The article develops marginal models for multivariate longitudinal responses. Overall, the model consists of five regression submodels, one for the mean and four for the covariance matrix, with the latter resulting by considering various matrix decompositions. The decompositions that we employ are intuitive, easy to understand, and they do not rely on any assumptions such as the presence of an ordering among the multivariate responses. The regression submodels are semiparametric, with unknown functions represented by basis function expansions. We use spike-slap priors for the regression coefficients to achieve variable selection and function regularization, and to obtain parameter estimates that account for model uncertainty. An efficient Markov chain Monte Carlo algorithm for posterior sampling is developed. The simulation studies presented investigate the effects of priors on posteriors, the gains that one may have when considering multivariate longitudinal analyses instead of univariate ones, and whether these gains can counteract the negative effects of missing data. We apply the methods on a highly unbalanced longitudinal dataset with four responses observed over of period of 20 years
Item nonresponse is frequently encountered in practice. Ignoring missing data can lose efficiency and lead to misleading inference. Fractional imputation is a frequentist approach of imputation for handling missing data. However, the parametric fractional imputation of \cite{kim2011parametric} may be subject to bias under model misspecification. In this paper, we propose a novel semiparametric fractional imputation method using Gaussian mixture models. The proposed method is computationally efficient and leads to robust estimation. The proposed method is further extended to incorporate the cat…
We consider the problem of multivariate density deconvolution when the interest lies in estimating the distribution of a vector-valued random variable but precise measurements of the variable of interest are not available, observations being contaminated with additive measurement errors. The existing sparse literature on the problem assumes the density of the measurement errors to be completely known. We propose robust Bayesian semiparametric multivariate deconvolution approaches when the measurement error density is not known but replicated proxies are available for each unobserved value of t…
We consider the problem of multivariate density deconvolution where the distribution of a random vector needs to be estimated from replicates contaminated with conditionally heteroscedastic measurement errors. We propose a conceptually straightforward yet fundamentally novel and highly robust approach to multivariate density deconvolution by stochastically rotating the replicates toward the corresponding true latent values. We also address the additionally significantly challenging problem of accommodating conditionally heteroscedastic measurement errors in this newly introduced framework. We …
Multivariate nonnegative orthant data are real vectors bounded to the left by the null vector, and they can be continuous, discrete or mixed. We first review the recent relative variability indexes for multivariate nonnegative continuous and count distributions. As a prelude, the classification of two comparable distributions having the same mean vector is done through under-, equi- and over-variability with respect to the reference distribution. Multivariate associated kernel estimators are then reviewed with new proposals that can accommodate any nonnegative orthant dataset. We focus on band…
This paper proposes a flexible Bayesian approach to multiple imputation using conditional Gaussian mixtures. We introduce novel shrinkage priors for covariate-dependent mixing proportions in the mixture models to automatically select the suitable number of components used in the imputation step. We develop an efficient sampling algorithm for posterior computation and multiple imputation via Markov Chain Monte Carlo methods. The proposed method can be easily extended to the situation where the data contains not only continuous variables but also discrete variables such as binary and count value…
In this paper, a Bayesian semiparametric copula approach is used to model the underlying multivariate distribution $F_{true}$. First, the Dirichlet process is constructed on the unknown marginal distributions of $F_{true}$. Then a Gaussian copula model is utilized to capture the dependence structure of $F_{true}$. As a result, a Bayesian multivariate normality test is developed by combining the relative belief ratio and the Energy distance. Several interesting theoretical results of the approach are derived. Finally, through several simulated examples and a real data set, the proposed approach…
Estimating the marginal and joint densities of the long-term average intakes of different dietary components is an important problem in nutritional epidemiology. Since these variables cannot be directly measured, data are usually collected in the form of 24-hour recalls of the intakes. The problem of estimating the density of the latent long-term average intakes from their observed but error contaminated recalls then becomes a problem of multivariate deconvolution of densities. The underlying densities could potentially vary with the subjects' demographic characteristics such as sex, ethnicity…
Semiparametric Copula Estimation for Spatially Correlated Multivariate Mixed Outcomes: Analyzing Visual Sightings of Fin Whales from a Line Transect Survey
Tokyo University of Science（东京理科大学）；Meisei University（明海大学）；Keio University（庆应义塾大学）；Institute of Marine Research（海洋研究所）；RIKEN Center for Brain Science（理化学研究所脑科学中心）
We describe a simple method for making inference on a functional of a multivariate distribution. The method is based on a copula representation of the multivariate distribution and it is based on the properties of an Approximate Bayesian Monte Carlo algorithm, where the proposed values of the functional of interest are weighed in terms of their empirical likelihood. This method is particularly useful when the "true" likelihood function associated with the working model is too costly to evaluate or when the working model is only partially specified.
