Strategies for Fitting Large,Geostatistical Data in MCMC Simulation

Models for geostatistical data introduce spatial dependence in the covariance matrix of location-specific random effects. This is usually defined to be a parametric function of the distances between locations. Bayesian formulations of such models overcome asymptotic inference and estimation problems involved in maximum likelihood-based approaches and can be fitted using Markov chain Monte Carlo (MCMC) simulation. The MCMC implementation, however, requires repeated inversions of the covariance matrix which makes the problem computationally intensive, especially for large number of locations. In the present work, we propose to convert the spatial covariance matrix to a sparse matrix and compare a number of numerical algorithms especially suited within the MCMC framework in order to accelerate large matrix inversion. The algorithms are assessed empirically on simulated datasets of different size and sparsity. We conclude that the band solver applied after ordering the distance matrix reduces the computational time in inverting covariance matrices substantially.     

Rank covariance matrix estimation of a partially known covariance matrix

Classical multivariate methods are often based on the sample covariance matrix, which is very sensitive to outlying observations. One alternative to the covariance matrix is the affine equivariant rank covariance matrix (RCM) that has been studied in Visuri et al. [2003. Affine equivariant multivariate rank methods. J. Statist. Plann. Inference 114, 161–185]. In this article we assume that the covariance matrix is partially known and study how to estimate the corresponding RCM. We use the properties that the RCM is affine equivariant and that the RCM is proportional to the inverse of the regular covariance matrix, and hence reduce the problem of estimating the original RCM to estimating marginal rank covariance matrices. This is a great computational advantage when the dimension of the original data vector is large.     

The Asymptotic Covariance Matrix of the Least Squares Estimator in the Stochastic Linear Regression Model: The Case of Elliptically Symmetric Distribution

This article considers the unconditional asymptotic covariance matrix of the least squares estimator in the linear regression model with stochastic explanatory variables. The asymptotic covariance matrix of the least squares estimator of regression parameters is evaluated relative to the standard asymptotic covariance matrix when the joint distribution of the dependent and explanatory variables is in the class of elliptically symmetric distributions. An empirical example using financial data is presented. Numerical examples and simulation experiments are given to illustrate the difference of the two asymptotic covariance matrices.     

Robust estimation of a high-dimensional integrated covariance matrix

In this article, we consider a robust method of estimating a realized covariance matrix calculated as the sum of cross products of intraday high-frequency returns. According to recent articles in financial econometrics, the realized covariance matrix is essentially contaminated with market microstructure noise. Although techniques for removing noise from the matrix have been studied since the early 2000s, they have primarily investigated a low-dimensional covariance matrix with statistically significant sample sizes. We focus on noise-robust covariance estimation under converse circumstances, that is, a high-dimensional covariance matrix possibly with a small sample size. For the estimation, we utilize a statistical hypothesis test based on the characteristic that the largest eigenvalue of the covariance matrix asymptotically follows a Tracy–Widom distribution. The null hypothesis assumes that log returns are not pure noises. If a sample eigenvalue is larger than the relevant critical value, then we fail to reject the null hypothesis. The simulation results show that the estimator studied here performs better than others as measured by mean squared error. The empirical analysis shows that our proposed estimator can be adopted to forecast future covariance matrices using real data.     

Classification Rules under Autoregressive and General Circulant Covariance

We develop classification rules for data that have an autoregressive circulant covariance structure under the assumption of multivariate normality. We also develop classification rules assuming a general circulant covariance structure. The new classification rules are efficient in reducing the misclassification error rates when the number of observations is not large enough to estimate the unknown variance–covariance matrix. The proposed classification rules are demonstrated by simulation study for their validity and illustrated by a real data analysis for their use. Analyses of both simulated data and real data show the effectiveness of our new classification rules.     

The Trace Restriction: An Alternative Identification Strategy for the Bayesian Multinomial Probit Model

Previous authors have made Bayesian multinomial probit models identifiable by fixing a parameter on the main diagonal of the covariance matrix. The choice of which element one fixes can influence posterior predictions. Thus, we propose restricting the trace of the covariance matrix, which we achieve without computational penalty. This permits a prior that is symmetric to permutations of the nonbase outcome categories. We find in real and simulated consumer choice datasets that the trace-restricted model is less prone to making extreme predictions. Further, the trace restriction can provide stronger identification, yielding marginal posterior distributions that are more easily interpreted.     

Adaptive robust estimation in joint mean–covariance regression model for bivariate longitudinal data

The estimation of the covariance matrix is important in the analysis of bivariate longitudinal data. A good estimator for the covariance matrix can improve the efficiency of the estimators of the mean regression coefficients. Furthermore, the covariance estimation itself is also of interest, but it is a challenging job to model the covariance matrix of bivariate longitudinal data due to the complex structure and positive definite constraint. In addition, most of existing approaches are based on the maximum likelihood, which is very sensitive to outliers or heavy-tail error distributions. In this article, an adaptive robust estimation method is proposed for bivariate longitudinal data. Unlike the existing likelihood-based methods, the proposed method can adapt to different error distributions. Specifically, at first, we utilize the modified Cholesky block decomposition to parameterize the covariance matrices. Secondly, we apply the bounded Huber's score function to develop a set of robust generalized estimating equations to estimate the parameters both in the mean and the covariance models simultaneously. A data-driven approach is presented to select the parameter c in the Huber's score function, which can ensure that the proposed method is robust and efficient. A simulation study and a real data analysis are conducted to illustrate the robustness and efficiency of the proposed approach.     

