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 distribution. We use an undirected graphical model to represent the correlation structure of the random components representing the clusters of observations for each response. This representation allows us to draw conclusions regarding unknown underlying determining factors related to the clusters of observations. Using a combination of an undirected graph and a directed acyclic graph (DAG), we jointly represent the correlation structure of the responses and the related random components. Applying the theory of graphical models allows us to describe and draw conclusions on the correlation and, in some cases, the dependence between responses of different statistical nature (\eg following different distributions, different linear predictors and link functions). We present some simulation studies illustrating the proposed methodology.
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…
When simultaneously testing multiple hypotheses, the usual approach in the context of confirmatory clinical trials is to control the familywise error rate (FWER), which bounds the probability of making at least one false rejection. In many trial settings, these hypotheses will additionally have a hierarchical structure that reflects the relative importance and links between different clinical objectives. The graphical approach of Bretz et al. (2009) is a flexible and easily communicable way of controlling the FWER while respecting complex trial objectives and multiple structured hypotheses. Ho…
Among Judea Pearl's many contributions to Causality and Statistics, the graphical d-separation} criterion, the do-calculus and the mediation formula stand out. In this chapter we show that d-separation} provides direct insight into an earlier causal model originally described in terms of potential outcomes and event trees. In turn, the resulting synthesis leads to a simplification of the do-calculus that clarifies and separates the underlying concepts, and a simple counterfactual formulation of a complete identification algorithm in causal models with hidden variables.
Graphical Principal Component Analysis of Multivariate Functional Time Series
School of Mathematics, Sun Yat-sen University（中山大学数学学院）；Center for Applied Statistics and School of Statistics, Renmin University of China（中国人民大学统计学院，应用统计中心）
We propose the use of a robust covariance estimator based on multivariate Winsorization in the context of the Tarr-Muller-Weber framework for sparse estimation of the precision matrix of a Gaussian graphical model. Likewise Croux-Ollerer's precision matrix estimator, our proposed estimator attains the maximum finite sample breakdown point of 0.5 under cellwise contamination. We conduct an extensive Monte Carlo simulation study to assess the performance of ours and the currently existing proposals. We find that ours has a competitive behavior, regarding the the estimation of the precision matri…
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 …
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 …
We describe an improved statistical downscaling method for Earth science applications using multivariate Basis Graphical Lasso (BGL). We demonstrate our method using a case study of sea surface temperature (SST) projections from CMIP6 Earth system models, which has direct applications for studies of multi-decadal projections of coral reef bleaching. We find that the BGL downscaling method is computationally tractable for large data sets, and that mean squared predictive error is roughly 8% lower than the current state-of-the-art interpolation-based statistical downscaling method. Finally, unli…
本文综述了利用图形模型表示多元依赖关系的缺失数据研究的最新进展。我们首先从三个不同的角度来研究传统框架的局限性：\textit{transparency, estimability and testability}。然后，我们将展示如何基于图形模型的程序可以克服这些限制，并提供有意义的性能保证，即使数据是非随机缺失（MNAR）。特别是，我们确定的条件，保证一致的估计，在广泛的类别的缺失数据问题，并推导出实施这种估计的程序。最后，我们推导出可检验的影响，缺失数据模型在MAR（随机缺失）和MNAR类别。
A Formal Graphical Inference Framework for Combining Effect Size with Statistical Significance: Application to Multivariate and Functional Linear Models
Faculty of Agriculture and Technology, University of South Bohemia（南波希米亚大学农业与技术学院）；Natural Resources Institute Finland (Luke)（芬兰自然资源研究所（Luke））；University of Helsinki（赫尔辛基大学）
Graphical models are a key class of probabilistic models for studying the conditional independence structure of a set of random variables. Circular variables are special variables, characterized by periodicity, arising in several contexts and fields. However, models for studying the dependence/independence structure of circular variables are under-explored. This paper analyses three multivariate circular distributions, the von Mises, the Wrapped Normal and the Inverse Stereographic distributions, focusing on their properties concerning conditional independence. For each one of these distributi…
Chair of Statistics, University of Göttingen, Germany（统计学系，哥廷根大学）；Campus Institute Data Science, University of Göttingen, Germany（数据科学校园研究所，哥廷根大学）；Department of Medical Bioinformatics, University Medical Center Göttingen, Germany（医学生物信息学系，哥廷根大学医学中心）
We introduce a new test for conditional independence which is based on what we call the weighted generalised covariance measure (WGCM). It is an extension of the recently introduced generalised covariance measure (GCM). To test the null hypothesis of X and Y being conditionally independent given Z, our test statistic is a weighted form of the sample covariance between the residuals of nonlinearly regressing X and Y on Z. We propose different variants of the test for both univariate and multivariate X and Y . We give conditions under which the tests yield the correct type I error rate. Finally,…
We propose a novel extremal dependence measure called the partial tail-correlation coefficient (PTCC), in analogy to the partial correlation coefficient in classical multivariate analysis. The construction of our new coefficient is based on the framework of multivariate regular variation and transformed-linear algebra operations. We show how this coefficient allows identifying pairs of variables that have partially uncorrelated tails given the other variables in a random vector. Unlike other recently introduced conditional independence frameworks for extremes, our approach requires minimal mod…