Modern datasets are often in the form of matrices or arrays,potentially having correlations along each set of data indices. For example, data involving repeated measurements of several variables over time may exhibit temporal correlation as well as correlation among the variables. A possible model for matrix-valued data is the class of matrix normal distributions, which is parametrized by two covariance matrices, one for each index set of the data. In this article we describe an extension of the matrix normal model to accommodate multidimensional data arrays, or tensors. We generate a class of…
In this work we present full Bayesian inference for a new flexible nonseparable class of cross-covariance functions for multivariate spatial data. A Bayesian test is proposed for separability of covariance functions which is much more interpretable than parameters related to separability. Spatial models have been increasingly applied in several areas, such as environmental science, climate science and agriculture. These data are usually available in space, time and possibly for several processes. In this context the modeling of dependence is crucial for correct uncertainty quantification and r…
This paper demonstrates the advantages of sharing information about unknown features of covariates across multiple model components in various nonparametric regression problems including multivariate, heteroscedastic, and semi-continuous responses. In this paper, we present methodology which allows for information to be shared nonparametrically across various model components using Bayesian sum-of-tree models. Our simulation results demonstrate that sharing of information across related model components is often very beneficial, particularly in sparse high-dimensional problems in which variabl…
For multivariate spatial Gaussian process (GP) models, customary specifications of cross-covariance functions do not exploit relational inter-variable graphs to ensure process-level conditional independence among the variables. This is undesirable, especially for highly multivariate settings, where popular cross-covariance functions such as the multivariate Matérn suffer from a "curse of dimensionality" as the number of parameters and floating point operations scale up in quadratic and cubic order, respectively, in the number of variables. We propose a class of multivariate "Graphical Gaussian…
Food and beverage authentication is the process by which foods or beverages are verified as complying with its label description, for example, verifying if the denomination of origin of an olive oil bottle is correct or if the variety of a certain bottle of wine matches its label description. The common way to deal with an authentication process is to measure a number of attributes on samples of food and then use these as input for a classification problem. Our motivation stems from data consisting of measurements of nine chemical compounds denominated Anthocyanins, obtained from samples of Ch…
Storm surge, the onshore rush of sea water caused by the high winds and low pressure associated with a hurricane, can compound the effects of inland flooding caused by rainfall, leading to loss of property and loss of life for residents of coastal areas. Numerical ocean models are essential for creating storm surge forecasts for coastal areas. These models are driven primarily by the surface wind forcings. Currently, the gridded wind fields used by ocean models are specified by deterministic formulas that are based on the central pressure and location of the storm center. While these equations…
Multivariate data that combine binary, categorical, count and continuous outcomes are common in the social and health sciences. We propose a semiparametric Bayesian latent variable model for multivariate data of arbitrary type that does not require specification of conditional distributions. Drawing on the extended rank likelihood method by Hoff [Ann. Appl. Stat. 1 (2007) 265-283], we develop a semiparametric approach for latent variable modeling with mixed outcomes and propose associated Markov chain Monte Carlo estimation methods. Motivated by cognitive testing data, we focus on bifactor mod…
We propose a structure of a semiparametric two-component mixture model when one component is parametric and the other is defined through linear constraints on its distribution function. Estimation of a two-component mixture model with an unknown component is very difficult when no particular assumption is made on the structure of the unknown component. A symmetry assumption was used in the literature to simplify the estimation. Such method has the advantage of producing consistent and asymptotically normal estimators, and identifiability of the semiparametric mixture model becomes tractable. S…
Novelty detection methods aim at partitioning the test units into already observed and previously unseen patterns. However, two significant issues arise: there may be considerable interest in identifying specific structures within the novelty, and contamination in the known classes could completely blur the actual separation between manifest and new groups. Motivated by these problems, we propose a two-stage Bayesian semiparametric novelty detector, building upon prior information robustly extracted from a set of complete learning units. We devise a general-purpose multivariate methodology tha…
