Many scientific applications involve mixed spatially indexed outcomes of heterogeneous types that are driven by shared latent mechanisms. Modeling such data is challenging due to complex, nonlinear, and potentially nonstationary spatial dependence, as well as the need for coherent joint inference across mixed outcome distributions. Existing multivariate mixed outcome models often rely on restrictive linear assumptions, while recent deep learning approaches emphasize predictive flexibility but typically lack coherent joint modeling and uncertainty quantification for spatial data. We develop MultiDeepGP, a scalable and statistically principled framework for joint modeling of multivariate mixed outcomes in spatial settings. The proposed approach introduces a shared latent spatial component that governs cross-outcome dependence while allowing outcome-specific distributions. Spatial dependence and nonlinear structure are captured through a deep latent representation, and uncertainty quantification is enabled via an efficient Monte Carlo-based inference strategy. This construction balances modeling flexibility with probabilistic interpretability and computational feasibility. The proposed method is evaluated through simulation studies designed to reflect key challenges in mixed outcome spatial modeling, as well as an application to georeferenced environmental and public health data from the African Great Lakes region. The results demonstrate that the proposed framework provides accurate joint prediction and reliable uncertainty quantification in complex spatial settings.
This paper focuses on the multivariate linear mixed-effects model, including all the correlations between the random effects when the marginal residual terms are assumed uncorrelated and homoscedastic with possibly different standard deviations. The random effects covariance matrix is Cholesky factorized to directly estimate the variance components of these random effects. This strategy enables a consistent estimate of the random effects covariance matrix which, generally, has a poor estimate when it is grossly (or directly) estimated, using the estimating methods such as the EM algorithm. By …
Many application domains such as ecology or genomics have to deal with multivariate non Gaussian observations. A typical example is the joint observation of the respective abundances of a set of species in a series of sites, aiming to understand the co-variations between these species. The Gaussian setting provides a canonical way to model such dependencies, but does not apply in general. We consider here the multivariate exponential family framework for which we introduce a generic model with multivariate Gaussian latent variables. We show that approximate maximum likelihood inference can be …
In epidemiological and clinical studies, identifying patients' phenotypes based on longitudinal profiles is critical to understanding the disease's developmental patterns. The current study was motivated by data from a Canadian birth cohort study, the CHILD Cohort Study. Our goal was to use multiple longitudinal respiratory traits to cluster the participants into subgroups with similar longitudinal respiratory profiles in order to identify clinically relevant disease phenotypes. To appropriately account for distinct structures and types of these longitudinal markers, we proposed a novel joint …
The identification of factors associated with mental and behavioral disorders in early childhood is critical both for psychopathology research and the support of primary health care practices. Motivated by the Millennium Cohort Study, in this paper we study the effect of a comprehensive set of covariates on children's emotional and behavioural trajectories in England. To this end, we develop a Quantile Mixed Hidden Markov Model for joint estimation of multiple quantiles in a linear regression setting for multivariate longitudinal data. The novelty of the proposed approach is based on the Multi…
In this article, we consider modeling ranked responses from a heterogeneous population. Specifically, we analyze data from the Eurobarometer 34.1 survey regarding public policy preferences towards drugs, alcohol and AIDS. Such policy preferences are likely to exhibit substantial differences within as well as across European nations reflecting a wide variety of cultures, political affiliations, ideological perspectives and common practices. We use a mixed membership model to account for multiple subgroups with differing preferences and to allow each individual to possess partial membership in m…
Mass cytometry technology enables the simultaneous measurement of over 40 proteins on single cells. This has helped immunologists to increase their understanding of heterogeneity, complexity, and lineage relationships of white blood cells. Current statistical methods often collapse the rich single-cell data into summary statistics before proceeding with downstream analysis, discarding the information in these multivariate datasets. In this article, our aim is to exhibit the use of statistical analyses on the raw, uncompressed data thus improving replicability, and exposing multivariate pattern…
It is important to characterize the temporal trajectories of disease-related biomarkers in order to monitor progression and identify potential points of intervention. This is especially important for neurodegenerative diseases, as therapeutic intervention is most likely to be effective in the preclinical disease stages prior to significant neuronal damage. Longitudinal neuroimaging allows for the measurement of structural, functional, and metabolic integrity of the brain over time at the level of voxels. However, commonly used longitudinal analysis approaches, such as linear mixed effects mode…
In probability and statistics, copulas play important roles theoretically as well as to address a wide range of problems in various application areas. We introduce the concept of multivariate discrete copulas, discuss their equivalence to stochastic arrays, and prove a multivariate discrete version of Sklar's theorem. These results provide the theoretical frame for multivariate statistical methods to postprocess weather forecasts made by ensemble systems, including the ensemble copula coupling approach and the Schaake shuffle.
