Probabilistic cellular automata with local transition matrices: synchronization, ergodicity, and inference

TL;DR

This paper introduces a new class of probabilistic cellular automata with local transition matrices, analyzing their synchronization, ergodicity, and proposing methods for inferring their parameters from data.

Contribution

It defines a novel probabilistic cellular automaton model, establishes conditions for key dynamical properties, and develops new estimators for parameter inference from various data types.

Findings

Conditions for synchronization and ergodicity are established.

New least squares estimators are proposed for local transition matrices.

Asymptotic normality and accuracy bounds of estimators are demonstrated.

Abstract

We introduce a new class of probabilistic cellular automata that are capable of exhibiting rich dynamics such as synchronization and ergodicity and can be easily inferred from data. The system is a finite-state locally interacting Markov chain on a circular graph. Each site's subsequent state is random, with a distribution determined by its neighborhood's empirical distribution multiplied by a local transition matrix. We establish sufficient and necessary conditions on the local transition matrix for synchronization and ergodicity. Also, we introduce novel least squares estimators for inferring the local transition matrix from various types of data, which may consist of either multiple trajectories, a long trajectory, or ensemble sequences without trajectory information. Under suitable identifiability conditions, we show the asymptotic normality of these estimators and provide non-asymptotic bounds for their accuracy.