Covariance Model Selection Research Articles

Approximate reference priors for Gaussian random fields

AbstractReference priors are theoretically attractive for the analysis of geostatistical data since they enable automatic Bayesian analysis and have desirable Bayesian and frequentist properties. But their use is hindered by computational hurdles that make their application in practice challenging. In this work, we derive a new class of default priors that approximate reference priors for the parameters of some Gaussian random fields. It is based on an approximation to the integrated likelihood of the covariance parameters derived from the spectral approximation of stationary random fields. This prior depends on the structure of the mean function and the spectral density of the model evaluated at a set of spectral points associated with an auxiliary regular grid. In addition to preserving the desirable Bayesian and frequentist properties, these approximate reference priors are more stable, and their computations are much less onerous than those of exact reference priors. Unlike exact reference priors, the marginal approximate reference prior of correlation parameter is always proper, regardless of the mean function or the smoothness of the correlation function. This property has important consequences for covariance model selection. An illustration comparing default Bayesian analyses is provided with a dataset of lead pollution in Galicia, Spain.

Investigation of multivariate seismic surrogate demand modeling for multi-response structural systems

Probabilistic seismic risk assessment of complex multi-response structural systems requires accurate estimations of multiple engineering demand parameters (EDPs) that are related to different failure modes. One conventional way is developing multiple separate univariate seismic demand models for each of the EDPs. However, such an approach lacks sufficient consideration of the correlations among different EDPs and also requires repetitive model tuning and training. Another alternative, which has not yet been extensively studied, is to develop a single multivariate surrogate demand model (MvSDM) that is able to jointly provide the estimates for all the EDPs. To facilitate more accurate and realistic seismic demand modeling of multi-response structural systems, the present study conducts a comprehensive investigation of different MvSDMs, which are constituted by a systematic trend model that characterizes the mean response hypersurface, and a covariance model that quantifies the correlated model errors. Three types of systematic trend models including multivariate linear regression (MvLR), partial least squares regression (PLSR) and artificial neural network (ANN); along with three model error covariance estimation methods including joint probabilistic demand model (JPSDM) method, sample regression residual covariance (SCOV) method and multi-output Gaussian process regression (MOGP), are considered. The influence of trend model and error covariance model selection on seismic demand and fragility modeling is then studied based on a typical three-span concrete-girder highway bridge structure in the Central and Southeastern US. The results suggest that kernel PLSR and ANN as the systematic trend models, and the SCOV method as the error covariance model are promising alternatives that can contribute to accurate and reliable multivariate surrogate demand modeling yet with good computational efficiency.

An R2 statistic for covariance model selection in the linear mixed model

ABSTRACTThe linear mixed model, sometimes referred to as the multi-level model, is one of the most widely used tools for analyses involving clustered data. Various definitions of have been proposed for the linear mixed model, but several limitations prevail. Presently, there is no method to compute for the linear mixed model that accommodates an interpretation based on variance partitioning, a method to quantify uncertainty and produce confidence limits for the statistic, and a capacity to use the statistic to conduct covariance model selection in a manner similar to information criteria. In this article, we introduce such an statistic. The proposed measures the proportion of generalized variance explained by fixed effects in the linear mixed model. Simulated and real longitudinal data are used to illustrate the statistical properties of the proposed and its capacity to be applied to covariance model selection.

Avoiding bias in mixed model inference for fixed effects

Analysis of a large longitudinal study of children motivated our work. The results illustrate how accurate inference for fixed effects in a general linear mixed model depends on the covariance model selected for the data. Simulation studies have revealed biased inference for the fixed effects with an underspecified covariance structure, at least in small samples. One underspecification common for longitudinal data assumes a simple random intercept and conditional independence of the within-subject errors (i.e., compound symmetry). We prove that the underspecification creates bias in both small and large samples, indicating that recruiting more participants will not alleviate inflation of the Type I error rate associated with fixed effect inference. Enumerations and simulations help quantify the bias and evaluate strategies for avoiding it. When practical, backwards selection of the covariance model, starting with an unstructured pattern, provides the best protection. Tutorial papers can guide the reader in minimizing the chances of falling into the often spurious software trap of nonconvergence. In some cases, the logic of the study design and the scientific context may support a structured pattern, such as an autoregressive structure. The sandwich estimator provides a valid alternative in sufficiently large samples. Authors reporting mixed-model analyses should note possible biases in fixed effects inference because of the following: (i) the covariance model selection process; (ii) the specific covariance model chosen; or (iii) the test approximation.

Working covariance model selection for generalized estimating equations

We investigate methods for data-based selection of working covariance models in the analysis of correlated data with generalized estimating equations. We study two selection criteria: Gaussian pseudolikelihood and a geodesic distance based on discrepancy between model-sensitive and model-robust regression parameter covariance estimators. The Gaussian pseudolikelihood is found in simulation to be reasonably sensitive for several response distributions and noncanonical mean-variance relations for longitudinal data. Application is also made to a clinical dataset. Assessment of adequacy of both correlation and variance models for longitudinal data should be routine in applications, and we describe open-source software supporting this practice.

