4 ± 0.1 | -22.2 ± 0.3 | -27.04 ± 0.02 | 12.0 ± 0.1 | 12.05 ± 0.01 | 0 |\n| Q4-33 | -18.6 ± 0.4 | -17.8 ± 0.7 | -20.06 ± 0.19 | 9.1 ± 0.1 | 8.76 ± 0.08 | 1 |\n| Q4-A6 | -19.3 ± 0.5 | -19.1 ± 0.5 | | 9.4 ± 0.2 | | 1 |\n| Q4-35 | -18.9 ± 0.5 | -18.2 ± 1.1 | -22.14 ± 0.15 | 9.3 ± 0.2 | 9.59 ± 0.06 | 1 |\n| Q4-36 | -23.5 ± 0.3 | -21.7 ± 0.5 | -26.43 ± 0.01 | 11.1 ± 0.1 | 11.30 ± 0.01 | 1 |\n| Q4-37 | -18.0 ± 0.5 | -16.2 ± 0.9 | | 8.9 ± 0.2 | | 1 |\n| Q4-38 | -21.8 ± 0.5 | -19.9 ± 0.9 | -24.80 ± 0.04 | 10.9 ± 0.2 | 11.15 ± 0.02 | 0 |\n| Q4-39 | -20.7 ± 0.5 | -18.8 ± 0.5 | | 10.5 ± 0.2 | | 0 |\n| Q4-40 | -22.2 ± 0.5 | -21.0 ± 0.5 | | 10.6 ± 0.2 | | 1 |\n| Q4-41 | -20.7 ± 0.7 | -18.7 ± 1.1 | -23.54 ± 0.13 | 10.5 ± 0.3 | 10.65 ± 0.05 | 0 |\n| Q4-42 | -21.2 ± 0.6 | -19.6 ± 1.0 | -23.85 ± 0.34 | 10.7 ± 0.2 | 10.77 ± 0.13 | 0 |\n| Q4-43 | -20.2 ± 0.6 | -18.5 ± 1.0 | -22.55 ± 0.32 | 10.3 ± 0.3 | 10.25 ± 0.13 | 0 |\n| Q4-44 | -22.3 ± 0.5 | -20.1 ± 0.9 | -25.73 ± 0.02 | 11.1 ± 0.2 | 11.52 ± 0.01 | 0 |\n| Q4-45 | -20.5 ± 0.6 | -18.6 ± 1.0 | -23.26 ± 0.21 | 10.4 ± 0.2 | 10.53 ± 0.08 | 0 |\n| Q4-46 | -21.7 ± 0.3 | -20.7 ± 0.6 | -23.60 ± 0.13 | 10.9 ± 0.1 | 10.67 ± 0.05 | 0 |\n| Q4-47 | -21.1 ± 0.7 | -19.2 ± 1.1 | -24.04 ± 0.09 | 10.6 ± 0.3 | 10.85 ± 0.04 | 0 |\n| Q4-48 | -21.5 ± 0.5 | -19.6 ± 0.8 | -24.65 ± 0.07 | 10.8 ± 0.2 | 11.09 ± 0.03 | 0 |\n| Q4-49 | -22.2 ± 0.5 | -20.0 ± 0.8 | -25.37 ± 0.04 | 11.1 ± 0.2 | 11.38 ± 0.02 | 0 |\n| Q4-50 | -19.2 ± 0.7 | -17.9 ± 1.2 | | 9.9 ± 0.3 | | 0 |\n| Q4-51 | -22.2 ± 0.4 | -20.2 ± 0.8 | -25.14 ± 0.03 | 11.1 ± 0.2 | 11.29 ± 0.01 | 0 |\n| Q4-52 | -20.6 ± 0.7 | -18.6 ± 1.1 | -23.55 ± 0.12 | 10.4 ± 0.3 | 10.65 ± 0.05 | 0 |\n| Q4-54 | -20.9 ± 0.5 | -19.5 ± 0.6 | -23.93 ± 0.17 | 10.6 ± 0.2 | 10.80 ± 0.07 | 0 |\n| Q4-55 | -20.2 ± 0.4 | -18.7 ± 0.5 | -23.39 ± 0.19 | 10.3 ± 0.2 | 10.59 ± 0.07 | 0 |\n| Q4-56 | -19.7 ± 0.5 | -18.8 ± 0.5 | | 9.6 ± 0.2 | | 1 |\n| Q4-57 | -20.9 ± 0.5 | -18.7 ± 0.5 | -22.93 ± 0.18 | 10.6 ± 0.2 | 10.40 ± 0.07 | 0 |\n| Q4-58 | -22.8 ± 0.5 | -18.7 ± 0.5 | -25.17 ± 0.03 | 11.3 ± 0.2 | 11.30 ± 0.01 | 0 |\n| Q4-59 | -20.5 ± 0.5 | -19.4 ± 0.5 | | 10.4 ± 0.2 | | 0 |\n| Q4-60 | -21.4 ± 0.5 | -19.4 ± 0.5 | -23.22 ± 0.13 | 10.8 ± 0.2 | 10.52 ± 0.05 | 0 |\n| Q4-61 | -22.7 ± 0.5 | -21.5 ± 0.5 | -25.92 ± 0.13 | 11.3 ± 0.2 | 11.6 ± 0.05 | 0 |\n| Q4-62 | -21.2 ± 0.5 | -19.3 ± 0.5 | -23.12 ± 0.21 | 10.7 ± 0.2 | 10.48 ± 0.08 | 0 |\n| Q4-63 | -21.4 ± 0.5 | -19.4 ± 0.5 | -23.80 ± 0.13 | 10.8 ± 0.2 | 10.75 ± 0.05 | 0 |\n| Q4-64 | -20.5 ± 0.7 | -18.5 ± 0.9 | -24.01 ± 0.11 | 10.4 ± 0.3 | 10.83 ± 0.04 | 0 |\n[table:3]\n| ID | 12+log(O/H)*O*3*N*2 |\n| --- | --- |\n| Q1-3 | 8.37 ± 0.03 |\n| Q1-A1 | 8.44 ± 0.04 |\n| Q1-5 | 8.27 ± 0.03 |\n| Q1-10 | 8.71 ± 0.04 |\n| Q1-12 | 8.56 ± 0.08 |\n| Q1-15 | 8.10 ± 0.06 |\n| Q1-18 | 8.70 ± 0.06 |\n| Q1-19 | 8.42 ± 0.03 |\n| Q1-25 | 8.58 ± 0.06 |\n| Q1-37 | 8.27 ± 0.05 |\n| Q1-40 | 8.68 ± 0.02 |\n| Q2-1 | 8.61 ± 0.08 |\n| Q2-3 | 8.35 ± 0.04 |\n| Q2-9 | 8.22 ± 0.05 |\n| Q2-11 | 8.31 ± 0.05 |\n| Q2-22 | 8.41 ± 0.05 |\n| Q2-A4 | 8.51 ± 0.02 |\n| Q2-32 | 8.49 ± 0.03 |\n| Q2-37 | 8.43 ± 0.05 |\n| Q2-38 | 8.34 ± 0.05 |\n| Q2-39 | 8.50 ± 0.02 |\n| Q3-1 | 8.26 ± 0.03 |\n| Q3-2 | 8.63 ± 0.03 |\n| Q3-4 | 8.49 ± 0.04 |\n| Q3-5 | 8.34 ± 0.04 |\n| Q3-8 | 8.73 ± 0.04 |\n| Q3-11 | 8.25 ± 0.05 |\n| Q3-12 | 8.64 ± 0.10 |\n| Q3-16 | 8.62 ± 0.11 |\n| Q3-18 | 8.71 ± 0.06 |\n| Q3-21 | 8.72 ± 0.09 |\n| Q3-30 | 8.15 ± 0.03 |\n| Q3-31 | 8.63 ± 0.10 |\n| Q3-34 | 8.32 ± 0.02 |\n| Q3-35 | 8.21 ± 0.10 |\n| Q3-36 | 8.52 ± 0.05 |\n| Q3-39 | 8.60 ± 0.04 |\n| Q3-A6 | 8.23 ± 0.05 |\n| Q3-44 | 8.57 ± 0.06 |\n| Q4-3 | 8.59 ± 0.09 |\n| Q4-5 | 8.54 ± 0.07 |\n| Q4-7 | 8.33 ± 0.03 |\n| Q4-8 | 8.17 ± 0.02 |\n| Q4-14 | 8.26 ± 0.02 |\n| Q4-15 | 8.09 ± 0.03 |\n| Q4-19 | 8.27 ± 0.04 |\n| Q4-22 | 8.75 ± 0.11 |\n| Q4-26 | 8.42 ± 0.03 |\n| Q4-A4 | 8.46 ± 0.06 |\n| Q4-33 | 8.37 ± 0.05 |\n| Q4-A6 | 8.49 ± 0.05 |\n| Q4-35 | 8.24 ± 0.02 |\n| Q4-40 | 8.77 ± 0.19 |\n| Q4-56 | 8.31 ± 0.03 |\n---\n1. https://www.eso.org/sci/software/pipelines/vimos/[↩](#fnref1)\n2. https://www.eso.org/rm/api/v1/public/releaseDescriptions/135[↩](#fnref2)\n3. Bagpipes is a state of the art Python code for modelling galaxy spectra and fitting spectroscopic and photometric observations. https://bagpipes.readthedocs.io/en/latest/[↩](#fnref3)\n4. IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.[↩](#fnref4)\n5. https://mfouesneau.github.io/pyphot/[↩](#fnref5)\n6. https://scikit-learn.org/stable/index.html[↩](#fnref6)\n"},"data_id":{"kind":"string","value":"arxiv_0183484"}}},{"rowIdx":45899,"cells":{"content":{"kind":"string","value":" equatorial coordinates for the position (*α*, *δ*), the proper motions ($\\mu\\_\\alpha\\equiv\\dot{\\alpha}\\cos\\delta, \\mu\\_{\\delta}$) and the parallax. Due to the high sensitivity of MeerKAT observations, the position of pulsars can be obtained extremely well, and we did not need to sample for the position parameter; so, these remained fixed. We performed three analyses using temponest sampling for (1) proper motion, (2) parallax, and (3) proper motion and parallax. Bayes factors produced by temponest showed that, for all pulsars, the proper motion and parallax were detectable (with B > 10) and superior to models without them. In addition, we used ecliptic coordinates for the position (*λ*, *β*) and proper motion ($\\mu\\_\\lambda\\equiv\\dot{\\lambda}\\cos\\beta, \\mu\\_{\\rm \\beta}$), and repeated the analyses and compared the Bayesian evidence for these solutions with the equatorial ones, and found them to be similar. Note that covariance between position and proper motion parameters was minimized by setting the epoch of the positions to be near the middle of the data set’s baseline in time. demonstrated that positional uncertainties reported in ecliptic coordinates are less than the equatorial coordinates. Therefore, we used timing analyses in ecliptic coordinates to derive the Galactic position (*l*, *b*) and Galactic proper motion ($\\mu\\_{l}\\equiv\\dot{l}\\cos b, \\mu\\_{b}$) of all pulsars using the Astropy package.\nDistances\n---------\n### Distances from parallax measurement\nThe most direct method for measuring distance is parallax $d = \\varpi^{-1}$, where $\\varpi$ is trigonometric parallax in mas and *d* is in kpc. This method does not depend on any intrinsic physical property of the object and is purely geometric, and thus model-independent. For pulsars, the two most common methods of measuring parallax are using pulsar timing solutions and VLBI observations. As will be seen, these two methods dominate the distances used in section [sec:results-analysis] for our transverse velocity analysis.\nVelocities\n----------\nThe full 3D velocity of a celestial object relative to the Sun can be broken into two components: a radial velocity component and a transverse (or tangential) velocity component. The radial velocity for stars in the Milky Way is typically estimated using the Doppler effect, although this is not possible for pulsars, as no spectral lines are available unless they are in a binary system with a star. The transverse velocity (in km s− 1), is the velocity perpendicular to the line of sight and is given by\n \n$$v\\_\\perp = 4.74\\,{\\rm km\\,s}^{-1} \\left( \\frac{\\mu}{\\rm mas\\,yr^{-1}}\\right) \\left( \\frac{d}{\\rm kpc} \\right),$$ \nwhere *μ* is the total proper motion, and *d* is the distance.\n### Correcting for the Local Standard of Rest and the Galactic differential rotation\nProper motions are usually measured relative to the Solar System’s barycentre, but if we want to understand their origins as Milky Way objects, we need to adjust for the reference frame. To obtain the velocity of pulsars with respect to the Galactic Center (GC), we need to correct the relative proper motions for two effects: (1) the peculiar velocity of the Sun with respect to the local standard of rest (LSR) and (2) differential rotation of the Galaxy. There are two types of LSR, both of which are defined as reference frames at the location of the Sun. The first is the ‘kinematic LSR’ which orbits the Galaxy with the mean velocity of the neighbouring stars ($\\bar{v}\\_{\\rm \\phi}$). The second is the ‘dynamic LSR’ which rotates around the Galaxy in a perfectly circular orbit with a velocity that is exerted by the gravitational potential of the Galaxy ($v\\_{\\rm c}$). The difference between the two ($v\\_{\\rm c} - \\bar{v}\\_{\\rm \\phi}$) is defined as the “asymmetric drift”.\nThe Galactic disk rotates differentially, and the Galactic orbital periods of stars are dependent on their distances from the GC. The circular rotation speed of stars at distance *R* from the GC defines the rotation curve of the Galaxy. In the last decade, there have been a number of studies to improve the measured rotation curve of the Galaxy using a variety of different methods. In this work, we used the results and rotation curve model from.\nFor every MSP, we transformed the coordinates of the pulsars from their ecliptic coordinates to the cartesian Galactocentric coordinates (*X*, *Y*, *Z*) using Astropy[4](#fn4). The distance from the Sun to the Galactic mid-plane was taken from (*Z*⊙ = 5.5(58) pc), and the distance from the Sun to the GC was taken from (*R*⊙ = 8.12(3) kpc). We then derived the Galactocentric velocity components ($V\\_{\\rm X}$, $V\\_{\\rm Y}$, $V\\_{\\rm Z}$). For subtracting the Sun’s peculiar velocity from these components, we used the values of (*U*, *V*, *W*)⊙ = (11.10(74), 12.24(47), 7.25(37)) km s− 1 from.\nTo correct for differential rotation of the Galaxy, we calculated the distances of the pulsars from the Galactic centre, and used the universal rotation curve of the Galaxy from to calculate the *X* and *Y* components of the Galactic rotation at the position of MSPs. Finally, we subtracted these from the velocities of MSPs and derived their transverse velocities with respect to their own LSR. All velocities reported in this work are thus corrected for the peculiar velocity of the Sun and the rotation curve of the Galaxy.\nResults: astrometry\n===================\nIn this section we report our astrometric results, including positions, proper motions, parallaxes for our sample pulsars, and compare with MSPs for which these have been previously measured in the literature.\nPositions and proper motions\n----------------------------\nIn Table [tab:positions], positions from timing analyses in equatorial and ecliptic coordinates for 77 MSPs are given. These were not sampled using temponest and only fitted using tempo2. In addition, the positions in Galactic coordinates are derived from ecliptic coordinates using Astropy. All uncertainties shown are computed via the median with 68.3% equally-tailed credible intervals.\nIn Table [tab:propermotions], we show proper motions from timing analyses in both equatorial and ecliptic coordinates. In this table, the proper motions in Galactic coordinates are derived from the ecliptic proper motions using Astropy. The reported values and uncertainties are the lower bound (16%), median (50%), and upper bound (84%) of marginalized posterior probability distributions. The timing baseline of observations for each pulsar, as well as the best previous measurements of proper motions in equatorial coordinates and their corresponding references, are listed. In our study, our limited timing baseline means that our proper motion results are not superior to all previous results, mainly because the relative error is proportional to the observing span *T*− 3/2, and some of the pulsars have been timed for over a decade with other instruments.\nOut of 77 pulsars, 69 had proper motions with statistical significance greater than 3*σ*, and are shown in the upper portion of Table [tab:propermotions]. The remaining 8 pulsars (i.e. those listed in the lower portion of the Table) consist of an isolated MSP (J1721 − 2457) with relatively poor rms timing residuals of 3.7*μ*s and less than two deg from the ecliptic, and seven binary MSPs (J0955 − 6150, J1022 + 1001, J1327 − 0755, J1653 − 2054, J1705 − 1903, J1802 − 2124, J1811 − 2405) that have either poor timing residuals or small proper motions (as low as 0.21(15) mas yr− 1 for J0955 − 6150). PSR J1022 + 1001 is near the ecliptic plane, and as expected had a very accurate proper motion in ecliptic longitude, but a poor constraint in ecliptic latitude. For this pulsar, we set the proper motion and orbital parameters from the high accuracy VLBI study of and derived a parallax of 1.2(4) mas.\nWe typically require at least two years of data in order to separate the annual parallax from the proper motion of a pulsar, due to the motion of the pulsar against the background of the sky and the orbit of the Earth around the Sun. We note that it is not always simple to break this degeneracy for the pulsars with high rms timing residuals, and worse for pulsars in (wide) binary systems. A solution to this is to implement VLBI techniques, which can often measure pulsar parallaxes significantly better, e.g. measured the parallax for PSR J0437 − 4715 to be 6.396(54) mas with the Australian Long Baseline Array.\nAll but two of the proper motions listed in Table [tab:propermotions] are consistent with previous results within 3*σ*. One component of the proper motion of PSRs J1446 − 4701 is just over 3*σ* inconsistent with the values in, but the total proper motion is the same to within 8%. The origin of this discrepancy is unclear. We checked the use of the Solar System ephemeris that used for their timing analysis, and our parallax remained the same to within errors. This inconsistency might be due to the different noise models that were adopted or be due to different noise properties at different epochs.