Estimation and testing for separable variance–covariance structures

Abstract

The statistical analysis of data for a p‐variate response observed repeatedly on q occasions or of spatiotemporal data recorded at p locations by q times for n individuals may require that constraints be imposed on the modeling of the variance–covariance structure of the underlying process, not because of the repeated‐measures or spatiotemporal nature of the data but because there is not enough data otherwise to estimate the model parameters. Besides stationarity and isotropy, separability is an interesting option for that purpose because it reduces the number of variance‐covariance parameters to estimate, from pq(pq + 1)/2 to the Kronecker product of two matrices with p(p + 1)/2 and q(q + 1)/2 parameters. Originally, in the late 1980s, separability of the variance–covariance structure was assumed. Under this model, combined with the normality assumption on the underlying distribution, novel theoretical developments were thus made. The question of estimation of the parameters of a separable variance–covariance structure, more particularly by maximum likelihood, was raised from the early 1990s on, the question of testing for this structure being effectively addressed several years later. The existence and uniqueness of maximum likelihood estimators for the matrix normal distribution (i.e., the doubly multivariate normal distribution characterized by a simply separable variance–covariance structure) have been and remain questions of interest, as shown by recent results. Below, the reader is guided throughout the field of study of the separable variance–covariance structures as the author provides a fair treatment of the topic, its components, extensions (e.g., double separability), and future perspectives. This article is categorized under Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms

Structured time‐dependent inverse regression (STIR)

“…If $$$ \boldsymbol{\Delta} $$$ has a Kronecker product structure, $$$ \boldsymbol{\Delta} ={\boldsymbol{\Delta}}_p\otimes {\boldsymbol{\Delta}}_T $$$, with $$$ {\boldsymbol{\Delta}}_p\in {\mathbb{R}}^{p\times p} $$$ and $$$ {\boldsymbol{\Delta}}_T\in {\mathbb{R}}^{T\times T} $$$, the number of variance‐covariance parameters is reduced from $$$ pT\left( pT+1\right)/2 $$$ to $$$ p\left(p+1\right)/2+T\left(T+1\right)/2 $$$. Here we estimate $$$ {\boldsymbol{\Delta}}_p $$$ and $$$ {\boldsymbol{\Delta}}_T $$$ based on the residuals of the regression models () or (), similarly to Pfeiffer et al, 2 but several other approaches have been proposed 13,14 . The $$$ p $$$‐vector of residual values for observation $$$ i $$$ at time $$$ t $$$ is …”

“…Here we estimate 𝚫 p and 𝚫 T based on the residuals of the regression models (4) or (22), similarly to Pfeiffer et al, 2 but several other approaches have been proposed. 13,14 The p-vector of residual values for observation i at time t is e j. is the T-vector of means of the jth marker, j = 1, … , p. Let n t be the number of observations at time t across all p markers and n j the number of observations of the jth marker across all time points. An estimate 𝚫 by Δ = Δp ⊗ ΔT , where…”

Two-sample intraclass correlation coefficient tests for matrix-valued data

“…In the Kronecker product of two matrices there are only 1 2 (p(p + 1) + q(q + 1)) parameters to be estimated as compared to the general case where we need to estimate pq(pq + 1)/2 parameters. Hence it is also preferable in many analysis when there is not enough data otherwise to estimate the model parameters (Dutilleul, 2018). From an inferential point of view, the Kronecker structure makes the estimation more complicated since the identification problem should be resolved and some restrictions have to be imposed on the parameter space.…”