Sparse Markov Models for High-dimensional Inference

TL;DR

This paper introduces a recursive method for efficiently identifying relevant lags in high-dimensional Mixture of Transition Distribution models, enabling consistent estimation despite the curse of dimensionality.

Contribution

It presents a novel recursive lag selection procedure for high-dimensional MTD models, improving estimation efficiency and interpretability.

Findings

Consistent recovery of relevant lags in high-dimensional MTD models.

Improved estimation efficiency with fewer parameters.

Successful application to weather data simulations.

Abstract

Finite order Markov models are theoretically well-studied models for dependent discrete data. Despite their generality, application in empirical work when the order is large is rare. Practitioners avoid using higher order Markov models because (1) the number of parameters grow exponentially with the order and (2) the interpretation is often difficult. Mixture of transition distribution models (MTD) were introduced to overcome both limitations. MTD represent higher order Markov models as a convex mixture of single step Markov chains, reducing the number of parameters and increasing the interpretability. Nevertheless, in practice, estimation of MTD models with large orders are still limited because of curse of dimensionality and high algorithm complexity. Here, we prove that if only few lags are relevant we can consistently and efficiently recover the lags and estimate the transition probabilities of high-dimensional MTD models. The key innovation is a recursive procedure for the selection of the relevant lags of the model. Our results are based on (1) a new structural result of the MTD and (2) an improved martingale concentration inequality. We illustrate our method using simulations and a weather data.