\nFigure [fig:aitoffprojection] presents the trajectories of the 69 MSPs since 5 Myr ago until the present in Galactic longitude (*l*) and Galactic latitude (*b*), assuming zero radial velocity for the MSPs. Both effects of the peculiar velocity of the Sun and the Galactic differential rotation are subtracted from the observed proper motions of MSPs. In addition, the Galactic gravitational potential is assumed to be zero, because the oscillation period in the galactic potential ( > 100Myr) is much larger than 5 Myr, and so the MSPs are simply moving along their corrected proper motion vectors. The area in gray indicates the part of the sky that the MeerKAT radio telescope is not able to reach. From the diagram it is obvious that the MSPs are both moving towards and away from the galactic plane, consistent with a relaxed ancient population.\n [fig:aitoffprojection]\nllllllllll & & & & & & \n Pulsar & *α* & *δ* & & *λ* & *β* & & *l* & *b* & Epoch \n& (hh:mm:ss) & (dd:mm:ss) & & (∘) & (∘) & & (∘) & (∘) & (MJD) \nJ0030 + 0451 & 00:30:27.42348(5) & + 04:51:39.7144(17) & & 8.910338462(13) & 1.4457050(5) & & 113.141 & − 57.611 & 59251 \nJ0125 − 2327 & 01:25:1.075151(3) & − 23:27:8.13167(5) & & 9.743403184(9) & − 29.830147598(15) & & 188.932 & − 81.569 & 59155 \nJ0437 − 4715 & 04:37:16.0492624(13) & − 47:15:10.027930(15) & & 50.469059178(11) & − 67.873319464(5) & & 253.395 & − 41.963 & 59187 \nJ0610 − 2100 & 06:10:13.602953(6) & − 21:00:27.74435(13) & & 93.34234093(3) & − 44.41732764(4) & & 227.746 & − 18.184 & 59152 \nJ0613 − 0200 & 06:13:43.977070(3) & − 02:00:47.34294(13) & & 93.799014159(15) & − 25.40716602(4) & & 210.413 & − 9.305 & 59152 \nJ0614 − 3329 & 06:14:10.348230(3) & − 33:29:54.13156(4) & & 95.410779861(16) & − 56.871120173(12) & & 240.501 & − 21.827 & 59155 \nJ0636 − 3044 & 06:36:34.398358(5) & − 30:44:1.10761(7) & & 103.35534034(3) & − 53.74983738(2) & & 239.526 & − 16.416 & 59163 \nJ0711 − 6830 & 07:11:54.153093(12) & − 68:30:47.23387(10) & & 204.06089121(19) & − 82.88867768(2) & & 279.531 & − 23.280 & 59149 \nJ0900 − 3144 & 09:00:43.95220(3) & − 31:44:30.8722(4) & & 150.54478942(11) & − 46.14670208(11) & & 256.162 & 9.486 & 59149 \nJ0931 − 1902 & 09:31:19.116246(6) & − 19:02:55.05445(13) & & 152.37696462(2) & − 31.77673163(4) & & 250.999 & 23.054 & 59149 \nJ0955 − 6150 & 09:55:20.847467(18) & − 61:50:16.89532(13) & & 197.40608543(8) & − 64.96064654(4) & & 283.685 & − 5.737 & 59138 \nJ1012 − 4235 & 10:12:12.938801(10) & − 42:35:53.40253(12) & & 176.78918824(4) & − 48.91085447(4) & & 274.218 & 11.225 & 59152 \nJ1017 − 7156 & 10:17:51.311783(15) & − 71:56:41.57398(6) & & 222.42774900(4) & − 67.736237859(16) & & 291.558 & − 12.553 & 59132 \nJ1022 + 1001 & 10:22:57.990(2) & + 10:01:52.92(8) & & 153.86581653(3) & − 0.06391(2) & & 231.794 & 51.101 & 59166 \nJ1024 − 0719 & 10:24:38.648440(3) & − 07:19:19.98201(9) & & 160.734318047(8) & − 16.04485601(3) & & 251.702 & 40.515 & 59152 \nJ1036 − 8317 & 10:36:40.59408(3) & − 83:17:56.14254(4) & & 253.01178441(3) & − 68.082219349(15) & & 298.943 & − 21.497 & 59187 \nJ1045 − 4509 & 10:45:50.179671(15) & − 45:09:54.05686(17) & & 186.51850583(6) & − 47.71477807(5) & & 280.851 & 12.254 & 59142 \nJ1101 − 6424 & 11:01:37.19154(3) & − 64:24:39.33316(16) & & 211.77519692(9) & − 60.54978942(5) & & 291.417 & − 4.023 & 59142 \nJ1103 − 5403 & 11:03:33.27734(4) & − 54:03:43.1296(4) & & 198.52939001(16) & − 53.10724535(11) & & 287.421 & 5.529 & 59149 \nJ1125 − 5825 & 11:25:44.352943(14) & − 58:25:16.85965(11) & & 207.30413917(5) & − 54.35535501(3) & & 291.893 & 2.602 & 59132 \nJ1125 − 6014 & 11:25:55.243299(4) & − 60:14:6.80942(3) & & 209.504093810(12) & − 55.659051349(9) & & 292.504 & 0.894 & 59149 \nJ1216 − 6410 & 12:16:7.321183(7) & − 64:10:9.16656(4) & & 221.617852389(19) & − 54.452089781(13) & & 299.096 & − 1.561 & 59132 \nJ1327 − 0755 & 13:27:57.58052(9) & − 07:55:29.918(3) & & 203.27816789(2) & 1.2046411(10) & & 318.383 & 53.848 & 59281 \nJ1421 − 4409 & 14:21:20.960491(9) & − 44:09:4.57445(14) & & 228.33837111(2) & − 28.29432179(4) & & 319.497 & 15.809 & 59142 \nJ1431 − 5740 & 14:31:3.49603(4) & − 57:40:11.6601(4) & & 236.41161578(9) & − 40.16383928(11) & & 315.962 & 2.660 & 59129 \nJ1435 − 6100 & 14:35:20.26018(2) & − 61:00:58.0019(2) & & 238.94141530(6) & − 42.97710038(6) & & 315.186 & − 0.641 & 59132 \nJ1446 − 4701 & 14:46:35.709721(8) & − 47:01:26.79678(14) & & 234.21533611(2) & − 29.41052857(4) & & 322.500 & 11.425 & 59129 \nJ1455 − 3330 & 14:55:47.976991(8) & − 33:30:46.40280(19) & & 231.347559978(19) & − 16.04479743(6) & & 330.722 & 22.562 & 59149 \nJ1525 − 5545 & 15:25:28.12719(4) & − 55:45:49.8854(6) & & 244.36826552(9) & − 35.71040433(14) & & 323.439 & 0.851 & 59149 \nJ1543 − 5149 & 15:43:44.145289(15) & − 51:49:54.7135(2) & & 246.13439909(4) & − 31.17770143(6) & & 327.921 & 2.479 & 59132 \nJ1545 − 4550 & 15:45:55.945403(5) & − 45:50:37.50813(11) & & 244.821544044(13) & − 25.29111285(3) & & 331.892 & 6.988 & 59142 \nJ1547 − 5709 & 15:47:24.121775(16) & − 57:09:17.5931(2) & & 248.44532730(4) & − 36.17005240(6) & & 325.076 & − 2.052 & 59162 \nJ1600 − 3053 & 16:00:51.902514(4) & − 30:53:49.45343(17) & & 244.347678961(9) & − 10.07185627(5) & & 344.090 & 16.451 & 59138 \nJ1603 − 7202 & 16:03:35.670781(16) & − 72:02:32.82328(9) & & 256.52028909(3) & − 49.96303799(2) & & 316.630 & − 14.496 & 59127 \nJ1614 − 2230 & 16:14:36.50974(3) & − 22:30:31.542(2) & & 245.788315625(16) & − 1.2568780(7) & & 352.636 & 20.192 & 59146 \nJ1629 − 6902 & 16:29:8.744211(5) & − 69:02:45.46922(3) & & 258.424015436(10) & − 46.517655909(9) & & 320.371 & − 13.926 & 59127 \nJ1643 − 1224 & 16:43:38.16612(2) & − 12:24:58.6334(14) & & 251.08723617(7) & 9.7783440(4) & & 5.670 & 21.218 & 59146 \nJ1652 − 4838 & 16:52:37.623911(18) & − 48:38:51.7486(4) & & 257.71020795(6) & − 25.92450458(11) & & 338.005 & − 2.920 & 59021 \nJ1653 − 2054 & 16:53:31.01268(7) & − 20:54:54.977(9) & & 254.49648386(6) & 1.633926(2) & & 359.969 & 14.261 & 59149 \nJ1658 − 5324 & 16:58:39.343556(7) & − 53:24:6.97315(11) & & 259.45266028(2) & − 30.52273666(3) & & 334.869 & − 6.625 & 59188 \nJ1705 − 1903 & 17:05:43.84864(4) & − 19:03:41.414(5) & & 257.16132332(7) & 3.7751398(10) & & 3.248 & 13.030 & 59198 \nJ1708 − 3506 & 17:08:17.616678(11) & − 35:06:22.6595(5) & & 259.21049750(4) & − 12.14991994(12) & & 350.470 & 3.124 & 59215 \nJ1713 + 0747 & 17:13:49.5367813(12) & + 07:47:37.45024(4) & & 256.668711830(6) & 30.700351152(10) & & 28.751 & 25.223 & 58957 \nJ1719 − 1438 & 17:19:10.076221(4) & − 14:38:1.0020(4) & & 260.016965101(16) & 8.45261155(11) & & 8.859 & 12.838 & 59150 \nJ1721 − 2457 & 17:21:5.50008(3) & − 24:57:6.226(6) & & 261.18398611(6) & − 1.8096426(17) & & 0.387 & 6.751 & 59205 \nJ1730 − 2304 & 17:30:21.68471(6) & − 23:04:31.259(16) & & 263.186095246(13) & 0.188838(4) & & 3.137 & 6.023 & 59150 \nJ1732 − 5049 & 17:32:47.766068(7) & − 50:49:0.31465(13) & & 265.16176864(2) & − 27.49163468(4) & & 340.029 & − 9.454 & 59115 \nJ1737 − 0811 & 17:37:47.109064(19) & − 08:11:8.9798(11) & & 264.30464763(7) & 15.1437208(2) & & 16.932 & 12.318 & 59150 \nJ1744 − 1134 & 17:44:29.4220844(14) & − 11:34:54.80132(10) & & 266.119458498(6) & 11.80517508(3) & & 14.794 & 9.180 & 59150 \nJ1747 − 4036 & 17:47:48.715690(13) & − 40:36:54.7995(5) & & 267.57913114(4) & − 17.20154481(13) & & 350.208 & − 6.412 & 59130 \nJ1751 − 2857 & 17:51:32.686848(5) & − 28:57:46.5713(7) & & 268.141987317(19) & − 5.5372894(2) & & 0.646 & − 1.124 & 59150 \nJ1757 − 5322 & 17:57:15.154960(7) & − 53:22:26.59604(12) & & 269.52715024(2) & − 29.93590243(4) & & 339.637 & − 13.980 & 59163 \nJ1801 − 1417 & 18:01:51.064902(5) & − 14:17:34.5498(6) & & 270.45421998(2) & 9.14561928(15) & & 14.546 & 4.162 & 59150 \nJ1802 − 2124 & 18:02:5.334393(11) & − 21:24:3.676(5) & & 270.48652542(4) & 2.0373763(13) & & 8.382 & 0.611 & 59127 \nJ1811 − 2405 & 18:11:19.854352(6) & − 24:05:18.428(4) & & 272.586046336(10) & − 0.6745983(11) & & 7.073 & − 2.559 & 59145 \nJ1825 − 0319 & 18:25:55.954707(11) & − 03:19:57.5837(5) & & 276.88733392(5) & 19.95175171(14) & & 27.045 & 4.138 & 59155 \nJ1832 − 0836 & 18:32:27.589497(5) & − 08:36:55.1454(3) & & 278.29199042(2) & 14.59068359(9) & & 23.109 & 0.256 & 59147 \nJ1843 − 1113 & 18:43:41.260405(4) & − 11:13:31.1031(3) & & 280.944529539(16) & 11.79998448(8) & & 22.055 & − 3.397 & 59150 \nJ1902 − 5105 & 19:02:2.844154(10) & − 51:05:57.0563(2) & & 280.99087040(3) & − 28.25008697(5) & & 345.650 & − 22.379 & 59152 \nJ1903 − 7051 & 19:03:38.783931(19) & − 70:51:43.58175(12) & & 277.69661567(4) & − 47.84766764(3) & & 324.391 & − 26.508 & 59136 \n[tab:positions]\nllllllllll & & & & & & \n Pulsar & *α* & *δ* & & *λ* & *β* & & *l* & *b* & Epoch \n& (hh:mm:ss) & (dd:mm:ss) & & (∘) & (∘) & & (∘) & (∘) & (MJD) \nJ1909 − 3744 & 19:09:47.4245244(7) & − 37:44:14.91930(3) & & 284.220813110(2) & − 15.155611509(9) & & 359.731 & − 19.596 & 59120 \nJ1918 − 0642 & 19:18:48.027685(2) & − 06:42:34.95735(12) & & 290.314613645(9) & 15.35104745(3) & & 30.027 & − 9.123 & 59155 \nJ1933 − 6211 & 19:33:32.413330(4) & − 62:11:46.69324(5) & & 283.961757663(10) & − 39.885817735(13) & & 334.431 & − 28.631 & 59132 \nJ1946 − 5403 & 19:46:34.4933990(19) & − 54:03:42.55867(3) & & 288.139717466(5) & − 32.287682135(10) & & 343.885 & − 29.580 & 59166 \nJ2010 − 1323 & 20:10:45.922635(4) & − 13:23:56.1327(3) & & 301.924490472(7) & 6.49093303(7) & & 29.446 & − 23.540 & 59155 \nJ2039 − 3616 & 20:39:16.580827(4) & − 36:16:17.23958(11) & & 302.723254857(9) & − 17.24602148(3) & & 6.331 & − 36.517 & 59166 \nJ2124 − 3358 & 21:24:43.834930(4) & − 33:58:45.49195(11) & & 312.738766685(11) & − 17.81894426(3) & & 10.925 & − 45.438 & 59155 \nJ2129 − 5721 & 21:29:22.781548(3) & − 57:21:14.33356(4) & & 303.827979841(7) & − 39.900000443(10) & & 338.005 & − 43.570 & 59145 \nJ2145 − 0750 & 21:45:50.453394(17) & − 07:50:18.5938(7) & & 326.02458223(2) & 5.3130387(2) & & 47.777 & − 42.084 & 59155 \nJ2150 − 0326 & 21:50:27.235330(10) & − 03:26:32.8318(4) & & 328.604922236(17) & 9.06674597(12) & & 53.573 & − 40.757 & 59199 \nJ2222 − 0137 & 22:22:5.996923(12) & − 01:37:15.7776(5) & & 336.73199735(2) & 7.97708619(14) & & 62.019 & − 46.075 & 59155 \nJ2229 + 2643 & 22:29:50.883815(5) & + 26:43:57.61393(13) & & 350.695629791(19) & 33.29016684(4) & & 87.693 & − 26.284 & 59251 \nJ2234 + 0944 & 22:34:46.856913(9) & + 09:44:30.0415(3) & & 344.11900953(3) & 17.31853087(9) & & 76.280 & − 40.438 & 59251 \nJ2241 − 5236 & 22:41:42.0400897(10) & − 52:36:36.272400(11) & & 318.696416363(3) & − 40.393437599(3) & & 337.457 & − 54.927 & 59132 \nJ2317 + 1439 & 23:17:9.235606(4) & + 14:39:31.29559(12) & & 356.129406224(12) & 17.68024246(4) & & 91.361 & − 42.360 & 59251 \nJ2322 + 2057 & 23:22:22.320041(6) & + 20:57:2.50080(14) & & 0.135924003(19) & 22.87835838(4) & & 96.515 & − 37.310 & 59251 \nJ2322 − 2650 & 23:22:34.638615(5) & − 26:50:58.38398(10) & & 340.443069768(14) & − 20.89690479(3) & & 28.637 & − 70.228 & 59155 \n@llllllllllllll & & & & & & & & \nPulsar & Span & $\\mu\\_\\alpha\\equiv\\dot{\\alpha}\\cos\\delta$ & *μ**δ* & & $\\mu\\_\\lambda\\equiv\\dot{\\lambda}\\cos\\beta$ & *μ**β* & & $\\mu\\_{l}\\equiv\\dot{l}\\cos b$ & *μ**b* & & *μ**α* & *μ**δ* & Ref \n& (yr) & (masyr− 1) & (masyr− 1) & & (masyr− 1) & (masyr− 1) & & (masyr− 1) & (masyr− 1) & & (masyr− 1) & (masyr− 1) & \n \nJ0030 + 0451 & 2.54 & − 6.4(11) & 1(2) & & − 5.51(7) & 3(3) & & − 6.2(6) & 2(3) & & − 6.2(1) & 0.5(3) & (1) \nJ0125 − 2327 & 3.06 & 37.15(5) & 10.75(6) & & 38.18(3) & − 6.17(6) & & 7.70(6) & 37.90(4) & & & & \nJ0437 − 4715 & 2.86 & 121.394(19) & − 71.489(19) & & 85.96(2) & − 111.60(2) & & 62.89(2) & 126.06(2) & & 121.4385(20) & − 71.4754(20) & (2) \nJ0610 − 2100 & 3.05 & 9.29(11) & 16.74(17) & & 8.86(11) & 16.97(17) & & − 11.81(17) & 15.08(15) & & 9.11(3) & 16.45(3) & (3) \nJ0613 − 0200 & 3.05 & 1.81(6) & − 10.31(17) & & 2.08(6) & − 10.28(17) & & 10.00(15) & − 3.11(11) & & 1.836(5) & − 10.349(13) & (6) \nJ0614 − 3329 & 3.06 & 0.60(4) & − 1.84(5) & & 0.68(4) & − 1.81(5) & & 1.93(4) & − 0.02(4) & & 0.61(3) & − 1.74(4) & (4) \nJ0636 − 3044 & 2.99 & 26.01(9) & − 19.26(10) & & 27.92(9) & − 16.37(10) & & 27.66(8) & 16.79(11) & & & & \nJ0711 − 6830 & 3.06 & − 15.54(5) & 14.24(6) & & − 12.11(5) & − 17.25(5) & & − 17.79(6) & − 11.21(5) & & − 15.568(10) & 14.175(11) & (6) \nJ0900 − 3144 & 3.06 & − 1.4(4) & 2.4(5) & & − 2.3(3) & 1.6(5) & & − 2.7(4) & 0.5(6) & & − 1.01(5) & 2.02(7) & (5) \nJ0931 − 1902 & 3.06 & − 2.44(11) & − 3.92(19) & & − 0.80(9) & − 4.5(2) & & 1.29(12) & − 4.47(19) & & − 2.4(2) & − 4.4(4) & (1) \nJ1012 − 4235 & 3.05 & − 4.15(15) & 5.20(16) & & − 6.30(14) & 2.14(17) & & − 6.40(15) & 1.82(17) & & & & \nJ1017 − 7156 & 3.15 & − 7.43(8) & 6.91(7) & & − 8.93(6) & − 4.84(6) & & − 10.03(8) & 1.49(9) & & − 7.411(12) & 6.870(11) & (6) \nJ1024 − 0719 & 3.05 & − 35.32(5) & − 48.39(12) & & − 14.37(4) & − 58.16(13) & & 8.62(7) & − 59.28(12) & & − 35.270(17) & − 48.22(3) & (6) \nJ1036 − 8317 & 2.86 & − 11.35(6) & 2.86(6) & & − 1.83(6) & − 11.56(6) & & − 11.15(6) & − 3.57(5) & & & & \nJ1045 − 4509 & 3.10 & − 6.25(19) & 4.9(2) & & − 7.9(2) & 0.6(2) & & − 7.8(2) & 1.4(2) & & − 6.07(3) & 5.19(4) & (6) \nJ1101 − 6424 & 3.10 & − 1.0(2) & 0.4(2) & & − 1.0(2) & − 0.5(3) & & − 1.1(3) & 0.0(3) & & & & \nJ1103 − 5403 & 3.06 & − 7.1(4) & 1.1(4) & & − 6.2(4) & − 3.8(4) & & − 7.1(4) & − 1.9(5) & & & & \nJ1125 − 5825 & 3.15 & − 8.95(13) & 1.79(13) & & − 7.81(13) & − 4.72(14) & & − 9.04(14) & − 1.21(15) & & − 10.0(3) & 2.4(3) & (7) \nJ1125 − 6014 & 3.06 & 11.09(3) & − 13.00(3) & & 17.01(3) & − 1.57(4) & & 14.71(4) & − 8.69(4) & & 11.106(13) & − 13.037(14) & (6) \nJ1216 − 6410 & 3.15 & − 7.80(5) & 2.69(5) & & − 7.54(5) & − 3.36(5) & & − 8.10(5) & 1.60(5) & & & & \nJ1421 − 4409 & 3.10 & − 10.27(13) & − 6.15(14) & & − 7.28(10) & − 9.50(15) & & − 11.78(15) & − 2.16(11) & & − 11.6(4) & − 7.9(8) & (8) \nJ1431 − 5740 & 3.17 & 0.0(3) & 2.5(4) & & − 1.0(3) & 2.3(5) & & 1.0(5) & 2.3(4) & & & & \nJ1435 − 6100 & 3.15 & − 5.6(2) & − 2.4(3) & & − 4.00(18) & − 4.5(3) & & − 6.1(3) & 0.0(3) & & & & \nJ1446 − 4701 & 3.17 & − 4.24(9) & − 2.45(16) & & − 3.15(8) & − 3.73(18) & & − 4.88(14) & − 0.32(14) & & − 