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weights valued at R(xs0). For these weights we will consider two alternatives, the first one is the classical Nadaraya-Watson kernel estimator wi,n(xs0) =K1 ||bR(xs0)−bR(Xsi)|| h1 Pn j=1K1 ||bR(xs0)−bR(Xsj)|| h1, (23) where K1is aKernel function and h1is the smoothing parameter. And, in the second alternative (see ... | https://arxiv.org/abs/2502.02781v1 |
7. Simulation studies To demonstrate the performance of the prediction methods, this section presents several sim- ulation studies where we vary the sample size ( n), the number of covariates ( p), and the hyperpa- rameters randd. 7.1. Settings First, we randomly generate sample point locations s= (s1, s2)∈R2within the... | https://arxiv.org/abs/2502.02781v1 |
these simulation results show that generating spatial data under SEM or SSCM does not always guarantee better predictive performance when assuming these respective models in inverse model estimates. However, even if a spatial model is not the correct one, assuming it generally tends to result in better predictions comp... | https://arxiv.org/abs/2502.02781v1 |
tend to decrease. The spatial reduction methodologies exhibit very comparable errors among themselves and with respect to the reduction method under the independence assumption. However, for the smaller sample size, spatial reduction methods yield lower predictive errors. In all cases, they outperform predictions witho... | https://arxiv.org/abs/2502.02781v1 |
400 0.000 0.005 0.010 0.015 0.020 nMean of MSE 225 400 625p = 16 0.000 0.005 0.010 0.015 0.020 pMean MSE 8 16 24n = 625 0.000 0.005 0.010 0.015 0.020 nMean of MSE 225 400 625p = 24 Methods 2k.FULL 2k.Ind 2k.SSCM 2k.SEMFigure 1: Average cross-validated MSE (computed over 100 replications) for the SSCM model using differ... | https://arxiv.org/abs/2502.02781v1 |
values are higher in the west compared to the east and with less dispersion. Additionally, the values tend to be higher at the north and south extremes compared to the center. As a conclusion, it seems appropriate to consider a spatial model for predicting zinc concentration at new locations. In this example, since pis... | https://arxiv.org/abs/2502.02781v1 |
Ohio taking 25 continuous variables about building and infrastructure characteristics of the schools, teacher characteristics, spending allocation per pupil, census information of the districts and educational attainments of past students, among others. This school characteristics index is used to predict the average p... | https://arxiv.org/abs/2502.02781v1 |
we can obtain better predictors to fit the data. Between the index built with SDR assuming independence (Figure 7 (a)) and the one built under the SSCM model (Figure 7 (b)), a clear im- provement is observed in the relationship to be modeled between the said index and the score. Under the SEM models (Figure 7 (c)), a f... | https://arxiv.org/abs/2502.02781v1 |
GDP growth is observed in the period but with lower rates than the first ones. This is in line with the so-called Beta-convergence (Barro and Sala-i Martin, 1992) in the economic development literature, where economies with low per capita GDP tends to grow faster than high-income countries. However, for most African co... | https://arxiv.org/abs/2502.02781v1 |
different dimensions depending on the prediction method used. 9. Discussion In this section, we discuss the application of our proposed sufficient dimension reduction methodologies for spatial prediction, highlighting the results obtained from experiments con- ducted with real data. As emphasized, the predictive gains ... | https://arxiv.org/abs/2502.02781v1 |
method. Therefore, as we could have hypothesized, for data of a geostatistical nature the SSCM gives better results. The other two application involve lattice data but exhibit very different spatial characteristics. For the case of proficiency scores prediction in Ohio elementary school, we encountered an increased den... | https://arxiv.org/abs/2502.02781v1 |
Adragni, K. and Cook, R. (2009). Sufficient dimension reduction and prediction in regression. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 367(1906):4385–4405. Amidi, S. and Majidi, A. F. (2020). Geographic proximity, trade and economic growth: a spatial... | https://arxiv.org/abs/2502.02781v1 |
classification: Principal components and par- tial least squares. Technical report, Working Paper 246, Red Nacional de Investigadores en Econom ´ıa (RedNIE). Forzani, L., Garc ´ıa-Arancibia, R., Llop, P., and Tomassi, D. (2018). Supervised dimension re- duction for ordinal predictors. Computational Statistics and Data ... | https://arxiv.org/abs/2502.02781v1 |
