Florian Angeletti

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F. Angeletti, E. Bertin, P. Abry

Random vector and time series definition and synthesis from matrix product representations: From Statistical Physics to Hidden Markov Models,

F. Angeletti, E. Bertin, P. Abry, IEEE Transactions on Signal Processing, (2013), 5389-5400,
doi:10.1109/TSP.2013.2278510, arxiv:1203.4500, BIB, PDF

Abstract

Inspired from modern out-of-equilibrium statistical physics models,a matrix product based framework is defined and studied,that permits the formal definition of random vectors and 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 probability densities by a product of matrices of probability densities. It is first shown that this matrix product model can be remapped onto the framework of Hidden Markov Models. Second,combining this double perspective enables us both to study the statistical properties of this model in terms of marginal distributions and dependencies (a stationarity condition is notably devised) and to devise an efficient and accurate numerical synthesis procedure. A design procedure is also described that permits the tuning of model parameters to attain targeted statistical 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 actually be prescribed.

Matrix Products for the Synthesis of Stationary Time Series with a priori Prescribed Joint Distributions,

F. Angeletti, E. Bertin, P. Abry, ICASSP conference, (2012), 3897-3900,
doi:10.1109/ICASSP.2012.6288769, arxiv:1204.3047, BIB, PDF

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 structure it has under independence while enabling to input controlled 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 properties can be actually prescribed.