Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies

TL;DR

This paper introduces Square Root Graphical Models (SQR), a new class of multivariate exponential family models that allows for positive and negative dependencies, addressing limitations of previous models especially in real-world datasets like airport delays.

Contribution

The paper presents SQR models that enable positive dependencies in multivariate exponential family distributions, with new estimation methods and demonstrated effectiveness on real and synthetic data.

Findings

SQR models successfully model positive dependencies in exponential and Poisson distributions.

Parameter estimation via node-wise regressions with $$ regularization is effective.

Real-world airport delay data is well modeled by the proposed SQR framework.

Abstract

We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models [Yang et al. 2015] did not allow positive dependencies for the exponential and Poisson generalizations. However, in many real-world datasets, variables clearly have positive dependencies. For example, the airport delay time in New York---modeled as an exponential distribution---is positively related to the delay time in Boston. With this motivation, we give an example of our model class derived from the univariate exponential distribution that allows for almost arbitrary positive and negative dependencies with only a mild condition on the parameter matrix---a condition akin to the positive definiteness of the Gaussian covariance matrix. Our Poisson generalization allows for both positive and negative dependencies without any constraints on the parameter values. We also develop parameter estimation methods using node-wise regressions with $\ell_1$ regularization and likelihood approximation methods using sampling. Finally, we demonstrate our exponential generalization on a synthetic dataset and a real-world dataset of airport delay times.