4.36(4) & − 3.00(7) & (6) \nJ1455 − 3330 & 3.06 & 7.66(13) & − 2.3(2) & & 8.01(9) & 0.1(2) & & 5.5(2) & − 5.82(17) & & 7.98(8) & − 2.0(2) & (1) \nJ1525 − 5545 & 3.06 & − 6.8(5) & − 3.3(8) & & − 5.5(5) & − 5.5(7) & & − 7.6(8) & 1.1(7) & & & & \nJ1543 − 5149 & 3.15 & − 4.33(18) & − 2.5(2) & & − 3.54(16) & − 3.5(3) & & − 4.9(3) & 0.68(18) & & − 4.3(14) & − 4(2) & (7) \nJ1545 − 4550 & 3.10 & − 0.51(7) & 2.50(13) & & − 1.09(6) & 2.29(12) & & 1.12(12) & 2.23(11) & & − 0.48(2) & 2.37(4) & (6) \nJ1547 − 5709 & 2.99 & − 6.59(18) & − 5.9(2) & & − 4.77(17) & − 7.4(3) & & − 8.8(3) & − 0.5(2) & & & & \nJ1600 − 3053 & 3.12 & − 1.04(6) & − 7.5(2) & & 0.48(4) & − 7.5(2) & & − 5.91(18) & − 4.79(12) & & − 0.960(7) & − 6.96(3) & (6) \nJ1603 − 7202 & 3.16 & − 2.55(10) & − 7.40(11) & & − 0.21(8) & − 7.83(11) & & − 6.92(12) & − 3.66(9) & & − 2.447(11) & − 7.356(13) & (6) \nJ1614 − 2230 & 3.05 & 4.4(5) & − 29(3) & & 9.52(7) & − 28(3) & & − 18(3) & − 23.1(17) & & 3.8(1) & − 32.5(7) & (1) \nJ1629 − 6902 & 3.16 & − 6.50(3) & − 8.44(4) & & − 4.45(3) & − 9.68(4) & & − 10.62(4) & − 0.78(3) & & & & \nJ1643 − 1224 & 3.05 & 6.1(4) & 3.4(17) & & 5.6(3) & 4.0(19) & & 6− 3+ 2 & − 3.2− 1.3+ 1.1 & & 5.970(18) & 3.77(8) & (6) \nJ1652 − 4838 & 2.33 & − 3.8(3) & − 9.7(7) & & − 2.5(3) & − 10.1(8) & & − 9.8(7) & − 3.2(5) & & & & \nJ1658 − 5324 & 2.82 & 0.02(10) & 2.86(17) & & − 0.33(11) & 2.84(17) & & 2.27(13) & 1.74(15) & & 0.02(10) & 4.90(23) & (9) \nJ1708 − 3506 & 2.67 & − 5.8(3) & − 0.4(8) & & − 5.8(2) & − 0.9(8) & & − 3.8(7) & 4.5(5) & & − 5.3(8) & − 2(3) & (7) \nJ1713+0747 & 1.98 & 4.94(4) & − 4.07(9) & & 5.30(4) & − 3.59(9) & & − 1.45(8) & − 6.24(7) & & 4.9254(14) & − 3.917(3) & (6) \nJ1719 − 1438 & 3.03 & 3.76(10) & − 3.3(5) & & 3.99(9) & − 3.0(5) & & − 0.8(6) & − 5.0(3) & & 1.9(4) & − 11(2) & (7) \nJ1730 − 2304 & 3.03 & 20.3(11) & 0(21) & & 20.29(6) & 1(21) & & 11(19) & − 17(11) & & 20.06(12) & − 4(2) & (6) \nJ1732 − 5049 & 3.22 & − 0.60(9) & − 10.0(2) & & − 0.07(9) & − 10.0(2) & & − 8.8(2) & − 4.78(13) & & − 0.41(9) & − 9.87(19) & (2) \nJ1737 − 0811 & 3.03 & − 3.9(3) & − 11.8(13) & & − 3.4(3) & − 11.4(10) & & − 12.5− 1.8+ 1.5 & − 2.6− 0.9+ 0.8 & & & & \nJ1744 − 1134 & 3.03 & 18.85(2) & − 9.73(13) & & 19.11(2) & − 9.21(13) & & 1.11(12) & − 21.18(7) & & 18.803(4) & − 9.390(18) & (6) \nJ1747 − 4036 & 3.14 & − 1.3(2) & 0.0(7) & & − 1.3(2) & − 0.1(6) & & − 0.8(10) & 1.2(7) & & − 1.30(13) & − 2.7(4) & (15) \nJ1751 − 2857 & 3.03 & − 7.37(10) & − 4.8(10) & & − 7.30(10) & − 5.0(10) & & − 7.8(9) & 3.9(5) & & − 7.4(1) & − 4.3(12) & (5) \nJ1757 − 5322 & 2.96 & − 2.48(8) & − 10.02(16) & & − 2.42(8) & − 10.04(16) & & − 10.07(17) & − 2.33(9) & & & & \nJ1801 − 1417 & 3.03 & − 10.67(10) & − 1.8(6) & & − 10.68(10) & − 1.9(6) & & − 6.8− 0.6+ 0.7 & 8.4− 0.3+ 0.4 & & − 10.89(12) & − 3.0(10) & (5) \nJ1825 − 0319 & 3.06 & 3.1(2) & − 1.6(6) & & 3.1(2) & − 1.8(6) & & 0.0(5) & − 3.5(4) & & & & \nJ1832 − 0836 & 3.05 & − 8.06(8) & − 21.5(4) & & − 9.30(8) & − 21.0(4) & & − 22.8(4) & − 2.8(2) & & − 8.06(5) & − 21.01(19) & (6) \nJ1843 − 1113 & 3.03 & − 2.02(7) & − 2.9(4) & & − 2.23(7) & − 2.7(4) & & − 3.5(3) & 0.5(2) & & − 1.91(7) & − 3.2(3) & (5) \nJ1902 − 5105 & 3.08 & − 3.90(11) & − 8.5(3) & & − 4.91(11) & − 8.0(3) & & − 9.2(3) & 1.43(14) & & − 4.8(13) & 4.4(16) & (9) \nJ1903 − 7051 & 3.17 & − 6.56(13) & − 17.68(18) & & − 9.35(13) & − 16.37(18) & & − 18.31− 0.19+ 0.20 & 4.45− 0.13+ 0.14 & & − 8.8(16) & − 16(2) & (9) \nJ1909 − 3744 & 3.26 & − 9.513(12) & − 35.70(4) & & − 13.852(11) & − 34.25(4) & & − 36.83(4) & − 3.06(2) & & − 9.5146(8) & − 35.776(3) & (6) \nJ1918 − 0642 & 3.06 & − 7.02(5) & − 6.37(17) & & − 7.84(4) & − 5.33(17) & & − 8.83(14) & 3.44(11) & & − 7.15(2) & − 5.97(5) & (1) \nJ1933 − 6211 & 3.19 & − 5.58(4) & 11.05(7) & & − 3.19(3) & 11.96(7) & & 10.58(7) & 6.43(4) & & − 5.54(7) & 10.7(2) & (10) \n[tab:propermotions]\n@llllllllllllll & & & & & & & & \nPulsar & Span & $\\mu\\_\\alpha\\equiv\\dot{\\alpha}\\cos\\delta$ & *μ**δ* & & $\\mu\\_\\lambda\\equiv\\dot{\\lambda}\\cos\\beta$ & *μ**β* & & $\\mu\\_{l}\\equiv\\dot{l}\\cos b$ & *μ**b* & & *μ**α* & *μ**δ* & Ref \n& (yr) & (masyr− 1) & (masyr− 1) & & (masyr− 1) & (masyr− 1) & & (masyr− 1) & (masyr− 1) & & (masyr− 1) & (masyr− 1) & \n \nJ1946 − 5403 & 3.01 & − 1.08(3) & − 4.75(5) & & − 2.06(2) & − 4.41(5) & & − 4.84(5) & 0.55(3) & & & & \nJ2010 − 1323 & 3.07 & 2.63(8) & − 6.1(4) & & 1.25(3) & − 6.5(4) & & − 4.5(3) & − 4.9(2) & & 2.4(3) & − 5.6(3) & (11) \nJ2039 − 3616 & 3.01 & − 10.11(5) & − 8.66(15) & & − 12.06(4) & − 5.64(16) & & − 10.17− 0.14+ 0.15 & 8.60(8) & & & & \nJ2124 − 3358 & 3.07 & − 14.17(6) & − 50.28(14) & & − 29.76(4) & − 42.93(15) & & − 51.44(15) & 9.11(8) & & − 14.109(19) & − 50.36(4) & (6) \nJ2129 − 5721 & 3.12 & 9.30(4) & − 9.61(5) & & 4.54(3) & − 12.59(5) & & − 12.06(6) & − 5.76(3) & & 9.30(1) & − 9.576(13) & (6) \nJ2145 − 0750 & 3.07 & − 9.7(3) & − 8.2(10) & & − 11.87(8) & − 4.5(10) & & − 11.9(7) & 4.4(8) & & − 9.48(2) & − 9.11(7) & (6) \nJ2150 − 0326 & 2.82 & 6.5(2) & − 11.1(6) & & 2.28(9) & − 12.7(6) & & − 5.5(5) & − 11.3(7) & & & & \nJ2222 − 0137 & 3.07 & 44.8(2) & − 5.7(6) & & 39.63(9) & − 21.7(7) & & 23.7(4) & − 38.5(6) & & 44.70(4) & − 5.69(8) & (12) \nJ2229 + 2643 & 2.54 & − 1.70(9) & − 5.75(14) & & − 4.05(8) & − 4.42(15) & & − 4.70(9) & − 3.74(15) & & − 2.1(6) & − 5.7(5) & (1) \nJ2234 + 0944 & 2.54 & 6.9(2) & − 33.2(5) & & − 6.48(13) & − 33.3(5) & & − 16.6(3) & − 29.6(5) & & 6.9(2) & − 32.0(4) & (1) \nJ2241 − 5236 & 3.19 & 18.875(12) & − 5.286(13) & & 13.830(10) & − 13.888(15) & & − 14.904(17) & − 12.736(10) & & 18.881(4) & − 5.294(5) & (6) \nJ2317 + 1439 & 2.54 & − 1.45(9) & 3.7(2) & & 0.20(6) & 4.0(2) & & 0.51(6) & 4.0(2) & & − 1.3(3) & 3.6(5) & (11) \nJ2322 + 2057 & 2.54 & − 18.06(10) & − 15.24(15) & & − 22.83(7) & − 6.09(17) & & − 22.82(8) & − 6.14(17) & & − 18.4(4) & − 15.4(5) & (5) \nJ2322 − 2650 & 3.06 & − 2.37(9) & − 8.20(17) & & − 5.59(6) & − 6.44(19) & & − 8.40(16) & 1.52(11) & & − 2.4(2) & − 8.3(4) & (13) \n \nJ0955 − 6150 & 3.12 & − 0.08− 0.14+ 0.15 & − 0.21− 0.15+ 0.16 & & 0.12− 0.17+ 0.16 & − 0.19− 0.14+ 0.15 & & 0.06− 0.16+ 0.17 & − 0.21(15) & & & & \nJ1022 + 1001 & 2.97 & − 27− 53+ 50 & − 27− 136+ 129 & & − 15.70(13) & − 36− 137+ 140 & & 9− 85+ 84 & − 38− 109+ 111 & & − 14.92− 0.03+ 0.05 & − 5.61(3) & (11) \nJ1327 − 0755 & 2.34 & − 4(2) & 5(6) & & − 5.06− 0.11+ 0.10 & 3(6) & & − 2(4) & 6(5) & & & & \nJ1653 − 2054 & 3.04 & − 10.5(13) & − 15(11) & & − 8.7− 0.3+ 0.4 & − 16− 12+ 11 & & − 19(10) & − 1(6) & & & & \nJ1705 − 1903 & 2.77 & − 1.7(9) & − 1− 6+ 7 & & − 1.5(5) & − 1(7) & & − 2(6) & 1− 3+ 4 & & & & \nJ1721 − 2457 & 2.73 & 3.4(6) & 5(8) & & 3.1(3) & 5(8) & & 6(7) & 0(4) & & & & \nJ1802 − 2124 & 3.16 & − 1.6(3) & − 14− 8+ 9 & & − 1.7(3) & − 14(8) & & − 13(7) & − 6(4) & & − 1.13(12) & − 3(4) & (5) \nJ1811 − 2405 & 3.12 & 0.53(9) & 3(5) & & 0.59(4) & 3(4) & & 3(4) & 1(2) & & 0.53(6) & 0 & (14) \nReferences: (1), (2), (3), (4), (5), (6), (7), (8), (9), (10), (11), (12), (13), (14), (15) \nParallaxes\n----------\nThe parallax measurements made in this study, as well as the best previous measurements of those pulsars in the literature and their corresponding measurement techniques (timing or VLBI), are listed in Table [tab:parallaxdistance] and shown in Figure [fig:parallaxloglog] ( > 3*σ* detections). The parallax values and their uncertainties are derived from the marginalized posterior probability distributions from temponest. There are 35 of our MSPs that had parallax measurements with > 3*σ* significance. 12 of these 35 are new (PSRs J0125 − 2327, J0614 − 3329, J0636 − 3044, J1421 − 4409, J1629 − 6902, J1652 − 4838, J1658 − 5324, J1757 − 5322, J1801 − 1417, J1811 − 2405, J1933 − 6211 and J1946 − 5403). We improved the precision of 4 other parallaxes (of the 35 MSPs) by a factor of 1.25 to 6 including PSRs J1446 − 4701, J1455 − 3330, J2124 − 3358 and J2322 − 2650. For the MSPs with less than 3*σ* significance in parallax, we calculated the 68% confidence upper level and listed them in the lower portion of Table [tab:parallaxdistance].\nTo gain confidence in our parallax measurements, we compared them with their values derived from other authors in Figure [fig:parallaxloglog] as 22 of our significant ( > 3*σ*) parallaxes have previous measurements. For PSR J1713 + 0747, we measured the parallax of 0.88(10) mas using only 1.98 yr of data as its subsequent timing was compromised by a sudden step-change in its pulse profile and gradual decay back to the original profile. In addition to the previously measured parallax value of 0.763(21) mas by using timing analysis of the Parkes pulsar timing array second data release, a parallax of 0.82(3) mas was obtained by using timing analysis of the NANOGrav 11-year data set. Using the VLBI technique, measured a value of 0.95− 0.05+ 0.06 mas. obtained a value of 0.90(3) mas, very close to our value. All measurements are in agreement with our value to within about 1*σ*.\nOf the 35 MSPs with significant measurements, only our parallaxes for PSRs J1643 − 1224 and J2322 − 2650 (shown in Figure [fig:parallaxloglog]) are markedly inconsistent with the best previous measurements. We discuss each of these two pulsars in turn below. Overall the standard deviation of the differences in relative parallaxes for the 23 pulsars with previous measurements in literature is only 10 percent, and if we exclude the two outliers discussed above, only 7 percent.\n### PSR J1643 − 1224\nFor PSR J1643 − 1224, we obtained the parallax value of $\\varpi = 2.6^{+0.5}\\_{-0.6}$ mas which is quite inconsistent with the VLBI value of $\\varpi = 1.1(1)$ mas obtained by and also has the lower formal error of the measurements.\n measured a value of 0.7(6) mas using the NANOGrav nine-year data set and a detailed model for DM variation that is known to be significant. They compared their value with the measurement of, who obtained a value of 2.2(4) mas which did not include a complex model for DM variation. hence argued that the limited DM variation modelling might be the possible source of the inconsistency between their measurements.\nTwo more measurements come from, who obtain the parallax to be 0.82(17) mas using the 24 yr observations of the Parkes Pulsar Timing Array data set and, who obtain a parallax of 1.17(26) mas. Clearly, the timing of this pulsar presents challenges.\nPSR J1643 − 1224 has been shown to be located behind the HII region Sh 2 − 27 by and undergoes annual DM variations. Its timing with MeerKAT also exhibits strong evidence for chromatic variations (Miles et al. in prep), probably due to time-dependent scattering in the HII region which are not modelled by our techniques. reported that there is a significant excess DM for PSR J1643 − 1224 consistent with its distance beyond the HII region.\nWe examined whether our parallax might be radio-frequency dependent, by measuring the parallax using just the top half of our band (1284–1712 MHz), and we obtained a smaller value of 1.7(4) mas, which is in better agreement (1.5*σ*) with the measurement of. In our noise analysis for this pulsar, using the entire band, the noise model with the white, red, and DM noise parameters had the highest Bayesian evidence; however, using only the top part of the band the noise model with the white and red noise parameters had the highest Bayesian evidence. We know that this pulsar is significantly affected by interstellar scattering, but this is not currently included in our noise model. Weaknesses in our noise model such as this may lead to small biases in the parallax measurement. This would be good topic for a new study. For the rest of this paper, we adopt the VLBI parallax.\n### PSR J2322 − 2650\nFor PSR J2322 − 2650, we have measured a parallax of 1.3(2) mas, which is very different from the previously measured value of 4.4(12) mas by but since it has 6 times lower uncertainty we favour our value. Our parallax distance for PSR J2322 − 2650 is thus 0.80− 0.12+ 0.18 kpc which is much nearer the YMW16 distance of 0.76 kpc than the 0.23 kpc suggested by. We speculate that the inconsistency between our values is due to the absence of any DM noise model in the earlier analysis of and their poorer timing residuals.