Nancy Heckman

Source: Statist. Surv., Volume 6, 113--141.

Abstract:
The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares $sum_{j}(Y_{j}-mu(t_{j}))^{2}+lambda int_{a}^{b}[mu''(t)]^{2},dt$, where the data are $t_{j},Y_{j}$, $j=1,ldots,n$. The minimization is taken over an infinite-dimensional function space, the space of all functions with square integrable second derivatives. But the calculations can be carried out in a finite-dimensional space. The reduction from minimizing over an infinite dimensional space to minimizing over a finite dimensional space occurs for more general objective functions: the data may be related to the function $mu$ in another way, the sum of squares may be replaced by a more suitable expression, or the penalty, $int_{a}^{b}[mu''(t)]^{2},dt$, might take a different form. This paper reviews the Reproducing Kernel Hilbert Space structure that provides a finite-dimensional solution for a general minimization problem. Particular attention is paid to the construction and study of the Reproducing Kernel Hilbert Space corresponding to a penalty based on a linear differential operator. In this case, one can often calculate the minimizer explicitly, using Green’s functions.

Jeffrey Picka

Source: Statist. Surv., Volume 6, 74--112.

Abstract:
This paper gives an overview of statistical inference for disordered sphere packing processes. These processes are used extensively in physics and engineering in order to represent the internal structure of composite materials, packed bed reactors, and powders at rest, and are used as initial arrangements of grains in the study of avalanches and other problems involving powders in motion. Packing processes are spatial processes which are neither stationary nor ergodic. Classical spatial statistical models and procedures cannot be applied to these processes, but alternative models and procedures can be developed based on ideas from statistical physics. Most of the development of models and statistics for sphere packings has been undertaken by scientists and engineers. This review summarizes their results from an inferential perspective.

Data-space inversion (DSI) and related procedures represent a family of methods applicable for data assimilation in subsurface flow settings. These methods differ from model-based techniques in that they provide only posterior predictions for quantities (time series) of interest, not posterior models with calibrated parameters. DSI methods require a large number of flow simulations to first be performed on prior geological realizations. Given observed data, posterior predictions can then be generated directly. DSI operates in a Bayesian setting and provides posterior samples of the data vector. In this work we develop and evaluate a new approach for data parameterization in DSI. Parameterization reduces the number of variables to determine in the inversion, and it maintains the physical character of the data variables. The new parameterization uses a recurrent autoencoder (RAE) for dimension reduction, and a long-short-term memory (LSTM) network to represent flow-rate time series. The RAE-based parameterization is combined with an ensemble smoother with multiple data assimilation (ESMDA) for posterior generation. Results are presented for two- and three-phase flow in a 2D channelized system and a 3D multi-Gaussian model. The RAE procedure, along with existing DSI treatments, are assessed through comparison to reference rejection sampling (RS) results. The new DSI methodology is shown to consistently outperform existing approaches, in terms of statistical agreement with RS results. The method is also shown to accurately capture derived quantities, which are computed from variables considered directly in DSI. This requires correlation and covariance between variables to be properly captured, and accuracy in these relationships is demonstrated. The RAE-based parameterization developed here is clearly useful in DSI, and it may also find application in other subsurface flow problems.

Beginning from a basic neural-network architecture, we test the potential benefits offered by a range of advanced techniques for machine learning, in particular deep learning, in the context of a typical classification problem encountered in the domain of high-energy physics, using a well-studied dataset: the 2014 Higgs ML Kaggle dataset. The advantages are evaluated in terms of both performance metrics and the time required to train and apply the resulting models. Techniques examined include domain-specific data-augmentation, learning rate and momentum scheduling, (advanced) ensembling in both model-space and weight-space, and alternative architectures and connection methods.

Following the investigation, we arrive at a model which achieves equal performance to the winning solution of the original Kaggle challenge, whilst being significantly quicker to train and apply, and being suitable for use with both GPU and CPU hardware setups. These reductions in timing and hardware requirements potentially allow the use of more powerful algorithms in HEP analyses, where models must be retrained frequently, sometimes at short notice, by small groups of researchers with limited hardware resources. Additionally, a new wrapper library for PyTorch called LUMINis presented, which incorporates all of the techniques studied.