We propose a Bayesian modeling framework for jointly analyzing multiple functional responses of different types (e.g. binary and continuous data). Our approach is based on a multivariate latent Gaussian process and models the dependence among the functional responses through the dependence of the latent process. Our framework easily accommodates additional covariates. We offer a way to estimate the multivariate latent covariance, allowing for implementation of multivariate functional principal components analysis (FPCA) to specify basis expansions and simplify computation. We demonstrate our m…
Multivariate functional data can be intrinsically multivariate like movement trajectories in 2D or complementary like precipitation, temperature, and wind speeds over time at a given weather station. We propose a multivariate functional additive mixed model (multiFAMM) and show its application to both data situations using examples from sports science (movement trajectories of snooker players) and phonetic science (acoustic signals and articulation of consonants). The approach includes linear and nonlinear covariate effects and models the dependency structure between the dimensions of the resp…
There are many data sources available that report related variables of interest that are also referenced over geographic regions and time; however, there are relatively few general statistical methods that one can readily use that incorporate these multivariate-spatio-temporal dependencies. As such, we introduce the multivariate-spatio-temporal mixed effects model (MSTM) to analyze areal data with multivariate-spatio-temporal dependencies. The proposed MSTM extends the notion of Moran's I basis functions to the multivariate-spatio-temporal setting. This extension leads to several methodologica…
Data reproducibility is a critical issue in all scientific experiments. In this manuscript, we consider the problem of quantifying the reproducibility of graphical measurements. We generalize the concept of image intra-class correlation coefficient (I2C2) and propose the concept of the graphical intra-class correlation coefficient (GICC) for such purpose. The concept of GICC is based on multivariate probit-linear mixed effect models. We will present a Markov Chain EM (MCEM) algorithm for estimating the GICC. Simulations results with varied settings are demonstrated and our method is applied to…
Information visualization significantly enhances human perception by graphically representing complex data sets. The variety of visualization designs makes it challenging to efficiently evaluate all possible designs catering to users' preferences and characteristics. Most of existing evaluation methods perform user studies to obtain multivariate qualitative responses from users via questionnaires and interviews. However, these methods cannot support online evaluation of designs as they are often time-consuming. A statistical model is desired to predict users' preferences on visualization desig…
We propose a Bayesian model for mixed ordinal and continuous multivariate data to evaluate a latent spatial Gaussian process. Our proposed model can be used in many contexts where mixed continuous and discrete multivariate responses are observed in an effort to quantify an unobservable continuous measurement. In our example, the latent, or unobservable measurement is wetland condition. While predicted values of the latent wetland condition variable produced by the model at each location do not hold any intrinsic value, the relative magnitudes of the wetland condition values are of interest. In…
We propose a method for inference in generalised linear mixed models (GLMMs) and several extensions of these models. First, we extend the GLMM by allowing the distribution of the random components to be non-Gaussian, that is, assuming an absolutely continuous distribution with respect to the Lebesgue measure that is symmetric around zero, unimodal and with finite moments up to fourth-order. Second, we allow the conditional distribution to follow a dispersion model instead of exponential dispersion models. Finally, we extend these models to a multivariate framework where multiple responses are …
