hdMTD: An R Package for High-Dimensional Mixture Transition Distribution Models

Abstract

Several natural phenomena exhibit long-range conditional dependencies. High-order mixture transition distribution (MTD) are parsimonious non-parametric models to study these phenomena. An MTD is a Markov chain in which the transition probabilities are expressed as a convex combination of lower-order conditional distributions. Despite their generality, inference for MTD models has traditionally been limited by the need to estimate high-dimensional joint distributions. In particular, for a sample of size n, the feasible order d of the MTD is typically restricted to d approximately O(log n). To overcome this limitation, Ost and Takahashi (2023) recently introduced a computationally efficient non-parametric inference method that identifies the relevant lags in high-order MTD models, even when d is approximately O(n), provided that the set of relevant lags is sparse. In this article, we introduce hdMTD, an R package allowing us to estimate parameters of such high-dimensional Markovian models. Given a sample from an MTD chain, hdMTD can retrieve the relevant past set using the BIC algorithm or the forward stepwise and cut algorithm described in Ost and Takahashi (2023). The package also computes the maximum likelihood estimate for transition probabilities and estimates high-order MTD parameters through the expectation-maximization algorithm. Additionally, hdMTD also allows for simulating an MTD chain from its stationary invariant distribution using the perfect (exact) sampling algorithm, enabling Monte Carlo simulation of the model. We illustrate the package's capabilities through simulated data and a real-world application involving temperature records from Brazil.