Dynamic Gaussian Graphical Models for Modelling Genomic Networks

Abstract

After sequencing the entire DNA for various organisms, the challenge has become understanding the functional interrelatedness of the genome. Only by understanding the pathways for various complex diseases can we begin to make sense of any type of treatment. Unfortunately, decyphering the genomic network structure is an enormous task. Even with a small number of genes the number of possible networks is very large. This problem becomes even more difficult, when we consider dynamical networks. We consider the problem of estimating a sparse dynamic Gaussian graphical model with $L_1$ penalized maximum likelihood of structured precision matrix. The structure can consist of specific time dynamics, known presence or absence of links in the graphical model or equality constraints on the parameters. The model is defined on the basis of partial correlations, which results in a specific class precision matrices. A priori $L_1$ penalized maximum likelihood estimation in this class is extremely difficult, because of the above mentioned constraints, the computational complexity of the $L_1$ constraint on the side of the usual positive-definite constraint. The implementation is non-trivial, but we show that the computation can be done effectively by taking advantage of an efficient maximum determinant algorithm developed in convex optimization.