We focus on the challenge of finding a diverse collection of quality solutions on complex continuous domains. While quality diver-sity (QD) algorithms like Novelty Search with Local Competition (NSLC) and MAP-Elites are designed to generate a diverse range of solutions, these algorithms require a large number of evaluations for exploration of continuous spaces. Meanwhile, variants of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) are among the best-performing derivative-free optimizers in single-objective continuous domains. This paper proposes a new QD algorithm called Covariance Matrix Adaptation MAP-Elites (CMA-ME). Our new algorithm combines the self-adaptation techniques of CMA-ES with archiving and mapping techniques for maintaining diversity in QD. Results from experiments based on standard continuous optimization benchmarks show that CMA-ME finds better-quality solutions than MAP-Elites; similarly, results on the strategic game Hearthstone show that CMA-ME finds both a higher overall quality and broader diversity of strategies than both CMA-ES and MAP-Elites. Overall, CMA-ME more than doubles the performance of MAP-Elites using standard QD performance metrics. These results suggest that QD algorithms augmented by operators from state-of-the-art optimization algorithms can yield high-performing methods for simultaneously exploring and optimizing continuous search spaces, with significant applications to design, testing, and reinforcement learning among other domains.

The random-effects or normal-normal hierarchical model is commonly utilized in a wide range of meta-analysis applications. A Bayesian approach to inference is very attractive in this context, especially when a meta-analysis is based only on few studies. The bayesmeta R package provides readily accessible tools to perform Bayesian meta-analyses and generate plots and summaries, without having to worry about computational details. It allows for flexible prior specification and instant access to the resulting posterior distributions, including prediction and shrinkage estimation, and facilitating for example quick sensitivity checks. The present paper introduces the underlying theory and showcases its usage.

The R package mvord implements composite likelihood estimation in the class of multivariate ordinal regression models with a multivariate probit and a multivariate logit link. A flexible modeling framework for multiple ordinal measurements on the same subject is set up, which takes into consideration the dependence among the multiple observations by employing different error structures. Heterogeneity in the error structure across the subjects can be accounted for by the package, which allows for covariate dependent error structures. In addition, different regression coefficients and threshold parameters for each response are supported. If a reduction of the parameter space is desired, constraints on the threshold as well as on the regression coefficients can be specified by the user. The proposed multivariate framework is illustrated by means of a credit risk application.

This paper is devoted to the R package JSM which performs joint statistical modeling of survival and longitudinal data. In biomedical studies it has been increasingly common to collect both baseline and longitudinal covariates along with a possibly censored survival time. Instead of analyzing the survival and longitudinal outcomes separately, joint modeling approaches have attracted substantive attention in the recent literature and have been shown to correct biases from separate modeling approaches and enhance information. Most existing approaches adopt a linear mixed effects model for the longitudinal component and the Cox proportional hazards model for the survival component. We extend the Cox model to a more general class of transformation models for the survival process, where the baseline hazard function is completely unspecified leading to semiparametric survival models. We also offer a non-parametric multiplicative random effects model for the longitudinal process in JSM in addition to the linear mixed effects model. In this paper, we present the joint modeling framework that is implemented in JSM, as well as the standard error estimation methods, and illustrate the package with two real data examples: a liver cirrhosis data and a Mayo Clinic primary biliary cirrhosis data.

Jianqing Fan, Dong Wang, Kaizheng Wang, Ziwei Zhu.

Source: The Annals of Statistics, Volume 47, Number 6, 3009--3031.

Abstract:
Principal component analysis (PCA) is fundamental to statistical machine learning. It extracts latent principal factors that contribute to the most variation of the data. When data are stored across multiple machines, however, communication cost can prohibit the computation of PCA in a central location and distributed algorithms for PCA are thus needed. This paper proposes and studies a distributed PCA algorithm: each node machine computes the top $K$ eigenvectors and transmits them to the central server; the central server then aggregates the information from all the node machines and conducts a PCA based on the aggregated information. We investigate the bias and variance for the resulting distributed estimator of the top $K$ eigenvectors. In particular, we show that for distributions with symmetric innovation, the empirical top eigenspaces are unbiased, and hence the distributed PCA is “unbiased.” We derive the rate of convergence for distributed PCA estimators, which depends explicitly on the effective rank of covariance, eigengap, and the number of machines. We show that when the number of machines is not unreasonably large, the distributed PCA performs as well as the whole sample PCA, even without full access of whole data. The theoretical results are verified by an extensive simulation study. We also extend our analysis to the heterogeneous case where the population covariance matrices are different across local machines but share similar top eigenstructures.