of Geo-Information , 10(9). Reinsel, G. and Velu, R. (1998). Multivariate Reduced-Rank Regression: Theory and Applica- tions . Springer. Sampson, P. D., Richards, M., Szpiro, A. A., Bergen, S., Sheppard, L., Larson, T. V ., and Kauf- man, J. D. (2013). A regionalized national universal kriging model using partial least... | https://arxiv.org/abs/2502.02781v1 |
the next step is to esti- mate the mean µ. Taking derivative with respect to µin (19), except for some terms independent ofµwe have that, ∂l(X|Y, θ;µ,A,B,∆) ∂µ= =∂ ∂µ −1 2tr Wθ(X−1nµT−FBTAT)∆−1(X−1nµT−FBTAT)TWθ =∆−1(XT−µ1T n−ABFT)W2 θ1n, 31 which is 0 if and only if eµ= (XT−ABFT)W2 θ1n(1T nW2 θ1n)−1. (B.1) Therefor... | https://arxiv.org/abs/2502.02781v1 |
arXiv:2502.02793v1 [math.ST] 5 Feb 2025Early Stopping in Contextual Bandits and Inferences Zihan Cui∗ Abstract Bandit algorithms sequentially accumulate data using adap tive sampling policies, offering flexibility for real-world applications. However, excessive sampling can becostly, motivatingthedevolopmentof early stop... | https://arxiv.org/abs/2502.02793v1 |
the conditional distribution of online data on previous history rathe r than treating them as independent variables. In addition, adaptive algorithms and online inferences may be sensitive to noise or outliers in ∗Department of Statistics, University of Michigan. Email: c hildhan@umich.edu 1 the datastream, potentially... | https://arxiv.org/abs/2502.02793v1 |
would be a truncated Gaussian. Under this scenario, we propose a general conditional inference procedure using Gibbs sampling and demonstrate its application on both simulated and real data. 1.1 Related Literature Linear Contextual Bandit. Our work focuses on the linear contextual bandit setting. Many pa pers have disc... | https://arxiv.org/abs/2502.02793v1 |
probability follows Pπt(at|xt) =πt(at|xt). After taking an action at= 1 or 0, we will observe a reward ytwith respect to the corresponding arm. The observed reward ythas a linear form yt=atxT tβ1+(1−a1)xT tβ0+et E(et|at) = 0,E(e2 t|at) =σ2 et⊥xt|at LetFt=σ(x1,a1,y1,···,xt,at,yt) be the sigma field generated by the histo... | https://arxiv.org/abs/2502.02793v1 |
plicity, we assume that the sample sizesntin each round are the same, i.e., nt=n,t= 1,2,···,T0. In order to guarantee enough exploration using our online algorithm s, we impose clipping on them. Clipping. In each batch, we force the online policy to explore all arms by introd ucing a clipping probability pts.t.pt≤P(at=... | https://arxiv.org/abs/2502.02793v1 |
probability at least 1−δ. By combining Theorem 1 and Lemma 1, we express the upper bound of regretRπ∗ tas a function of time in both non-batched and batched settings, as stated in Corollary 1 and Corollary 2 respectively. Corollary 1. Under non-batched setting, the regret Rπ∗ t≤(2L√ K)1+λM(/radicalBig 1 tp2 t)1+λ=K′(/r... | https://arxiv.org/abs/2502.02793v1 |
are usually too conservative and prioritize safety, leading t o the result that the algorithm may stop either too early or too late. Besides, off-data stopping could b ring additional challenges in precision, robustness, and interpretability. Therefore, it is crucial to deve lop online stopping rules to address these is... | https://arxiv.org/abs/2502.02793v1 |
the same unbiased estimator of β but uses data up to batch t−1, with estimated variances St−1. If/bardblSt−1/bardbl − /bardblSt/bardbl ≤c′n, we stop the experiment at the end of batch t. Here, c′is the transformed opportunity cost in terms of variances, which is different from sampling costs c. In fact, this opportunity... | https://arxiv.org/abs/2502.02793v1 |
I(T=t) is measurable with respect to the sufficient statistics T1:t, then we can conduct the conditional inference totally based on T1:t. If we define the new filtration F′t=σ(T1:t), thenA1:t,1:n,Y1:t,1:n|Ft∼T1:t|F′t. 4.2.3 Online Estimators and Stopping rules Though Lemma 3 gives the joint distribution of the batched OLS ... | https://arxiv.org/abs/2502.02793v1 |
rules with threshold or opportunity cost based on the estimated variance of the statistics ˆβIV W t,1andˆβIV W t,0are: Online Stopping with Threshold. Stop the experiment when /bardblˆΣt,1/bardbl ≤k,/bardblˆΣt,0/bardbl ≤k. OnlineStoppingwith Opportunity Cost. Stoptheexperimentwhen /bardblˆΣt−1,1/bardbl−/bardblˆΣt,1/bar... | https://arxiv.org/abs/2502.02793v1 |
