Matrix-correlated random variables: a statistical physics and signal processing dialogue,
F. Angeletti, Séminaire LPT Toulouse, Toulouse, (2015), Abstract
Finding statistical descriptions of non-equilibrium stationary state is often an arduous task. For specific systems,like ASEP or 1D diffusion-reaction systems,stationary solutions have been characterized using matrix product representations. These representations generalize the product structure of independent random variable to matrices; the non-commutativity of matrices generates correlation while preserving many of the algebraic properties of expectation. Depending on the matrix structure,the correlation can vary from short-range to long-range correlation. Moreover,from a signal processing perspective,these matrix-correlated random variables can be recast as specific Hidden Markov Models. In this talk,we propose to investigate the general statistical properties of this mathematical framework,with the long-term hope to improve our understanding of related physical systems. In particular,we shall focus on the statistical properties of sums of such random variables. Do we have a large deviation principle for this sum? Can we find analogous of the law of large number or the central limit theorem?
Matrix-correlated random variables: a statistical physics and signal processing dialogue,
F. Angeletti, Séminaire LPTMS Orsay, Orsay, (2015), Abstract
Finding statistical descriptions of non-equilibrium stationary state is often an arduous task. For specific systems,like ASEP or 1D diffusion-reaction systems,stationary solutions have been characterized using matrix product representations. These representations generalize the product structure of independent random variable to matrices; the non-commutativity of matrices generates correlation while preserving many of the algebraic properties of expectation. Depending on the matrix structure,the correlation can vary from short-range to long-range correlation. Moreover,from a signal processing perspective,these matrix-correlated random variables can be recast as specific Hidden Markov Models. In this talk,we propose to investigate the general statistical properties of this mathematical framework,with the long-term hope to improve our understanding of related physical systems. In particular,we shall focus on the statistical properties of sums of such random variables. Do we have a large deviation principle for this sum? Can we find analogous of the law of large number or the central limit theorem?
Critical order for moment estimation : insights from statistical physics.,
F. Angeletti, Club journal LPT Orsay, Orsay, (2015), Abstract
Moment estimation is one of the most basic question in statistical signal processing.
For i.i.d. random variables and a infinite number of observations,the law of large numbers implies that the classical moment estimator converges towards the theoretical moment. However,these hypothesis are not really satisfied in practical settings where only a finite number of observations is available and most data are correlated.
It is therefore quite natural to ask ourselves: up to which order can moments be reliably estimated from a given number of observation? In this presentation,I will show how the notion of glassy phase transition developed for studying disordered systems in statistical physics can help to answer such a question.