Aryeh Kontorovich, Iosif Pinelis.

Source: The Annals of Statistics, Volume 47, Number 5, 2822--2854.

Abstract:
We provide an exact nonasymptotic lower bound on the minimax expected excess risk (EER) in the agnostic probably-approximately-correct (PAC) machine learning classification model and identify minimax learning algorithms as certain maximally symmetric and minimally randomized “voting” procedures. Based on this result, an exact asymptotic lower bound on the minimax EER is provided. This bound is of the simple form $c_{infty}/sqrt{ u}$ as $ u oinfty$, where $c_{infty}=0.16997dots$ is a universal constant, $ u=m/d$, $m$ is the size of the training sample and $d$ is the Vapnik–Chervonenkis dimension of the hypothesis class. It is shown that the differences between these asymptotic and nonasymptotic bounds, as well as the differences between these two bounds and the maximum EER of any learning algorithms that minimize the empirical risk, are asymptotically negligible, and all these differences are due to ties in the mentioned “voting” procedures. A few easy to compute nonasymptotic lower bounds on the minimax EER are also obtained, which are shown to be close to the exact asymptotic lower bound $c_{infty}/sqrt{ u}$ even for rather small values of the ratio $ u=m/d$. As an application of these results, we substantially improve existing lower bounds on the tail probability of the excess risk. Among the tools used are Bayes estimation and apparently new identities and inequalities for binomial distributions.

Joshua Cape, Minh Tang, Carey E. Priebe.

Source: The Annals of Statistics, Volume 47, Number 5, 2405--2439.

Abstract:
The singular value matrix decomposition plays a ubiquitous role throughout statistics and related fields. Myriad applications including clustering, classification, and dimensionality reduction involve studying and exploiting the geometric structure of singular values and singular vectors. This paper provides a novel collection of technical and theoretical tools for studying the geometry of singular subspaces using the two-to-infinity norm. Motivated by preliminary deterministic Procrustes analysis, we consider a general matrix perturbation setting in which we derive a new Procrustean matrix decomposition. Together with flexible machinery developed for the two-to-infinity norm, this allows us to conduct a refined analysis of the induced perturbation geometry with respect to the underlying singular vectors even in the presence of singular value multiplicity. Our analysis yields singular vector entrywise perturbation bounds for a range of popular matrix noise models, each of which has a meaningful associated statistical inference task. In addition, we demonstrate how the two-to-infinity norm is the preferred norm in certain statistical settings. Specific applications discussed in this paper include covariance estimation, singular subspace recovery, and multiple graph inference. Both our Procrustean matrix decomposition and the technical machinery developed for the two-to-infinity norm may be of independent interest.

Yen-Chi Chen, Adrian Dobra.

Source: The Annals of Applied Statistics, Volume 14, Number 1, 409--432.

Abstract:
Activity spaces are fundamental to the assessment of individuals’ dynamic exposure to social and environmental risk factors associated with multiple spatial contexts that are visited during activities of daily living. In this paper we survey existing approaches for measuring the geometry, size and structure of activity spaces, based on GPS data, and explain their limitations. We propose addressing these shortcomings through a nonparametric approach called density ranking and also through three summary curves: the mass-volume curve, the Betti number curve and the persistence curve. We introduce a novel mixture model for human activity spaces and study its asymptotic properties. We prove that the kernel density estimator, which at the present time, is one of the most widespread methods for measuring activity spaces, is not a stable estimator of their structure. We illustrate the practical value of our methods with a simulation study and with a recently collected GPS dataset that comprises the locations visited by 10 individuals over a six months period.

Ian L. Dryden, Kwang-Rae Kim, Charles A. Laughton, Huiling Le.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2213--2234.

Abstract:
Molecular dynamics simulations produce huge datasets of temporal sequences of molecules. It is of interest to summarize the shape evolution of the molecules in a succinct, low-dimensional representation. However, Euclidean techniques such as principal components analysis (PCA) can be problematic as the data may lie far from in a flat manifold. Principal nested spheres gives a fundamentally different decomposition of data from the usual Euclidean subspace based PCA [ Biometrika 99 (2012) 551–568]. Subspaces of successively lower dimension are fitted to the data in a backwards manner with the aim of retaining signal and dispensing with noise at each stage. We adapt the methodology to 3D subshape spaces and provide some practical fitting algorithms. The methodology is applied to cluster analysis of peptides, where different states of the molecules can be identified. Also, the temporal transitions between cluster states are explored.