The semiparametric accelerated failure time model is not as widely used as the Cox relative risk model mainly due to computational difficulties. Recent developments in least squares estimation and induced smoothing estimating equations provide promising tools to make the accelerate failure time models more attractive in practice. For semiparametric multivariate accelerated failure time models, we propose a generalized estimating equation approach to account for the multivariate dependence through working correlation structures. The marginal error distributions can be either identical as in seq…
Recently, Su and Cook proposed a dimension reduction technique called the inner envelope which can be substantially more efficient than the original envelope or existing dimension reduction techniques for multivariate regression. However, their technique relied on a linear model with normally distributed error, which may be violated in practice. In this work, we propose a semiparametric variant of the inner envelope that does not rely on the linear model nor the normality assumption. We show that our proposal leads to globally and locally efficient estimators of the inner envelope spaces. We a…
In this work, we estimate extreme sea surface temperature (SST) hotspots, i.e., high threshold exceedance regions, for the Red Sea, a vital region of high biodiversity. We analyze high-resolution satellite-derived SST data comprising daily measurements at 16703 grid cells across the Red Sea over the period 1985-2015. We propose a semiparametric Bayesian spatial mixed-effects linear model with a flexible mean structure to capture spatially-varying trend and seasonality, while the residual spatial variability is modeled through a Dirichlet process mixture (DPM) of low-rank spatial Student-$t$ pr…
The covariance structure of multivariate functional data can be highly complex, especially if the multivariate dimension is large, making extensions of statistical methods for standard multivariate data to the functional data setting challenging. For example, Gaussian graphical models have recently been extended to the setting of multivariate functional data by applying multivariate methods to the coefficients of truncated basis expansions. However, a key difficulty compared to multivariate data is that the covariance operator is compact, and thus not invertible. The methodology in this paper …
This article presents an approach to Bayesian semiparametric inference for Gaussian multivariate response regression. We are motivated by various small and medium dimensional problems from the physical and social sciences. The statistical challenges revolve around dealing with the unknown mean and variance functions and in particular, the correlation matrix. To tackle these problems, we have developed priors over the smooth functions and a Markov chain Monte Carlo algorithm for inference and model selection. Specifically, Dirichlet process mixtures of Gaussian distributions are used as the bas…
We introduce semiparametric Bayesian networks that combine parametric and nonparametric conditional probability distributions. Their aim is to incorporate the advantages of both components: the bounded complexity of parametric models and the flexibility of nonparametric ones. We demonstrate that semiparametric Bayesian networks generalize two well-known types of Bayesian networks: Gaussian Bayesian networks and kernel density estimation Bayesian networks. For this purpose, we consider two different conditional probability distributions required in a semiparametric Bayesian network. In addition…
In this manuscript, we consider a finite multivariate nonparametric mixture model where the dependence between the marginal densities is modeled using the copula device. Pseudo EM stochastic algorithms were recently proposed to estimate all of the components of this model under a location-scale constraint on the marginals. Here, we introduce a deterministic algorithm that seeks to maximize a smoothed semiparametric likelihood. No location-scale assumption is made about the marginals. The algorithm is monotonic in one special case, and, in another, leads to ``approximate monotonicity'' -- where…
This paper investigates the cross-correlations across multiple climate model errors. We build a Bayesian hierarchical model that accounts for the spatial dependence of individual models as well as cross-covariances across different climate models. Our method allows for a nonseparable and nonstationary cross-covariance structure. We also present a covariance approximation approach to facilitate the computation in the modeling and analysis of very large multivariate spatial data sets. The covariance approximation consists of two parts: a reduced-rank part to capture the large-scale spatial depen…
When facing multivariate covariates, general semiparametric regression techniques come at hand to propose flexible models that are unexposed to the curse of dimensionality. In this work a semiparametric copula-based estimator for conditional quantiles is investigated for complete or right-censored data. In spirit, the methodology is extending the recent work of Noh et al. (2013) and Noh et al. (2015), as the main idea consists in appropriately defining the quantile regression in terms of a multivariate copula and marginal distributions. Prior estimation of the latter and simple plug-in lead to…
