Regressions in Covariances, Dependencies and Graphs

Mohsen Pourahmadi and Aramayis Dallakyan

The book is about modeling dependencies: linear and nonlinear, cross-sectional, temporal, and spatial in multivariate data. Its unifying idea is to bring the success of regression on means (first moments) to the harder problem of modeling covariance matrices (second moments), and eventually full distributions. Two complementary strategies anchor the treatment. The first, covariance regression, models a covariance matrix (or a transform of it) as a function of covariates, moving toward generalized linear models for second moments. The second, hidden regression, is the column-by-column iterative idea behind the regularized Gaussian likelihood and the Graphical Lasso. Copulas extend the framework to nonlinearity, extreme values, and long-tail behavior, and regularization makes it work in high dimensions. Every chapter ends with hands-on R scripts, and the companion package ships real and simulated datasets along with the functions to reproduce every example.