Anton Thalmaier, James Thompson.

Source: Bernoulli, Volume 26, Number 3, 2202--2225.

Abstract:
In this article, we derive moment estimates, exponential integrability, concentration inequalities and exit times estimates for canonical diffusions firstly on sub-Riemannian limits of Riemannian foliations and secondly in the nonsmooth setting of $operatorname{RCD}^{*}(K,N)$ spaces. In each case, the necessary ingredients are Itô’s formula and a comparison theorem for the Laplacian, for which we refer to the recent literature. As an application, we derive pointwise Carmona-type estimates on eigenfunctions of Schrödinger operators.

Peter Bank, Yan Dolinsky.

Source: Bernoulli, Volume 26, Number 3, 2176--2201.

Abstract:
We prove a scaling limit theorem for the super-replication cost of options in a Cox–Ross–Rubinstein binomial model with transient price impact. The correct scaling turns out to keep the market depth parameter constant while resilience over fixed periods of time grows in inverse proportion with the duration between trading times. For vanilla options, the scaling limit is found to coincide with the one obtained by PDE-methods in ( Math. Finance 22 (2012) 250–276) for models with purely temporary price impact. These models are a special case of our framework and so our probabilistic scaling limit argument allows one to expand the scope of the scaling limit result to path-dependent options.

Galatia Cleanthous, Athanasios G. Georgiadis, Gerard Kerkyacharian, Pencho Petrushev, Dominique Picard.

Source: Bernoulli, Volume 26, Number 3, 1832--1862.

Abstract:
We consider the problem of estimating the density of observations taking values in classical or nonclassical spaces such as manifolds and more general metric spaces. Our setting is quite general but also sufficiently rich in allowing the development of smooth functional calculus with well localized spectral kernels, Besov regularity spaces, and wavelet type systems. Kernel and both linear and nonlinear wavelet density estimators are introduced and studied. Convergence rates for these estimators are established and discussed.

Binyan Jiang, Ziqi Chen, Chenlei Leng.

Source: Bernoulli, Volume 26, Number 2, 1234--1268.

Abstract:
High-dimensional data that evolve dynamically feature predominantly in the modern data era. As a partial response to this, recent years have seen increasing emphasis to address the dimensionality challenge. However, the non-static nature of these datasets is largely ignored. This paper addresses both challenges by proposing a novel yet simple dynamic linear programming discriminant (DLPD) rule for binary classification. Different from the usual static linear discriminant analysis, the new method is able to capture the changing distributions of the underlying populations by modeling their means and covariances as smooth functions of covariates of interest. Under an approximate sparse condition, we show that the conditional misclassification rate of the DLPD rule converges to the Bayes risk in probability uniformly over the range of the variables used for modeling the dynamics, when the dimensionality is allowed to grow exponentially with the sample size. The minimax lower bound of the estimation of the Bayes risk is also established, implying that the misclassification rate of our proposed rule is minimax-rate optimal. The promising performance of the DLPD rule is illustrated via extensive simulation studies and the analysis of a breast cancer dataset.

Jing Lei.

Source: Bernoulli, Volume 26, Number 1, 767--798.

Abstract:
We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence on the dimensionality. Our method also covers the important case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper bounds for functional data distributions whose coordinates decay geometrically or polynomially. Moreover, our bounds of the expected value can be combined with mean-concentration results to yield improved exponential tail probability bounds for the Wasserstein error of empirical measures under Bernstein-type or log Sobolev-type conditions.

Zhuang Ma, Xiaodong Li.

Source: Bernoulli, Volume 26, Number 1, 432--470.

