In various applications with large spatial regions, the relationship between the response variable and the covariates is expected to exhibit complex spatial patterns. We propose a spatially clustered varying coefficient model, where the regression coefficients are allowed to vary smoothly within each cluster but change abruptly across the boundaries of adjacent clusters, and we develop a unified approach for simultaneous coefficient estimation and cluster identification. The varying coefficients are approximated by penalized splines, and the clusters are identified through a fused concave penalty on differences in neighboring locations, where the spatial neighbors are specified by the minimum spanning tree (MST). The optimization is solved efficiently based on the alternating direction method of multipliers, utilizing the sparsity structure from MST. Furthermore, we establish the oracle property of the proposed method considering the structure of MST. Numerical studies show that the proposed method can efficiently incorporate spatial neighborhood information and automatically detect possible spatially clustered patterns in the regression coefficients. An empirical study in oceanography illustrates that the proposed method is promising to provide informative results.
Spatial regression or geographically weighted regression models have been widely adopted to capture the effects of auxiliary information on a response variable of interest over a region. In contrast, relationships between response and auxiliary variables are expected to exhibit complex spatial patterns in many applications. This paper proposes a new approach for spatial regression, called spatially clustered regression, to estimate possibly clustered spatial patterns of the relationships. We combine K-means-based clustering formulation and penalty function motivated from a spatial process know…
Small Area Estimation under Spatial Regimes: Spatially Clustered Fay-Herriot Models for Agricultural Indicators
University of Milano-Bicocca（米兰比可卡大学）；Fondazione Eni Enrico Mattei (FEEM)（埃尼·恩里科·马泰伊基金会）；University of Campania "Luigi Vanvitelli"（坎帕尼亚路易吉·范维特利大学）；Keio University（庆应义塾大学）
In this work, we present an additive model for space-time data that splits the data into a temporally correlated component and a spatially correlated component. We model the spatially correlated portion using a time-varying Gaussian graphical model. Under assumptions on the smoothness of changes in covariance matrices, we derive strong single sample convergence results, confirming our ability to estimate meaningful graphical structures as they evolve over time. We apply our methodology to the discovery of time-varying spatial structures in human brain fMRI signals.
Rare populations, such as endangered animals and plants, drug users and individuals with rare diseases, tend to cluster in regions. Adaptive cluster sampling is generally applied to obtain information from clustered and sparse populations since it increases survey effort in areas where the individuals of interest are observed. This work aims to propose a unit-level model which assumes that counts are related to auxiliary variables, improving the sampling process, assigning different weights to the cells, besides referring them spatially. The proposed model fits rare and grouped populations, di…
In medical studies, the collected covariates usually contain underlying outliers. For clustered /longitudinal data with censored observations, the traditional Gehan-type estimator is robust to outliers existing in response but sensitive to outliers in the covariate domain, and it also ignores the within-cluster correlations. To take account of within-cluster correlations, varying cluster sizes, and outliers in covariates, we propose weighted Gehan-type estimating functions for parameter estimation in the accelerated failure time model for clustered data. We provide the asymptotic properties of…
This study develops a spatially varying coefficient model by extending the random effects eigenvector spatial filtering model. The developed model has the following properties: its coefficients are interpretable in terms of the Moran coefficient; each of its coefficients can have a different degree of spatial smoothness; and it yields a variant of a Bayesian spatially varying coefficient model. Also, parameter estimation of the model can be executed with a relatively small computationally burden. Results of a Monte Carlo simulation reveal that our model outperforms a conventional eigenvector s…
Motivated by recent work on studying massive imaging data in various neuroimaging studies, we propose a novel spatially varying coefficient model (SVCM) to spatially model the varying association between imaging measures in a three-dimensional (3D) volume (or 2D surface) with a set of covariates. Two key features of most neuorimaging data are the presence of multiple piecewise smooth regions with unknown edges and jumps and substantial spatial correlations. To specifically account for these two features, SVCM includes a measurement model with multiple varying coefficient functions, a jumping s…
Two-stage hierarchical models have been widely used in small area estimation to produce indirect estimates of areal means. When the areas are treated exchangeably and the model parameters are assumed to be the same over all areas, we might lose the efficiency in the presence of spatial heterogeneity. To overcome this problem, we consider a two-stage area-level model based on natural exponential family with spatially varying model parameters. We employ geographically weighted regression approach to estimating the varying parameters and suggest a new empirical Bayes estimator of the areal mean. …