Semiparametric estimation for multivariate Hawkes processes using dependent Dirichlet processes: An application to order flow data in financial markets
The prevalence of spatially referenced multivariate data has impelled researchers to develop a procedure for the joint modeling of multiple spatial processes. This ordinarily involves modeling marginal and cross-process dependence for any arbitrary pair of locations using a multivariate spatial covariance function. However, building a flexible multivariate spatial covariance function that is nonnegative definite is challenging. Here, we propose a semiparametric approach for multivariate spatial covariance function estimation with approximate Matérn marginals and highly flexible cross-covarianc…
We propose a new class of multivariate survival models based on archimedean copulas with margins modeled by the Yang and Prentice (YP) model. The Ali-Mikhail-Haq (AMH), Clayton, Frank, Gumbel-Hougaard (GH), and Joe copulas are employed to accommodate the dependency among marginal distributions. Baseline distributions are modeled semiparametrically by the piecewise exponential (PE) distribution and the Bernstein polynomials. The new class of models possesses some attractive features: i) the ability to take into account survival data with crossing survival curves; ii) the inclusion of the well-k…
Estimation of the covariance structure of spatial processes is of fundamental importance in spatial statistics. In the literature, several non-parametric and semi-parametric methods have been developed to estimate the covariance structure based on the spectral representation of covariance functions. However,they either ignore the high frequency properties of the spectral density, which are essential to determine the performance of interpolation procedures such as Kriging, or lack of theoretical justification. We propose a new semi-parametric method to estimate spectral densities of isotropic s…
It is possible to approach regression analysis with random covariates from a semiparametric perspective where information is combined from multiple multivariate sources. The approach assumes a semiparametric density ratio model where multivariate distributions are "regressed" on a reference distribution. A kernel density estimator can be constructed from many data sources in conjunction with the semiparametric model. The estimator is shown to be more efficient than the traditional single-sample kernel density estimator, and its optimal bandwidth is discussed in some detail. Each multivariate d…
To better understand the spatial structure of large panels of economic and financial time series and provide a guideline for constructing semiparametric models, this paper first considers estimating a large spatial covariance matrix of the generalized $m$-dependent and $β$-mixing time series (with $J$ variables and $T$ observations) by hard thresholding regularization as long as ${\log J \, \cx^*(\ct)}/{T} = \Co(1)$ (the former scheme with some time dependence measure $\cx^*(\ct)$) or $\log J /{T} = \Co(1)$ (the latter scheme with some upper bounded mixing coefficient). We quantify the interpl…
In this paper we propose a new approach for constructing \emph{multivariate} Gaussian random fields (GRFs) with oscillating covariance functions through systems of stochastic partial differential equations (SPDEs). We discuss how to build systems of SPDEs that introduces oscillation characteristics in the covariance functions of the multivariate GRFs. By choosing different parametrization of the equations, some GRFs can be made with oscillating covariance functions but other fields can have Matérn covariance functions or close to Matérn covariance functions. The multivariate GRFs constructed b…
Multivariate spatial-statistical models are often used when modeling environmental and socio-demographic processes. The most commonly used models for multivariate spatial covariances assume both stationarity and symmetry for the cross-covariances, but these assumptions are rarely tenable in practice. In this article we introduce a new and highly flexible class of nonstationary and asymmetric multivariate spatial covariance models that are constructed by modeling the simpler and more familiar stationary and symmetric multivariate covariances on a warped domain. Inspired by recent developments i…
Spatial extent inference (SEI) is widely used across neuroimaging modalities to study brain-phenotype associations that inform our understanding of disease. Recent studies have shown that Gaussian random field (GRF) based tools can have inflated family-wise error rates (FWERs). This has led to fervent discussion as to which preprocessing steps are necessary to control the FWER using GRF-based SEI. The failure of GRF-based methods is due to unrealistic assumptions about the covariance function of the imaging data. The permutation procedure is the most robust SEI tool because it estimates the co…
Multivariate space-time data are increasingly available in various scientific disciplines. When analyzing these data, one of the key issues is to describe the multivariate space-time dependencies. Under the Gaussian framework, one needs to propose relevant models for multivariate space-time covariance functions, i.e. matrix-valued mappings with the additional requirement of non-negative definiteness. We propose a flexible parametric class of cross-covariance functions for multivariate space-time Gaussian random fields. Space-time components belong to the (univariate) Gneiting class of space-ti…
