Matrix product representation and synthesis for random vectors: Insight from statistical physics

TL;DR

This paper introduces a matrix product framework inspired by statistical physics to define and synthesize random vectors and time series with prescribed joint distributions and dependencies, offering theoretical insights and practical tools.

Contribution

It presents a novel matrix product approach for modeling dependent random vectors and time series, linking it to Hidden Markov Models for efficient synthesis and parameter tuning.

Findings

Thorough depiction of dependence structures achievable with the framework

Derivation of a stationarity condition for time series

Development of an efficient synthesis procedure

Abstract

Inspired from modern out-of-equilibrium statistical physics models, a matrix product based framework permits the formal definition of random vectors (and random time series) whose desired joint distributions are a priori prescribed. Its key feature consists of preserving the writing of the joint distribution as the simple product structure it has under independence, while inputing controlled dependencies amongst components: This is obtained by replacing the product of distributions by a product of matrices of distributions. The statistical properties stemming from this construction are studied theoretically: The landscape of the attainable dependence structure is thoroughly depicted and a stationarity condition for time series is notably obtained. The remapping of this framework onto that of Hidden Markov Models enables us to devise an efficient and accurate practical synthesis procedure. A design procedure is also described permitting the tuning of model parameters to attain targeted properties. Pedagogical well-chosen examples of times series and multivariate vectors aim at illustrating the power and versatility of the proposed approach and at showing how targeted statistical properties can be actually prescribed.