IDGM: an approach to estimate the graphical model of interval-valued data

Graphical models describe the conditional dependence structure among random variables via vertices and edges and have attracted increasing attention in recent years. However, when the variable is interval-valued instead of a scalar, it remains unclear how the graphical model can be estimated since interval-valued data impose additional complexity, including the lower bound should not be greater than the upper bound and each interval is itself a two-dimensional object. In this paper, we propose an algorithm, named the interval-valued data graphical model (IDGM), to realize such estimation, extending the graphical model concept to interval-valued data modeling. To address the complexity of interval-valued data, we apply the midpoints and log-ranges transformation to engage the center and range information of an interval. Then, we identify the network structure based on a variant 2x2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 2 \times 2 $$\end{document} block-wise sparsity graphical lasso that incorporates the penalty term of the precision matrix. The numerical simulations along with two real-world applications in the fields of macroeconomics and finance show the advantages of IDGM over the competing methods and demonstrate the effectiveness of IDGM in graphical model estimation for interval-valued data.