This paper introduces a method for studying the correlation structure of a range of responses modelled by a multivariate generalised linear mixed model (MGLMM). The methodology requires the existence of clusters of observations and that each of the several responses studied is modelled using a generalised linear mixed models (GLMM) containing random components representing the clusters. We construct a MGLMM by assuming that the distribution of each of the random components representing the clusters is the marginal distribution of a (sufficiently regular) multivariate elliptically contoured dis…
Causal analysis based on non-uniform embedding schemes is an important way to detect the underlying interactions between dynamic systems. However, there are still some obstacles to estimate high-dimensional conditional mutual information and form optimal mixed embedding vector in traditional non-uniform embedding schemes. In this study, we present a new non-uniform embedding method framed in information theory to detect causality for multivariate time series, named LM-PMIME, which integrates the low-dimensional approximation of conditional mutual information and the mixed search strategy for t…
Computational efficient evaluation of penalized estimators of multivariate exponential family distributions is sought. These distributions encompass among others Markov random fields with variates of mixed type (e.g. binary and continuous) as special case of interest. The model parameter is estimated by maximization of the pseudo-likelihood augmented with a convex penalty. The estimator is shown to be consistent. With a world of multi-core computers in mind, a computationally efficient parallel Newton-Raphson algorithm is presented for numerical evaluation of the estimator alongside conditions…
Multivariate outcomes are common in pragmatic cluster randomized trials. While sample size calculation procedures for multivariate outcomes exist under parallel assignment, none have been developed for a stepped wedge design. In this article, we present computationally efficient power and sample size procedures for stepped wedge cluster randomized trials (SW-CRTs) with multivariate outcomes that differentiate the within-period and between-period intracluster correlation coefficients (ICCs). Under a multivariate linear mixed model, we derive the joint distribution of the intervention test stati…
Mixed Membership Models (MMMs) are a popular family of latent structure models for complex multivariate data. Instead of forcing each subject to belong to a single cluster, MMMs incorporate a vector of subject-specific weights characterizing partial membership across clusters. With this flexibility come challenges in uniquely identifying, estimating, and interpreting the parameters. In this article, we propose a new class of Dimension-Grouped MMMs (Gro-M$^3$s) for multivariate categorical data, which improve parsimony and interpretability. In Gro-M$^3$s, observed variables are partitioned into…
Univariate regression models have rich literature for counting data. However, this is not the case for multivariate count data. Therefore, we present the Multivariate Generalized Linear Mixed Models framework that deals with a multivariate set of responses, measuring the correlation between them through random effects that follows a multivariate normal distribution. This model is based on a GLMM with a random intercept and the estimation process remains the same as a standard GLMM with random effects integrated out via Laplace approximation. We efficiently implemented this model through the TM…
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…
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）
Dynamic statistical process monitoring methods have been widely studied and applied in modern industrial processes. These methods aim to extract the most predictable temporal information and develop the corresponding dynamic monitoring schemes. However, measurement noise is widespread in real-world industrial processes, and ignoring its effect will lead to sub-optimal modeling and monitoring performance. In this article, a probabilistic predictable feature analysis (PPFA) is proposed for multivariate time series modeling, and a multi-step dynamic predictive monitoring scheme is developed. The …
Many real-world datasets contain missing entries and mixed data types including categorical and ordered (e.g. continuous and ordinal) variables. Imputing the missing entries is necessary, since many data analysis pipelines require complete data, but this is challenging especially for mixed data. This paper proposes a probabilistic imputation method using an extended Gaussian copula model that supports both single and multiple imputation. The method models mixed categorical and ordered data using a latent Gaussian distribution. The unordered characteristics of categorical variables is explicitl…
