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max j∈[p], k∈[q]|Rem 1(j, k)|=Op{sn−7/10log3/2(˜dn)}+Op{s1/2n−13/20(logn)−3/4log(˜dn)} provided that s≲n3/10(log˜d)1/2and log ˜d≪n1/10(logn)−1/2. We complete the proof of Lemma 9. 2 N.1 Proof of Lemma N1 Notice that 1 nnX i=1ˆWi,lδi,k=1 nnX i=1Wi,lδi,k |{z} L1(l,k)+1 nnX i=1(ˆWi,l−Wi,l)δi,k | {z } L2(l,k). (N.4) As we ... | https://arxiv.org/abs/2504.02233v1 |
n+ 1ˆFZ,l(Zi,l) −Φ−1{FZ,l(Zi,l)} δi,kI(M2<|Wi,l| ≤M1)I(|δi,k| ≤Q) | {z } L22(l,k) +1 nnX i=1(ˆWi,l−W∗ i,l)δi,kI(|Wi,l|> M 1)I(|δi,k| ≤Q) | {z } L23(l,k)+1 nnX i=1(W∗ i,l−Wi,l)δi,kI(|δi,k| ≤Q) | {z } L24(l,k) S117 +1 nnX i=1(ˆWi,l−Wi,l)δi,kI(|δi,k|> Q) | {z } L25(l,k)+√ 2π(n−1) n(n+ 1)nX s=1˜δ3,k(Ws,l) | {z } L26(l,k)... | https://arxiv.org/abs/2504.02233v1 |
2×1 nnX i=1eW2 i,l/4I(M2<|Wi,l| ≤M1) =C15Qlog1/2(˜dn) n1/2M1/2 2×1 nnX i=1 eW2 i,l/4I(M2<|Wi,l| ≤M1)−E eW2 i,l/4I(M2<|Wi,l| ≤M1) | {z } L221(l,k) +C15Qlog1/2(˜dn) n1/2M1/2 2×E eW2 i,l/4I(M2<|Wi,l| ≤M1) | {z } L222(l,k). Recall Wi,l∼ N (0,1). Since max i∈[n], l∈[m]E{eW2 i,l/4I(M2<|Wi,l| ≤M1)}≲M−1 2e−M2 2/4and max i... | https://arxiv.org/abs/2504.02233v1 |
+ 2Z∞ QxP(|δi,k|> x) dx≲Q2e−˜cQ2. Due to Wi,l∼ N(0,1), then E eW2 i,l/2 I(Ws,l≤Wi,l)−Φ(Wi,l) I(|Wi,l| ≤M2)δi,kI(|δi,k|> Q) Ws,l ≤E eW2 i,l/2I(|Wi,l| ≤M2)|δi,k|I(|δi,k|> Q) ≤ E{eW2 i,lI(|Wi,l| ≤M2)}1/2 E{δ2 i,kI(|δi,k|> Q)}1/2 ≲M−1/2 2QeM2 2/4e−˜cQ2/2, (N.17) which implies max l∈[m], k∈[q]|L26(l, k)|=O(M−1/2 2Qe... | https://arxiv.org/abs/2504.02233v1 |
that s≪n1/2(logn)−1{log(˜dn)}−1and log ˜d≪n1/10(logn)−1/2. We complete the proof of Lemma N3. 2 N.4 Proofs of (N.1) and (N.2) Recall εi,j=Ui,j−α⊤ jWiand ˆεi,j=ˆUi,j−ˆα⊤ jˆWi. Then T1(j, k) =1 nnX i=1 ˆUi,j−ˆα⊤ jˆWi−Ui,j+α⊤ jWi δi,k = (αj−ˆαj)⊤1 nnX i=1Wiδi,k | {z } T11(j,k)−ˆα⊤ j1 nnX i=1(ˆWi−Wi)δi,k | {z } T12(j,... | https://arxiv.org/abs/2504.02233v1 |
where ˜ c= (1∧c7)/4. Hence, we have max j∈[p], k∈[q]|T136(j, k)|=O{n3/20(logn)−1/4Qe−˜cQ2/2}. With selecting Q=Clog1/2(˜dn) for some sufficiently large constant C >p 5/(2˜c), it holds that T13(j, k) =√ 2π(n−1) n(n+ 1)nX s=1˜δ4,k(Us,j) + Rem 13(j, k) with max j∈[p], k∈[q]|Rem 13(j, k)| ≤ max j∈[p], k∈[q]|T131(j, k)|+ ma... | https://arxiv.org/abs/2504.02233v1 |
to the derivation of (N.25), we can show max j∈[p], l∈[m] 1 nnX i=1(ˆUi,j−Ui,j)(ˆWi,l−Wi,l) =Op{n−7/10(logn)1/2}+Op{n−1(log˜d) log( ˜dn)} provided that log ˜d≲n1/8(logn). By Lemma N3, it holds that T321≤max k∈[q]|ˆβk|1max j∈[p], l∈[m] 1 nnX i=1(ˆUi,j−Ui,j)(ˆWi,l−Wi,l) =Op{s1/2n−7/10(logn)1/2}(N.30) provided that s≪n1/2... | https://arxiv.org/abs/2504.02233v1 |
Us,j, δs,k) are independent for any s̸=i, and Us,j∼ N(0,1), then E{˜δ43,k(Us,j)}=E eU2 i,j/2I{|Ui,j| ≤p 3(log n)/5}δi,kI(˜M <|δi,k| ≤Q) ×E I(Us,j≤Ui,j)−Φ(Ui,j) Ui,j, δi,k = 0. Analogous to the derivation of (N.17) in Section N.1 for the proof of Lemma N1, we have max s∈[n], j∈[p], k∈[q]|˜δ43,k(Us,j)|≲n3/20(logn)−1/... | https://arxiv.org/abs/2504.02233v1 |
for R 11defined in (I.7), it holds that S211=Op{n−1/2(logn)(log ˜d)1/2log3/2(˜dn)} (P.9) provided that log ˜d≲n5/12(logn)−1/2. Analogously, we can also show such convergence rate holds for max v∈[m], k∈[p], l,t∈[q]|n−1Pn i=1(ˆWi,v−Wi,v)εi,kδi,lδi,t|. By Lemma N3, it holds that S212=Op{s1/2n−1/2(logn)(log ˜d)1/2log3/2(˜... | https://arxiv.org/abs/2504.02233v1 |
the convergence rate of R 14in Section I.1.3 for R 14defined in (I.7), we have S241=Op{n−1/2(logn)(log ˜d)1/2log3/2(˜dn)} (P.15) provided that log ˜d≲n5/12(logn)−1/2. As we will show later, S242=Op{s1/2n−1/2(logn)(log ˜d)1/2log3/2(˜dn)}= S 243, (P.16) S244=Op{sn−1/2(logn)(log ˜d)1/2log3/2(˜dn)}= S 245, (P.17) S246=Op{s... | https://arxiv.org/abs/2504.02233v1 |
holds that S2466≤max j∈[p]|ˆαj−αj|2 1·max k∈[q]|ˆβk−βk|1· max r1,r2,r3∈[m], t∈[q] 1 nnX i=1Wi,r1Wi,r2Wi,r3δi,t S144 =Op{s3n−3/2(log˜d)3/2} provided that s≪n1/2(logn)−1{log(˜dn)}−1and log ˜d≪n1/10(logn)−1/2. By (P.22), we have (P.18) holds. 2 P.1.4 Convergence rate of S25 Recall εi,j=Ui,j−α⊤ jWi, ˆεi,j=ˆUi,j−ˆα⊤ jˆWi,δi... | https://arxiv.org/abs/2504.02233v1 |
nnX i=1ˆα⊤ k(ˆWi−Wi)ˆβ⊤ l(ˆWi−Wi)(ˆβt−βt)⊤Wi(ˆUi,j−Ui,j) | {z } S2562 + max j,k∈[p], l,t∈[q] 1 nnX i=1ˆα⊤ k(ˆWi−Wi)(ˆβl−βl)⊤Wi(ˆβt−βt)⊤Wi(ˆUi,j−Ui,j) | {z } S2563(P.29) + max j,k∈[p], l,t∈[q] 1 nnX i=1(ˆαk−αk)⊤Wiˆβ⊤ l(ˆWi−Wi)ˆβ⊤ t(ˆWi−Wi)(ˆUi,j−Ui,j) | {z } S2564 + 2 max j,k∈[p], l,t∈[q] 1 nnX i=1(ˆαk−αk)⊤Wiˆβ⊤ l(ˆWi−W... | https://arxiv.org/abs/2504.02233v1 |
