arXiv subjects

Patrice Abry

Publications and source records attributed to Patrice Abry.

At least 55 records · Page 3Linked to original sources

arXiv2016

Non-Linear Wavelet Regression and Branch & Bound Optimization for the Full Identification of Bivariate Operator Fractional Brownian Motion

Self-similarity is widely considered the reference framework for modeling the scaling properties of real-world data. However, most theoretical studies and their practical use have remained univariate. Operator Fractional Brownian Motion (OfBm) was recently proposed as a multivariate model for self-similarity. Yet it has remained seldom used in applications because of serious issues that appear in the joint estimation of its numerous parameters. While the univariate fractional Brownian motion requires the estimation of two parameters only, its mere bivariate extension already involves 7 parameters which are very different in nature. The present contribution proposes a method for the full identification of bivariate OfBm (i.e., the joint estimation of all parameters) through an original formulation as a non-linear wavelet regression coupled with a custom-made Branch & Bound numerical scheme. The estimation performance (consistency and asymptotic normality) is mathematically established and numerically assessed by means of Monte Carlo experiments. The impact of the parameters defining OfBm on the estimation performance as well as the associated computational costs are also thoroughly investigated.

math.PR↗

arXiv2016

p-exponent and p-leaders, Part II: Multifractal Analysis. Relations to Detrended Fluctuation Analysis

Multifractal analysis studies signals, functions, images or fields via the fluctuations of their local regularity along time or space, which capture crucial features of their temporal/spatial dynamics. It has become a standard signal and image processing tool and is commonly used in numerous applications of different natures. In its common formulation, it relies on the Hölder exponent as a measure of local regularity, which is by nature restricted to positive values and can hence be used for locally bounded functions only. In this contribution, it is proposed to replace the Hölder exponent with a collection of novel exponents for measuring local regularity, the $p$-exponents. One of the major virtues of $p$-exponents is that they can potentially take negative values. The corresponding wavelet-based multiscale quantities, the $p$-leaders, are constructed and shown to permit the definition of a new multifractal formalism, yielding an accurate practical estimation of the multifractal properties of real-world data. Moreover, theoretical and practical connections to and comparisons against another multifractal formalism, referred to as multifractal detrended fluctuation analysis, are achieved. The performance of the proposed $p$-leader multifractal formalism is studied and compared to previous formalisms using synthetic multifractal signals and images, illustrating its theoretical and practical benefits. The present contribution is complemented by a companion article studying in depth the theoretical properties of $p$-exponents and the rich classification of local singularities it permits.

math.CA↗

arXiv2016

Combining local regularity estimation and total variation optimization for scale-free texture segmentation

Texture segmentation constitutes a standard image processing task, crucial to many applications. The present contribution focuses on the particular subset of scale-free textures and its originality resides in the combination of three key ingredients: First, texture characterization relies on the concept of local regularity ; Second, estimation of local regularity is based on new multiscale quantities referred to as wavelet leaders ; Third, segmentation from local regularity faces a fundamental bias variance trade-off: In nature, local regularity estimation shows high variability that impairs the detection of changes, while a posteriori smoothing of regularity estimates precludes from locating correctly changes. Instead, the present contribution proposes several variational problem formulations based on total variation and proximal resolutions that effectively circumvent this trade-off. Estimation and segmentation performance for the proposed procedures are quantified and compared on synthetic as well as on real-world textures.

cs.CV↗

arXiv2016

A Primal-Dual Algorithm for Link Dependent Origin Destination Matrix Estimation

Origin-Destination Matrix (ODM) estimation is a classical problem in transport engineering aiming to recover flows from every Origin to every Destination from measured traffic counts and a priori model information. In addition to traffic counts, the present contribution takes advantage of probe trajectories, whose capture is made possible by new measurement technologies. It extends the concept of ODM to that of Link dependent ODM (LODM), keeping the information about the flow distribution on links and containing inherently the ODM assignment. Further, an original formulation of LODM estimation, from traffic counts and probe trajectories is presented as an optimisation problem, where the functional to be minimized consists of five convex functions, each modelling a constraint or property of the transport problem: consistency with traffic counts, consistency with sampled probe trajectories, consistency with traffic conservation (Kirchhoff's law), similarity of flows having close origins and destinations, positivity of traffic flows. A primal-dual algorithm is devised to minimize the designed functional, as the corresponding objective functions are not necessarily differentiable. A case study, on a simulated network and traffic, validates the feasibility of the procedure and details its benefits for the estimation of an LODM matching real-network constraints and observations.