Researchers are often interested in understanding the relationship between a set of covariates and a set of response variables. To achieve this goal, the use of regression analysis, either linear or generalized linear models, is largely applied. However, such models only allow users to model one response variable at a time. Moreover, it is not possible to directly calculate from the regression model a correlation measure between the response variables. In this article, we employed the Multivariate Generalized Linear Mixed Models framework, which allows the specification of a set of response va…
Dynamic prediction of causal effects under different treatment regimes conditional on an individual's characteristics and longitudinal history is an essential problem in precision medicine. This is challenging in practice because outcomes and treatment assignment mechanisms are unknown in observational studies, an individual's treatment efficacy is a counterfactual, and the existence of selection bias is often unavoidable. We propose a Bayesian framework for identifying subgroup counterfactual benefits of dynamic treatment regimes by adapting Bayesian g-computation algorithm (J. Robins, 1986; …
CIRAD（法国农业研究与国际合作组织）；Inria（法国国家信息与自动化研究所）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Laboratoire Jean Kuntzmann（让·库茨曼实验室）
Multivariate count data are defined as the number of items of different categories issued from sampling within a population, which individuals are grouped into categories. The analysis of multivariate count data is a recurrent and crucial issue in numerous modelling problems, particularly in the fields of biology and ecology (where the data can represent, for example, children counts associated with multitype branching processes), sociology and econometrics. We focus on I) Identifying categories that appear simultaneously, or on the contrary that are mutually exclusive. This is achieved by ide…
We consider the problem of predicting several response variables using the same set of explanatory variables. This setting naturally induces a group structure over the coefficient matrix, in which every explanatory variable corresponds to a set of related coefficients. Most of the existing methods that utilize this group formation assume that the similarities between related coefficients arise solely through a joint sparsity structure. In this paper, we propose a procedure for constructing an estimator of a multivariate regression coefficient matrix that directly models and captures the within…
Considerable interest has recently been focused on studying multiple phenotypes simultaneously in both epidemiological and genomic studies, either to capture the multidimensionality of complex disorders or to understand shared etiology of related disorders. We seek to identify {\em multiple regulators} or predictors that are associated with multiple outcomes when these outcomes may be measured on very different scales or composed of a mixture of continuous, binary, and not-fully-observed elements. We first propose an estimation technique to put all effects on similar scales, and we induce spar…
We propose a new method for multivariate response regression and covariance estimation when elements of the response vector are of mixed types, for example some continuous and some discrete. Our method is based on a model which assumes the observable mixed-type response vector is connected to a latent multivariate normal response linear regression through a link function. We explore the properties of this model and show its parameters are identifiable under reasonable conditions. We impose no parametric restrictions on the covariance of the latent normal other than positive definiteness, there…
I begin my discussion by summarizing the methodology proposed and new distributional results on multivariate log-Gamma derived in the paper. Then, I draw an interesting connection between their work with mean field variational Bayes. Lastly, I make some comments on the simulation results and the performance of the proposed Poisson multivariate spatio-temporal mixed effects model (P-MSTM).