Robert Schapire. Contextua l bandits with linear payoff functions. In Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics , pages 208–214. JMLR Workshop and Conference Proceedings, 2 011. [7] Eyal Even-Dar, Shie Mannor, Yishay Mansour, and Sridhar Maha devan. Action eliminatio... | https://arxiv.org/abs/2502.02793v1 |
arXiv:2502.02848v1 [math.ST] 5 Feb 2025Kronecker sum covariance models for spatio-temporal data Shuheng Zhou Seyoung Park Kerby Shedden University of California, Riverside Yonsei University Uni versity of Michigan, Ann Arbor Abstract In this paper, we study the subgaussian matrix variate model , where we observe the ma... | https://arxiv.org/abs/2502.02848v1 |
can be decorrelated from the estimated cov ariance matrices, we cannot directly eliminate the noise matrix Wfrom the observed data X. However, statistical methods that utilize the second-ord er statistics of X, such as regression models and graph estimation, can still b enefit from the Kronecker sum model. Our approach ... | https://arxiv.org/abs/2502.02848v1 |
2.1, a reduction from Proposition 2.3. in [27] , illuminates this zero correlation condition for the genera l sub-gaussian matrix variate model (1), as well as the explicit relations between the regression coefficients βj∗ k, error variances for {V0,j,j∈[n]}, and the inverse covariance Θ = (θij)≻0. Proposition 2.1. [Zho... | https://arxiv.org/abs/2502.02848v1 |
εjandX0,−j. Proposition 2.2 describes the correlation between the res idual error vectorV0,jfor eachj∈[m]and them−1column vectors X0,i,i∈[m],i/ne}ationslash=jin the regression function (9). This result is new to the best of our knowledge. See [27] for re sults related to matrix-variate data with missing values. Proposi... | https://arxiv.org/abs/2502.02848v1 |
whereb0satisfiesφb2 0≤/vextenddouble/vextenddoubleβi∗/vextenddouble/vextenddouble2 2≤b2 0for some 0< φ <1. LetD′ 0=/ba∇dblB/ba∇dbl21/2+a1/2 max andC0be some positive absolute constant. Let ∀i,λi≥4ψi/radicalbig logm/n whereψi:=C0D′ 0K2/parenleftBig τ+/2 B/vextenddouble/vextenddoubleβi∗/vextenddouble/vextenddouble 2+σVi/p... | https://arxiv.org/abs/2502.02848v1 |
and some absolute constant C0, ∀i,/vextenddouble/vextenddouble/vextenddouble/hatwideγ(i)−/hatwideΓ(i)βi∗/vextenddouble/vextenddouble/vextenddouble ∞≤ψi/radicalbigg logm n,where (20) ψi:=C0K2D′ 0/parenleftBig τ+/2 B/vextenddouble/vextenddoubleβi∗/vextenddouble/vextenddouble 2+σVi/parenrightBig ≤C0K2D′ 0(τ+/2 Bκ(A)+σV). ... | https://arxiv.org/abs/2502.02848v1 |
have focused on the design of estim ators and computationally efficient algo- rithms while establishing sharp rates of convergence on the inverse covariance estimation. [27] developed multiple regression methods for estimating the inverse cov ariance matrices in a subgaussian matrix variate model, where the matrix varia... | https://arxiv.org/abs/2502.02848v1 |
by Lemma 4.2, we hav e for some absolute constant C, for allℓ∈[n], /vextenddouble/vextenddouble/vextenddoubleXℓ 0,iVℓ 0,j/vextenddouble/vextenddouble/vextenddouble ψ1≤/vextenddouble/vextenddouble/vextenddoubleXℓ 0,i/vextenddouble/vextenddouble/vextenddouble ψ2/vextenddouble/vextenddouble/vextenddoubleVℓ 0,j/vextenddoub... | https://arxiv.org/abs/2502.02848v1 |
∆ :=/hatwideΓ−A:=1 nXTX−1 n/hatwidetr(B)Im−A = (1 nXT 0X0−A)+1 n/parenleftbig WTX0+XT 0W/parenrightbig +1 n/parenleftbig WTW−/hatwidetr(B)Im/parenrightbig . First notice that /vextendsingle/vextendsingle/vextendsingle/hatwideΓA−A/vextendsingle/vextendsingle/vextendsingle max≤/vextendsingle/vextendsingle1 nXT 0X0−A/vext... | https://arxiv.org/abs/2502.02848v1 |
The 30th International Conference on Machin e Learning ICML- 13. [7] D UTILLEUL , P. (1999). The mle algorithm for the matrix normal distribu tion. Journal of Statistical Computation and Simulation 64105–123. [8] E FRON , B. (2009). Are a set of microarrays independ of each other? Annals of Applied Statistics 3 922–942... | https://arxiv.org/abs/2502.02848v1 |
Tech. rep. ArXiv preprint arXiv:2309.15355. [27] Z HOU , S. (2024). Concentration of measure bounds for matrix-var iate data with missing values. Bernoulli 30198–226. [28] Z HOU , S. and G REENEWALD , K. (2024). Sharper rates of convergence for the tensor grap hical lasso estimator. In Proceedings of the IEEE Internati... | https://arxiv.org/abs/2502.02848v1 |
subgaussian colum n vectors. Before we continue, we need to define some parameters related to the restricted and sparse eigenvalue conditions that are needed to state our main results on EIV re gression. We first state Definitions B.2- 3.5. For more details of these, see Rudelson and Zhou [18]. Definition B.2. (Restricted ... | https://arxiv.org/abs/2502.02848v1 |