math.OC↗

arXiv2015

Wavelet estimation for operator fractional Brownian motion

Operator fractional Brownian motion (OFBM) is the natural vector-valued extension of the univariate fractional Brownian motion. Instead of a scalar parameter, the law of an OFBM scales according to a Hurst matrix that affects every component of the process. In this paper, we develop the wavelet analysis of OFBM, as well as a new estimator for the Hurst matrix of bivariate OFBM. For OFBM, the univariate-inspired approach of analyzing the entry-wise behavior of the wavelet spectrum as a function of the (wavelet) scales is fraught with difficulties stemming from mixtures of power laws. The proposed approach consists of considering the evolution along scales of the eigenstructure of the wavelet spectrum. This is shown to yield consistent and asymptotically normal estimators of the Hurst eigenvalues, and also of the coordinate system itself under assumptions. A simulation study is included to demonstrate the good performance of the estimators under finite sample sizes.

math.PR↗

arXiv2015

Multifractal analysis based on p-exponents and lacunarity exponents

Many examples of signals and images cannot be modeled by locally bounded functions, so that the standard multifractal analysis, based on the Hölder exponent, is not feasible. We present a multifractal analysis based on another quantity, the p-exponent, which can take arbitrarily large negative values. We investigate some mathematical properties of this exponent, and show how it allows us to model the idea of "lacunarity" of a singularity at a point. We finally adapt the wavelet based multifractal analysis in this setting, and we give applications to a simple mathematical model of multifractal processes: Lacunary wavelet series.

math.FA↗

arXiv2015

Bayesian estimation of the multifractality parameter for image texture using a Whittle approximation

Texture characterization is a central element in many image processing applications. Multifractal analysis is a useful signal and image processing tool, yet, the accurate estimation of multifractal parameters for image texture remains a challenge. This is due in the main to the fact that current estimation procedures consist of performing linear regressions across frequency scales of the two-dimensional (2D) dyadic wavelet transform, for which only a few such scales are computable for images. The strongly non-Gaussian nature of multifractal processes, combined with their complicated dependence structure, makes it difficult to develop suitable models for parameter estimation. Here, we propose a Bayesian procedure that addresses the difficulties in the estimation of the multifractality parameter. The originality of the procedure is threefold: The construction of a generic semi-parametric statistical model for the logarithm of wavelet leaders; the formulation of Bayesian estimators that are associated with this model and the set of parameter values admitted by multifractal theory; the exploitation of a suitable Whittle approximation within the Bayesian model which enables the otherwise infeasible evaluation of the posterior distribution associated with the model. Performance is assessed numerically for several 2D multifractal processes, for several image sizes and a large range of process parameters. The procedure yields significant benefits over current benchmark estimators in terms of estimation performance and ability to discriminate between the two most commonly used classes of multifractal process models. The gains in performance are particularly pronounced for small image sizes, notably enabling for the first time the analysis of image patches as small as 64x64 pixels.

physics.data-an↗

arXiv2014

General limit distributions for sums of random variables with a matrix product representation

The general limit distributions of the sum of random variables described by a finite matrix product ansatz are characterized. Using a mapping to a Hidden Markov Chain formalism, non-standard limit distributions are obtained, and related to a form of ergodicity breaking in the underlying non-homogeneous Hidden Markov Chain. The link between ergodicity and limit distributions is detailed and used to provide a full algorithmic characterization of the general limit distributions.

cond-mat.stat-mech↗

arXiv2014

Large deviations for correlated random variables described by a matrix product ansatz

We study the large deviations of sums of correlated random variables described by a matrix product ansatz, which generalizes the product structure of independent random variables to matrices whose non-commutativity is the source of correlations. We show with specific examples that different large deviation behaviors can be found with this ansatz. In particular, it is possible to construct sums of correlated random variables that violate the Law of Large Numbers, the Central Limit Theorem, as well as sums that have nonconvex rate functions or rate functions with linear parts or plateaux.