Spatial sampling design for parameter estimation of the covariance function

We study the spatial optimal sampling design for covariance parameter estimation. The spatial process is modeled as a Gaussian random field and maximum likelihood (ML) is used to estimate the covariance parameters. We use the log determinant of the inverse Fisher information matrix as the design criterion and run simulations to investigate the relationship between the inverse Fisher information matrix and the covariance matrix of the ML estimates. A simulated annealing algorithm is developed to search for an optimal design among all possible designs on a fine grid. Since the design criterion depends on the unknown parameters, we define relative efficiency of a design and consider minimax and Bayesian criteria to find designs that are robust for a range of parameter values. Simulation results are presented for the Matérn class of covariance functions.     

On discrimination and classification with multivariate repeated measures data

We study the problem of classification with multiple q-variate observations with and without time effect on each individual. We develop new classification rules for populations with certain structured and unstructured mean vectors and under certain covariance structures. The new classification rules are effective when the number of observations is not large enough to estimate the variance–covariance matrix. Computational schemes for maximum likelihood estimates of required population parameters are given. We apply our findings to two real data sets as well as to a simulated data set.     

Modified linear projection for large spatial datasets

Recent developments in engineering techniques for spatial data collection such as geographic information systems have resulted in an increasing need for methods to analyze large spatial datasets. These sorts of datasets can be found in various fields of the natural and social sciences. However, model fitting and spatial prediction using these large spatial datasets are impractically time-consuming, because of the necessary matrix inversions. Various methods have been developed to deal with this problem, including a reduced rank approach and a sparse matrix approximation. In this article, we propose a modification to an existing reduced rank approach to capture both the large- and small-scale spatial variations effectively. We have used simulated examples and an empirical data analysis to demonstrate that our proposed approach consistently performs well when compared with other methods. In particular, the performance of our new method does not depend on the dependence properties of the spatial covariance functions.     

Modeling a mixture of ordinal and continuous repeated measures

We study the correlation structure for a mixture of ordinal and continuous repeated measures using a Bayesian approach. We assume a multivariate probit model for the ordinal variables and a normal linear regression for the continuous variables, where latent normal variables underlying the ordinal data are correlated with continuous variables in the model. Due to the probit model assumption, we are required to sample a covariance matrix with some of the diagonal elements equal to one. The key computational idea is to use parameter-extended data augmentation, which involves applying the Metropolis-Hastings algorithm to get a sample from the posterior distribution of the covariance matrix incorporating the relevant restrictions. The methodology is illustrated through a simulated example and through an application to data from the UCLA Brain Injury Research Center.     

Singular Gaussian graphical models: Structure learning

ABSTRACT

The goal of this article is to introduce singular Gaussian graphical models and their conditional independence properties. In fact, we extend the concept of Gaussian Markov Random Field to the case of a multivariate normally distributed vector with a singular covariance matrix. We construct, then, the associated graph’s structure from the covariance matrix’s pseudo-inverse on the basis of a characterization of the pairwise conditional independence. The proposed approach can also be used when the covariance matrix is ill-conditioned, through projecting data on a smaller subspace. In this case, our method ensures numerical stability and consistency of the constructed graph and significantly reduces the inference problem’s complexity. These aspects are illustrated using numerical experiments.     

Covariance structure approximation via gLasso in high-dimensional supervised classification

Recent work has shown that the Lasso-based regularization is very useful for estimating the high-dimensional inverse covariance matrix. A particularly useful scheme is based on penalizing the ?₁ norm of the off-diagonal elements to encourage sparsity. We embed this type of regularization into high-dimensional classification. A two-stage estimation procedure is proposed which first recovers structural zeros of the inverse covariance matrix and then enforces block sparsity by moving non-zeros closer to the main diagonal. We show that the block-diagonal approximation of the inverse covariance matrix leads to an additive classifier, and demonstrate that accounting for the structure can yield better performance accuracy. Effect of the block size on classification is explored, and a class of asymptotically equivalent structure approximations in a high-dimensional setting is specified. We suggest a variable selection at the block level and investigate properties of this procedure in growing dimension asymptotics. We present a consistency result on the feature selection procedure, establish asymptotic lower an upper bounds for the fraction of separative blocks and specify constraints under which the reliable classification with block-wise feature selection can be performed. The relevance and benefits of the proposed approach are illustrated on both simulated and real data.     

