Regularized Principal Component Analysis for Spatial Data

In many atmospheric and earth sciences, it is of interest to identify dominant spatial patterns of variation based on data observed at $p$ locations and $n$ time points with the possibility that $p>n$. While principal component analysis (PCA) is commonly applied to find the dominant patterns, the eigenimages produced from PCA may exhibit patterns that are too noisy to be physically meaningful when $p$ is large relative to $n$. To obtain more precise estimates of eigenimages, we propose a regularization approach incorporating smoothness and sparseness of eigenimages, while accounting for their orthogonality. Our method allows data taken at irregularly spaced or sparse locations. In addition, the resulting optimization problem can be solved using the alternating direction method of multipliers, which is easy to implement, and applicable to a large spatial dataset. Furthermore, the estimated eigenfunctions provide a natural basis for representing the underlying spatial process in a spatial random-effects model, from which spatial covariance function estimation and spatial prediction can be efficiently performed using a regularized fixed-rank kriging method. Finally, the effectiveness of the proposed method is demonstrated by several numerical example

Spatial modeling using graphical Markov models and wavelets

Graphical Markov models use graphs to represent possible dependencies among random variables. This class of models is extremely rich and includes inter alia causal Markov models and Markov random fields. In this dissertation, we develop a very efficient optimal-prediction algorithm for graphical Markov models. The algorithm is a generalization of the Kalman-filter algorithm for temporal processes, and it can in principle be applied to any Gaussian undirected graphical model and any Gaussian acyclic directed graphical model;We also propose a new class of multiscale models for stochastic processes in terms of scale-recursive dynamics defined on acyclic directed graphs. The models are an extension of multiscale tree-structured models. The optimal prediction can be obtained using the newly developed generalized Kalman-filter algorithm referred to above, and the parameters can be estimated by maximum likelihood via the EM algorithm. A subclass of these models are multiscale wavelet models, for which we show that the optimal predictors of hidden state variables can be obtained by a level-dependent (scale-dependent) wavelet shrinkage rule;In a series of papers, D. Donoho and I. Johnstone develop wavelet shrinkage methods to solve statistical problems. We propose a new rationale for wavelet shrinkage, based on the assumption that the underlying process can be decomposed into a large-scale deterministic trend plus a small-scale Gaussian process. Our approach has several advantages over current shrinkage methods. It takes the dependencies of empirical wavelet coefficients, both within scales and across scales, into account. Moreover, it does not rely on asymptotic properties for its justification so that it is also appropriate when the sample size is small;Finally, we introduce partially ordered Markov models, which are acyclic directed graphical models for spatial problems. The model can be regarded as a Markov random field with neighborhood structures derivable from an associated partially ordered set. We use a martingale approach to derive the asymptotic properties of maximum (composite) likelihood estimators for partially ordered Markov models. We prove that the maximum (composite) likelihood estimators are consistent, asymptotically normal, and also asymptotically efficient under checkable conditions

An Exploration of Sedentary Behavior, Physical Activity, and Quality of Life During the COVID-19 Outbreak

Objectives: Staying physically active is a cost-efficient strategy for disease prevention during a pandemic. The purposes of this study were to explore precautionary behaviors, psychological factors associated with physical activity and sedentary behavior, and impacts of active and sedentary lifestyles on the quality of life in the early stage of the coronavirus disease 2019 (COVID-19) outbreak.Methods: Participants were community-dwelling adults aged over 20 years who had not been infected with COVID-19 and who lived in the United States. A study with a cross-sectional design was conducted between July and October 2020. Quantitative data were collected by a self-reported questionnaire.Results: In total, 467 valid responses were obtained. Participants who engaged in an active lifestyle had significantly higher scores on all domains of quality of life compared to those who engaged in an inactive lifestyle. Participants with a non-sedentary lifestyle had significantly higher scores of psychological and social domains of quality of life than those with a sedentary lifestyle.Conclusion: Engaging in an active lifestyle and avoiding a sedentary lifestyle are recommended when facing future, unpredictable pandemics similar to COVID-19