cond-mat.stat-mech↗

arXiv2014

Statistics of sums of correlated variables described by a matrix product ansatz

We determine the asymptotic distribution of the sum of correlated variables described by a matrix product ansatz with finite matrices, considering variables with finite variances. In cases when the correlation length is finite, the law of large numbers is obeyed, and the rescaled sum converges to a Gaussian distribution. In constrast, when correlation extends over system size, we observe either a breaking of the law of large numbers, with the onset of giant fluctuations, or a generalization of the central limit theorem with a family of nonstandard limit distributions. The corresponding distributions are found as mixtures of delta functions for the generalized law of large numbers, and as mixtures of Gaussian distributions for the generalized central limit theorem. Connections with statistical physics models are emphasized.

cond-mat.stat-mech↗

arXiv2013

On the existence of a glass transition in a Random Energy Model

We consider a generalized version of the Random Energy Model in which the energy of each configuration is given by the sum of $N$ independent contributions ("local energies") with finite variances but otherwise arbitrary statistics. Using the large deviation formalism, we find that the glass transition generically exists when local energies have a smooth distribution. In contrast, if the distribution of the local energies has a {Dirac mass} at the minimal energy (e.g., if local energies take discrete values), the glass transition ceases to exist if the number of energy levels grows sufficiently fast with system size. This shows that statistical independence of energy levels does not imply the existence of a glass transition.

cond-mat.stat-mech↗

arXiv2012

Hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic textures

Global and local regularities of functions are analyzed in anisotropic function spaces, under a common framework, that of hyperbolic wavelet bases. Local and directional regularity features are characterized by means of global quantities constructed upon the coefficients of hyperbolic wavelet decompositions. A multifractal analysis is introduced, that jointly accounts for scale invariance and anisotropy. Its properties are studied in depth.

math.FA↗

arXiv2012

Irregularities and Scaling in Signal and Image Processing: Multifractal Analysis

B. Mandelbrot gave a new birth to the notions of scale invariance, selfsimilarity and non-integer dimensions, gathering them as the founding corner-stones used to build up fractal geometry. The first purpose of the present contribution is to review and relate together these key notions, explore their interplay and show that they are different facets of a same intuition. Second, it will explain how these notions lead to the derivation of the mathematical tools underlying multifractal analysis. Third, it will reformulate these theoretical tools into a wavelet framework, hence enabling their better theoretical understanding as well as their efficient practical implementation. B. Mandelbrot used his concept of fractal geometry to analyze real-world applications of very different natures. As a tribute to his work, applications of various origins, and where multifractal analysis proved fruitful, are revisited to illustrate the theoretical developments proposed here.

math.FA↗

arXiv2012

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

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.

cond-mat.stat-mech↗

arXiv2012

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

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.

cond-mat.stat-mech↗

arXiv2012

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

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 k.q*(n), non-standard limit distributions emerge.

cond-mat.stat-mech↗

arXiv2011

Linearization effect in multifractal analysis: Insights from the Random Energy Model

The analysis of the linearization effect in multifractal analysis, and hence of the estimation of moments for multifractal processes, is revisited borrowing concepts from the statistical physics of disordered systems, notably from the analysis of the so-called Random Energy Model. Considering a standard multifractal process (compound Poisson motion), chosen as a simple representative example, we show: i) the existence of a critical order $q^*$ beyond which moments, though finite, cannot be estimated through empirical averages, irrespective of the sample size of the observation; ii) that multifractal exponents necessarily behave linearly in $q$, for $q > q^*$. Tayloring the analysis conducted for the Random Energy Model to that of Compound Poisson motion, we provide explicative and quantitative predictions for the values of $q^*$ and for the slope controlling the linear behavior of the multifractal exponents. These quantities are shown to be related only to the definition of the multifractal process and not to depend on the sample size of the observation. Monte-Carlo simulations, conducted over a large number of large sample size realizations of compound Poisson motion, comfort and extend these analyses.

cond-mat.stat-mech↗

arXiv2011

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

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 $e^{-y^ρ}$. 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 \emph{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.

math.ST↗