Copulas allow a flexible and simultaneous modeling of complicated dependence structures together with various marginal distributions. Especially if the density function can be represented as the product of the marginal density functions and the copula density function, this leads to both an intuitive interpretation of the conditional distribution and convenient estimation procedures. However, this is no longer the case for copula models with mixed discrete and continuous marginal distributions, because the corresponding density function cannot be decomposed so nicely. In this paper, we introdu…
Multivariate data occurs in a wide range of fields, with ever more flexible model specifications being proposed, often within a multivariate generalised linear mixed effects (MGLME) framework. In this article, we describe an extended framework, encompassing multiple outcomes of any type, each of which could be repeatedly measured (longitudinal), with any number of levels, and with any number of random effects at each level. Many standard distributions are described, as well as non-standard user-defined non-linear models. The extension focuses on a complex linear predictor for each outcome mode…
Mandarin Chinese is characterized by being a tonal language; the pitch (or $F_0$) of its utterances carries considerable linguistic information. However, speech samples from different individuals are subject to changes in amplitude and phase which must be accounted for in any analysis which attempts to provide a linguistically meaningful description of the language. A joint model for amplitude, phase and duration is presented which combines elements from Functional Data Analysis, Compositional Data Analysis and Linear Mixed Effects Models. By decomposing functions via a functional principal co…
In genome wide association studies (GWAS), researchers are often dealing with non-normally distributed traits or a mixture of discrete-continuous traits. However, most of the current region-based methods rely on multivariate linear mixed models (mvLMMs) and assume a multivariate normal distribution for the phenotypes of interest. Hence, these methods are not applicable to disease or non-normally distributed traits. Therefore, there is a need to develop unified and flexible methods to study association between a set of (possibly rare) genetic variants and non-normal multivariate phenotypes. Cop…
Data collected in clinical trials are often composed of multiple types of variables. For example, laboratory measurements and vital signs are longitudinal data of continuous or categorical variables, adverse events may be recurrent events, and death is a time-to-event variable. Missing data due to patients' discontinuation from the study or as a result of handling intercurrent events using a hypothetical strategy almost always occur during any clinical trial. Imputing these data with mixed types of variables simultaneously is a challenge that has not been studied extensively. In this article, …
Multivariate Functional Principal Component Analysis for Mixed-Type mHealth Data: An Application to Mood Disorders
Texas A&M University（德克萨斯农工大学）；University of South Carolina（南卡罗来纳大学）；National Institute of Mental Health（美国国家精神卫生研究所）；Johns Hopkins Bloomberg School of Public Health（约翰霍普金斯大学布隆伯格公共卫生学院）
We introduce a general approach for modeling the dynamic of multivariate time series when the data are of mixed type (binary/count/continuous). Our method is quite flexible and conditionally on past values, each coordinate at time $t$ can have a distribution compatible with a standard univariate time series model such as GARCH, ARMA, INGARCH or logistic models whereas past values of the other coordinates play the role of exogenous covariates in the dynamic. The simultaneous dependence in the multivariate time series can be modeled with a copula. Additional exogenous covariates are also allowed…
We define a multivariate medial correlation coefficient that extends the probabilistic interpretation and properties of Blomqvist's $β$ coefficient, incorporates multivariate marginal dependencies and it preserves a stronger multivariate concordance relation. We determine the maximum and minimum values attainable and illustrate the results in some models. We end with an application on real datasets.
Modeling of longitudinal data often requires diffusion models that incorporate overall time-dependent, nonlinear dynamics of multiple components and provide sufficient flexibility for subject-specific modeling. This complexity challenges parameter inference and approximations are inevitable. We propose a method for approximate maximum-likelihood parameter estimation in multivariate time-inhomogeneous diffusions, where subject-specific flexibility is accounted for by incorporation of multidimensional mixed effects and covariates. We consider $N$ multidimensional independent diffusions $X^i = (X…
In this paper, we propose a mixture of probabilistic partial canonical correlation analysis (MPPCCA) that extracts the Causal Patterns from two multivariate time series. Causal patterns refer to the signal patterns within interactions of two elements having multiple types of mutually causal relationships, rather than a mixture of simultaneous correlations or the absence of presence of a causal relationship between the elements. In multivariate statistics, partial canonical correlation analysis (PCCA) evaluates the correlation between two multivariates after subtracting the effect of the third …
Linear mixed-effect models with two variance components are often used when variability comes from two sources. In genetics applications, variation in observed traits can be attributed to biological and environmental effects, and the heritability coefficient is a fundamental quantity that measures the proportion of total variability due to the biological effect. We propose a new inferential model approach which yields exact prior-free probabilistic inference on the heritability coefficient. In particular we construct exact confidence intervals and demonstrate numerically our method's efficienc…