we have 1 nnX i=1 ˆεi,jˆδi,k−εi,jδi,k =√ 2π(n−1) n(n+ 1)nX s=1˜δ44,k(Us,j) +˜δ54,j(Vs,k) + Rem 1(j, k) + Rem 2(j, k), with max j∈[p], k∈[q]|Rem 1(j, k)|=Op{sn−7/10log3/2(˜dn)}+Op{s1/2n−13/20(logn)−3/4log(˜dn)}, max j∈[p], k∈[q]|Rem 2(j, k)|=Op{n−4/5(logn)1/4(log˜d)1/2} provided that s≲n3/10(log˜d)1/2and log ˜d≪n1/10... | https://arxiv.org/abs/2504.02233v1 |
Independent Power and Renewable Electricity Producers Insurance IT Information TechnologyIT Services RE Real EstateDiversified REITs Software Industrial REITs Communications Equipment Hotel & Resort REITs Technology Hardware, Storage & Peripherals Office REITs Electronic Equipment, Instruments & Components Health Care ... | https://arxiv.org/abs/2504.02233v1 |
0.0796 0.0822 <0.0001 0.0238 0.0940 0.0018 0.4532 Uti 0.1278 0.4956 0.0144 0.0644 0.4522 0.1062 0.0316 0.2754 0.1300 0.0010 Table S4: The degrees of nodes associated with the 11 sectors in the networks constructed based on the proposed conditional independence tests with Rademacher multiplier and the three competing me... | https://arxiv.org/abs/2504.02233v1 |
arXiv:2504.02292v1 [math.ST] 3 Apr 2025Unifying Different Theories of Conformal Prediction Rina Foygel Barber∗Ryan J. Tibshirani† Abstract This paper presents a unified framework for understanding th e methodology and theory behind several different methods in the conformal pred iction literature, which includes standard ... | https://arxiv.org/abs/2504.02292v1 |
Podkopaev and Ramdas (2021) study the problem of label shift. •Nonexchangeable conformal prediction ( Barber et al. ,2023), which applies a different sort of weighting to improve the robustness of the prediction sets to mild but unknown violations of the exchangeability assumption. •Localized conformal prediction ( Guan... | https://arxiv.org/abs/2504.02292v1 |
can be known exactly or approximately), then it is possible to derive theory which ensures that the conformal prediction method is competitive with model-based metho ds; see, e.g., Chapter 5 ofAngelopoulos et al. (2024b) for an overview of results of this type. Second, in the presence of arbitrary (possibly large and d... | https://arxiv.org/abs/2504.02292v1 |
condition on s, in the example of the loss-based score function as in ( 1), this is equivalent to requiring the algorithm Ato be symmetric in the training data—that is, the function ˆf=A(z1,...,z n+1) is unchanged if we permute the data points z1,...,z n+1in the training set. The bag of data. For a given z= (z1,...,z n... | https://arxiv.org/abs/2504.02292v1 |
al. 2005 ).The conformal prediction set C(Xn+1)defined in (3)can be written in terms of the conformal p-value p(y)defined in (5), as C(Xn+1) =/braceleftbig y∈ Y:p(y)> α/bracerightbig . In order to prove the proposition, we need a result relating p-value s and quantiles. Its proof is given Appendix A.1. Lemma 1. LetQbe a ... | https://arxiv.org/abs/2504.02292v1 |
settings th at are more general than exchangeability, and second, we will allow for conditioning on mor e information than the bag of data /lbagZ/rbag. To begin, we will need the following two ingredients: •Partial information about the data, encoded by a random variable U∈ Uon which we will condition to perform infere... | https://arxiv.org/abs/2504.02292v1 |
as convenient f or each example. 8 Remark 2. LetPZ|Udenote the conditional distribution associated with the joint PZ,U. If QZ|U=PZ|Uholds almost surely, i.e., our choice for the conditional distribution of Z|Uis exactly correct in implementing the unified conformal method, then by taking QU=PU, we obtainQZ,U=PZ,Uand the... | https://arxiv.org/abs/2504.02292v1 |
for completeness before describing its reformulation under the unified framework. 4.1.1 Method and theory Fix any score function of the form s((x,y),/lbagz/rbag), which assigns a value to a single data point (its first argument), based on a bag of data points (its second arg ument). For arbitrary y∈ Y, we define the augme... | https://arxiv.org/abs/2504.02292v1 |
prediction to train a model on Z(0), then compute scores for all remaining data points in order to build a p rediction set for the test point. Concretely, we will work with a score function of the for m s((x,y),z(0)). This accommodates choices of the form s((x,y),z(0)) =ℓ(y,ˆf(x)), forˆf=A(z(0)), as in (1). Notice that... | https://arxiv.org/abs/2504.02292v1 |
additionally assume that we have knowledge of thedistribution shift relating the likelihood of GtoF, w∗(x,y) =dG dF(x,y). (12) As our knowledge of this shift might be only approximate, we will work w ith a user-specified weight function wthat approximates w∗. WCP was introduced by Tibshirani et al. (2019), who focused o... | https://arxiv.org/abs/2504.02292v1 |