\nllllllllllll & & & & & & & \n Pulsar & & Parallax $\\varpi$ & Distance *d* & & Parallax $\\varpi$ & Technique & Ref. & & NE2001 & YMW16 & \n& & (mas) & (kpc) & & (mas) & & & & (kpc) & (kpc) \n \nJ0030 + 0451 & & 2.91(18) & 0.34(2) & & 3.04(5) & VLBI & (1) & & 0.32 & 0.34 \nJ0125 − 2327 & & 0.84(11) & 1.19− 0.13+ 0.17 & & & & & & 0.44 & 0.87 \nJ0437 − 4715 & & 6.8(4) & 0.148− 0.008+ 0.009 & & 6.396(54) & VLBI & (2) & & 0.14 & 0.16 \nJ0613 − 0200 & & 0.86(17) & 1.16− 0.19+ 0.28 & & 1.01(9) & Timing & (3) & & 1.71 & 1.02 \nJ0614 − 3329 & & 1.5(4) & 0.67− 0.14+ 0.25 & & < 2.2 & Timing & (4) & & 1.90 & 2.69 \nJ0636 − 3044 & & 4.3(7) & 0.23− 0.03+ 0.04 & & & & & & 1.00 & 0.68 \nJ1024 − 0719 & & 0.97− 0.13+ 0.12 & 1.03− 0.11+ 0.15 & & 0.93(5) & VLBI & (1) & & 0.39 & 0.38 \nJ1125 − 6014 & & 1.2− 0.2+ 0.3 & 0.86− 0.16+ 0.22 & & 0.6(3) &Timing & (3) & & 1.49 & 0.99 \nJ1421 − 4409 & & 1.2(4) & 0.8− 0.2+ 0.5 & & & & & & 1.57 & 2.09 \nJ1446 − 4701 & & 0.7(2) & 1.4− 0.3+ 0.7 & & 0.6(3) & Timing & (3) & & 1.46 & 1.57 \nJ1455 − 3330 & & 1.1(3) & 0.94− 0.19+ 0.33 & & 0.99(22) & Timing & (4) & & 0.53 & 0.68 \nJ1600 − 3053 & & 0.51(11) & 2.0− 0.3+ 0.5 & & 0.53(6) & Timing & (3) & & 1.63 & 2.53 \nJ1614 − 2230 & & 1.3(2) & 0.77− 0.11+ 0.14 & & 1.54(10) & Timing & (5) & & 1.27 & 1.39 \nJ1629 − 6902 & & 0.9(2) & 1.1− 0.2+ 0.3 & & & & & & 0.96 & 0.96 \nJ1643 − 1224 & & 2.6− 0.6+ 0.5 & 0.39− 0.07+ 0.11 & & 1.1(1) & VLBI & (1) & & 2.40 & 0.79 \nJ1652 − 4838 & & 2.4− 0.7+ 0.8 & 0.41− 0.10+ 0.19 & & & & & & 3.38 & 4.33 \nJ1658 − 5324 & & 1.3(3) & 0.75− 0.12+ 0.17 & & & & & & 0.93 & 0.88 \nJ1713 + 0747 & & 0.88(10) & 1.14− 0.11+ 0.15 & & 0.763(21) & Timing & (3) & & 0.89 & 0.92 \nJ1730 − 2304 & & 2.3(2) & 0.44− 0.04+ 0.05 & & 2.0(1) & VLBI & (1) & & 0.53 & 0.51 \nJ1744 − 1134 & & 2.61(9) & 0.384− 0.013+ 0.014 & & 2.44(5) & Timing & (3) & & 0.41 & 0.15 \nJ1757 − 5322 & & 1.2(3) & 0.81− 0.15+ 0.25 & & & & & & 0.96 & 0.94 \nJ1801 − 1417 & & 0.6(2) & 1.7− 0.5+ 1.1 & & & & & & 1.52 & 1.10 \nJ1811 − 2405 & & 0.72(15) & 1.4− 0.2+ 0.4 & & < 0.4 & Timing & (4) & & 1.77 & 1.83 \nJ1832 − 0836 & & 0.7(2) & 1.4− 0.3+ 0.6 & & 0.48(14) & Timing & (5) & & 1.11 & 0.81 \nJ1909 − 3744 & & 0.92(3) & 1.09(4) & & 0.86(1) & Timing & (3) & & 0.46 & 0.56 \nJ1918 − 0642 & & 0.84(15) & 1.20− 0.18+ 0.26 & & 0.71(7) & VLBI & (1) & & 1.24 & 1.03 \nJ1933 − 6211 & & 0.7(2) & 1.5− 0.3+ 0.6 & & & & & & 0.52 & 0.65 \nJ1946 − 5403 & & 0.81− 0.11+ 0.10 & 1.23− 0.14+ 0.18 & & & & & & 0.87 & 1.15 \nJ2010 − 1323 & & 0.32(11) & 3.1− 0.8+ 1.7 & & 0.41(12) & Timing & (10) & & 1.02 & 1.16 \nJ2124 − 3358 & & 2.06− 0.15+ 0.16 & 0.48(4) & & 2.3(2) & Timing & (3) & & 0.27 & 0.36 \nJ2145 − 0750 & & 1.6(3) & 0.61− 0.09+ 0.14 & & 1.603− 0.009+ 0.063 & VLBI & (9) & & 0.57 & 0.69 \nJ2222 − 0137 & & 3.7(3) & 0.27(2) & & 3.730(16) & VLBI & (6) & & 0.31 & 0.27 \nJ2241 − 5236 & & 0.91(6) & 1.10− 0.06+ 0.07 & & 0.96(4) & Timing & (3) & & 0.51 & 0.96 \nJ2322 + 2057 & & 1.2(3) & 0.80− 0.15+ 0.21 & & 0.98(26) & Timing & (5) & & 0.80 & 1.01 \nJ2322 − 2650 & & 1.3(2) & 0.80− 0.12+ 0.18 & & 4.4(12) & Timing & (7) & & 0.32 & 0.76 \n[tab:parallaxdistance]\nllllllllllll & & & & & & & \n Pulsar & Measurement & 1*σ* limit & 1*σ* limit & & Parallax $\\varpi$ & Technique & Ref. & & NE2001 & YMW16 & \n& (mas) & (mas) & (kpc) & & (mas) & & & & (kpc) & (kpc) \n \nJ0610 − 2100 & 0.5− 0.4+ 0.5 & < 1.0 & > 1.0 & & 0.72(11) & VLBI & (1) & & 3.54 & 3.26 \nJ0711 − 6830 & 4− 3+ 6 & < 10.0 & > 0.1 & & & & & & 0.86 & 0.11 \nJ0900 − 3144 & 1.1− 0.8+ 1.3 & < 2.4 & > 0.4 & & 0.77(44) & Timing & (8) & & 0.54 & 0.38 \nJ0931 − 1902 & 0.5− 0.3+ 0.4 & < 0.9 & > 1.2 & & 0.7(7) & Timing & (5) & & 1.88 & 3.72 \nJ0955 − 6150 & 0.6− 0.4+ 0.8 & < 1.4 & > 0.6 & & & & & & 4.04 & 2.17 \nJ1012 − 4235 & 1.3− 0.8+ 0.9 & < 2.2 & > 0.5 & & & & & & 2.51 & 0.37 \nJ1017 − 7156 & 1.5− 0.8+ 0.9 & < 2.4 & > 0.5 & & 0.6(6) & Timing & (3) & & 2.98 & 1.81 \nJ1022 + 1001 & 1.1(4) & < 1.5 & > 0.7 & & 1.38− 0.03+ 0.04 & VLBI & (9) & & 0.45 & 0.83 \nJ1036 − 8317 & 0.6− 0.4+ 0.7 & < 1.3 & > 0.7 & & & & & & 1.03 & 0.93 \nJ1045 − 4509 & 1.0− 0.7+ 1.0 & < 2.0 & > 0.5 & & 1.7(7) & Timing & (3) & & 1.96 & 0.33 \nJ1101 − 6424 & 1.0− 0.7+ 1.3 & < 2.3 & > 0.4 & & & & & & 4.48 & 2.18 \nJ1103 − 5403 & 1.4− 1.0+ 1.9 & < 3.3 & > 0.3 & & & & & & 2.55 & 1.68 \nJ1125 − 5825 & 1.3− 0.7+ 0.8 & < 2.1 & > 0.5 & & & & & & 2.62 & 1.74 \nJ1216 − 6410 & 0.9(4) & < 1.3 & > 0.8 & & 0.6(3) & Timing & (3) & & 1.33 & 1.10 \nJ1327 − 0755 & 0.13− 0.09+ 0.17 & < 0.3 & > 3.4 & & & & & & 1.73 & 25.00 \nJ1431 − 5740 & 2.4− 1.3+ 1.4 & < 3.8 & > 0.3 & & & & & & 2.55 & 3.55 \nJ1435 − 6100 & 0.4− 0.3+ 0.5 & < 0.9 & > 1.1 & & & & & & 2.16 & 2.82 \nJ1525 − 5545 & 1.5− 1.0+ 1.3 & < 2.8 & > 0.3 & & & & & & 2.37 & 3.14 \nJ1543 − 5149 & 0.6− 0.4+ 0.5 & < 1.1 & > 0.9 & & & & & & 2.43 & 1.15 \nJ1545 − 4550 & 0.38(18) & < 0.6 & > 1.8 & & 0.45(14) & Timing & (3) & & 2.13 & 2.25 \nJ1547 − 5709 & 0.6− 0.4+ 0.6 & < 1.2 & > 0.8 & & & & & & 1.87 & 2.70 \nJ1603 − 7202 & 1.1− 0.6+ 0.7 & < 1.8 & > 0.6 & & 0.3(3) & Timing & (3) & & 1.17 & 1.13 \nJ1653 − 2054 & 0.24− 0.17+ 0.34 & < 0.6 & > 1.7 & & & & & & 1.67 & 2.63 \nJ1705 − 1903 & 0.6− 0.4+ 0.6 & < 1.2 & > 0.8 & & & & & & 1.65 & 2.34 \nJ1708 − 3506 & 0.5− 0.3+ 0.4 & < 0.9 & > 1.1 & & & & & & 2.80 & 3.32 \nJ1719 − 1438 & 0.30− 0.18+ 0.23 & < 0.6 & > 1.9 & & & & & & 1.21 & 0.34 \nJ1721 − 2457 & 0.24− 0.16+ 0.29 & < 0.6 & > 2.0 & & & & & & 1.30 & 1.39 \nJ1732 − 5049 & 0.7(4) & < 1.1 & > 0.9 & & & & & & 1.41 & 1.87 \nJ1737 − 0811 & 0.4− 0.3+ 0.4 & < 0.8 & > 1.2 & & & & & & 1.71 & 0.21 \nJ1747 − 4036 & 0.17− 0.12+ 0.25 & < 0.5 & > 2.3 & & 0.1(7) & Timing & (5) & & 3.39 & 7.15 \nJ1751 − 2857 & 0.6(3) & < 0.9 & > 1.2 & & & & & & 1.10 & 1.09 \nJ1802 − 2124 & 0.4− 0.3+ 0.5 & < 0.9 & > 1.2 & & 1.24(57) & Timing & (8) & & 2.94 & 3.03 \nJ1825 − 0319 & 0.5− 0.3+ 0.4 & < 0.9 & > 1.1 & & & & & & 3.07 & 3.86 \nJ1843 − 1113 & 0.32− 0.16+ 0.17 & < 0.5 & > 2.0 & & 0.69(33) & Timing & (8) & & 1.70 & 1.71 \nJ1902 − 5105 & 0.20− 0.14+ 0.25 & < 0.5 & > 2.3 & & & & & & 1.18 & 1.65 \nJ1903 − 7051 & 0.8− 0.5+ 0.8 & < 1.6 & > 0.6 & & & & & & 0.76 & 0.93 \nJ2039 − 3616 & 0.25− 0.13+ 0.14 & < 0.4 & > 2.5 & & & & & & 0.91 & 1.70 \nJ2129 − 5721 & 0.12− 0.08+ 0.13 & < 0.3 & > 4.1 & & 0.26(17) & Timing & (3) & & 1.36 & 6.16 \nJ2150 − 0326 & 0.30− 0.19+ 0.28 & < 0.6 & > 1.7 & & & & & & 1.05 & 1.98 \nJ2229 + 2643 & 0.6(3) & < 0.9 & > 1.1 & & 0.2(3) & Timing & (5) & & 1.43 & 1.80 \nJ2234 + 0944 & 1.0(4) & < 1.4 & > 0.7 & & 0.7(3) & Timing & (5) & & 1.00 & 1.58 \nJ2317 + 1439 & 0.29− 0.15+ 0.16 & < 0.5 & > 2.2 & & 0.60(8) & Timing & (5) & & 0.83 & 2.16 \nReferences: (1), (2), (3), (4), (5), (6), (7), (8), (9), (10) \n [fig:parallaxloglog]\nllllllc & & & & & \n Pulsar & $v\\_{\\rm \\perp,LSR}$ & & $v\\_{\\rm \\perp,obs}$ & & *v*⊥ & Ref. \n& (km s− 1) & & (km s− 1) & & (km s− 1) & \nJ0030 + 0451 & 25− 3+ 4 & & 11− 2+ 3 & & 15.4(2) & (1) \nJ0125 − 2327 & 204− 24+ 31 & & 218− 24+ 31 & & & \nJ0437 − 4715 & 99(6) & & 99(6) & & 104.72(14) & (2) \nJ0610 − 2100† & 113− 17+ 23 & & 126− 17+ 23 & & 120− 17+ 25 & (1) \nJ0613 − 0200 & 58− 9+ 14 & & 58− 9+ 14 & & 49(5) & (3) \nJ0614 − 3329 & 22.4− 1.3+ 2.3 & & 6.1− 1.3+ 2.3 & & & \nJ0636 − 3044 & 43− 5+ 6 & & 36− 5+ 6 & & & \nJ1024 − 0719 & 278− 32+ 44 & & 293− 33+ 44 & & 300(20) & (1) \nJ1045 − 4509† & 16− 2+ 4 & & 22− 6+ 15 & & 22− 7+ 16 & (3) \nJ1125 − 6014 & 101− 12+ 17 & & 70− 13+ 18 & & 124− 40+ 90 & (3) \nJ1421 − 4409 & 21− 13+ 27 & & 48− 13+ 27 & & & \nJ1446 − 4701 & 7− 5+ 8 & & 33− 8+ 15 & & 39− 14+ 40 & (3) \nJ1455 − 3330 & 53− 7+ 12 & & 36− 7+ 13 & & & \nJ1545 − 4550† & 56− 5+ 10 & & 27− 7+ 12 & & 26− 7+ 12 & (3) \nJ1600 − 3053 & 57− 12+ 20 & & 71− 13+ 20 & & 62− 7+ 8 & (3) \nJ1614 − 2230 & 98− 16+ 21 & & 108− 17+ 22 & & & \nJ1629 − 6902 & 26− 10+ 16 & & 56− 11+ 17 & & & \nJ1643 − 1224† & 39− 5+ 6 & & 30− 4+ 5 & & 41− 4+ 5 & (1) \nJ1652 − 4838 & 5− 2+ 7 & & 20− 5+ 9 & & & \nJ1658 − 5324 & 31.3− 1.8+ 2.7 & & 10.1− 1.8+ 2.7 & & & \nJ1713 + 0747 & 28− 3+ 4 & & 35− 3+ 4 & & 39.1(11) & (3) \nJ1730 − 2304 & 54− 11+ 26 & & 53− 10+ 25 & & 54− 2+ 3 & (1) \nJ1744 − 1134 & 35.0(13) & & 38.6(14) & & 40.9− 0.8+ 0.9 & (3) \nJ1757 − 5322 & 21− 7+ 12 & & 39− 7+ 12 & & & \nJ1801 − 1417 & 84− 24+ 57 & & 85− 24+ 57 & & & \nJ1832 − 0836 & 134− 33+ 61 & & 149− 33+ 61 & & 170− 40+ 80 & (3) \nJ1843 − 1113† & 14− 5+ 18 & & 24− 8+ 20 & & & \nJ1909 − 3744 & 179(7) & & 191(7) & & 203(3) & (3) \nJ1918 − 0642 & 44− 8+ 11 & & 54− 8+ 11 & & 44− 5+ 6 & (1) \nJ1933 − 6211 & 117− 20+ 37 & & 88− 20+ 37 & & & \nJ1946 − 5403 & 21− 2+ 3 & & 28− 3+ 4 & & & \nJ2010 − 1323 & 83− 22+ 47 & & 99− 25+ 49 & & 58.7− 16+ 27 & (4) \nJ2124 − 3358 & 114− 9+ 10 & & 120− 9+ 10 & & 110− 9+ 11 & (3) \nJ2145 − 0750 & 37− 5+ 8 & & 37− 6+ 8 & & 38.9− 1.9+ 0.5 & (4) \nJ2222 − 0137 & 47− 4+ 5 & & 58− 4+ 5 & & & \nJ2234 + 0944† & 207− 68+ 162 & & 227− 69+ 162 & & & \nJ2241 − 5236 & 79− 6+ 7 & & 103− 6+ 7 & & 97(5) & (3) \nJ2322 + 2057 & 82− 17+ 23 & & 90− 17+ 23 & & & \nJ2322 − 2650 & 32− 4+ 7 & & 32− 5+ 8 & & & \nWe used our proper motions and other author’s (usually parallax) distances (listed in Table [tab:parallaxdistance]) to calculate the velocities of the MSPs that are marked by a “”.\nReferences: (1), (2), (3), (4),\n[tab:velocities]\nKinematics of the MSP sample\n============================\nIn this section we study the kinematics of our MSP sample, using only those with parallax-based distances and velocities. While models of the ISM can be used to estimate distances to pulsars, it is well known that this can lead to large uncertainties. We choose here to base our kinematic analysis on MSPs with high quality distances ( > 3*σ* significance) only.\nFor a detailed analysis of the uncertainties involved in the use of DM-based distances via ISM models, we refer the reader to Appendix [sec:distcomparisondistratio].\nDistances\n---------\nWe used the posterior samples of the parallax measurements to derive the posterior distribution of the distances. The distances of the 35 MSPs with > 3*σ* significance are listed in the upper portion of Table [tab:parallaxdistance]. For the MSPs with < 3*σ* significance in parallax, we calculated 68% confidence lower limit on distances and listed them in the lower portion of Table [tab:parallaxdistance].\n pointed out a bias in the parallax of stars and showed that the bias depends on the significance of the parallax with low-significance parallaxes more likely to be over-estimated due to volume effects. This has the effect of pushing low-significance pulsar parallax objects further from the Sun. confirmed the existence of the Lutz-Kelker (LK) bias in observations, and they provided a method in order to correct the values of parallaxes and distances. For our discussions concerning pulsar distances and velocities, we will not adopt the LK corrections for individual objects to ease comparison with other recent authors, but will discuss what effects it would have on the median velocities of the population. One problem with the LK correction is that the magnitude of the correction depends upon the assumed model for the spatial (i.e. Galactic) distribution of the pulsars, and also the luminosity function of MSPs. For high-precision parallaxes, the effects are negligible.\nTangential Velocities\n---------------------\nFrom the proper motions and distances of the MSPs, we calculated the barycentric transverse velocities, *v*⊥, and then using the method explained in Section [sec:methodvelcorr] we corrected them for the peculiar velocity of the Sun and the rotation curve of the Galaxy, assuming zero radial velocity with respect to our line of sight. In Table [tab:velocities], a summary of 39 velocities for two subsets of our MSPs is given. The first subset is those MSPs with > 3*σ* significance for their parallax and proper motion measurements. The second subset is those MSPs with > 3*σ* significance for their proper motions measurements but < 3*σ* significance for their parallax measurements. For MSPs in the second subset namely: PSRs J0610 − 2100, J1045 − 4509, J1545 − 4550, J1643 − 1224, J1843 − 1113, and J2234 + 0944, we used parallaxes measured by other authors listed in the fifth column of Table [tab:parallaxdistance] to calculate their transverse velocities (these velocities are denoted by a † after the pulsar name). We used the positions, proper motions, and parallaxes of other MSPs in the pulsar catalogue and selected those with significant parallaxes ( > 3*σ*) and calculated their transverse velocities using our method described in Section [sec:methods] in order to increase our MSP velocity sample.\n### Isolated versus Binary Millisecond Pulsars\nWe divided the MSPs into two subsets depending upon whether they are isolated or binary MSPs. The velocity distributions for the isolated and binary MSPs are presented in Figure [fig:velhistisolatedbinary]. We obtained a mean velocity of $67(12) \\, {\\rm km \\, s^{-1}}$ for 16 isolated MSPs, and a mean velocity of $72(8) \\, {\\rm km \\, s^{-1}}$ for the 49 binary MSPs. Between these MSPs, PSR J1024 − 0719 is a special case with an extremely wide binary with the orbital period of 2–20 kyr (presented with a black bin in Figure [fig:velhistisolatedbinary]). Also, PSR J1300 + 1240 is a planetary system (presented with an olive bin in Figure [fig:velhistisolatedbinary]). We did not include them for calculating the mean velocity of the binary MSPs as their evolutionary history is probably quite different from the other systems. The two-sample Kolmogorov-Smirnov test was performed for comparing the distributions of velocity samples. The maximum absolute difference between the cumulative distribution functions of the two samples was 0.17 and the *p*-value was 0.87. This means with our limited sample that there is no statistical evidence that the two samples are drawn from different distributions. Whatever causes isolated MSPs to be created does not have an appreciably different effect on their velocities from that of the binaries.\n [fig:velhistisolatedbinary]\nDispersion of the Velocity Components\n-------------------------------------\nWe obtained cylindrical Galactocentric velocity components of $v\\_{\\rm \\rho}$ (radial), $v\\_{\\rm \\phi}$ (rotational), and $v\\_{\\rm z}$ (perpendicular) and then calculated the components that are at least 70∘ from the line of sight (i.e. for the components that are nearly perpendicular to the line of sight direction). This condition satisfies the 68% confidence level condition for the measures of the velocity components. The angles with respect to the line of sight are:\n \n$$\\cos(a\\_{\\rm \\rho}) = \\frac{d^{2} + \\rho^{2} - \\sqrt{(z - z\\_{\\odot})^{2} + R^{2}\\_{\\odot}}}{2 \\, \\rho \\, d},$$ \n \n$$\\cos (a\\_{\\rm z}) = \\frac{z - z\\_{\\odot}}{d},$$ \nand\n \n$$\\cos(a\\_{\\rm \\phi}) = \\sqrt{1 - \\cos(a\\_{\\rm \\rho})^{2} - \\cos (a\\_{\\rm z})^{2}},$$ \nwhere $a\\_{\\rm \\rho}$, $a\\_{\\rm \\phi}$, and $a\\_{\\rm z}$ are the angles between the line of sight vector and the *ρ*, *ϕ*, and *z* components, respectively. When the condition is satisfied, we have $\\left | \\cos a\\_{\\rm \\rho, \\phi, z} \\right | \\leq 0.34$. We implemented the direction cosines of the line of sight vector for calculating $\\cos(a\\_{\\rm \\phi})$. We have plotted them in Figures [fig:velocityrho], [fig:velocityphi], and [fig:velocityz].\nAfter identifying the desired components for every pulsar, we aimed to find the dispersion (the standard deviation) for each velocity component. We are not able to measure 3D space velocities of MSPs due to the lack of line-of-sight radial velocities for many of them. We found the dispersion of the components as follows:\n \n$$\\sigma\\_{\\rho} = 101(17) \\, {\\rm km \\, s^{-1}} ~~~~ \\sigma\\_{\\phi} = 56(9) \\, {\\rm km \\, s^{-1}} ~~~~ \\sigma\\_{z} = 37(5) \\, {\\rm km \\, s^{-1}}.