on a class of generative models w hich rely on the sum of Kronecker product covariance matrices to model complex trial-wise dependenc ies as well as to provide a general statistical framework for dealing with signal and noise decomposition. It can be challenging to handle this type of data because there are often depen... | https://arxiv.org/abs/2502.02848v1 |
Loh and Wainwright [14]. Let L(β) :=1 2βT/hatwideΓ(i)β− /an}b∇acketle{t/hatwideγ(i),β/an}b∇acket∇i}ht. The gradient of the loss function is ∇L(β) =/hatwideΓ(i)β−/hatwideγ(i). The composite gradient descent algorithm produces a sequence of iterates {β(t), t= 0,1,2,···,}by β(t+1)=arg min /bardblβ/bardbl1≤b1L(β(t))+/an}b∇... | https://arxiv.org/abs/2502.02848v1 |
arXiv:2502.02887v2 [cs.IT] 27 Apr 2025Variations on the Expectation due to Changes in the Probability Measure Samir M. Perlaza∗†‡and Gaetan Bisson‡ ∗INRIA, Centre Inria d’Université Côte d’Azur, Sophia Antip olis, France. †ECE Dept. Princeton University, Princeton, 08544 NJ, USA. ‡Laboratoire GAATI, University of Frenc... | https://arxiv.org/abs/2502.02887v2 |
mutual and lautum information induced by the joint distribution /u1D443/u1D44C|/u1D44B/u1D443/u1D44B. Although these results were originally discovered in the analysis of generalization error of machine learning algo- rithms, see for instance [4]–[8], where the function ℎin (1) was assumed to represent an empirical ris... | https://arxiv.org/abs/2502.02887v2 |
probability measure, the measure /u1D443(ℎ,/u1D444,/u1D706) /u1D44C|/u1D44B=/u1D465, obtained by con- ditioning it upon a given vector /u1D465∈R/u1D45B, is referred to as an(ℎ,/u1D444,/u1D706)-Gibbs probability measure. The condition in (8) is easily met under certain assump- tions. For instance, if ℎis a nonnegative f... | https://arxiv.org/abs/2502.02887v2 |
uni fied statement of such results is presented hereunder. Lemma 3. Consider an (ℎ,/u1D444,/u1D706)-Gibbs probability measure, denoted by /u1D443(ℎ,/u1D444,/u1D706) /u1D44C|/u1D44B=/u1D465∈ △(R/u1D45A), with/u1D706≠0and/u1D465∈R. For all/u1D443∈ △/u1D444(R/u1D45A), Gℎ/parenleftBig /u1D465,/u1D443,/u1D443(ℎ,/u1D444,/u1D7... | https://arxiv.org/abs/2502.02887v2 |
to /u1D4431, the reference measure /u1D444in Theorem 4 can always be chosen as a convex combination of /u1D4431and/u1D4432. That is, for all Borel sets A ∈ℬ(R/u1D45A),/u1D444(A)=/u1D6FC/u1D4431(A)+ (1−/u1D706)/u1D4432(A), with/u1D6FC∈ (0,1). Theorem 4 can be especialized to the specific cases in which/u1D444is the Lebes... | https://arxiv.org/abs/2502.02887v2 |
that the terms −∫ /u1D437/parenleftBig /u1D443(1) /u1D44C|/u1D44B=/u1D465/bardblex/bardblex/bardblex/u1D444/parenrightBig d/u1D443/u1D44B(/u1D465) and−∫ /u1D437/parenleftBig /u1D443(2) /u1D44C|/u1D44B=/u1D465/bardblex/bardblex/bardblex/u1D444/parenrightBig d/u1D443/u1D44B(/u1D465)in (28) both become Shannon’s different... | https://arxiv.org/abs/2502.02887v2 |
Perlaza, G. Bisson, I. Esnaola, A. Jean-Marie, and S . Rini, “Empirical risk minimization with relative entropy regula rization,” IEEE Transactions on Information Theory , vol. 70, no. 7, pp. 5122 – 5161, Jul. 2024. [6] X. Zou, S. M. Perlaza, I. Esnaola, and E. Altman, “General ization Analysis of Machine Learning Algo... | https://arxiv.org/abs/2502.02887v2 |
DATA DENOISING WITH SELF CONSISTENCY, VARIANCE MAXIMIZATION, AND THE KANTOROVICH DOMINANCE BYJOSHUA ZOEN-GITHIEW1,b, TONGSEOK LIM*2,c, BRENDAN PASS1,a,AND MARCELO CRUZ DE SOUZA3,d 1Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB Canada,apass@ualberta.ca; bjoshuazo@ualberta.ca 2Mi... | https://arxiv.org/abs/2502.02925v1 |
order, Kantorovich dominance, Variance maximization, Optimal transport, Principal curve. 1arXiv:2502.02925v1 [stat.ME] 5 Feb 2025 2 of a martingale coupling π∈ M(µ,ν)defined in (4) below) can naturally be interpreted as, conditional on the signal being X, the average of the noise around Xvanishing, it also arises (roug... | https://arxiv.org/abs/2502.02925v1 |
but must still be in convex order with it) [21]. Neither the definition in [10] nor the one in [21] had a variational aspect analogous to (3), although the idea of minimizing the distance to the data was clearly present in the formulation of the self-consistency condition as discussed above. Heuristically, when Dis tak... | https://arxiv.org/abs/2502.02925v1 |