=EZ∼Fn+1[¯w(Zn+1)· /BD{Z∈A}] =EZ∼Fn×(F◦¯w)[ /BD{Z∈A}], where the next-to-last line holds by exchangeability of Fn+1. Meanwhile, the true distribution on ZisPZ=Fn×G=Fn×(F◦w∗). Theorem 2thus gives the Type I error bound, P{p≤α} ≤α+dTV(PS,T,QS,T)≤α+dTV(PZ,QZ), where (recalling information monotonicity, in Remark 1) the la... | https://arxiv.org/abs/2504.02292v1 |
calculate p=/summationdisplay k∈[n]\{K}wk· /BD/braceleftbig s(Zk,ZK)≥s(Zn+1,ZK)/bracerightbig +wK+wn+1· /BD/braceleftbig s(ZK,ZK)≥s(Zn+1,ZK)/bracerightbig , using the fact that ( ZK)k=Zkfor allk∈[n]\{K}, whereas ( ZK)K=Zn+1and (ZK)n+1= ZK. Sincewn+1≥wKby assumption, we therefore have p≤/summationdisplay k∈[n]\{K}wk· /B... | https://arxiv.org/abs/2504.02292v1 |
e.g., in the case X=Rd, we can use the Gaussian kernel H(x,x′) =e−/bardblx−x′/bardbl2 2/2σ2/(2πσ2)d/2. We assume that, for any x∈ X, H(x,·) is a density (relative to some base measure), with H(x,x)>0. We then sample ˜Xn+1|Xn+1∼H(Xn+1,·), which is then used to define weights in the RLCP set at coverage level 1−α, defined ... | https://arxiv.org/abs/2504.02292v1 |
let P∗ Zbe the distribution of Z= ((X1,Y1),...,(Xn+1,Yn+1)) (which we assume is exchangeable), and let PZbe the distribution of Zconditional on the event Xn+1∈B. In other words, the conditional coverage probability can be written a s PP∗ Z{Yn+1∈ C(Xn+1)|Xn+1∈B}=PPZ{Yn+1∈ C(Xn+1)}. Letu= (/lbagz/rbag,˜x), and write z= (... | https://arxiv.org/abs/2504.02292v1 |
an important application of this generalized view is the problem of feedback covariate shift (FCS), where data is collected sequentia lly. At time t, the observed data from past times 1 ,...,t−1 determines a distribution from which the next covariate Xt is sampled—for example, this can arise if our aim is to identify r... | https://arxiv.org/abs/2504.02292v1 |
≤α. 5 Extensions of the unified framework In this section, we develop several extensions of the unified frame work. These extensions will be used to derive some of the new results on conformal given in th e next section. All proofs for this section are deferred until the appendix. 5.1 Monte Carlo p-values For our first ex... | https://arxiv.org/abs/2504.02292v1 |
the p-value ˜ pin (19) can be obtained by simply letting RZ|U=QZ|U. We now present our theoretical guarantee for the importance sa mpling p-value ˜ p. This result is proved in Appendix A.3. Theorem 9. Suppose(Z,U)∼PZ,U. Then the importance sampling p-value defined in (20) satisfies P{˜p≤α} ≤α+inf QUdTV(PZ,U,QZ,U), where ... | https://arxiv.org/abs/2504.02292v1 |
for conformal prediction. We will focus on developing extensions of the conformal approaches and theory in Section 4; our intention to provide a flavor of the types of extensions possib le and not to derive an exhaustive set of new results. Some of these e xamples will rely on the extensions of the unified theory from Se... | https://arxiv.org/abs/2504.02292v1 |
Choices of UandQZ|U.DefineU=/lbagZ/rbag, and write the score function for u=/lbagz/rbagas s(z,u) =s(zn+1,/lbagz/rbag). Next define QZ|U=/lbagz/rbag=/summationtext σ∈Sn+1w(zσ(n+1))·δzσ (n+1)!, which is a measure (not necessarily a distribution), supported on pe rmutations of z. P-value. Under these choices, the p-value in... | https://arxiv.org/abs/2504.02292v1 |
in Theorem 4 can be derived as a special case with Fi=Fand consequently w∗ i=w, for each i. 6.2.2 View from the unified framework We describe how WCP under training drift fits into the unified conform al framework. Choices of UandQZ|U.The unified view here is the same in Section 4.3, for WCP. For concreteness, we set U=/lb... | https://arxiv.org/abs/2504.02292v1 |
lattice. In contrast, the modified form of RLCP presented here will always reuse an existing data point instead of sampling a new value, and therefore avoids this issue. 6.3.1 Method and theory Fix any score function of the form s((x,y),/lbagz/rbag), and a localizing kernel H:X ×X → R+, as in RLCP in Section 4.5. Now, g... | https://arxiv.org/abs/2504.02292v1 |
The scorefunctionnowtakestheform s((x,y),z), comparingadatapoint( x,y)toanordered datasetz∈ Zn+1(note that sis no longer required to be symmetric in its second argument). This accommodates, e.g., a score that can depend on the time-orde ring of data collected in the FCS application (with likelihood ratio in ( 17)). For... | https://arxiv.org/abs/2504.02292v1 |