We consider the inference problem for high-dimensional linear models, when covariates have an underlying spatial organization reflected in their correlation. A typical example of such a setting is high-resolution imaging, in which neighboring pixels are usually very similar. Accurate point and confidence intervals estimation is not possible in this context with many more covariates than samples, furthermore with high correlation between covariates. This calls for a reformulation of the statistical inference problem, that takes into account the underlying spatial structure: if covariates are lo…
Spatially varying coefficient (SVC) models are a type of regression model for spatial data where covariate effects vary over space. If there are several covariates, a natural question is which covariates have a spatially varying effect and which not. We present a new variable selection approach for Gaussian process-based SVC models. It relies on a penalized maximum likelihood estimation (PMLE) and allows variable selection both with respect to fixed effects and Gaussian process random effects. We validate our approach both in a simulation study as well as a real world data set. Our novel appro…
Isotropic covariance structures can be unreasonable for phenomena in three-dimensional spaces such as the ocean. In the ocean, the variability of the response may vary with depth, and ocean currents may lead to spatially varying anisotropy. We construct a class of non-stationary anisotropic Gaussian random fields (GRFs) in three dimensions through stochastic partial differential equations (SPDEs) where computations are done using Gaussian Markov random field approximations. The approach is proven in a simulation study where the amount of data required to estimate these models is explored. Then…
Consider a population consisting of clusters of sampling units, evolving temporally, spatially, or according to other dynamics. We wish to monitor the evolution of its means, medians, or other parameters. For administrative convenience and informativeness, clustered data are often collected via a rotating plan. Under rotating plans, the observations in the same clusters are correlated, and observations on the same unit collected on different occasions are also correlated. Ignoring this correlation structure may lead to invalid inference procedures. Accommodating cluster structure in parametric…
Gaussian processes (GPs) are well-known tools for modeling dependent data with applications in spatial statistics, time series analysis, or econometrics. In this article, we present the R package varycoef that implements estimation, prediction, and variable selection of linear models with spatially varying coefficients (SVC) defined by GPs, so called GP-based SVC models. Such models offer a high degree of flexibility while being relatively easy to interpret. Using varycoef, we show versatile applications of (spatially) varying coefficient models on spatial and time series data. This includes m…
Q-Learning with Clustered-SMART (cSMART) Data: Examining Moderators in the Construction of Clustered Adaptive Interventions
Department of Biostatistics（生物统计学系）；School of Public Health University of Michigan（公共卫生学院）；University of Michigan（密歇根大学）；Department of Learning Health Sciences University of Michigan（学习健康科学系）；Office of Research and Development U.S. Department of Veterans Affairs（美国退伍军人事务部研发办公室）；Department of Family Medicine and Community Health University of Wisconsin Madison WI USA（家庭医学与社区健康系）；University of Wisconsin Madison WI USA（威斯康星大学麦迪逊分校）；Department of Statistics University of Michigan（统计学系）；Survey Research Center Institute for Social Research Ann Arbor MI USA（调查研究中心社会研究研究所）
While spatially varying coefficient (SVC) modeling is popular in applied science, its computational burden is substantial. This is especially true if a multiscale property of SVC is considered. Given this background, this study develops a Moran's eigenvector-based spatially varying coefficients (M-SVC) modeling approach that estimates multiscale SVCs computationally efficiently. This estimation is accelerated through a (i) rank reduction, (ii) pre-compression, and (iii) sequential likelihood maximization. Steps (i) and (ii) eliminate the sample size N from the likelihood function; after these …
In regression models for spatial data, it is often assumed that the marginal effects of covariates on the response are constant over space. In practice, this assumption might often be questionable. In this article, we show how a Gaussian process-based spatially varying coefficient (SVC) model can be estimated using maximum likelihood estimation (MLE). In addition, we present an approach that scales to large data by applying covariance tapering. We compare our methodology to existing methods such as a Bayesian approach using the stochastic partial differential equation (SPDE) link, geographical…
Parameter estimation is a growing area of interest in statistical signal processing. Some parameters in real-life applications vary in space as opposed to those that are static. Most common methods in estimating parameters involve solving an optimization problem where the cost function is assembled variously; for example, maximum likelihood and maximum a posteriori methods. However, these methods do not have exact solutions to most real-life problems. It is for this reason that Monte Carlo methods are preferred. In this paper, we treat the estimation of parameters which vary with space. We use…
In this paper, we study the problem of inferring spatially-varying Gaussian Markov random fields (SV-GMRF) where the goal is to learn a network of sparse, context-specific GMRFs representing network relationships between genes. An important application of SV-GMRFs is in inference of gene regulatory networks from spatially-resolved transcriptomics datasets. The current work on inference of SV-GMRFs are based on the regularized maximum likelihood estimation (MLE) and suffer from overwhelmingly high computational cost due to their highly nonlinear nature. To alleviate this challenge, we propose a…
