The Convex Mixture Distribution: Granger Causality for Categorical Time Series

Area of Science:

• Statistics
• Machine Learning
• Time Series Analysis

Background:

• Mixture Transition Distribution (MTD) models are used for categorical time series but suffer from non-convexity and local optima.
• Existing methods struggle with high-dimensional data and identifiability issues.

Purpose of the Study:

• To present a novel convex formulation for MTD to enable robust Granger causality network inference.
• To compare the proposed convex MTD with a multi-output logistic autoregressive model (mLTD) for categorical time series.
• To establish theoretical guarantees for the convex MTD in high-dimensional settings.

Main Methods:

• Recasting MTD inference as a convex optimization problem.
• Developing efficient optimization algorithms for the convex MTD formulation.
• Formulating and comparing a multi-output logistic autoregressive model (mLTD) as a baseline.
• Establishing identifiability conditions and consistency for MTD and mLTD.

Main Results:

• The convex MTD formulation successfully addresses traditional limitations, allowing application to high-dimensional data.
• Comparative experiments on simulated and real data demonstrate the efficacy of the convex MTD.
• Identifiability conditions for MTD and mLTD were established, and consistency of convex MTD in high dimensions was proven.

Conclusions:

• The proposed convex MTD framework offers a significant advancement for Granger causality network inference in multivariate categorical time series.
• This work facilitates modern, regularized inference techniques for MTD models.
• The study provides a valuable comparison of network inference methods for categorical time series data.