permutations, Sy σ=s/parenleftbig Zy σ(n+1),Zy σ/parenrightbig , σ∈ Sn+1. FixingM≥1, letgXbe a user-chosen proposal density, where fX,gXare assumed to be absolutely continuous with respect to one another. Conditional on /lbagX/rbag, we can think of gX as inducing a distribution on permutations, which places probability... | https://arxiv.org/abs/2504.02292v1 |
e unified framework. Many others should also be possible to derive. For example, two exist ing methods, WCP and NexCP, are both weighted variants of conformal prediction but th ey appear quite different in their assumptions and technical motivations. Our work provides a unified explanation for why they both work; this sug... | https://arxiv.org/abs/2504.02292v1 |
framework for conformal prediction. Biometrika , 110(1):33–50, 2023. Laszlo Gyˆ orfi and Harro Walk. Nearest neighbor based confor mal prediction. Annales de L’Institut de Statistique de L’Universit´ e de Paris , 63(2–3):173–190, 2019. 40 Matthew T. Harrison. Conservative hypothesis tests and con fidence intervals using ... | https://arxiv.org/abs/2504.02292v1 |
/BD{xi≥x} ≥m/summationdisplay i=1pi /BD{xi≥xk∗}=/summationdisplay i≥k∗pi= 1−/summationdisplay i<k∗pi> α, since/summationtext i<k∗pi<1−αby definition of k∗. And similarly, if x > xk∗then PQ{X≥x}=m/summationdisplay i=1pi /BD{xi≥x} ≤m/summationdisplay i=1pi /BD{xi> xk∗}=/summationdisplay i>k∗pi= 1−/summationdisplay i≤k∗pi≤... | https://arxiv.org/abs/2504.02292v1 |
construction, we have V(k)⊥ ⊥(Z,U), and 1−V(k)∼Beta(M+1−k,k) = Beta M,α. In other words, setting ˜T=tA(U) where we define A= 1−V(k)∼BetaM,α, so far we we have shown that PpPZ,U×(QZ|U)M˜p≤α=PPS,˜T/braceleftBig S >˜T/bracerightBig . But by an identical argument, it also holds that PQZ,U×(QZ|U)M{˜p≤α}=PQS,˜T/braceleftBig S... | https://arxiv.org/abs/2504.02292v1 |
and the las t step holds by definition ofw∗ i. Returning to the definition of Term i, we then calculate Termi= sup A/braceleftBig PZn+1,U(A)−EF1×···×FnׯF[¯w(Zi)· /BD{(Zi,/lbagZ/rbag)∈A}]/bracerightBig ≤sup A/braceleftbigg/parenleftBig EF1×···×FnׯF[w∗ i(Zi)· /BD{(Zi,/lbagZ/rbag)∈A}]+dTV(Fi,¯F)/parenrightBig −EF1×···×Fn×... | https://arxiv.org/abs/2504.02292v1 |
Fermat Distance-to-Measure: a robust Fermat-like metric J´ erˆ ome Taupin∗†Fr´ ed´ eric Chazal∗ Abstract Given a probability measure with density, Fermat distances and density-driven met- rics are conformal transformation of the Euclidean metric that shrink distances in high density areas and enlarge distances in low d... | https://arxiv.org/abs/2504.02381v1 |
Eq. (1) by the DTM. Thanks to regularity and stability properties of the DTM, the FDTM presents several benefits compared to the classical Fermat distance. First, it is well-defined for any measure µ, regardless of the existence of a density. Second, we provide quantitative stability results for the FDTM with respect t... | https://arxiv.org/abs/2504.02381v1 |
of the FDTM between two points as the integral of the DTM – possibly elevated at some power greater than 1 – minimized over paths between xandy. Definition 2.1 (Fermat distance-to-measure (FDTM)) .Letµbe a probability measure over Rd,Fbe a closed subset of Rdandβ≥1 be a power parameter. Given a rectifiable path γ: [0,1... | https://arxiv.org/abs/2504.02381v1 |
Then min dµ>0. About the role of F.The domain F⊂Rdis introduced to achieve interesting convergence rates for the estimation of the FDTM in a way that allows to evaluate all admissible paths with reasonably computational complexity independent from the ambient dimension d- see Section 4. In this case Fwill be chosen to ... | https://arxiv.org/abs/2504.02381v1 |
from classical arguments [6, Lemma 1.12]. The proof is adapted in to account for domain constraints – see Section 3.1. Theorem 2.2 (Existence of geodesics) .Letµbe a probability measure over RdandF⊂Rd a closed domain. Then for all x, y∈Rd, there exists a path γ∈ΓF(x, y) such that Dµ(x, y) =Dµ(γ). γis called a (minimizi... | https://arxiv.org/abs/2504.02381v1 |
bounds is detailed in Section 3.2. Stability of the FDTM Beyond the fact that the FDTM metric is defined for any prob- ability measure on Rd, one of the main motivations to introduce it as a variant of the Fermat distance is that it turns out to be more stable w.r.t. the measure. This property follows from the bound in... | https://arxiv.org/abs/2504.02381v1 |
Assumption (A).(iii), recall that it is satisfied according to Lemma 2.1 by any measure without atoms of mass at least m. Assumption (A).(iii) and .(iv) are technical assumptions that do not have a fundamental impact on the results. We denote MK,a,b,σ the set of measures satifying Assumption (A) for given K,a,bandσ. Th... | https://arxiv.org/abs/2504.02381v1 |
