High-Dimensional Covariance Decomposition into Sparse Markov and Independence Domains

In this paper, we present a novel framework incorporating a combination of sparse models in different domains. We posit the observed data as generated from a linear combination of a sparse Gaussian Markov model (with a sparse precision matrix) and a sparse Gaussian independence model (with a sparse covariance matrix). We provide efficient methods for decomposition of the data into two domains, \viz Markov and independence domains. We characterize a set of sufficient conditions for identifiability and model consistency. Our decomposition method is based on a simple modification of the popular $\ell_1$-penalized maximum-likelihood estimator ($\ell_1$-MLE). We establish that our estimator is consistent in both the domains, i.e., it successfully recovers the supports of both Markov and independence models, when the number of samples $n$ scales as $n = \Omega(d^2 \log p)$, where $p$ is the number of variables and $d$ is the maximum node degree in the Markov model. Our conditions for recovery are comparable to those of $\ell_1$-MLE for consistent estimation of a sparse Markov model, and thus, we guarantee successful high-dimensional estimation of a richer class of models under comparable conditions. Our experiments validate these results and also demonstrate that our models have better inference accuracy under simple algorithms such as loopy belief propagation.Comment: Appears in Proceedings of the 29th International Conference on Machine Learning (ICML 2012

High-Dimensional Bayesian Geostatistics

With the growing capabilities of Geographic Information Systems (GIS) and user-friendly software, statisticians today routinely encounter geographically referenced data containing observations from a large number of spatial locations and time points. Over the last decade, hierarchical spatiotemporal process models have become widely deployed statistical tools for researchers to better understand the complex nature of spatial and temporal variability. However, fitting hierarchical spatiotemporal models often involves expensive matrix computations with complexity increasing in cubic order for the number of spatial locations and temporal points. This renders such models unfeasible for large data sets. This article offers a focused review of two methods for constructing well-defined highly scalable spatiotemporal stochastic processes. Both these processes can be used as "priors" for spatiotemporal random fields. The first approach constructs a low-rank process operating on a lower-dimensional subspace. The second approach constructs a Nearest-Neighbor Gaussian Process (NNGP) that ensures sparse precision matrices for its finite realizations. Both processes can be exploited as a scalable prior embedded within a rich hierarchical modeling framework to deliver full Bayesian inference. These approaches can be described as model-based solutions for big spatiotemporal datasets. The models ensure that the algorithmic complexity has $\sim n$ floating point operations (flops), where $n$ the number of spatial locations (per iteration). We compare these methods and provide some insight into their methodological underpinnings

Network Psychometrics

This chapter provides a general introduction of network modeling in psychometrics. The chapter starts with an introduction to the statistical model formulation of pairwise Markov random fields (PMRF), followed by an introduction of the PMRF suitable for binary data: the Ising model. The Ising model is a model used in ferromagnetism to explain phase transitions in a field of particles. Following the description of the Ising model in statistical physics, the chapter continues to show that the Ising model is closely related to models used in psychometrics. The Ising model can be shown to be equivalent to certain kinds of logistic regression models, loglinear models and multi-dimensional item response theory (MIRT) models. The equivalence between the Ising model and the MIRT model puts standard psychometrics in a new light and leads to a strikingly different interpretation of well-known latent variable models. The chapter gives an overview of methods that can be used to estimate the Ising model, and concludes with a discussion on the interpretation of latent variables given the equivalence between the Ising model and MIRT.Comment: In Irwing, P., Hughes, D., and Booth, T. (2018). The Wiley Handbook of Psychometric Testing, 2 Volume Set: A Multidisciplinary Reference on Survey, Scale and Test Development. New York: Wile

Functional Regression

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article will focus on functional regression, the area of FDA that has received the most attention in applications and methodological development. First will be an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: [1] functional predictor regression (scalar-on-function), [2] functional response regression (function-on-scalar) and [3] function-on-function regression. For each, the role of replication and regularization will be discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. At the end is a brief discussion describing potential areas of future development in this field

Localization for MCMC: sampling high-dimensional posterior distributions with local structure

We investigate how ideas from covariance localization in numerical weather prediction can be used in Markov chain Monte Carlo (MCMC) sampling of high-dimensional posterior distributions arising in Bayesian inverse problems. To localize an inverse problem is to enforce an anticipated "local" structure by (i) neglecting small off-diagonal elements of the prior precision and covariance matrices; and (ii) restricting the influence of observations to their neighborhood. For linear problems we can specify the conditions under which posterior moments of the localized problem are close to those of the original problem. We explain physical interpretations of our assumptions about local structure and discuss the notion of high dimensionality in local problems, which is different from the usual notion of high dimensionality in function space MCMC. The Gibbs sampler is a natural choice of MCMC algorithm for localized inverse problems and we demonstrate that its convergence rate is independent of dimension for localized linear problems. Nonlinear problems can also be tackled efficiently by localization and, as a simple illustration of these ideas, we present a localized Metropolis-within-Gibbs sampler. Several linear and nonlinear numerical examples illustrate localization in the context of MCMC samplers for inverse problems.Comment: 33 pages, 5 figure