Mixed and Covariate Dependent Graphical Models

Mixed and Covariate Dependent Graphical Models by Jie Cheng Co-Chairs: Assoc. Prof. Elizaveta Levina and Prof. Ji Zhu Graphical models have proven to be a useful tool in understanding the conditional dependency structure of multivariate distributions. In Chapters II and III of the thesis, we consider two types of undirected graphical models that are motivated by particular types of applications. The first model we consider is a mixed graphical model, linking both continuous and discrete variables. The proposed model is simple enough to be suitable for high-dimensional data, yet flexible enough to represent all possible graph structures for mixed types of data. We develop a computationally efficient regression-based algorithm for fitting the model by focusing on the conditional log-likelihood of each variable given the rest. The parameters have a natural group structure, and sparsity in the fitted graph is attained by incorporating a group lasso penalty, approximated by a weighted `1 penalty for computational efficiency. We demonstrate the effectiveness of our method through an extensive simulation study and apply it to a music annotation data set (CAL500), obtaining a sparse and interpretable graphical model relating the continuous features of the audio signal to categorical variables such as genre, emotions, and usage associated with particular songs.