that Dµ(γ)≤lim inf k→+∞Dµ(γk) =Dµ,F(x, y). It remains to show that γ∈ΓF(x, y). It is immediate from the uniform convergence that γ(0) = xandγ(1) = y. Now, given any open interval Isuch that γ(I)⊂Rd\F, it suffices to show that ˙ γhas constant direction over all segments J⊂I. First, γ(J) is compact and at positive distan... | https://arxiv.org/abs/2504.02381v1 |
n≥0 :Aadmits a r-packing of size n and the covering number as cov(A, r) = min n≥0 :Aadmits a r-covering of size n . Ar-packing Fismaximal if no point can be added to it while preserving its packing property. Note that a maximal r-packing is also a r-covering. In particular, cov( A, r)≤pack( A, r). 11 3 GEODESICS AND S... | https://arxiv.org/abs/2504.02381v1 |
outer sections of γthat intersect Lµ,δwhile not being too long outside of this sub-level. ˜ ηis a modification of ηsatisfying |˜η| ≤c1δ and Dµ(˜η)≤c2δβ+1. (ii)χcontains the remaining outer sections of γintersecting Lµ,δand satisfies [χ]δ ≤1 2 [γ]ρδ δ . ˜χ⊂χis a subset of these outer sections. (iii)ω⊂[γ]δcontains the re... | https://arxiv.org/abs/2504.02381v1 |
the straight path [ x, y]∈ΓG(x, y), for which the FDTM is upper bounded using Eq. (8) and the fact that λ≥5 2√β(see Appendix B.2): Dµ,G(x, y)≤Dµ([x, y]) ≤ ∥x−y∥ max Kdµβ ≤diam( K)β+1 ≤diam( K)β+1 2·25βp dH(F, G) ≤10p βλdiam( K)β+1 2p dH(F, G). We now describe the main reasoning behind Theorem 3.4. Given a geodesic γw... | https://arxiv.org/abs/2504.02381v1 |
with a′=ab′/b, one has ( a′)−1/b′=a−1/bandarb≥a′rb′forr∈(0,(a′)−1/b′]. 15 4 ESTIMATING THE FDTM This would however lead to a term of the form n−1/2pin the final bound, which is slower when p > b . Equations (22), (23) and (24) put together with Theorems 2.8 and 3.3 allow for the following convergence rate which is a sl... | https://arxiv.org/abs/2504.02381v1 |
(1−m)ρ, ν=µ−mεbδry+mλ, where ρis the uniform probability measure over [3 ry,4ry] and λhas density z7→b∥z−(r− ε)y∥b−1with regard to the Lebesgue measure on [( r−ε)y, ry], amounting to a total mass λ [(r−ε)y, ry] =εb. Then, there exists a choice of α≤1 2andr≤1 4depending on pandβ along with positive constants a,candCde... | https://arxiv.org/abs/2504.02381v1 |
procedure doesn’t affect the Fermat distance as ngrows larger [3]. This trick however cannot be performed in the context of FDTM without altering the limiting object as it would effectively restrict paths to remain constrained within the support, which may exclude geodesics. If it is known that no geodesic exits the su... | https://arxiv.org/abs/2504.02381v1 |
between two endpoints on the circle are always made 20 5 NUMERICAL ILLUSTRATIONS Figure 3: Convergence of the empirical FDTM on the unit circle compared to the theoretical rate 1 /√n. FDTM parameters: m= 0.1,p= 2 and β= 2. The empirical FDTM is averaged over 500 iterations. up of finite number of equally sized cords – ... | https://arxiv.org/abs/2504.02381v1 |
based on the ideas developed in Section 4.3. The question of optimal minimax bounds for the convergence of the empirical FDTM is also of interest. Moreover, in the optic of further adapting results coming from the Fermat literature, objects such as the Fermat graph laplacians [2] may be adapted to the FDTM model, hopef... | https://arxiv.org/abs/2504.02381v1 |
subset K⊂Rd. The second part details the fact that the FDTM and Euclidean metrics share the same topology over the domain F. A.1 Restriction to a Compact Set As long as we restrict the study of the FDTM to endpoints belonging in a compact set A, the fact that dµis a Lipschitz and proper function ensures that geodesic p... | https://arxiv.org/abs/2504.02381v1 |
be lower bounded by some ε=f(r, δ, β )>0 by Lipchitz property of the DTM. Therefore, for all small enough r >0 there exists a ε-FDTM ball included in the r-Euclidean ball of radius r. This implies that the FDTM between two distinct points is positive – hence the FDTM is a metric – and coincidently shows that the FDTM t... | https://arxiv.org/abs/2504.02381v1 |
sum of at most 2 |F|paths of the form [ w, z] or [z, w] where z∈ F ⊂ Lµ,δand w∈ B(z,2δ) is either an endpoint xioryi, or a point halfway to another z′∈ F. It follows immediately that |˜η| ≤2|F| ·4δ≤16 mδ. Moreover, ˜ ηremains at any point at distance at most 4 δfromF ⊂ Lµ,δ, hence ˜ η⊂Lµ,5δby Lipschitz property of dµ. ... | https://arxiv.org/abs/2504.02381v1 |
