High Dimensional Semiparametric Latent Graphical Model for Mixed Data
This addresses a statistical modeling problem for researchers dealing with mixed data, offering a novel method for latent graph and principal component analysis, but it is incremental as it builds on existing latent variable frameworks.
The paper tackles the challenge of modeling mixed data (continuous and discrete) in graphical models by proposing a semiparametric latent Gaussian copula model, achieving convergence rates for precision matrix and eigenvector estimation comparable to having observed latent variables, with performance validated through simulations and a genetic dataset.
Graphical models are commonly used tools for modeling multivariate random variables. While there exist many convenient multivariate distributions such as Gaussian distribution for continuous data, mixed data with the presence of discrete variables or a combination of both continuous and discrete variables poses new challenges in statistical modeling. In this paper, we propose a semiparametric model named latent Gaussian copula model for binary and mixed data. The observed binary data are assumed to be obtained by dichotomizing a latent variable satisfying the Gaussian copula distribution or the nonparanormal distribution. The latent Gaussian model with the assumption that the latent variables are multivariate Gaussian is a special case of the proposed model. A novel rank-based approach is proposed for both latent graph estimation and latent principal component analysis. Theoretically, the proposed methods achieve the same rates of convergence for both precision matrix estimation and eigenvector estimation, as if the latent variables were observed. Under similar conditions, the consistency of graph structure recovery and feature selection for leading eigenvectors is established. The performance of the proposed methods is numerically assessed through simulation studies, and the usage of our methods is illustrated by a genetic dataset.