Florian Angeletti

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Estimation, Phase transition

Critical moment definition and estimation,for finite size observation of log-exponential-power law random variables,

F. Angeletti, E. Bertin, P. Abry, Signal Processing, (2012), 2848-2865,
doi:10.1016/j.sigpro.2012.05.007, arxiv:1103.5033, BIB, PDF


This contribution aims at studying the behaviour of the classical sample moment estimator,$S(n,q)=\sum_k=1^n X_k^q/n $,as a function of the number of available samples $n$,in the case where the random variables $X$ are positive,have finite moments at all orders and are naturally of the form $X=\exp Y$ with the tail of $Y$ behaving like $\exp(-y^\rho)$. This class of laws encompasses and generalizes the classical example of the log-normal law. This form is motivated by a number of applications stemming from modern statistical physics or multifractal analysis. Borrowing heuristic and analytical results from the analysis of the Random Energy Model in statistical physics,a critical moment $q_c(n)$ is defined as the largest statistical order $q$ up to which the sample mean estimator $S(n,q)$ correctly accounts for the ensemble average $\E X^q$,for a given $n$. A practical estimator for the critical moment $q_c(n)$ is then proposed. Its statistical performance are studied analytically and illustrated numerically in the case of i.i.d. samples. A simple modification is proposed to explicitly account for correlation amongst the observed samples. Estimation performance are then carefully evaluated by means of Monte-Carlo simulations in the practical case of correlated time series.

Matrix representation, Random variable synthesis

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


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.

Extreme values, Renormalization

Renormalization flow for extreme value statistics of random variables raised to a varying power,

F. Angeletti, E. Bertin, P. Abry, J. Phys. A, (2012), 115004,
doi:10.1088/1751-8113/45/11/115004, arxiv:1112.2965, BIB, PDF


Using a renormalization approach,we study the asymptotic limit distribution of the maximum value in a set of independent and identically distributed random variables raised to a power $q_n$ that varies monotonically with the sample size $n$. Under these conditions,a non-standard class of max-stable limit distributions,which mirror the classical ones,emerges. Furthermore a transition mechanism between the classical and the non-standard limit distributions is brought to light. If $q_n$ grows slower than a characteristic function $q^*_n$,the standard limit distributions are recovered,while if $q_n$ behaves asymptotically as $\lambda q^*_n$,non-standard limit distributions emerge.