figures, orange dashed lines) to make up ˜η(right figure, orange). The χandωsections (left figure, purple and green respectively) are left untouched in this example. The main issue when dealing with the domain constraints is that when modifying the sub- level of a path, if a long outer section crosses the sub-level the... | https://arxiv.org/abs/2504.02381v1 |
so that ˜ γ∈ΓF(x, y) and each point z∈ F ∪ G is visited at most once by ˜ γ. Moreover, we write ˜ γ= ˜η+ ˜χ+ ˜ωwhere ˜ η= ˜η1+ ˜η2and ˜η1, ˜η2, ˜χand ˜ωare respectively the result of the removal of the loops inP iη1 i,P jη2 j,χandω. This concludes the construction of the modified path ˜ γand we now analyze the properti... | https://arxiv.org/abs/2504.02381v1 |
all k≥0 by Eq. (32), hence +∞X k=0 [χk]δk ≤1 2+∞X k=0 [γ]δk−1 δk =1 2 [γ]ρδ0 (39) On the other hand, for all k≥0, δβ k+1 [ηk]δk+1 ≤Dµ [ηk]δk+1 (dβ µ≥δβ k+1over [ ηk]δk+1) ≤Dµ(ηk) =Dµ(γ)−Dµ(χk+ωk) ≤Dµ(˜γk)−Dµ(˜χk+ ˜ωk) ( γis optimal and ˜ χk+ ˜ωk⊂χk+ωk) =Dµ(˜ηk) ≤c2δβ+1 k (Eq. (31)). Therefore, +∞X k=0 [ηk]δk+1 ≤c2+∞X... | https://arxiv.org/abs/2504.02381v1 |
either case, it always holds that |γ| ≤(2β+2+ 1)λ∥x−y∥. 35 C STABILITY OF THE FDTM C Stability of the FDTM This section is devoted to the proof of Theorem 3.4. In this section we fix K⊂Rd,F⊂K andµ∈ M K. In the following, assume without loss of generality that diam( K) = 1 (see Appendix A.1), which allows to upper bound... | https://arxiv.org/abs/2504.02381v1 |
sections of length close to r, excluding long outer sections that cannot be broken down, and with the exception of a possible small section between two long outer sections. This lemma will be applied to a geodesic γ∈Γ⋆ µ,F(x, y) to construct a path in Γ G(x, y) that approximates γ. From now on, the convention used for ... | https://arxiv.org/abs/2504.02381v1 |
that |C|= 0 or |A| ≥1, which is true as long as the path is not composed of a unique type (c) section. In this case, |γ|=X i∈A⊔B⊔C(ti+1−ti) ≥X i∈A(ti+1−ti) +1 2|B|r (ti+1−ti≥r 2ifi∈B) ≥1 2X i∈A(ti+1−ti) +1 2|A|r+1 2|B|r (ti+1−ti≥rifi∈A) ≥1 4 2X i∈A(ti+1−ti) + 2|B|r+|C|r! (|A| ≥|C| 2) Given a geodesic γ, Theorem 3.4 is ... | https://arxiv.org/abs/2504.02381v1 |
This section is devoted to proving Theorem 4.1: Theorem 4.1. Assume that µ∈ M K,a,b,σ satisfies Assumption (A) and m≤1 2. Then for all n≥1 m, Eh Dµ−Dˆµn ∞,Ki ≲diam( K)β+1 diam( K) σ(p−β)∨0dlog(n)√n+p log(n) n1 2b! (26) where≲hides a multiplicative constant depending on m,p,β,aandb. This result derives from convergenc... | https://arxiv.org/abs/2504.02381v1 |
Fis a maximal r-packing of S(µ) (and therefore a r-covering), 1≥µ G x∈FB x,r 2! =X x∈Fµ B x,r 2 ≥ |F| ar 2b ≥cov(S(µ), r)ar 2b , which implies Eq. (58). In other words, Lemma D.3 essentially states that the intrinsic dimension of a ( a, b)-standard measure is at most b. Together, Lemmas D.2 and D.3 allow to u... | https://arxiv.org/abs/2504.02381v1 |
D.3 Convergence of the Empirical FDTM Proof of Theorem 4.1. Recall that we assume diam( K) = 1 without loss of generality. In order to use Proposition D.1, assume that m≤1 2andn≥1 m. Decompose the offset into the offset w.r.t. the DTM and the offset w.r.t. the domain: Dµ−Dˆµn ∞,K≤ Dµ,S(µ)−Dˆµn,S(µ) ∞,K+ Dˆµn,S(µ)−Dˆµn,... | https://arxiv.org/abs/2504.02381v1 |
0: (i)f(·, u) is non-decreasing when β≥pand non-increasing when β≤p. (ii)f(x,·) is convex when β≥pand concave when β≤p. (iii)f(x,0) = 0. Moreover, dµis lower bounded by dµ(0)p≥1 3 m 6p=Cmand upper bounded by 2 over [0 ,1]. Therefore, when β≥p, Dν(0,1)−Dµ(0,1)≥f (Cm)1 p, Cε (using (i)) =Cβ pf m1 p, ε ≥Cβ p f m1 p... | https://arxiv.org/abs/2504.02381v1 |
that Dα,r(0) = A(r) and Dα,r(r) =B(r) +Gr(α) where Gris some continuous function with Gr(0) = 0. Moreover, one can see that both AandBareC1onR∗ +and that for all r >0, A′(r) r=βˆ1 0(r2+t2)β 2−1dt− − → r→0βˆ1 0tβ−2dt=β β−1 with the convention thatβ β−1= +∞when β= 1, and B′(r) r= (1 + r2)β−1 2− − → r→01. It follows that ... | https://arxiv.org/abs/2504.02381v1 |
is given by ∂ ∂αˆ1 0dα,r(r, t)β−p∂ ∂udα,r(r, t)p α=0dt =−ˆ1 0F(p, r, t )β p−1 ϕ(r, t)β−2p ψ(r, t)p−ϕ(r, t)p dt+ˆ1 0G(p, r, t )ϕ(r, t)β−pdt =−(β−p)rˆ1 0(1−t)tϕ(r, t)β−p−2 ψ(r, t)p−ϕ(r, t)p dt +prˆ1 0(1−t)ϕ(r, t)β−p (2−t)ψ(r, t)p−2+tϕ(r, t)p−2 dt and, since ϕ(r,·) and ψ(r,·) both converge uniformly to the ident... | https://arxiv.org/abs/2504.02381v1 |
( Dµ(γ0), Dµ(γr)) (Eqs. (73) and (74)) = 2 min ( Dα,r(0),Dα,r(r)) (Eq. (68) and invariance by τy) = 2Dα,r(r) (Eq. (69)) ≥2Dα,r(r−ε) + 2Cε (Eq. (69)) =Dµ(γr−ε) + 2Cε (Eq. (68) and invariance by τy). Finally, for all zon the path γr−ε,νbrings a small amount of mass closer to zcompared to µ, hence dν(z)≤dµ(z). Therefore, ... | https://arxiv.org/abs/2504.02381v1 |