cones, we show that the new problem is equivalent to the relaxed problem. In addition, we show that, like (3), (20) enjoys quantifiable robustness properties as the noise becomes small, but, in contrast to (3), (20) is stable with respect to perturbations in the data distribution ν. We illustrate the properties of our ... | https://arxiv.org/abs/2502.02925v1 |
reduction, as it involves optimizing over µ∈ D, rather than (µ,π)withµ∈ D and π∈ M(µ,ν). The catch, of course, is that the constraint µ⪯Cνinvolves νin a sophisticated way and is not straightforward to check. We also consider the following relaxed version of (3), where the delicate self-consistency condition is removed:... | https://arxiv.org/abs/2502.02925v1 |
will follow if W2(µk,µ)→0. By [22, Theorem 7.12], it suffices to show that for any ϵ >0, there is R >0such thatR |x|≥R|x|2dµk(x)< ϵfor all k. By monotone convergence, there is a >0withR (2|x|2−a2)+dν < ϵ . On the other hand, Z (2|x|2−a2)+dν≥Z (2|x|2−a2)+dµk≥Z |x|≥a(2|x|2−a2)+dµk≥Z |x|≥a|x|2dµk, since µk⪯Cν. This comple... | https://arxiv.org/abs/2502.02925v1 |
proposed in [15]. EXAMPLE 2 (Das the set of measures on monotone increasing curves). A set Γ⊂R2is said to be monotone if, for any (x1,x2),(y1,y2)∈Γ, the following condition holds: (10) (y1−x1)(y2−x2)≥0. Let MON denote the collection of all monotone sets in R2. We define the search space as (11) DMON= µ∈ P(R2)|spt(µ)⊂Γ... | https://arxiv.org/abs/2502.02925v1 |
−11 xy µ∞ ν∞ FIG2. Support of µ∞=ν∞ SetMν,MON={µ∈ D MON|µ⪯Cν}. For each n∈N∪ {∞} , letµnbe the solution to (5) with respect to νn, which aims to maximize the variance by optimally spreading out the mass. This results in µn=µfor all n∈Ndue to the monotonicity constraint. However, in the limit n→ ∞ ,ν∞itself is monotone,... | https://arxiv.org/abs/2502.02925v1 |
some of the computational and theoretical difficulties associated with the convex order relation. Let⟨·,·⟩denote the inner product. For µ,ν∈ P2(Rd), we say that µis less than νin the Kantorovich dominance relation (KDR), and write µ⪯Kν, if there is π∈Π(µ,ν)such that (15)Z ⟨x,y−x⟩dπ(x,y) = 0. REMARK 4. If µ⪯Cν, then the... | https://arxiv.org/abs/2502.02925v1 |
topol- ogy. Since W2convergence is equivalent to weak convergence when Ωis compact (see Re- mark 2.8 of [1]), and MK ν,Ωis closed, the result follows. However, MK νis not necessarily compact in the W2topology, even if spt(ν)is compact. This motivates the consideration of the domain MK ν,Ωwith a compact Ωinstead of MK ν... | https://arxiv.org/abs/2502.02925v1 |
all domains we have considered. DEFINITION 4.2. Let ν∈ P2,0(Rd). We say that Disapproachable from the interior with respect to νif, for any µ∈ D withµ̸=δ0andµ⪯Kν, there exists a sequence {µn} ⊂ D such that W2(µn,µ)→0andD(µn,ν)>0in (16) for all n∈N. The class of domains that are closed under contractions serves as a key... | https://arxiv.org/abs/2502.02925v1 |
IfDis a cone, then for any µ∗∈argmaxµ∈D∩MK νVar(µ)andπ∗∈ argmaxπ∈Π(µ∗,ν)R ⟨x,y⟩dπ(x,y), the following equality holds: (22)Z ⟨x,y⟩dπ∗(x,y) =Z |x|2dµ∗(x). If, in addition, Dis translation invariant, then µ∗must be centered;R xdµ∗(x) = 0 . PROOF . By the KDR (16), for any µ∈ D ∩ MK νand for any corresponding optimal cou- ... | https://arxiv.org/abs/2502.02925v1 |
(26)Z ⟨x,y⟩dπ(x,y)≥Z |x|2dµ(x)and cπ#(µ)∈ D. For instance, the domains DmandDm uin Example 3 are closed under weak optimizers. THEOREM 4.11. IfDis closed under weak optimizers, then every optimizer µfor the problem (20) satisfies µ⪯Cν, and consequently, µsolves the original problem (5). PROOF . Assume that µsolves (20)... | https://arxiv.org/abs/2502.02925v1 |
ν. This result cannot hold if the full convex order constraint is imposed. In many practical cases, such as when νnis discrete (e.g., νnis an empirical measure sampled from ν), the convex order condition µ⪯Cνnfails for any Gaussian µ∈ D unless µ=δ0. As a result, the domain D ∩ { µ⪯Cνn}reduces to {δ0}ifνnis centered; ot... | https://arxiv.org/abs/2502.02925v1 |
is known for its speed and computational efficiency. To illustrate our example, we consider νas a discrete measure, defined over either n= 2000 or5000 points, where Y= (Z,Z2) +ε, with Zbeing a one-dimensional variable uniformly distributed over [−1,1], and εrepresenting Gaussian noise. The initial measure µ is supporte... | https://arxiv.org/abs/2502.02925v1 |
standard deviation. Since rescaling µdoes not change the ratio L(µ)/SD(µ), we see that D2is a cone. For each value of B, we set m= 300 points x1,...,x 300and linearly connect them. Figure 6 illustrates the resulting curves. We can see that as Bincreases, so does the curve length. Supplementary Material C describes the ... | https://arxiv.org/abs/2502.02925v1 |
