Complex Normal Graphical Models

Abstract

Graphical models are used to examine conditional independence among random variables. In this chapter we take graphical models for the multivariate complex normal distribution w.r.t. simple undirected graphs into consideration. This is the first published presentation of these models. Graphical models for the multivariate real normal distribution, also called covariance selection models, have already been studied in the literature. The initial work on covariance selection models is done by Dempster (1972) and Wermuth (1976) and a presentation of these models is given in Eriksen (1992). Graphical models for contingency tables are introduced in statistics by Darroch et al. (1980) and further these are well-described in Lauritzen (1989). Graphical association models are treated in general in Whittaker (1990) and Lauritzen (1993). The complex normal graphical models are quite similar to the covariance selection models. We have chosen to develop this chapter without use of exponential families. We study definition of the model, maximum likelihood estimation and hypothesis testing. To verify some of the results in the chapter we use results from mathematical analysis. These can be found in e.g. Rudin (1987). In graphical models one uses the concentration matrix instead of the variance matrix as it is more advantageous. Therefore we define this matrix and derive a relation which is basic for complex normal graphical models. Afterwards we formally define a complex normal graphical model w.r.t. a simple undirected graph. As these models are used to examine conditional independence of selected pairs of variables given the remaining ones we are mainly interested in inference on the concentration matrix. It is possible to base the maximum likelihood estimation of the concentration matrix on a complex random matrix with mean zero.