The Markov approximation of the periodic multivariate Poisson autoregression Mahmoud Khabou∗, Edward A. K. Cohen and Almut E. D. Veraart Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, United Kingdom Abstract This paper introduces a periodic multivariate Poisson autoregression wit... | https://arxiv.org/abs/2504.02649v1 |
squares. For the observation driven models, Fokianos et al. [19] studied both the stability and estimation of multivariate Poisson autoregressions using a Markov chain perturbation approach. The stability condition for such autoregressions has been significantly improved by Debaly and Truquet [9]. We refer the reader t... | https://arxiv.org/abs/2504.02649v1 |
24 for the hourly number of posts on a social network. Because of this variation, classical concepts such as stationarity and ergodicity no longer apply and are replaced by the analogous periodic counterparts. Such concepts have been initially studied in the signal processing literature [53, 52] under the name of “cycl... | https://arxiv.org/abs/2504.02649v1 |
this article, Z,NandN∗denote the sets integers, non-negative integers and positive integers, respectively. Superscripts are denoted between brackets to be distinguished from powers. Let (Nt(·))t∈N∗= N(1) t(·),···, N(d) t(·) t∈N∗be a family of independent and identically distributed (iid) unit intensity Poisson proces... | https://arxiv.org/abs/2504.02649v1 |
here with a periodic and infinite memory extension of the aforementioned autoregression. We consider a network with a fixed adjacency matrix M= (mij)i,j=1,···,d, that is mij= 1 if there is a directed edge from node ito node j, and mij= 0 otherwise. We impose that a node is not connected to itself, that is mii= 0 for al... | https://arxiv.org/abs/2504.02649v1 |
periodicity, the network remembers its past activity weighted by coefficients that depend the past seasons, then aggregates it. To the best of our knowledge, the difference between the two types of periodicity has not been explicitly studied in the literature. However, we point out that Maillard and Wintenberger [66] m... | https://arxiv.org/abs/2504.02649v1 |
E[|fv(0,0···;ζv)|]<+∞. We follow the proofs in [9], generalising them for the case of a periodic process with infinite memory. We start by proving periodic stationarity for the finite memory approximation. The m−truncated regression is defined by the equation X(m) t=ft X(m) t−1, X(m) t−2,···, X(m) t−mp,0,···;ζt . (7)... | https://arxiv.org/abs/2504.02649v1 |
the convolution product ( a∗b)t=Pt−1 k=1akbt−kfor the nonnegative sequences of matrices ( ak) and ( bk) defined on N∗. By associativity, we can define recursively a∗1=aanda∗(n+1)=a∗a∗n. Given the matrix sequence (Ak)k∈N∗from Assumption 3.1, we define B=P n≥1A∗n, which is in ℓ1(N∗). We also define the matrix remainder s... | https://arxiv.org/abs/2504.02649v1 |
at each season ϕ(v)is an attenuation of a global kernel A,e.g. ϕ(v) k=sin(2πv/p) LAk. Proposition 3.8. Let(ϕ(t) k)t∈Z,k∈N∗be a family of kernel matrices periodic in tand let ψbe an L−Lipschitz non-negative jump-rate function. Assume that (Yt)t∈Nis a multivariate Poisson autore- gression that follows the recursion (2)or... | https://arxiv.org/abs/2504.02649v1 |
Similarly, after a transitory period the curve E[Y(3) t] joins the strictly periodic curve E[˜Y(3) t]. For our example, the kernel matrices are dominated by Ce−3kfor some positive matrix C, hence we expect that the difference between the empty history time series Yand its periodically stationary version ˜Ydecay at leas... | https://arxiv.org/abs/2504.02649v1 |
The Markov properties of the Poisson autoregression with an exponen- tial polynomial kernel To fix the ideas, we consider a linear Poisson autoregression with a constant baseline intensity and kernel, that is( Yt=Nt(λt) λt=µ+Pt−1 k=1ϕt−kYk,(12) where µ∈Rd +andϕis a family of non-negative matrices such that ρ(P k≥1ϕk)<1... | https://arxiv.org/abs/2504.02649v1 |
q << T leads to a significant reduction in computation time. We now state our universal approximation result for periodic Poisson autoregressions with Markov chains. Theorem 4.3. Let(ϕ(t) k)t∈Z,k∈N∗be a family of matrix kernels satisfying the stability Assumption 3.7. Letτ >0be a fixed characteristic time and let ε >0.... | https://arxiv.org/abs/2504.02649v1 |
or directly numerically. The approximation on Figure 5 is obtained by numerically minimising the ℓ1(N∗)using the method COBYLA inscipy.optimize.minimize . In Figure 6 we simulate a Type I periodic Poisson autoregression with kernel ϕ(in blue) as well as a trajectory with the same underlying randomness Nwith the kernel ... | https://arxiv.org/abs/2504.02649v1 |