$$ \nThere are multiple outliers in the data points of the velocity components in the Figures. To remove the outliers to see any underlying trends, we used the median absolute deviation (MAD) as a robust measure of dispersion, assuming a Gaussian velocity distribution, and followed the criteria discussed by. Similar to, for every MSP with the velocity component of $v\\_{\\rm i}$ we obtained $\\left | v\\_{\\rm i} - M \\right | / \\rm MAD$, where *M* is the global median of the i-th velocity component. To be consistent with analysis, we chose $\\left | v\\_{\\rm i} - M \\right | / \\rm MAD > 3$ to find outliers. After excluding the outliers, the dispersions of the components changed to:\n \n$$\\label{eq:dispersions\\_px\\_cleaned}\n \\sigma\\_{\\rho} = 63(11) \\, {\\rm km \\, s^{-1}} ~~~~ \\sigma\\_{\\phi} = 48(8) \\, {\\rm km \\, s^{-1}} ~~~~ \\sigma\\_{z} = 19(3) \\, {\\rm km \\, s^{-1}}.$$ \nBefore interpreting the velocities we should point out that there are multiple selection effects at play in Figure [fig:velocityz]. This figure tends to imply that *σ**z* of nearby MSPs is much lower compared to that of further MSPs. This means it is more likely to find MSPs with low *v**z* near the Sun. MSPs that are near us are either moving very slowly and will always be near us, or they are moving fast and just happen to be passing near us. MSPs rising out of the Galactic plane will slow down eventually (e.g. a pulsar with a vertical velocity of ≈ 80 kms− 1 will rise ≈ 2 kpc) and then return back towards the plane. The amount of time such pulsars spend around 2 kpc is much more than the time they spend nearby the Sun when they are passing quickly through the plane. Therefore, for a short amount of time they are expected to be observed with high luminosity, and so it is easier to measure their parallaxes and proper motions: this leads to a strong selection effect in the sample. Another selection effect is that the more distant pulsars in the sample are likely to be drawn from the luminous end of the pulsar luminosity function and could be seen at greater *z*-distances. Finally, the proper motions of fast moving pulsars are easier to determine than those of slow moving pulsars, which will likely boost the mean tangential velocity of the sample at larger distances. All these selection effects are at play, and simple comparisons of the observed MSP velocities as a function of *z* or distance needs to be performed with caution and best done by simulations as was attempted for the slow pulsars by.\n [fig:velocityrho]\n [fig:velocityphi]\n [fig:velocityz]\nAsymmetric drift in the Galaxy\n------------------------------\nAn asymmetric drift is predicted to be observed in normal pulsars with characteristic ages greater than 107 yr. MSPs have characteristic ages greater than 108 yr and are expected to be old enough to have reached a virialized state; so they are also expected to demonstrate such a drift. predicted an asymmetric drift of $13 \\, {\\rm km \\, s^{-1}}$ for the case of a uniform surface density model, and $25 \\, {\\rm km \\, s^{-1}}$ for the case of an exponential surface density model, both with the MSP velocity dispersion of $60 \\, {\\rm km \\, s^{-1}}$. In Figure [fig:velocityphi], we can clearly see most of the MSPs moving opposite to the direction of Galactic rotation, which is the signature of the asymmetric drift. Utilising the rotational velocities, *V**ϕ*, we obtained a median rotational velocity of $38(16) \\, {\\rm km \\, s^{-1}}$ and the mean rotational velocity of $45(13) \\, {\\rm km \\, s^{-1}}$. By removing the outliers (as defined in [sec:velocitydispersion]), we obtained the median value of $33(14) \\, {\\rm km \\, s^{-1}}$ and the mean value of $38(11) \\, {\\rm km \\, s^{-1}}$. Our values are higher than the $25(12) \\, {\\rm km \\, s^{-1}}$ obtained by. Our median value is closer to $25 \\, {\\rm km \\, s^{-1}}$, the predicted value corresponding to the exponential surface density model of although the MSPs in our sample have higher mean velocities than theirs.\nComparison with previous work\n=============================\nMillisecond Pulsars vs. Normal Pulsars\n--------------------------------------\nSoon after their discovery, a population study by, based on data from just 41 young pulsars, argued that pulsars are probably born in a disk distribution with an initial scale height of ∼ 80 pc (similar to the normal OB stars), and move with average velocities of $\\sim 100 \\, {\\rm km \\, s^{-1}}$. In the first large scale study of pulsar proper motions, studied a sample of 26 and showed that the pulsars with Galactic scale height greater than a few tens of parsecs tend to be moving away from the Galactic plane, probably due to a velocity kick at birth. They found pulsars had an rms velocity of about 210 km s− 1. Later, studied the youngest pulsars with proper motions, and found they had a mean space velocity at birth of 450(90) km s− 1.\nBy modelling the kinematics of the spatial distribution of MSPs, estimated that such pulsars on average receive a *z*-velocity kick at birth of just $52^{+17}\\_{-11} \\, {\\rm km \\, s^{-1}}$, much less that than of the normal pulsars. They suggested that the rms speed of young pulsars is ∼ 5 time larger than that of MSPs, and a significant contribution to the observed *z*-velocity (the velocity component along Galactocentric *z*) of MSPs originates from the diffusive processes that affect all old stars in the disk.\nWe plotted the histograms of the transverse velocities of the normal pulsars from the pulsar catalogue and the MSPs in Figure [fig:velhistmspnormal], to highlight how different the two distributions are. In this Figure, the top, middle, and bottom panels show the distributions of MSPs from this work, MSPs from the combination of this work and the pulsar catalogue, and normal pulsars from the pulsar catalogue, respectively. For the 87 normal pulsars in the pulsar catalogue with significant parallaxes and proper motions ( > 3*σ*), we found a mean velocity of $246(21) \\, {\\rm km \\, s^{-1}}$. Comparing the mean velocities showed that the normal pulsars seem to be faster than MSPs by a factor of ∼ 3.2. The characteristic age of MSPs is generally higher than that of normal pulsars, and accordingly, the old pulsars seem to be slower than young pulsars. If we restrict our attention to the pulsars with characteristic ages less than 3 Myr, we find a mean transverse velocity of 283(44) km s− 1, ∼ 3.7 times faster than the MSPs.\nThe velocities of the MSPs and normal pulsars as a function of Galactocentric *z* are shown in Figure [fig:velglat]. This figure demonstrates that pulsars with velocities < 100 km s− 1 are more concentrated in the Galactic plane and are less scattered compared to the pulsars with velocities > 100 km s− 1. In addition, on average, normal pulsars seem to have a wider *z*-distribution around the Galactic plane (i.e. larger *z* scale height) compared to the MSPs consistent with their higher velocities. This figure provides additional evidence for the results from the numerical studies of ; that suggested MSPs have lower velocities compared to the young, long-period pulsars. The higher velocities of normal pulsars reinforce the idea that the high recoil velocity that pulsars receive during their birth in supernova explosions, and are in agreement with some of the core-collapse supernovae simulations performed by. On the other hand, some of the low velocities of normal pulsars might be due to the weak supernova kicks for a sub-population of the pulsars. Some caution in interpreting Figure [fig:velglat] is required, as many selection effects are at play.\n [fig:velhistmspnormal]\n [fig:velglat]\nVelocity dispersions for millisecond pulsars\n--------------------------------------------\nThe dispersions of the 3 velocity components that obtained from the analysis of NANOGrav nine-year MSP timing data set, after excluding outliers, are $\\sigma\\_{\\rm \\rho} = 46 \\, {\\rm km \\, s^{-1}}$, $\\sigma\\_{\\rm \\phi} = 40 \\, {\\rm km \\, s^{-1}}$, and $\\sigma\\_{\\rm z} = 24 \\, {\\rm km \\, s^{-1}}$. We obtained $\\sigma\\_{\\rm \\rho} = 63(11) \\, {\\rm km \\, s^{-1}}$, $\\sigma\\_{\\rm \\phi} = 48(8) \\, {\\rm km \\, s^{-1}}$, and $\\sigma\\_{\\rm z} = 19(3) \\, {\\rm km \\, s^{-1}}$. Our $\\sigma\\_{\\rm \\phi}$ and $\\sigma\\_{\\rm z}$ are thus comparable with their values. Also, they calculated the dispersion of velocity components, using the fit equations to the local stellar data provided by, to be $\\sigma\\_{\\rm \\rho} = 34 \\, {\\rm km \\, s^{-1}}$, $\\sigma\\_{\\rm \\phi} = 22 \\, {\\rm km \\, s^{-1}}$, and $\\sigma\\_{\\rm z} = 18 \\, {\\rm km \\, s^{-1}}$ for the characteristic age of *τ* ∼ 5 Gyr, and $\\sigma\\_{\\rm \\rho} = 42 \\, {\\rm km \\, s^{-1}}$, $\\sigma\\_{\\rm \\phi} = 28 \\, {\\rm km \\, s^{-1}}$, and $\\sigma\\_{\\rm z} = 24 \\, {\\rm km \\, s^{-1}}$ for *τ* ∼ 10 Gyr. This is showing that our $\\sigma\\_{\\rm z}$ is consistent with the model fit corresponding to the characteristic age of *τ* ∼ 5 Gyr, but our $\\sigma\\_{\\rm \\rho}$ and $\\sigma\\_{\\rm \\phi}$ are closer to the model fit corresponding to the characteristic age of *τ* ∼ 10 Gyr. Both comparisons show that the MSPs are consistent with having been drawn from the old disk stellar population, and that they are subject to kick velocities at birth, as discussed in the next section.\nThe fact that the *z*-components are the lowest is perhaps easiest to understand. MSPs with low *z*-velocities spend more time near that galactic disk (and hence the Sun) and are preferentially detected by MSP surveys, plus as mentioned before, the *z*-velocity is on average lower than the birth *z*-velocity as it exhibits simple harmonic motion in the Galactic potential.\n [fig:velocityvtpb]\nVelocity distributions for millisecond pulsars\n----------------------------------------------\n studied the kinematics of 233 pulsars and found the mean two-dimensional speeds of $211(18) \\, {\\rm km \\, s^{-1}}$ for all pulsars in their sample, $307(47) \\, {\\rm km \\, s^{-1}}$ for pulsars with characteristic ages less than 3 Myr, $87(13) \\, {\\rm km \\, s^{-1}}$ for recycled pulsars, and $209(19) \\, {\\rm km \\, s^{-1}}$ for the normal pulsars with characteristic ages greater than 3 Myr. Our mean velocity of $78(8) \\, {\\rm km \\, s^{-1}}$ for MSPs is consistent with their mean two-dimensional speed of $87(13) \\, {\\rm km \\, s^{-1}}$ for recycled pulsars. Also, obtained the mean 2D speeds of $77(16) \\, {\\rm km \\, s^{-1}}$ for seven isolated MSPs and $89(15) \\, {\\rm km \\, s^{-1}}$ for 28 binary MSPs. These are comparable with the mean velocities of $67(12) \\, {\\rm km \\, s^{-1}}$ and $77(9) \\, {\\rm km \\, s^{-1}}$ for the 16 solitary and 49 binary MSPs in our sample, respectively, excluding PSR J1024 − 0719., by studying scintillation parameters for a sample of 49 pulsars, suggested that binary MSPs have higher velocities compared with the isolated MSPs. calculated velocities for a sample of 23 MSPs and presented that, on average, the binary MSPs are one-third faster than isolated MSPs. However, reported that there is no significant difference between the mean velocities of both. Our results are also showing that the mean velocities of both are not significantly different.\n [fig:velocityvtminmass]\nUsing simulations, predicted a mild inverse correlation between the recoil velocity (the post-explosion velocity that a pulsar receives at birth) and the orbital period of binary MSPs, depending upon the birth component masses of the binary, the separation between the two objects, and the evolution of the system during the common-envelope and mass transfer stages. did not find any significant correlation with orbital period using a sample of 23 binary MSPs. In addition, looked for this correlation using the binary MSPs in their pulsar sample, but they did not find any either. We also investigated the relationship between transverse velocity and orbital period of binary MSPs in Figure [fig:velocityvtpb], and measured the correlation coefficient to be − 0.24 (excluding the very long orbital period pulsar PSRs J1024 − 0719, the planet pulsar J1300 + 1240, and the eccentric MSP J2234 + 0611 - which probably has a complex evolutionary history) showing a weak anti-correlation. We need to keep in mind that many selection effects are influencing our observed MSP distribution. For instance, binary MSPs with short orbital periods and heavy companions might be less likely to appear in pulsar surveys unless acceleration searches have been undertaken. Nevertheless, predicted an approximate velocity range of 25–270 km s− 1 for binary MSPs and that those above 300 km s− 1 should be very rare. This is borne out by our observations of our sample’s transverse velocities. In the standard model MSPs obtain their velocities as the vector sum of three distinct mechanisms. First, MSP progenitors, the binaries containing OB stars, are formed in large stellar nurseries where they obtain velocities of 10s of km s− 1 due to stellar interactions. Second, when the neutron star is produced there is a Blaauw momentum kick to the binary as a result of the mass loss of the supernova, and possibly an asymmetric kick imparted to the neutron star. To get very high velocities requires very compact binaries, large mass loss and asymmetric retrograde kicks that do not disrupt the orbit. These must be rare in MSP formation. As we only measure the 2D velocities the presence of some very low velocities is not unexpected and could be a projection effect, but MSP velocities can help constrain models for neutron star formation in population synthesis codes such as COMPAS and STARTRACK.