Bayesian geostatistical design: Task‐driven optimal site investigation when the geostatistical model is uncertain

Geostatistical optimal design optimizes subsurface exploration for maximum information toward task‐specific prediction goals. Until recently, most geostatistical design studies have assumed that the geostatistical description (i.e., the mean, trends, covariance models and their parameters) is given a priori. This contradicts, as emphasized by Rubin and Dagan (1987a), the fact that only few or even no data at all offer support for such assumptions prior to the bulk of exploration effort. We believe that geostatistical design should (1) avoid unjustified a priori assumptions on the geostatistical description, (2) instead reduce geostatistical model uncertainty as secondary design objective, (3) rate this secondary objective optimal for the overall prediction goal, and (4) be robust even under inaccurate geostatistical assumptions. Bayesian Geostatistical Design follows these guidelines by considering uncertain covariance model parameters. We transfer this concept from kriging‐like applications to geostatistical inverse problems. We also deem it inappropriate to consider parametric uncertainty only within a single covariance model. The Matérn family of covariance functions has an additional shape parameter. Controlling model shape by a parameter converts covariance model selection to parameter identification and resembles Bayesian model averaging over a continuous spectrum of covariance models. This is appealing since it generalizes Bayesian model averaging from a finite number to an infinite number of models. We illustrate how our approach fulfills the above four guidelines in a series of synthetic test cases. The underlying scenarios are to minimize the prediction variance of (1) contaminant concentration or (2) arrival time at an ecologically sensitive location by optimal placement of hydraulic head and log conductivity measurements. Results highlight how both the impact of geostatistical model uncertainty and the sampling network design vary according to the choice of objective function.

Analysis of variance concepts and computations

AbstractThe analysis of variance (ANOVA) compares means among three or more groups of observations defined by a categorical variable. Any discussion of ANOVA involves intertwined considerations of (1) scientifically informed study design, (2) computational methods, and (3) statistical theory. Our understanding advances at roughly at the same rate in all three aspects. The earliest ANOVA methods were developed in 1925 and required independent Gaussian observations with common variance. The years 1925–1950 saw the flowering of complex variations for evaluating two or more grouping variables (factors) and the possibility of interaction among factors. An interaction occurs whenever the effect of one factor differs across levels of the other. Requiring equal numbers of observations in each group allowed developing sums‐of‐squares formulas practical for electromechanical calculators. The widespread adoption of digital electronic computers during 1950–1975 stimulated a shift to regression‐based calculations allowing unequal group sizes and the use of multivariate methods. During 1975–2000, statisticians generalized ANOVA based on mixed models to allow missing data. The more general methods require iterative solutions of nonlinear systems of equations. The dominance of the mixed‐model approach creates vulnerabilities to computing and modeling errors in practice. Basic numerical analysis concerns (variable scaling, location shift, and error accumulation) can interact badly with some types of coding dummy variables and covariance model selection. Future directions include automation of access to computation and assumption diagnostics. The proliferation of data with large numbers of variables or observations presents challenges at all three interfaces of scientific modeling, computing, and statistical theory. Copyright © 2009 John Wiley & Sons, Inc.This article is categorized under: Statistical Models > Linear Models Statistical Models > Classification Models

Spatial Analysis of Precision Agriculture Treatments in Randomized Complete Blocks: Guidelines for Covariance Model Selection

Failure to account for spatially correlated errors when present in the classical randomized complete block (RCB) analysis may cause inefficient estimation of treatment significance. Covariance model selection is a necessary component for spatial adjustment to estimate treatment significance. We discuss methods for selecting a covariance model in RCB analyses in the presence of spatial correlation and demonstrate one procedure in detail. The procedure uses three models: the randomized complete block with independent and identically distributed errors (RCBiid), RCB with correlated errors, and models with correlated errors but no block effects. The semivariogram of the residuals from fitting a model with just fixed effects, the likelihood ratio test, and Akaike Information Criterion are used for model selection. To illustrate the procedure, we analyzed winter wheat (Triticum aestivum L.) forage and corn (Zea mays L.) grain yield in the presence of spatial heterogeneity within blocks from a site‐specific N management study. We compared the selected covariance models to the RCBiid models and to other spatial models with respect to the estimation of treatment significance. The procedure can be extended to any experiment with fixed effects, or with both fixed and random effects, and which may potentially have spatially correlated errors. The procedure is systematic and readily implemented; however, it remains difficult to evaluate whether an adequate covariance model has been selected.

Generalized cross-validation for covariance model selection: D. Marcotte, Mathematical Geology, 27(5), 1995, pp 659–672

Generalized cross-validation for covariance model selection: D. Marcotte, Mathematical Geology, 27(5), 1995, pp 659–672

Generalized cross-validation for covariance model selection

A weighted cross-validation technique known in the spline literature as generalized cross-validation (GCV), is proposed for covariance model selection and parameter estimation. Weights for prediction errors are selected to give more importance to a cluster of points than isolated points. Clustered points are estimated better by their neighbors and are more sensitive to model parameters. This rational weighting scheme also provides a simplifying significantly the computation of the cross-validation mean square error of prediction. With small- to medium-size datasets, GCV is performed in a global neighborhood. Optimization of usual isotropic models requires only a small number of matrix inversions. A small dataset and a simulation are used to compare performances of GCV to ordinary cross-validation (OCV) and least-squares filling (LS).