a similar way, mutatis mutandis . 5.1 Properties of the Markov maximum likelihood estimator In Section 4.2 we showed that the Poisson autoregression with exponential polynomial kernels can be represented as the Markov chain of order q ( Yt =Nt ψ µt+Pq m=1G(m) tξ(m) t ξ(m) t =e−2m+1 τξ(m) t−1+e−2m+1 τYt−1, m∈ {1,···... | https://arxiv.org/abs/2504.02649v1 |
Positivity: There exists ε >0such that ψ(x)≥εfor all x∈R. 3. Compactness: Γis a compact subset of Rp(1+d2q)and contains the true parameter γ∗. 4. Identifiability: If for γandγ′we have ˜λv(γ) =˜λv(γ′)forv∈ {1,···, p},then γ=γ′. Building on the seminal work of Ahmad and Francq [28], we now prove the strong consistency of... | https://arxiv.org/abs/2504.02649v1 |
the Markov likelihood is maximised using the method BFGS in scipy.optimize.minimize . For the initial values, we take a(v) m=b(v) m= 0 for all v= 1,···, pand m= 1,···, qandµ(v)=ψ−1 1 TpdPTp k=1Pd j=1Y(j) k . The period is set to p= 7 and for the baseline pre-intensity we take µv=1v≤3. 5.2.1 Estimation for the well sp... | https://arxiv.org/abs/2504.02649v1 |
coefficients ( a(v) q) and ( b(v) q), nor a ground truth characteristic time τnor an order q. Assuming that we have the prior knowledge that the network “forgets” its state after one period that is Tc=p= 7, we choose the characteristic time to be τ= 73 5≃4, in accordance with Remark 4.5. The order of the Markov approxi... | https://arxiv.org/abs/2504.02649v1 |
slowly compared to β’s extinction time ( Tc≃15vs.Tc≃5). The average kernel (lower panel, in orange) still captures the shape of the ground truth kernel quite well. 5.2.3 Estimation for the misspecified heavy-tailed model Heavy-tailed kernels are any element of ℓ1(N∗) whose decay is slower than that of an exponential, e... | https://arxiv.org/abs/2504.02649v1 |
the Bayes Information Criterion (BIC). Then we use both models to forecast the weekly number of cases of Rotavirus for each of the 12 districts and compare how they perform compared to each other. 6.1 Model comparison Throughout this section, we only consider linear Poisson autoregressions, that is, ψ(x) =x. Linearity ... | https://arxiv.org/abs/2504.02649v1 |
the tower property of the conditional expectation E[Yt+j|FY t] =E E[Yt+j|FY t+j−1]|FY t =E" µt+j+t+jX k=1ϕ(t+j) t+j−kYk FY t# =µt+j+tX k=1ϕ(t+j) t+j−kYk+t+jX k=t+1ϕ(t+j) t+j−kE[Yk|FY t]. The conditional value will play the role of our predictor in this section. After fitting both our model (18) and PNAR(1) (19) by ma... | https://arxiv.org/abs/2504.02649v1 |
The underlined value corresponds to the only district (Mitte) in which the PNAR(1) model predicts the weekly number of cases significantly better than our model. We can then conclude that our model (18) performs significantly better than the PNAR(1) process in 5 of the 12 districts and that it performs significantly wo... | https://arxiv.org/abs/2504.02649v1 |
depend on the “severity” of the claims: λt=ψ µt+t−1X k=1(α(t) t−kId+β(t) t−kW)Rk! . The stability condition becomes LE[Q]ρP k≥1max v=1,···,p(|α(v) k|Id+|β(v) k|W) <1 and the periodic stationarity results as well as those of the Markov approximation generalise naturally. The model can also be extended to include exoge... | https://arxiv.org/abs/2504.02649v1 |
innovations ( ζt)t∈Z, that is, an independent sequence such that ζt+phas the same distribution as ζt, for all t∈Z. Let q≥mbe two positive integers and let ˜X(m)and ˜X(q)be the periodically stationary solutions of ˜X(m) t=ft(˜X(m) t−1,˜X(m) t−2, . . . , ˜X(m) t−mp,0,···;ζt) and ˜X(q) t=ft(˜X(q) t−1,˜X(q) t−2, . . . , ˜X... | https://arxiv.org/abs/2504.02649v1 |
k=tk−(1+β)C ⪯t−(1+2β)C, which means that Uis in ℓ1(N). For a given n∈N∗, define Mn=P+∞ k=1kA∗n k∈ M d([0,+∞]). For a given n∈N∗we have that Mn+1=+∞X k=1kA∗(n+1) k =+∞X k=1k+∞X j=1A∗n jAk−j1j≤k−1 =+∞X j=1A∗n j+∞X k=1(k+j)Ak = +∞X j=1Aj n M1+Mn +∞X k=1Ak! , and since ρP+∞ k=1Ak <1, we conclude that ( Mn)n∈N∗∈ℓ1(N∗)... | https://arxiv.org/abs/2504.02649v1 |
x2,···;Nt) =Nt ψ µt++∞X k=1ϕ(t−k) kxk!! . A.6 Proof of Proposition 4.1 We start by proving the continuity result on Type I periodicity. Given t∈N∗we have E |Yt−¯Yt| =E Nt(λt)−Nt(¯λt) =E" Nt ψ µt+t−1X k=1ϕ(t) t−kYk!! −Nt ψ µt+t−1X k=1¯ϕ(t) t−k¯Yk!! # . 33 By conditioning on Ft−1and using the fact that ψisL−Lipschit... | https://arxiv.org/abs/2504.02649v1 |
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