From empirical data to continuous Markov processes: a systematic approach

TL;DR

This paper introduces a systematic method for testing and estimating continuous Markov process generators from discrete data, extending existing approaches to handle time-inhomogeneous cases with high accuracy.

Contribution

The authors develop a new mathematical framework and computational algorithm for identifying continuous generators of discrete stochastic matrices, including time-inhomogeneous cases, with proven effectiveness.

Findings

Detection algorithm succeeds in over 80% of cases, typically 90-95%

Provides estimates of non-homogeneous generator matrices

Analytically solves embedding problem for 3D circulant matrices

Abstract

We present an approach for testing for the existence of continuous generators of discrete stochastic transition matrices. Typically, the known approaches to ascertain the existence of continuous Markov processes are based in the assumption that only time-homogeneous generators exist. Here, a systematic extension to time-inhomogeneity is presented, based in new mathematical propositions incorporating necessary and sufficient conditions, which are then implemented computationally and applied to numerical data. A discussion concerning the bridging between rigorous mathematical results on the existence of generators to its computational implementation. Our detection algorithm shows to be effective in more than $80\%$ of tested matrices, typically $90\%$ to $95\%$, and for those an estimate of the (non-homogeneous) generator matrix follows. We also solve the embedding problem analytically for the particular case of three-dimensional circulant matrices. Finally, a discussion of possible applications of our framework to problems in different fields is briefly addressed.