\nAny correlation between the minimum masses of companions of binary MSPs and their transverse velocities of binary MSPs was explored. We used the pulsar catalogue to find the minimum mass of the companions and plotted them as a function of transverse velocity in Figure [fig:velocityvtminmass]. The minimum masses were calculated assuming the orbital inclination angle to be 90∘ and the mass of the pulsar to be 1.35 M⊙. We measured the correlation coefficient between the minimum mass and the transverse velocity (excluding PSRs J1024 − 0719, J1300 + 1240, and J2234 + 0611) to be only − 0.10, showing at best a very weak anti-correlation.\nConclusions\n===========\nWe have undertaken a study on the parallaxes and proper motions of 77 MSPs, observed with the MeerKAT radio telescope, over a time span of ∼ 3 years. Out of the 77 MSPs, 35 and 69 had significant parallaxes and proper motions (above 3*σ*), respectively. We calculated the transverse velocities for MSPs that had both significant parallaxes and proper motions. For the MSPs for which we did not have significant parallaxes, we used parallaxes taken from the literature. Pulsars with significant parallaxes and proper motions from the pulsar catalogue psrcat were also added to our MSP sample to produce the largest MSP velocity sample to date. We found that the transverse velocity of MSPs has a mean of 78(8) km s− 1 and an rms scatter of 101 km s− 1.\nThe Lutz-Kelker effect on the derived transverse velocities was investigated for a sample of 35 pulsars with parallaxes. The median transverse velocities increased by ∼ 2% and ∼ 13% for parallaxes with > 10*σ* and < 10*σ* significance, respectively. The median of the whole population increased by ∼ 7%.\nOut of our 69 MSPs with significant ( > 3*σ*) proper motions, the long orbital period pulsar J1024 − 0719 had the highest transverse velocity of 278− 32+ 44 km s− 1, and the 12.4 d orbital period binary pulsar PSR J1652 − 4838 had the lowest transverse velocity of just 5− 2+ 7 km s− 1.\nAlthough the mean velocity of 72(8) km s− 1 for 49 binary MSPs is slightly faster than that of 67(12) km s− 1 for 16 isolated ones, there was no evidence that their distributions are statistically significantly different. In comparison to the normal pulsars in the pulsar catalogue with the mean transverse velocity of 246(21) km s− 1, MSPs had a much lower mean transverse velocity of 78(8) km s− 1.\nThe velocity dispersions of the pulsars in the (cylindrical) Galactocentric velocity components (radial, rotational, and perpendicular) were measured to be $\\sigma\\_{\\rm \\rho} = 63(11) \\, {\\rm km \\, s^{-1}}$, $\\sigma\\_{\\rm \\phi} = 48(8) \\, {\\rm km \\, s^{-1}}$, and $\\sigma\\_{\\rm z} = 19(3) \\, {\\rm km \\, s^{-1}}$ after removal of a small number of outliers. The lower *z*-velocity component is likely a consequence of the fact that high velocity MSPs will spend less time near the Sun suppressing their representation in pulsar surveys.\nThe expected asymmetric drift was clearly seen in the rotational component of the velocities, and the mean value was found to be $38(11) \\, \\rm km~s^{-1}$ after removing outliers. This is consistent with the number predicted by, who model the formation of MSPs in our galaxy. They predicted an asymmetric drift of approximately 25 km s− 1 for their more realistic (exponential surface density) Milky Way model.\nThe substantially increased sample of MSPs with good parallaxes and proper motions presented in this study bolsters the case that MSPs arise out of the old disk of the Milky Way and, are subject to kick velocities at birth significantly smaller than those seen for young pulsars. We explored the distribution of MSP velocities versus orbital period and found a weak anti-correlation, in agreement with the predictions of, based on simulations of the recoil velocities of MSPs in various binary stellar evolutionary scenarios.\nAcknowledgements\n================\nWe thank Dr. Simon Stevenson and Dr. Hao Ding for their suggestions and comments that improved the manuscript. The MeerKAT telescope is operated by the South African Radio Astronomy Observatory, which is a facility of the National Research Foundation, an agency of the Department of Science and Innovation. MS, MB, CF, MTM, DJR, and RMS acknowledge support through the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav), through project number CE17010004. RMS acknowledges support through Australian Research Council Future Fellowship FT190100155. MK acknowledges significant support from the Max-Planck Society (MPG) and the MPIfR contribution to the PTUSE hardware. This work used the OzSTAR national facility at Swinburne University of Technology. OzSTAR is funded by Swinburne University of Technology and the National Collaborative Research Infrastructure Strategy (NCRIS) that also supports the pulsars.org.au data portal that was used extensively for this work. This research has made use of NASA’s Astrophysics Data System and software such as: psrchive, tempo2, temponest, psrcat, pulseportraiture, pygedm, numpy, scipy, matplotlib, ipython, astropy.\nData Availability\n=================\nData is available from the Swinburne pulsar portal: https://pulsars.org.au\nOn the use of Galactic electron density models for MSP distances and velocities\n===============================================================================\nThe integrated electron column density ($n\\_{\\rm e}$) along the line of sight from the Earth to a pulsar at distance *d* relates to the DM of the pulsar as\n \nDM = ∫0*d**n*e(*l*)*d**l*. \nThe distance to a pulsar can be derived from an observational DM by having a model which describes the distribution of electrons throughout the Galaxy. An inaccurate electron-density model might result in overestimating or underestimating distances. By measuring new pulsar distances via pulsar parallaxes, as in this work, we are able to provide an improved basis for refining the ISM models for different lines of sight. There are two widely used DM models, namely NE2001 and YMW16. The comparison of the two has been carried out by many authors. They found that the different models have performance that is dependent on where one is looking on the sky, and thus one model works better than others depending on which part of the Galactic ISM is being probed by any given pulsar.\nDistances\n---------\nUsing the two Galactic electron-density models: NE2001 and YMW16, implemented in PyGEDM, we have calculated DM-based distances of the 35 MSPs with > 3*σ* significance in their parallaxes. In addition, for MSPs with < 3*σ* significance in their parallaxes, we added those MSPs whose DM distances were less than the lower distance limits obtained from the probability distributions (the third column in Table [tab:parallaxdistance]). We also added other MSPs from the pulsar catalogue with > 3*σ* significance in parallax into our MSP sample and calculated their DM-based distances. We excluded all MSPs in globular clusters as they have different origins and their velocities are dominated by their host clusters.\nThe top and bottom panels of Figure [fig:dmpxdistanceratio] show the comparison of parallax distances with DM distances obtained from the NE2001 and YMW16 models, respectively. In this Figure, points in red show a DM distance overestimation (i.e. MSPs are closer than what the models predict) and points in blue show a DM distance underestimation (i.e. MSPs are further than what the models predict). The ratios of $d\\_{\\rm DM} / d\\_{\\rm \\varpi}$ and $d\\_{\\rm \\varpi} / d\\_{\\rm DM}$ are represented by the area of the circles in red and blue for the cases of overestimation and underestimation, respectively. The black circles in the legend of Figure [fig:dmpxdistanceratio] provide the standard circle sizes for the ratios of 10, 5, 2, and 1. We found the maximum underestimation based on the YMW16 model for PSR J1737 − 0811 with the distance ratio of $d\\_{\\varpi} / d\\_{\\rm DM} \\approx 5.9$ (considering the lower limit of its parallax distance), and the maximum overestimation was for PSR J1652 − 4838 with the distance ratio of $d\\_{\\rm DM} / d\\_{\\varpi} \\approx 10.5$.\nThe 16%, 50%, and 84% of the underestimation and overestimation of the YMW16 distances ratios were obtained to be 1.4− 0.3+ 1.1 and 1.2− 0.1+ 1.1, respectively. In addition, the underestimation and overestimation of the NE2001 distances were obtained to be 1.6− 0.5+ 1.1 and 1.3− 0.2+ 1.8, respectively. Also, the distance ratios of $d\\_{\\varpi} / d\\_{\\rm DM}$ were obtained for YMW16 model to be 1.1− 0.3+ 0.7 and for NE2001 to be 1.2− 0.4+ 1.3. The comparison of distance ratios highlighted that there are obviously patches in Figure [fig:dmpxdistanceratio] that the DM-based distances are systematically overestimated or underestimated by the two models.\n [fig:dmpxdistanceratio]\nVelocities\n----------\nThe top panel of Figure [fig:velhistpxdmdist] presents the histogram of transverse velocities for the total sample that possesses parallax measurements. The mean transverse velocity is $78(8) \\, {\\rm km \\, s^{-1}}$. The fastest moving MSP for these parallax-based velocities is the planet pulsar PSR J1300 + 1240, which has a velocity of $330^{+20}\\_{-18} \\, {\\rm km \\, s^{-1}}$, and the slowest-moving, PSR J1652 − 4838, has a velocity of just $5^{+7}\\_{-2} \\, {\\rm km \\, s^{-1}}$. Using the code [5](#fn5) provided by, we explored whether the LK corrections might affect our velocities by breaking our sample into two groups, those with > 10*σ* parallaxes and those with lower significance. The median velocities of the first group before and after the LK correction were 67 and 68 km s− 1 respectively and the second group 43 and 49 km s− 1. Clearly, the LK correction raises the velocities of the second group and brings their median closer to that of the (more accurate) group, as might be predicted by the LK selection effect. Overall, the median velocities of the entire population before and after the LK correction were 46 and 50 km s− 1, an increase of ∼ 7 %.\nTo illustrate the effect of using DM-based distances rather than our preferred parallax-based distances, we have derived transverse velocities using DM-based distances obtained from NE2001 and YMW16 models. The middle and the bottom panels of Figure [fig:velhistpxdmdist] present the histograms of transverse velocities for the MSPs that their velocities are calculated from YMW16 and NE2001 models, respectively.\nUsing the YMW16 model, the mean transverse velocity is $71(8) \\, {\\rm km \\, s^{-1}}$. The fastest moving MSP, PSR J1300 + 1240, would have a velocity of $407(100) \\, {\\rm km \\, s^{-1}}$ in the DM model, and the slowest-moving, PSR J1446 − 4701, has a velocity of $6^{+6}\\_{-4} \\, {\\rm km \\, s^{-1}}$, assuming 25% distance errors. Using the NE2001 model, the mean transverse velocity is $65(6) \\, {\\rm km \\, s^{-1}}$. The fastest moving MSP in this model, PSR J0610 − 2100, has a velocity of $292(80) \\, {\\rm km \\, s^{-1}}$, and the slowest-moving, PSR J1446 − 4701, has a velocity of $6^{+6}\\_{-4} \\, {\\rm km \\, s^{-1}}$, again assuming 25% distance errors.\nIt is curious to note that distance errors in YMW16 are responsible for a marked increase in the number of very low velocity MSPs (see Figure [fig:velhistpxdmdist], lower two panels). In reality there are much fewer low velocity MSPs in the sample based on parallax distances only (Figure [fig:velhistpxdmdist], upper panel), and this is merely an artifact of moving the MSPs too close to the Sun, reducing their apparent speeds.\nWe have performed a two-sample Kolmogorov-Smirnov test for comparing the continuous distributions of the parallax-based velocities and either of the two DM-based velocities to see if their macroscopic properties would differ. For the YMW16 model, the maximum absolute differences between the cumulative distribution functions were 0.11 and the p-value was 0.86. For the NE2001 model, the maximum absolute differences between the cumulative distribution functions were 0.17 and the p-values was 0.32. There is thus no statistical evidence that the parallax-based distances and DM-based distances differ that markedly in *v*⊥.\nFor the 35 pulsars with significant parallaxes ( > 3*σ*), ∼ 2/3 of their YMW16 DM-based distances are more than 1*σ* different from the parallax distances. This affects any velocities calculated from DM-based distances and makes the distribution of the velocities broader. It is beyond the scope of this current work to examine each DM model and search for reasons why they predict the wrong distances but using either the NE2001 or YMW16 models for calculating distances for finding velocities of individual objects is subject to large uncertainties and for the rest of the paper we only consider the transverse velocities of pulsars that have reliable parallax measurements. It is very difficult to calibrate distance models at high galactic latitudes in the absence of pulsars with parallaxes.\n[lastpage]\n---\n1. E-mail:msh.ph.ir@gmail.com[↩](#fnref1)\n2. <https://github.com/pennucci/PulsePortraiture/tree/py3>[↩](#fnref2)\n3. <https://webtai.bipm.org/ftp/pub/tai/ttbipm/TTBIPM.2020>[↩](#fnref3)\n4. <https://docs.astropy.org/en/stable/api/astropy.coordinates.Galactocentric.html>[↩](#fnref4)\n5. <https://github.com/jantoniadis/GAIA_pulsars_eDR3/blob/main/GAIA_pulsars_eDR3.ipynb>[↩](#fnref5)\n"},"data_id":{"kind":"string","value":"arxiv_0183485"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":458,"numItemsPerPage":100,"numTotalItems":46503,"offset":45800,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NjE5MTMwNywianRpIjoiNmZhODgyNDMtOGMxOC00ZDEwLThhMjktZTVjNzg1YWZhNGM5Iiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3ODYxOTQ5MDcsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.wAClL4i2pIbau-jrRmXCCv5sDQnJWptuu4g6pd1W4Jy3YN5jH0YoYFwPMykSiPzGzyAQKd6hvdsj1MtHSmKTDg","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