Principal curves. Journal of the American Statistical Association , 84(406):502–516, 1989. [11] Pierre Henry-Labordère. Model-free hedging: A martingale optimal transport viewpoint . Chapman and Hall/CRC, 2017. [12] I.T. Jolliffe. Principal Component Analysis . Springer Series in Statistics. Springer, 2nd ed. edition, ... | https://arxiv.org/abs/2502.02925v1 |
for each constraint ensures that the Lagrange mul- tipliers only contribute when their respective constraints are active. Specifically: λ1X iui−1 = 0, λ 1,iui= 0∀i, λ 2 B−m−1X i=1∥xi+1−xi∥ = 0,and λ3X j∥yj∥2/n−X i∥xi∥2ui− W2 2(µ,ν) = 0. Letπbe an optimal coupling corresponding to W2 2(µ,ν). The gradients with res... | https://arxiv.org/abs/2502.02925v1 |
to be optimized as (x1 1,...,xd 1,...,x1 m,...,xd m,a1,...,a m−1) where the superscript in xidenotes the dimensional component and a1,...,a m−1is an aux- iliary variable. The second-moment constraint on xis replaced by the equivalent vuutmX i=1dX k=1(xk i)2≤1. The constraints are equivalently rewritten as xk 1+...+xk m... | https://arxiv.org/abs/2502.02925v1 |
Bayesian estimation of Unit-Weibull distribution based on dual generalized order statistics with application to the Cotton Production Data Qazi J. Azhada, Abdul Nasir Khanb1, Bhagwati Devic, Jahangir Sabbir Khand, Ayush Tripathie aDepartment of Mathematics, Shiv Nadar Institution of Eminence, Dadri, India bDepartment o... | https://arxiv.org/abs/2502.02927v1 |
al. (2020) delved into the analysis of multicomponent stress strength reliability using both classical and Bayesian approaches, focus- ing on the UW distribution. Alotaibi et al. (2021) obtained estimates of system reliability under known and unknown parameters using classical and Bayesian approaches when the data were... | https://arxiv.org/abs/2502.02927v1 |
generalized lower record values and when we set mi=−1along with k= 1, then the obtained model is known as ordinary lower record values, etc. To know more about models of ordered random variables readers are advised to go through the articles/books: David and Nagaraja (2004), Ahsanullah (2004), Arnold et al. (2008). Let... | https://arxiv.org/abs/2502.02927v1 |
L12=−"n−1X i=1(mi+ 1) (−lnxi)βln (−lnxi) +k(−lnxn)βln (−lnxn)# L122=−"n−1X i=1(mi+ 1) (−lnxi)β(ln (−lnxi))2+k(−lnxn)β(ln (−lnxn))2# L222=2n β3−α"n−1X i=1(mi+ 1) (−lnxi)β(ln (−lnxi))3+k(−lnxn)β(ln (−lnxn))3# L11=−n α2,L112= 0,L111=2n α3, ϕ1=a1−1 α−b1, ϕ2=a2−1 β−b2 .(3.5) Uti... | https://arxiv.org/abs/2502.02927v1 |
for α,βandR(t)under SELF. For Bayes estimator ofα, we have ζ(α, β) =αthis implies ψ∗(α, β) =ψ(α, β) +1 nlnα. (3.9) Now, differentiating (3.9) with respect to α,β, we get ∂ψ(α, β) ∂α+1 nα= 0 &∂ψ(α, β) ∂β= 0. Thus, the solution of first derivatives given above produces the MLE’s for αandβ. Now, determinant for the negati... | https://arxiv.org/abs/2502.02927v1 |
distribution (Geman and Geman (1987)). But for βit is not easy as it does not have a nice analytical form of any known probability distribution. So for this, we employ the idea of Metropolis Hasting (MH) algorithm with normal distribution (see Gelman et al. (2013)) as the proposal density. The readers are referred to A... | https://arxiv.org/abs/2502.02927v1 |
of Bayes estimates obtained using MCMC method for lower record values. From the table, it is observed that risks based on asymmetric loss functions (LINEX and GELF) are much smaller than symmetric loss function. It is also observed that mostly risks of estimators based on MCMC method are not smaller than risk of estima... | https://arxiv.org/abs/2502.02927v1 |
0.0569 0.0000 0.0177 100.0992 0.0397 0.0120 0.0116 0.0046 0.0015 0.0291 0.0000 0.0161 150.1127 0.0260 0.0130 0.0131 0.0031 0.0017 0.0323 0.0001 0.0161 (1.5,1)50.2304 0.1645 0.0315 0.0292 0.0188 0.0037 0.1231 0.0000 0.0273 100.2245 0.0721 0.0300 0.0297 0.0085 0.0036 0.0299 0.0034 0.0251 150.2122 0.0472 0.0272 0.0277 0.0... | https://arxiv.org/abs/2502.02927v1 |
b1, b2) = (2 ,2,2,2) (1,1)50.0399 0.0642 0.0079 0.0052 0.0079 0.0010 0.0077 0.0103 0.0017 100.0626 0.0643 0.0083 0.0078 0.0081 0.0011 0.0087 0.0078 0.0013 150.0621 0.0449 0.0075 0.0075 0.0055 0.0009 0.0076 0.0053 0.0011 (1.5,1)50.2937 0.0861 0.0284 0.0404 0.0111 0.0032 0.0419 0.0150 0.0034 100.0903 0.0572 0.0107 0.0105... | https://arxiv.org/abs/2502.02927v1 |