We propose a general framework for non-normal multivariate data analysis called multivariate covariance generalized linear models (McGLMs), designed to handle multivariate response variables, along with a wide range of temporal and spatial correlation structures defined in terms of a covariance link function combined with a matrix linear predictor involving known matrices. The method is motivated by three data examples that are not easily handled by existing methods. The first example concerns multivariate count data, the second involves response variables of mixed types, combined with repeate…
Many multivariate statistical methods rely heavily on the sample covariance matrix. It is well known though that the sample covariance matrix is highly non-robust. One popular alternative approach for "robustifying" the multivariate method is to simply replace the role of the covariance matrix with some robust scatter matrix. The aim of this paper is to point out that in some situations certain properties of the covariance matrix are needed for the corresponding robust "plug-in" method to be a valid approach, and that not all scatter matrices necessarily possess these important properties. In …
We propose a Bayesian procedure for simultaneous variable and covariance selection using continuous spike-and-slab priors in multivariate linear regression models where q possibly correlated responses are regressed onto p predictors. Rather than relying on a stochastic search through the high-dimensional model space, we develop an ECM algorithm similar to the EMVS procedure of Rockova & George (2014) targeting modal estimates of the matrix of regression coefficients and residual precision matrix. Varying the scale of the continuous spike densities facilitates dynamic posterior exploration and …
Compositional data, which is data consisting of fractions or probabilities, is common in many fields including ecology, economics, physical science and political science. If these data would otherwise be normally distributed, their spread can be conveniently represented by a multivariate normal distribution truncated to the non-negative space under a unit simplex. Here this distribution is called the simplex-truncated multivariate normal distribution. For calculations on truncated distributions, it is often useful to obtain rapid estimates of their integral, mean and covariance; these quantiti…
We propose, for multivariate Gaussian copula models with unknown margins and structured correlation matrices, a rank-based, semiparametrically efficient estimator for the Euclidean copula parameter. This estimator is defined as a one-step update of a rank-based pilot estimator in the direction of the efficient influence function, which is calculated explicitly. Moreover, finite-dimensional algebraic conditions are given that completely characterize efficiency of the pseudo-likelihood estimator and adaptivity of the model with respect to the unknown marginal distributions. For correlation matri…
In applied research, it is often sensible to account for one or several covariates when testing for differences between multivariate means of several groups. However, the "classical" parametric multivariate analysis of covariance (MANCOVA) tests (e.g., Wilks' Lambda) are based on quite restrictive assumptions (homoscedasticity and normality of the errors), which might be difficult to justify in small sample size settings. Furthermore, existing potential remedies (e.g., heteroskedasticity-robust approaches) become inappropriate in cases where the covariance matrices are singular. Nevertheless, …
Interval censoring arises frequently in clinical, epidemiological, financial, and sociological studies, where the event or failure of interest is known only to occur within an interval induced by periodic monitoring. We formulate the effects of potentially time-dependent covariates on the interval-censored failure time through a broad class of semiparametric transformation models that encompasses proportional hazards and proportional odds models. We consider nonparametric maximum likelihood estimation for this class of models with an arbitrary number of monitoring times for each subject. We de…
This paper describes a generalization of the Hellinger distance which we call the S -Hellinger distance; this general family connects the Hellinger distance smoothly with the $L_2$-divergence by a tuning parameter $α$ and is indeed a subfamily of the S -Divergence family of Ghosh et al. (2013 a, b). We use this general divergence in the context of estimating the location and covariances under (continuous) multivariate models and show that the proposed minimum S -Hellinger distance estimator is affine equivariant, asymptotically consistent and have high breakdown point under suitable conditions…
It is well recognised that animal and plant pathogens form complex ecological communities of interacting organisms within their hosts. Although community ecology approaches have been applied to determine pathogen interactions at the within-host scale, methodologies enabling robust inference of the epidemiological impact of pathogen interactions are lacking. Here we developed a novel statistical framework to identify statistical covariances from the infection time-series of multiple pathogens simultaneously. Our framework extends Bayesian multivariate disease mapping models to analyse multivari…
