Block matrix approximation via entropy loss function

Abstract

The aim of the paper is to present a procedure for the approximation of a symmetric positive definite matrix by symmetric block partitioned matrices with structured off-diagonal blocks. The entropy loss function is chosen as approximation criterion. This procedure is applied in a simulation study of the statistical problem of covariance structure identification.

Structure identification for a linearly structured covariance matrix

“…The identification procedure can be also used for a non-linear structure -such as the autoregressive structure of order one; cf. Filipiak et al (2021), Janiszewska et al (2020).…”

Structures of the covariance matrix: An overview

“…The entropy loss function was used in the paper Janiszewska et al (2020) to identify covariance structure, where consideration was given to a block covariance structure with off-diagonal blocks corresponding to the off-diagonal blocks of CS and AR(1) structures. The entropy loss function is also known as Kullback-Leibler divergence between two probability distributions; cf.…”

“…In the literature, some subblocks of the structures given in the previous sections have been considered. In the paper Janiszewska et al (2020), the authors study the following structure:…”

“…Szatrowski (1976), Szatrowski (1982), . In the papers Janiszewska et al (2020) and the authors study block covariance matrices, where off-diagonal blocks with compound symmetry or first-order autoregression structures were considered. In the paper , banded Toeplitz structures in blocks were used.…”

“…Devijver and Gallopin 2018). Algebraic techniques, based on the identification of a structure from a set of potential structures, are widely used in the literature; see for example Cui et al (2016), , Janiszewska et al (2020), Lin et al (2014). After choosing the most suitable structure for the considered covariance matrix, the natural procedure is to test hypotheses about covariance structure.…”