b2) = (2 ,2,2,2) (1,1)50.1697 0.1478 0.0145 0.0188 0.0167 0.0019 0.0178 0.0105 0.0136 100.0953 0.0894 0.0102 0.0108 0.0108 0.0012 0.0102 0.0071 0.0074 150.0737 0.0511 0.0087 0.0087 0.0058 0.0010 0.0079 0.0045 0.0051 (1.5,1)50.1812 0.1342 0.0176 0.0228 0.0125 0.0026 0.0194 0.0097 0.0128 100.1645 0.0758 0.0121 0.0156 0.0... | https://arxiv.org/abs/2502.02927v1 |
Exponential α= 0.27 -16.13396 34.26793 34.21384 Now, we will show that the transformed data supports UW distribution. For this we apply KS test and it is found that for α= 6.89, β= 7.67×10−5the KS statistics is 0.1958 with p− value 0.9525. As this article provides generalized nature of results for ordered random variab... | https://arxiv.org/abs/2502.02927v1 |
andPathak, A.K.(2023). BayesianinferenceofUnitGom- pertz distribution based on dual generalized order statistics. Communications in Statistics- Simulation and Computation , 52(8):3657–3675. Burkschat, M., Cramer, E., and Kamps, U. (2003). Dual generalized order statistics. Metron- International Journal of Statistics , ... | https://arxiv.org/abs/2502.02927v1 |
arXiv:2502.02986v1 [math.ST] 5 Feb 2025Matching Criterion for Identifiability in Sparse Factor Ana lysis Nils Sturma∗Miriam Kranzlmüller†Irem Portakal‡Mathias Drton§ Abstract Factor analysis models explain dependence among observed v ariables by a smaller number of unobserved factors. A main challenge in confirmatory fac... | https://arxiv.org/abs/2502.02986v1 |
analyses has been subj ect to much controversy, due to the difficulties in determining model identifiability ( Long,1983). A factor analysis model is identifiable if the loading matrix Λcan be recovered from the covariance matrix Σin (1). IfΛis not identifiable, then its estimates are to some degree arbitrary and standard i... | https://arxiv.org/abs/2502.02986v1 |
the remaining loadings have a large abso lute value. Developing methods for recovering a sparse factor loading matrix remains a very active field of research. Examples include regularization techniques ( Ning and Georgiou ,2011;Hirose and Konishi ,2012;Lan et al. , 2014;Trendafilov et al. ,2017;Scharf and Nestler ,2019;G... | https://arxiv.org/abs/2502.02986v1 |
efficient algorithms (Appendix B), all technical proofs (Appendix C), and an explanation of how to decide identifiability using algebraic tools (Appendix D). An implementation of the algorithms and code for reproduc ing the experiments is available at https://github.com/MiriamKranzlmueller/id-factor-an alysis . 2 Graphica... | https://arxiv.org/abs/2502.02986v1 |
matrixΛis zero, then existing criteria may also yield generic sign- identifiability. Definition 3.1. A factor analysis graph G= (V∪H,D)satisfies the Zero Upper Triangular Assumption (ZUTA) if there exists an ordering ≺on the latent nodes Hsuch that ch (h)is not contained in/uniontext ℓ≻hch(ℓ)for allh∈H. In this case, we s... | https://arxiv.org/abs/2502.02986v1 |
des. Theorem 3.7 (BB-identifiability, Bekker and ten Berge ,1997).LetG= (V∪H,D)be a full-ZUTA graph. Then, Gis generically sign-identifiable if |V|+|D|</parenleftbig|V|+1 2/parenrightbig . If a full-ZUTA graph is generic sign-identifiability by Theo rem3.7, then we term the graph BB- identifiable . Note that|V|+|D|=|V|(|H|... | https://arxiv.org/abs/2502.02986v1 |
generalize AR- and BB- identifiability for ZUTA graphs, and are capable to certify i dentifiability of models not covered by the AR- nor BB-criterion. 4.1 Matching Criterion Our first criterion takes the form of a recursive procedure an d is based on a graphical extension of the Anderson-Rubin criterion that can be applie... | https://arxiv.org/abs/2502.02986v1 |
h, then Condition (iii) ensures det([ΛΛ⊤]W,U)/\e}atio\slash= 0, and Condition (iv) ensures det([ΛΛ⊤]{v}∪W,{v}∪U) = 0 after removing the nodes in Sfrom the graph. The Laplace expansion of determinants then a llows us to find a rational formula for λ2 vhin terms of the entries of the covariance matrix. We can thus i denti... | https://arxiv.org/abs/2502.02986v1 |
graphs that are always M-identifiable. Corollary 4.13. LetG= (V∪H,D)be a factor analysis graph such that |ch(h)| ≥3for all h∈H. Moreover, assume that there exist relabelings of the laten t and observed nodes such that H={h1,...,h m}andV={v1,...,vp}withv2i−1,v2i∈ch(hi)andv2i−1,v2i/\e}atio\slash∈/uniontext j>ich(hj)for al... | https://arxiv.org/abs/2502.02986v1 |
the largest node according ≺h1. Then the ordering≺h1is aB-first-ordering and v1≺h1v7. Moreover, observe that jpa ({v1,v7}) ={h1}. Similarly, we can take any ordering ≺h2on ch(h2)such that v6is the largest node according ≺h2. Since jpa ({v1,v6}) ={h2}, we conclude that Condition (ii) is satisfied. h3,h4:TakeU={v5,...,v9}a... | https://arxiv.org/abs/2502.02986v1 |