Abstract:
Canonical correlation analysis (CCA) is a fundamental statistical tool for exploring the correlation structure between two sets of random variables. In this paper, motivated by the recent success of applying CCA to learn low dimensional representations of high dimensional objects, we propose two losses based on the principal angles between the model spaces spanned by the sample canonical variates and their population correspondents, respectively. We further characterize the non-asymptotic error bounds for the estimation risks under the proposed error metrics, which reveal how the performance of sample CCA depends adaptively on key quantities including the dimensions, the sample size, the condition number of the covariance matrices and particularly the population canonical correlation coefficients. The optimality of our uniform upper bounds is also justified by lower-bound analysis based on stringent and localized parameter spaces. To the best of our knowledge, for the first time our paper separates $p_{1}$ and $p_{2}$ for the first order term in the upper bounds without assuming the residual correlations are zeros. More significantly, our paper derives $(1-lambda_{k}^{2})(1-lambda_{k+1}^{2})/(lambda_{k}-lambda_{k+1})^{2}$ for the first time in the non-asymptotic CCA estimation convergence rates, which is essential to understand the behavior of CCA when the leading canonical correlation coefficients are close to $1$.

Luca M. Giordano, Maria Jolis, Lluís Quer-Sardanyons.

Source: Bernoulli, Volume 26, Number 1, 352--386.

Abstract:
In this article, we consider the quasi-linear stochastic wave and heat equations on the real line and with an additive Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index $Hin (0,1)$. The drift term is assumed to be globally Lipschitz. We prove that the solution of each of the above equations is continuous in terms of the index $H$, with respect to the convergence in law in the space of continuous functions.

The infections and deaths are spread across roughly two farms and 189 meat and processed food factories.

Sohan Seth, Iain Murray, Christopher K. I. Williams.

Source: Bayesian Analysis, Volume 14, Number 3, 703--725.

Abstract:
Model criticism is usually carried out by assessing if replicated data generated under the fitted model looks similar to the observed data, see e.g. Gelman, Carlin, Stern, and Rubin (2004, p. 165). This paper presents a method for latent variable models by pulling back the data into the space of latent variables, and carrying out model criticism in that space. Making use of a model's structure enables a more direct assessment of the assumptions made in the prior and likelihood. We demonstrate the method with examples of model criticism in latent space applied to factor analysis, linear dynamical systems and Gaussian processes.

Anna L. Smith, Dena M. Asta, Catherine A. Calder.

Source: Statistical Science, Volume 34, Number 3, 428--453.

Abstract:
We review the class of continuous latent space (statistical) models for network data, paying particular attention to the role of the geometry of the latent space. In these models, the presence/absence of network dyadic ties are assumed to be conditionally independent given the dyads’ unobserved positions in a latent space. In this way, these models provide a probabilistic framework for embedding network nodes in a continuous space equipped with a geometry that facilitates the description of dependence between random dyadic ties. Specifically, these models naturally capture homophilous tendencies and triadic clustering, among other common properties of observed networks. In addition to reviewing the literature on continuous latent space models from a geometric perspective, we highlight the important role the geometry of the latent space plays on properties of networks arising from these models via intuition and simulation. Finally, we discuss results from spectral graph theory that allow us to explore the role of the geometry of the latent space, independent of network size. We conclude with conjectures about how these results might be used to infer the appropriate latent space geometry from observed networks.

Gregorio Quintana-Ortí, Amelia Simó.

Source: Statistical Science, Volume 34, Number 2, 236--252.

Abstract:
This paper is focused on kernel regression when the response variable is the shape of a 3D object represented by a configuration matrix of landmarks. Regression methods on this shape space are not trivial because this space has a complex finite-dimensional Riemannian manifold structure (non-Euclidean). Papers about it are scarce in the literature, the majority of them are restricted to the case of a single explanatory variable, and many of them are based on the approximated tangent space. In this paper, there are several methodological innovations. The first one is the adaptation of the general method for kernel regression analysis in manifold-valued data to the three-dimensional case of Kendall’s shape space. The second one is its generalization to the multivariate case and the addressing of the curse-of-dimensionality problem. Finally, we propose bootstrap confidence intervals for prediction. A simulation study is carried out to check the goodness of the procedure, and a comparison with a current approach is performed. Then, it is applied to a 3D database obtained from an anthropometric survey of the Spanish child population with a potential application to online sales of children’s wear.

Stefan Klöppel
Mar 3, 2010; 30:3271-3275
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The following is a guest post by Martha H. Kennedy, Curator of Popular & Applied Graphic Art, Prints & Photographs Division. During Will Eisner Week, March 1-7, 2020, the Library of Congress joins art schools, libraries, universities, and museums in a global celebration of this legendary creator’s contributions to the world of comic art and […]

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