content stringlengths 34.5k 188k | data_id stringlengths 13 13 |
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Designing and Embedding Reliable Virtual Infrastructures
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### Wednesday 31st March, 2010
In a virtualized infrastructure where physical resources are shared, a single physical server failure will terminate several virtual servers and crippling the virtual infrastr... | arxiv_0183131 |
{1}{T} \int\_0^T E^{\mathbb{Q}}(e^{X\_s} | \mathcal{F}\_t)\, ds \nonumber\\
&= \frac{1}{T} \left(\int\_t^T E^{\mathbb{Q}}(e^{X\_s} | \mathcal{F}\_t)\, ds + \int\_0^t E^{\mathbb{Q}}(e^{X\_s} | \mathcal{F}\_t)\, ds \right) \nonumber \\
&= \frac{1}{T} \left(\int\_t^T E^{\mathbb{Q}}(e^{X\_s} | \mathcal{F}\_t)\, ds + \int\_... | arxiv_0183132 |
A Survey on Graph-Based Deep Learningfor Computational Histopathology
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With the remarkable success of representation learning for prediction problems, we have witnessed a rapid expansion of the use of machine learning and deep learning for the analysi... | arxiv_0183137 |
0.85
**Universality for Random Matrix Flows with Time-dependent Density**
0.5
László Erdos
*lerdos@ist.ac.at*
0.45
Kevin Schnelli
*kevin.schnelli@ist.ac.at*
0.9 We show that the Dyson Brownian Motion exhibits local universality after a very short time assuming that local rigidity and level repulsion hold. Th... | arxiv_0183141 |
. ), 1714 from STELLA (see also Strassmeier et al. ), and 802 historical values dating back ≈ 130 years (individually referenced in Stefanik et al. ). An orbit combined with the mid times of the eclipses from photometry was given and favored by Stefanik et al. (). This resulted in an orbital period longer by 12d (0.12... | arxiv_0183146 |
of the body. For retarded motion, he believed that the measure of force was the amount of momentum, *m**v*, lost in each interval.
Today, we use two formulae: *m**i**v**i* for momentum and $\frac{1}{2} mv^2$ for kinetic energy.[33](#fn33) Did Châtelet understand the difference between mass (*m*) and weight (*m**g*), o... | arxiv_0183154 |
– (1/2,0.02); (-0.02, 1/4) node[left] $\frac{1}{4}$ – (0.02, 1/4); (-0.02, 1/2) node[left] $\frac{1}{2}$ – (0.02, 1/2); (1/2, 1/2) circle[radius=0.05ex]; (1/2, 0) circle[radius=0.05ex]; (1/6, 1/2) circle[radius=0.05ex]; at (1/6, 1/2) $(\frac{1}{6}, \frac{1}{2})$; at (1/2 + 0.1,1/2) $( \frac{1}{2}, \frac{1}{2}) $ ; (1/... | arxiv_0183155 |
only *after* the peak ram pressure, as gas parcels are first pushed away from the galactic plane but then require more work and time to escape.
Given the diverse uncertainties involved in the ram pressure estimates, discussed in Section [s:uncertain], it is quite remarkable how well the galaxies of different HI morpho... | arxiv_0183161 |
Interpretation of the new Ω*c*0 states via their mass and width
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The mass and pole residue of the ground and first radially excited Ω*c*0 states with spin-parities *J**P* = 1/2+, 3/2+, as well as P-wave Ω*c*0 with *J**P* = 1/2−, 3/2− are calculated by mea... | arxiv_0183164 |
Observation of The Far-Ultraviolet Continuum Background with SPEAR/FIMS
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We present the general properties of the far-ultraviolet (FUV; 1370–1710 Å) continuum background over most of the sky, obtained with the *Spectroscopy of Plasma Evolution from ... | arxiv_0183167 |
Model atmospheres for X-ray bursting neutron stars
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The hydrogen and helium accreted by X-ray bursting neutron stars is periodically consumed in runaway thermonuclear reactions that cause the entire surface to glow brightly in X-rays for a few seconds. With models of th... | arxiv_0183169 |
features associated with individual metabolites to infer the chemical composition of a sample or tissue voxel. There are many ways to tackle quantification, such as basis set fitting, singular value decomposition, principal component analysis or peak integration, focusing on either time or frequency domain analysis. I... | arxiv_0183171 |
The *R*-Process Alliance: Chemo-Dynamically Tagged Groups of Halo *r*-Process-Enhanced Stars Reveal a Shared Chemical-Evolution History
=======================================================================================================================================
We derive dynamical parameters for a large sampl... | arxiv_0183177 |
A Search for L/T Transition Dwarfs With and *WISE*:Discovery of Seven Nearby Objects Including Two Candidate Spectroscopic Variables
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We present initial results from a wide-field (30,000... | arxiv_0183178 |
On the complexity of computing Gröbner bases for weighted homogeneous systems
=============================================================================
,
,
Solving polynomial systems arising from applications is frequently made easier by the structure of the systems. Weighted homogeneity (or quasi-homogeneity) is o... | arxiv_0183181 |
`UPR-1309-T`
Root Bundles and
Towards Exact Matter Spectra of F-theory MSSMs
*1Department of Mathematics, University of Pennsylvania,
Philadelphia, PA 19104-6396, USA*
Motivated by the appearance of fractional powers of line bundles in studies of vector-like spectra in 4d F-theory compactifications, we analyze th... | arxiv_0183184 |
Monotone subsequences in locally uniform random permutations
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### March, 2023
A locally uniform random permutation is generated by sampling *n* points independently from some absolutely continuous distribution *ρ* on the plane and interpreting them as a permut... | arxiv_0183185 |
games admit a saddle-point equilibrium, at which the jammer’s optimal policy is to randomize in a region of the plant’s state space, thus requiring the controller to undertake a nontrivial response which is different from what one would expect in a standard stochastic control problem over a packet dropping link. The p... | arxiv_0183186 |
the $2^{{\textnormal}{nd}}$ continuum (*N**γ*) relative to the number of excited states obtained in simulation (*N**e**x*), a ratio that we conveniently refer to as the ‘scintillation probability’, P*s**c**i**n*. They are represented by continuous and dashed lines in Figs., respectively. The scintillation probability ... | arxiv_0183187 |
Adaptive Pressure Control for Use in Variable-Thrust Rocket Development
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The precise control of gas generator pressure is important to obtain variable thrust in ducted rockets. A delay resistant closed loop reference model adaptive control is propos... | arxiv_0183188 |
where *m**k* is the multiplicity of a given irreducible representation Rep*k*. The block corresponding to this irreducible representation then has dimensions *m**k* × *m**k*. It is noteworthy that the final dimension of the positivity matrix is ∑*k**m**k*, rather than dim*V* = ∑*k**d**k**m**k*.[13](#fn13)
Implementi... | arxiv_0183189 |
that $G(\ba,\hat{\y})$ can be reformulated by $\bar{G}(\ba,\bM\_{\y})=\1'\ba - \frac{1}{2}\ba'\Big(\K\odot \bM\_{\y}
\Big) \ba$, where *Ḡ* is concave in $\ba$ and linear in $\bM\_{\y}$. Hence, according to Theorem 1, WellSVMis at least as tight as the SDP relaxation in.
### Algorithm
The cutting plane algorithm can s... | arxiv_0183191 |
Regularity of Local times associated to Volterra-Lévy processes and path-wise regularization of stochastic differential equations
=================================================================================================================================
We investigate the space-time regularity of the local time a... | arxiv_0183198 |
SAT-Based PAC Learning of Description Logic Concepts
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We propose *bounded fitting* as a scheme for learning description logic concepts in the presence of ontologies. A main advantage is that the resulting learning algorithms come with theoretical guarantees regarding ... | arxiv_0183199 |
Casimir preserving stochastic Lie-Poisson integrators
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#### Keywords
Stochastic Lie-Poisson integration ⋅ Hamiltonian mechanics ⋅ stochastic differential equations ⋅ geometric integration ⋅ structure preservation ⋅ Lie group ⋅ Lie algebra ⋅ coadjoint or... | arxiv_0183213 |
Finite tidal effects in GW170817: Observational evidence or model assumptions?
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After the detection of gravitational waves caused by the coalescence of compact objects in the mass range of neutron stars, GW170817, several studies have searche... | arxiv_0183214 |
presented in Table 1 differs from that published by L15. As part of the Gemini North program GN-2013A-Q-21, W1738 was observed for several hours with the Gemini near infrared imager (NIRI; Hodapp et al. 2003) in *Y* and *J* in a search for variability. The variability result will be published elsewhere; here we presen... | arxiv_0183219 |
Spectral statistics in constrained many-body quantum chaotic systems
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10000
We study the spectral statistics of spatially-extended many-body quantum systems with on-site Abelian symmetries or local constraints, focusing primarily on those with conserve... | arxiv_0183227 |
Catalogs of Hot White Dwarfs in the Milky Way from GALEX’s Ultraviolet Sky Surveys. Constraining Stellar Evolution.
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### Accepted —- Received
[firstpage]
We present comprehensive catalogs of hot star candi... | arxiv_0183229 |
**The iterative proportional fitting algorithm and the NM-method: solutions for two different sets of problems**
================================================================================================================
[1](#fn1)
Introduction
============
The iterative proportional fitting (IPF) algorithm, o... | arxiv_0183230 |
Scheduling and Power Allocation to Optimize Service and Queue-Waiting Times in Cognitive Radio Uplinks
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Cognitive Radio (CR) is an emerging wireless communication paradigm to improve the spectrum utilization. Cognitive... | arxiv_0183232 |
.94 & -83.97 & -2090.02 & -647.79 & -85.90 \end{bmatrix}\;, \\
K\_{H} = \begin{bmatrix} -159.82 & -49.22 & -48.09 & -1714.38 & -547.06 & -94.92 \end{bmatrix}\;.\end{aligned}$$
Solving the LMIs for this larger system takes a few hours, but appears to be numerically stable.
It turns out that an exponential decay rate o... | arxiv_0183237 |
inkowski spacetime through variation of the action $\mathcal{S}=\int{\left(-\sum{m\_{i}}\sqrt{\eta\_{ab}\dot{x}\_{(i)}^{a}\dot{x}\_{(i)}^{b}}-\frac{1}{4}F\_{cd}F^{cd}-\sum{e\_{i}}A\_{a}\dot{x}\_{(i)}^{a}\right)\,d\tau}$ with respect to the particles’ coordinates *x**a*(*τ*) (*τ* is the particle’s proper–time, its worl... | arxiv_0183241 |
Sushma Kumari
A Dissertation Submitted to
Department of Mathematics, Graduate School of Science
Kyoto University
In Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy in
Mathematics
Kyoto, July 2018
Abstract
========
This thesis consists of two independent parts: random matr... | arxiv_0183242 |
& 3.04 & 8.92 & 22.63 & 10.83 & 13.45 & 5.09 & 29.9 & 1.6 & 2007
ome Ori & 05 39 11.14 & +04 07 17.3 & 37490 & 4.6 & B3Ve & 2.36 & 0.29 & 0.84 & 0.28 & 0.00 & 0.16 & 20.5 & 0.9 &
125 Tau & 05 39 44.20 & +25 53 49.5 & 37438 & 5.2 & B3IV & 7.63 & 0.33 & 11.04 & 0.36 & -23.59 & 0.20 & 16.5 & 1.9 &
133 Tau & 05 47 4... | arxiv_0183243 |
106. <http://stacks.iop.org/0954-3899/45/i=1/a=015106>
N. J. Upadhyay, A. Bhagwat, [Taylor approximation to treat nonlocality in the scattering process](https://link.aps.org/doi/10.1103/PhysRevC.98.024605), Phys. Rev. C 98 (2018) 024605.. <https://link.aps.org/doi/10.1103/PhysRevC.98.024605>
H. F. Arellano, G. Blanchon... | arxiv_0183247 |
On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete *k*-Center for Small *k*
======================================================================================================
We study the time complexity of the discrete *k*-center problem and related (exact) geometric set cover problems w... | arxiv_0183251 |
]{cap.eps}} \endxy}}\\
\\
\mathrm{Double Dot Removal} & {\vcenter{\xy (0,0)\*{\includegraphics[scale=.3]{doubleDot.eps}} \endxy}} & \quad
\end{array} &
\begin{array}{ccc}
\mathrm{Move} & \mathrm{Start} & \mathrm{Finish}\\
\\
\mathrm{Needle} & {\vcenter{\xy (0,0)\*{\includegraphics[scale=.3]{needle2.eps}} \endxy}} & {\v... | arxiv_0183256 |
≃ 0.1 eV. This is the only setup of the parameters for which, within the T2K range, ${\rm BR}(\mu\to e\gamma)$ can be suppressed with respect to the hierarchical case up to 2 orders of magnitude.[↩](#fnref10)
11. Note that the LHCb collaboration has recently carried out a combined analysis of *B**s* → *ψ**ϕ* and *B**s... | arxiv_0183257 |
A stochastic target approach to Ricci flow on surfaces
======================================================
./style/arxiv-general.cfg
We develop a stochastic target representation for Ricci flow and normalized Ricci flow on smooth, compact surfaces, analogous to Soner and Touzi’s representation of mean curvature flow... | arxiv_0183260 |
})}{1\!+\!{\tilde{z}\_\textit{I}}(\epsilon; \Delta\mu\_\text{coex})}\right]-\beta\epsilon. \tag{D2}$$
In homogeneous systems, this prediction reduces to *β**ε̃* = *β**ε*, meaning that the effective adatom interaction strength does not change no matter how far the system is driven out of equilibrium. However, in the c... | arxiv_0183261 |
i.e. a contradiction. We have no solutions in this case.
The theorem is proved.