5.2. Extended M-identifiability of a factor analysis graph G= (V∪H,D)is decidable in timeO(|H|2|V|max{k,l}+1(|V|+|H|)3)if we only allow sets Wwith|W|≤kin the matching criterion and only sets Bwith|B|≤ℓin the local BB-criterion. Proof. See Theorem B.5and Algorithm 6in Appendix B. If we allow the cardinality of the sets t... | https://arxiv.org/abs/2502.02986v1 |
14 3829 3192 221 366 368 15 4890 4147 404 768 768 16 5963 4972 759 1430 1435 17 6599 5490 1299 2187 2204 18 6937 5519 1927 2861 2913 19 6599 5047 2385 3164 3273 20 5963 4191 2509 3037 3179 21 4890 3157 2231 2520 2656 22 3829 2139 1705 1833 1913 23 2726 1310 1128 1177 1215 24 1880 710 651 665 683 25 1165 343 328 331 344... | https://arxiv.org/abs/2502.02986v1 |
the graphs from an Erdös-Renyi model with edge probabilities 0.2,0.25,0.3,0.35,0.4,0.45, where we fix the upper triangle of the adjacency matrix to zero to increase the prob ability of satisfying ZUTA. Moreover, we only consider graphs with at most 10children per latent node such that the maximal cardinality of a setBsa... | https://arxiv.org/abs/2502.02986v1 |
up to sign via the formula /radicalbiggσ23σ14 σ12σ34=/radicalbigg λ21φ12λ32λ11φ12λ42 λ11λ21λ32λ42=|φ12|. 18 Given|φ12|, we obtain generic identifiability of |λ11|via /radicalbiggσ13σ14 σ34|φ12|2=/radicaligg λ11φ12λ32λ11φ12λ42 λ32λ42φ2 12=|λ11|, and all other parameters can be recovered similarly. Notabl y, since each l... | https://arxiv.org/abs/2502.02986v1 |
the diagonal for a subset U⊆Vwith|U|=|H|. We can choose the subset Usuch that, for generically chosen Λ, the lower triangular matrix ΛU,His nonsingular. Observe that ΛQ∈RDimplies that the matrix T:= ΛU,HQalso has to be lower triangular. Hence, Q= Λ−1 U,HTis also lower triangular. But then Q−1=Q⊤has to be lower triangul... | https://arxiv.org/abs/2502.02986v1 |
tis a target node. The set of edges is given by the union D(iii)={s→w:w∈W} ∪{w→h:h∈H\S,w∈W,h→w∈D} ∪{h→u:h∈H\S,u∈U,h→u∈D} ∪{u→t:u∈U}. 21 (a)h1 h2h3 v1v2v3v4v5v6 (b)h1 h2 h3s tv3 v4v2 v6 (c)h1 h2 h3s tv1 v3 v4v2 v6v′ 1 Figure 13: Using maximum flow to verify whether the tuple (v,W,U,S ) = (v1,{v3,v4},{v2,v6},∅) satisfies C... | https://arxiv.org/abs/2502.02986v1 |
and W∩U=∅. Now, suppose for the other direction that MaxFlow(G(iii) flow(v,W,U,S )) =|W|. Based on the properties of the max-flow problem with integer capacities ( Ford and Fulkerson ,1962), this implies that there are|W|directed path from stot, each having a flow of size 1. Let/tildewideΠbe the collection of these paths.... | https://arxiv.org/abs/2502.02986v1 |
to see that the algorithm is sound and complete. Under the same constraints as in Theorem B.2, the complexity is at most O(|H|2|V|k+1(|V|+|H|)3). 24 Algorithm 3 Deciding full-ZUTA. Input: Factor analysis graph G= (V∪H,D)with|H|≤|V|. Initialize: p←|V|andm←|H| . 1:RelabelH←{h1,...,h m}such that ch (hi)≤ch(hi+1)for alli∈[... | https://arxiv.org/abs/2502.02986v1 |
check Condition (ii). Lemma B.4. LetG= (V∪H,D)be a factor analysis graph and consider a tuple (B,S)∈2V×2H such that G[B∪(jpa(B)\S)]is a full-ZUTA graph. Then, the tuple (B,S)satisfies Condition (ii) of the local BB-criterion if and only if Algorithm 4returns “yes”. Moreover, the algorithm has complexity at most O(|H|3|V... | https://arxiv.org/abs/2502.02986v1 |
p≥⌊m+1 2√8m+1+1 2⌋+1 forp=|U|andm=|jpa(B)\S|. Sincem≥1, it follows that|U|≥4. We conclude this section by providing a procedure for decidi ng extended M-identifiability in Algorithm 6. It is easy to see that the algorithm is sound and complete. Un der the same constraints as in Theorem B.2and in Theorem B.5, the complex... | https://arxiv.org/abs/2502.02986v1 |
al. (2010, Lemma 3.2). By the Cauchy-Binet determinant expansion formula, we have det([ΛΛ⊤]A,B) =/summationdisplay S⊆Hdet(ΛA,S)det(Λ B,S), where the sum runs over subsets S⊆Hwith|S|=|A|=|B|. LetM(S,A)be the set of all pair- ings ofSandA. By the Lindström-Gessel–Viennot lemma ( Gessel and Viennot ,1985;Lindström , 1973)... | https://arxiv.org/abs/2502.02986v1 |
In particular, det(/hatwideΣW,U)is not the zero polynomial. Since Λ∈RD was generically chosen, we conclude that det(/hatwideΣW,U)/\e}atio\slash= 0and we obtain λ2 v,h=B/det(/hatwideΣW,U). (6) 30 Now, define /tildewideAequivalent as Abut replace λ2 v,hin the upper left corner with /tildewideλ2 v,h. Recalling Equation ( 5... | https://arxiv.org/abs/2502.02986v1 |
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