New HEAVY Models for Fat-Tailed Realized Covariances and Returns

ABSTRACT

We develop a new score-driven model for the joint dynamics of fat-tailed realized covariance matrix observations and daily returns. The score dynamics for the unobserved true covariance matrix are robust to outliers and incidental large observations in both types of data by assuming a matrix-F distribution for the realized covariance measures and a multivariate Student's t distribution for the daily returns. The filter for the unknown covariance matrix has a computationally efficient matrix formulation, which proves beneficial for estimation and simulation purposes. We formulate parameter restrictions for stationarity and positive definiteness. Our simulation study shows that the new model is able to deal with high-dimensional settings (50 or more) and captures unobserved volatility dynamics even if the model is misspecified. We provide an empirical application to daily equity returns and realized covariance matrices up to 30 dimensions. The model statistically and economically outperforms competing multivariate volatility models out-of-sample. Supplementary materials for this article are available online.     

Multivariate mixtures of normals with unknown number of components

We present full Bayesian analysis of finite mixtures of multivariate normals with unknown number of components. We adopt reversible jump Markov chain Monte Carlo and we construct, in a manner similar to that of Richardson and Green (1997), split and merge moves that produce good mixing of the Markov chains. The split moves are constructed on the space of eigenvectors and eigenvalues of the current covariance matrix so that the proposed covariance matrices are positive definite. Our proposed methodology has applications in classification and discrimination as well as heterogeneity modelling. We test our algorithm with real and simulated data.     

Unbiased Estimator for a Covariance Matrix Under Two-Step Monotone Incomplete Sample

In this article, we consider an inference for a covariance matrix under two-step monotone incomplete sample. The maximum likelihood estimator of the mean vector is unbiased but that of the covariance matrix is biased. We derive an unbiased estimator for the covariance matrix using some fundamental properties of the Wishart matrix. The properties of the estimators are investigated and the accuracies are checked by a numerical simulation.     

Detection and Interpretation of a Multivariate Signal Using Combined Charts

A control procedure is presented in this article that is based on jointly using two separate control statistics in the detection and interpretation of signals in a multivariate normal process. The procedure detects the following three situations: (i) a mean vector shift without a shift in the covariance matrix; (ii) a shift in process variation (covariance matrix) without a mean vector shift; and (iii) both a simultaneous shift in the mean vector and covariance matrix as the result of a change in the parameters of some key process variables. It is shown that, following the occurrence of a signal on either of the separate control charts, the values from both of the corresponding signaling statistics can be decomposed into interpretable elements. Viewing the two decompositions together helps one to specifically identify the individual components and associated variables that are being affected. These components may include individual means or variances of the process variables as well as the correlations between or among variables. An industrial data set is used to illustrate the procedure.     

A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects

This article studies a general joint model for longitudinal measurements and competing risks survival data. The model consists of a linear mixed effects sub-model for the longitudinal outcome, a proportional cause-specific hazards frailty sub-model for the competing risks survival data, and a regression sub-model for the variance–covariance matrix of the multivariate latent random effects based on a modified Cholesky decomposition. The model provides a useful approach to adjust for non-ignorable missing data due to dropout for the longitudinal outcome, enables analysis of the survival outcome with informative censoring and intermittently measured time-dependent covariates, as well as joint analysis of the longitudinal and survival outcomes. Unlike previously studied joint models, our model allows for heterogeneous random covariance matrices. It also offers a framework to assess the homogeneous covariance assumption of existing joint models. A Bayesian MCMC procedure is developed for parameter estimation and inference. Its performances and frequentist properties are investigated using simulations. A real data example is used to illustrate the usefulness of the approach.     

Testing diagonality of high-dimensional covariance matrix under non-normality

Under non-normality, this article is concerned with testing diagonality of high-dimensional covariance matrix, which is more practical than testing sphericity and identity in high-dimensional setting. The existing testing procedure for diagonality is not robust against either the data dimension or the data distribution, producing tests with distorted type I error rates much larger than nominal levels. This is mainly due to bias from estimating some functions of high-dimensional covariance matrix under non-normality. Compared to the sphericity and identity hypotheses, the asymptotic property of the diagonality hypothesis would be more involved and we should be more careful to deal with bias. We develop a correction that makes the existing test statistic robust against both the data dimension and the data distribution. We show that the proposed test statistic is asymptotically normal without the normality assumption and without specifying an explicit relationship between the dimension p and the sample size n. Simulations show that it has good size and power for a wide range of settings.     

Estimation and Test for Multi-Dimensional Regression Models

This work is concerned with the estimation of multi-dimensional regression and the asymptotic behavior of the test involved in selecting models. The main problem with such models is that we need to know the covariance matrix of the noise to get an optimal estimator. We show in this article that if we choose to minimize the logarithm of the determinant of the empirical error covariance matrix, then we get an asymptotically optimal estimator. Moreover, under suitable assumptions, we show that this cost function leads to a very simple asymptotic law for testing the number of parameters of an identifiable and regular regression model. Numerical experiments confirm the theoretical results.