Matrix products for the synthesis of stationary time series with a priori prescribed joint distributions

TL;DR

This paper introduces a matrix product framework inspired by physics to synthesize stationary time series with customizable joint distributions, preserving independence structure while controlling dependencies.

Contribution

It develops a novel matrix product approach for generating stationary time series with prescribed joint distributions, extending the applicability of Hidden Markov Models for controlled dependency modeling.

Findings

Framework allows precise control of joint distributions in time series.

Efficient synthesis procedure based on Hidden Markov Models.

Examples demonstrate the ability to prescribe specific statistical properties.

Abstract

Inspired from non-equilibrium statistical physics models, a general framework enabling the definition and synthesis of stationary time series with a priori prescribed and controlled joint distributions is constructed. Its central feature consists of preserving for the joint distribution the simple product struc- ture it has under independence while enabling to input con- trolled and prescribed dependencies amongst samples. To that end, it is based on products of d-dimensional matrices, whose entries consist of valid distributions. The statistical properties of the thus defined time series are studied in details. Having been able to recast this framework into that of Hidden Markov Models enabled us to obtain an efficient synthesis procedure. Pedagogical well-chosen examples (time series with the same marginal distribution, same covariance function, but different joint distributions) aim at illustrating the power and potential of the approach and at showing how targeted statistical prop- erties can be actually prescribed.