The proof of Theorem [lemmazerocurrent]
=======================================
**Case 1.** In the first case ([psi]), we calculate *J**μ*, *μ* = 0, 1, 2, 3:
$$\begin{aligned}
&&J^0=i\Psi^\T \Psi-\frac{1}{2}\tr (i\Psi^\T \Psi)=\left(... | arxiv_0183262 |
and in 2014 (2 in May and June 2014) to put constraints on the mass of KOI-12.01. The 2014 observations are of better quality than the 2011 ones, which may be affected by a systematic shift reminiscent of previous SOPHIE observations in High-Efficiency mode (). Contrary to the 2014 dataset, no radial velocity standard... | arxiv_0183268 |
Constructing quantum many-body scar Hamiltonians from Floquet automata
======================================================================
Ø
=
=
==
We provide a systematic approach for constructing approximate quantum many-body scars (QMBS) starting from two-layer Floquet automaton circuits that exhibit trivial many... | arxiv_0183269 |
Relating relative entropy, optimal transportand Fisher information: a quantum HWI inequality.
=============================================================================================
Quantum Markov semigroups characterize the time evolution of an important class of open quantum systems. Studying convergence proper... | arxiv_0183270 |
8% are DB white dwarfs, 27% are DC white dwarfs, 11% show metal-pollution, 6% are DAH white dwarfs, and 4% are DQ white dwarfs. Our sample is more similar to a magnitude-limited sample; in, 65% are DA white dwarfs, 4.7% are DB white dwarfs, 2.8% are DC white dwarfs, 2% are metal-polluted, 3.2% are DAH white dwarfs, and... | arxiv_0183274 |
Analytic regularity of strong solutions for the complexified stochastic non-linear Poisson Boltzmann Equation
=============================================================================================================
Semi-linear elliptic Partial Differential Equations (PDEs) such as the non-linear Poisson Boltzmann ... | arxiv_0183275 |
Local and non-local quantum transport due to Andreev bound statesin finite Rashba nanowires with superconducting and normal sections
=====================================================================================================================================
We analyze Andreev bound states (ABSs) that form in ... | arxiv_0183276 |
Tensor-triangular fields: Ruminations
=====================================
We examine the concept of field in tensor-triangular geometry. We gather examples and discuss possible approaches, while highlighting open problems. As the construction of residue tt-fields remains elusive, we instead produce suitable homologic... | arxiv_0183277 |
Global well-posedness and scattering for the defocusing quintic NLS in three dimensions
=======================================================================================
We revisit the proof of global well-posedness and scattering for the defocusing energy-critical NLS in three space dimensions in light of recent... | arxiv_0183289 |
Intrinsic activation energy for twin wall motion
================================================
[1](#fn1)
Even in a topologically perfect crystal, a moving twin wall will experience forces due to the discrete nature of the lattice. The potential energy landscape can be described in terms of one of two parameters: the... | arxiv_0183294 |
, TE and EE angular spectra. This parameter is expected to be *A*lens = 1 in the standard ΛCDM model. However the most recent Planck data analysis shows a statistically significant preference for values *A*lens > 1 (in particular, *A*lens = 1.15− 0.12+ 0.13 at ± 2*σ*). While the physical motivations behind this result... | arxiv_0183300 |
An a posteriori error analysis for an optimal control problem with point sources
================================================================================
[1](#fn1)
We propose and analyze a reliable and efficient a posteriori error estimator for a control–constrained linear–quadratic optimal control problem invo... | arxiv_0183312 |
Change-point detection in a linear model by adaptive fused quantile method
==========================================================================
A novel approach to quantile estimation in multivariate linear regression models with change-points is proposed: the change-point detection and the model estimation are b... | arxiv_0183317 |
arsity, *s*Δ. This quantity measures the ratio of halo masses at two different overdensities:
*s*Δ1 Δ2 = *M*Δ1/*M*Δ2
and has been recently proposed as new cosmological probe for galaxy clusters. If the halo follows a NFW profile, the sparsity and concentration are directly related. However, halo sparsity has the k... | arxiv_0183320 |
High-order short-time expansions for ATM option prices of exponential Lévy models
=================================================================================
### April 5, 2014
The short-time asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In the... | arxiv_0183328 |
t.$
0.45
0.45
[fig:numericsone]
The density function is then
$$\left\vert \psi(x,t) \right\vert^2=\frac{e^{\Im(\mathcal B\_0/A\_0) \frac{\vert x-x\_c(t) \vert^2 }{ \vert A(t)/A\_0 \vert^2}}}{\vert A(t)/A\_0 \vert}.$$
Thus, we see that although the state disperses over time, it remains exponentially localized. ... | arxiv_0183335 |
*Ab initio* molecular dynamics calculations of ion hydration free energies
==========================================================================
epsf
We apply *ab initio* molecular dynamics (AIMD) methods in conjunction with the thermodynamic integration (TI) or “*λ*-path” technique to compute the intrinsic hydrat... | arxiv_0183340 |
galaxies with less dispersion in their UV-optical colors than that of ellipticals as a whole, and their colors are consistent with colors of inactive ellipticals. Whether this uniformity means that they have much less contamination from star formation, or are more uniform in their metallicity and age than their lower-... | arxiv_0183341 |
A New QoS-Guarantee Strategy for NOMA Assisted Semi-Grant-Free Transmission
=============================================================================
Semi-grant-free (SGF) transmission has recently received significant attention due to its capability to accommodate massive connectivity and reduce access delay by ... | arxiv_0183342 |
A Survey on Explainable Artificial Intelligence (XAI): towards Medical XAI
==========================================================================
Recently, artificial intelligence and machine learning in general have demonstrated remarkable performances in many tasks, from image processing to natural language proce... | arxiv_0183348 |
cal E}\_{\sigma,{\rm rel}}$ and $W = - \int d^3r \sum\_\sigma n\_{\sigma 0} \delta {\cal E}\_{\sigma,{\rm rel}}$. [Note, the relative entropy term differs from the nonlinear term used in Ref. that reproduces Eq. ([eq:kappaint]) when linearized; theirs is related to *M̄**K**P* rather than the relative energy term.] The... | arxiv_0183349 |
discontinuous method of modelling the interface, where the stress jump is prescribed at the coupling porous-fluid boundary (see e.g. for this method applied to the channel flow). In such an approach the filtering size is different on both sides of the boundary. Assuming that ℓ*V* changes continuously over such an int... | arxiv_0183351 |
$\begin{array}{r}\text{Gothenburg preprint} \\ \text{December,
2012}\\$$\end{array}$ -3mm
Loop amplitudes in maximal supergravity
with manifest supersymmetry
We present a description for amplitude diagrams in maximal supergravities obtained by dimensional reduction from *D* = 11, derived from a field theor... | arxiv_0183354 |
radius=0.15];
\draw[very thick, -] (-0.75,0) -- (0,0);
\node[below] at (0,-0.15) {$n\_i$};
\node[below] at (2,-0.15) {$n\_{i+1}$};
\node[above] at (0,1.55) {$r\_i$};
\node[above] at (2,1.55) {$r\_{i+1}$};
\draw[very thick,fill=white] (0,0) circle [radius=0.15];
\draw[very thick,fill=white] (2,0) circle [radius=0.15];
\... | arxiv_0183357 |
Estimation under group actions: recovering orbits from invariants
=================================================================
In memory of Amelia Perry.
We study a class of *orbit recovery* problems in which we observe independent copies of an unknown element of $\RR^p$, each linearly acted upon by a random eleme... | arxiv_0183362 |
**Towards Positive Geometry of Multi Scalar Field Amplitudes
Accordiohedron and Effective Field Theory**
.6cm
Mrunmay Jagadale*a* and Alok Laddha*b*
**a* California Institute of Technology, Pasadena, CA 91125, USA*
**b*Chennai Mathematical Institute, Siruseri, Chennai, India*
*Siruseri, SIPCOT IT Park*
E-mail: mjagad... | arxiv_0183366 |
1). S. Horiuchi, Y. Okimoto, R. Kumai, Y. Tokura, J. Phys. Soc. Jpn. **69**, 1302 (2000)
Y. Okimoto, S. Horiuchi, E. Saitoh, R. Kumai, Y. Tokura, Phys. Rev. Lett. **87**, 187401 (2001)
A. Girlando, A. Painelli, S. A. Bewick, Z. G. Soos, Synth. Metals **141**, 129 (2004)
S. Horiuchi, Y. Okimoto, R. Kumai, Y. Tokura, Sci... | arxiv_0183374 |
V1369 Cen high resolution panchromatic late nebular spectra in the context of a unified picture for nova ejecta
===============================================================================================================
Nova Cen 2013 (V1369 Cen) is the fourth bright nova observed panchromatically through high resol... | arxiv_0183375 |
TRAPPIST Habitable Atmosphere Intercomparison (THAI)workshop report
===================================================================
The era of atmospheric characterization of terrestrial exoplanets is just around the corner. Modeling prior to observations is crucial in order to predict the observational challenges ... | arxiv_0183377 |
Exploring the Interplay of Intrinsic Fluctuation and Complexity inIntracellular Calcium Dynamics
================================================================================================
The concentration of intracellular calcium ion (Ca2 +) exhibits complex oscillations, including bursting and chaos, as observe... | arxiv_0183379 |
Minimax and adaptive tests for detecting abrupt and possibly transitory changes in a Poisson process
====================================================================================================
Motivated by applications in cybersecurity and epidemiology, we consider the problem of detecting an abrupt change in ... | arxiv_0183384 |
math{\sigma\left[{H(\hat{{\ensuremath{\mathcal{P}}}}'||\hat{{\ensuremath{\mathcal{P}}}})}\right]}}}
{{\ensuremath{\mathbb{E}\left[{H(\hat{{\ensuremath{\mathcal{P}}}}'||\hat{{\ensuremath{\mathcal{P}}}})}\right]}}}$$
where *ε*P normalizes the posterior deviation to a desired scale. For the evidence Z, this is just the... | arxiv_0183386 |
Perturbed interpolation formulae and applications
=================================================
We employ functional analysis techniques in order to deduce that some classical and recent interpolation results in Fourier analysis can be suitably perturbed. As an application of our techniques, we obtain generalizatio... | arxiv_0183389 |
Mean-field dynamics for mixture condensates via Fock space methods
==================================================================
We consider a mean-field model to describe the dynamics of *N*1 bosons of species one and *N*2 bosons of species two in the limit as *N*1 and *N*2 go to infinity. We embed this model int... | arxiv_0183393 |
objectives or calibration of ranking scores. Alternatively, we envision extensions of our approach to more complicated settings with, e.g., more advanced operational constraints, multiple or continuous treatments, or more complex objectives (e.g., incremental return on investment). Additionally, it would be insightfu... | arxiv_0183394 |
Dimension Formulae and Generalised Deep Holes of the Leech Lattice Vertex Operator Algebra
==========================================================================================
[1](#fn1) [2](#fn2) [3](#fn3)
We prove a dimension formula for the weight-1 subspace of a vertex operator algebra *V*orb(*g*) obtained by ... | arxiv_0183398 |
Sharp and strong non-uniqueness for the magneto-hydrodynamic equations
======================================================================
In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the three-dimensional magneto-hydrodynamic (MHD) system. More precisely, we show that a... | arxiv_0183399 |
Automated construction of symmetrized Wannier-like tight-binding models from *ab initio* calculations
=====================================================================================================
[1](#fn1)
[2](#fn2)
Wannier tight-binding models are effective models constructed from first-principles calculations... | arxiv_0183400 |
*β* Cephei stars in the ASAS-3 data
===================================
I. Long-term variations of periods and amplitudes
=================================================
### Received / Accepted
We analyse *V*-filter ASAS-3 photometry of 41 known *β* Cephei-type stars. The ASAS-3 photometry is combined with the archiv... | arxiv_0183405 |
Shock Breakout in Type II Plateau Supernovae: Prospects for High Redshift Supernova Surveys
===========================================================================================
Shock breakout is the brightest radiative phenomenon in a supernova (SN) but is difficult to be observed owing to the short duration and... | arxiv_0183415 |
1021 cm− 2) column densities. While the FeK lines again indicated abundances of only 0.22 ± 0.05Z⊙ other Fe lines arising from the L-shell give abundances which were consistent with the solar value. The strengths of other lines were also consistent with solar abundances.
Observations and data reduction
===============... | arxiv_0183418 |
Succinct Choice Dictionaries
============================
The *choice dictionary* is introduced as a data structure that can be initialized with a parameter $n\in\TbbbN=\{1,2,\ldots\}$ and subsequently maintains an initially empty subset *S* of {1, …, *n*} under insertion, deletion, membership queries and an operation ... | arxiv_0183436 |
Shortest *k*-Disjoint Paths via Determinants
============================================
The well-known *k*-disjoint path problem (*k*-DPP) asks for pairwise vertex-disjoint paths between *k* specified pairs of vertices (*s**i*, *t**i*) in a given graph, if they exist. The decision version of the shortest *k*-DPP asks... | arxiv_0183440 |
Almost sure scattering for the defocusing cubic nonlinear Schrödinger equation on R3 × T
========================================================================================
We consider the Cauchy problem for the defocusing cubic nonlinear Schrödinger equation (NLS) on the waveguide manifold R3 × T and establish al... | arxiv_0183442 |
Explainable Convolutional Neural Networks for Retinal Fundus Classification and Cutting-Edge Segmentation Models for Retinal Blood Vessels from Fundus Images
=============================================================================================================================================================
Intr... | arxiv_0183449 |
maximum likelihood estimator (MLE), in terms of acceptance rate.
Application of TMCMC to the Challenger dataset
==============================================
In 1986, the space shuttle Challenger exploded during take off, killing the seven astronauts aboard. The explosion was the result of an O-ring failure, a splitt... | arxiv_0183451 |
On the scaling limits of Galton Watson processes in varying environment
=======================================================================
We establish a general sufficient condition for a sequence of Galton Watson branching processes in varying environment to converge weakly. This condition extends previous resul... | arxiv_0183453 |
ERRATUM: “STUDIES OF VARIABILITY IN PROTO-PLANETARY NEBULAE: II. LIGHT AND VELOCITY CURVE ANALYSES OF IRAS 22272+5435 and 22223+4327” (2013, ApJ, 766, 116)
===========================================================================================================================================================
We have ... | arxiv_0183455 |
& & \Pi\_{ij} \nabla\_{i} W\_{ij} \Bigg) + \bmath{g}\_{i},\end{aligned}$$
where $f\_{i} = \left(1 + \frac{h\_{i}}{3 \rho\_{i}} \frac{\partial \rho\_{i}}{\partial h\_{i}}\right)^{-1}$, *W**i**j*, *i* = *W*(*r**i**j*, *h**i*) and Π*i**j* refers to the artificial viscosity contribution. Smoothing length *h* was compute... | arxiv_0183461 |
Driving Solar Giant Cells through the Self-Organization of Near-Surface Plumes
==============================================================================
Global 3D simulations of solar giant-cell convection have provided significant insight into the processes which yield the Sun’s observed differential rotation and... | arxiv_0183466 |
Comparing and Counting Logs in Direct and Effective Methods of QCD Resummation
==============================================================================
LA-UR-13-28363, YITP-SB-14-02
Introduction
============
The theory Quantum Chromodynamics (QCD) is remarkably successful in describing the strong interaction. T... | arxiv_0183471 |
Capacity Bounds for State-Dependent Broadcast Channels
======================================================
In this paper, we derive information-theoretic performance limits for three classes of two-user state-dependent discrete memoryless broadcast channels, with noncausal side-information at the encoder. The first ... | arxiv_0183472 |
Hospital management in the COVID-19 emergency: Abelian Sandpile paradigmand beyond
==================================================================================
In this article, we propose a mathematical model –based on a cellular automaton– for the redistribution of patients within a network of hospitals with lim... | arxiv_0183474 |
by a clump of 16 sungrazers discovered over 12 days, December 9–20. This rate is equivalent to nearly 500 objects per year. A third, double spike is located symmetrically to the largest one, in late 2009. Thus, comparing only December spikes in Figure 3 — in order to eliminate much of the annual selection effect — we ... | arxiv_0183476 |
Nonparametric Bayesian estimation of a Hölder continuous diffusion coefficient
==============================================================================
We consider a nonparametric Bayesian approach to estimate the diffusion coefficient of a stochastic differential equation given discrete time observations over a ... | arxiv_0183477 |
Does Financial Literacy Impact Investment Participation and Retirement Planning in Japan?[1](#fn1)
==================================================================================================
**Abstract**
**Keywords:** financial literacy, investment participation, retirement planning, causal discovery, Fast Causa... | arxiv_0183481 |
{I}\_{4\times 4}\} \\
\mbox{Case 2: } & x\_{100} \sim \mathcal{N}\{0,0.25\times\mathbf{I}\_{4\times 4}\} \end{aligned}$$
[fig:Scenario]
where the target state is position and velocity in two dimensions. Snapshots of one dimension of both cases are shown in Fig. [fig:Scenario]. It is well-known that JPDA and related a... | arxiv_0183482 |
( \pm 0.007)$, with *z* ∈ [0.1, 0.7]. At the cluster distance *D**L* = 1670 ± 17 Mpc, the corresponding scale is ∼ 4.6 Kpc/ʺ, and at the cluster range 0.28 < *z* < 0.34 the scale spans values from ∼ 4.3Kpc/ʺ to ∼ 4.9Kpc/ʺ.
AC114 has a nonspherical shape, being elongated in the southeast-northwest direction (, ). ... | arxiv_0183484 |
equatorial coordinates for the position (*α*, *δ*), the proper motions ($\mu\_\alpha\equiv\dot{\alpha}\cos\delta, \mu\_{\delta}$) and the parallax. Due to the high sensitivity of MeerKAT observations, the position of pulsars can be obtained extremely well, and we did not need to sample for the position parameter; so, ... | arxiv_0183485 |
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