arXiv subjects

Eric Moulines

Publications and source records attributed to Eric Moulines.

At least 181 records · Page 10Linked to original sources

arXiv2009

Error exponents for Neyman-Pearson detection of a continuous-time Gaussian Markov process from noisy irregular samples

This paper addresses the detection of a stochastic process in noise from irregular samples. We consider two hypotheses. The \emph{noise only} hypothesis amounts to model the observations as a sample of a i.i.d. Gaussian random variables (noise only). The \emph{signal plus noise} hypothesis models the observations as the samples of a continuous time stationary Gaussian process (the signal) taken at known but random time-instants corrupted with an additive noise. Two binary tests are considered, depending on which assumptions is retained as the null hypothesis. Assuming that the signal is a linear combination of the solution of a multidimensional stochastic differential equation (SDE), it is shown that the minimum Type II error probability decreases exponentially in the number of samples when the False Alarm probability is fixed. This behavior is described by \emph{error exponents} that are completely characterized. It turns out that they are related with the asymptotic behavior of the Kalman Filter in random stationary environment, which is studied in this paper. Finally, numerical illustrations of our claims are provided in the context of sensor networks.

cs.IT↗

arXiv2009

On adaptive stratification

This paper investigates the use of stratified sampling as a variance reduction technique for approximating integrals over large dimensional spaces. The accuracy of this method critically depends on the choice of the space partition, the strata, which should be ideally fitted to thesubsets where the functions to integrate is nearly constant, and on the allocation of the number of samples within each strata. When the dimension is large and the function to integrate is complex, finding such partitions and allocating the sample is a highly non-trivial problem. In this work, we investigate a novel method to improve the efficiency of the estimator "on the fly", by jointly sampling and adapting the strata and the allocation within the strata. The accuracy of estimators when this method is used is examined in detail, in the so-called asymptotic regime (i.e. when both the number of samples and the number of strata are large). We illustrate the use of the method for the computation of the price of path-dependent options in models with both constant and stochastic volatility. The use of this adaptive technique yields variance reduction by factors sometimes larger than 1000 compared to classical Monte Carlo estimators.

math.PR↗

arXiv2009

On the Forward Filtering Backward Smoothing particle approximations of the smoothing distribution in general state spaces models

A prevalent problem in general state-space models is the approximation of the smoothing distribution of a state, or a sequence of states, conditional on the observations from the past, the present, and the future. The aim of this paper is to provide a rigorous foundation for the calculation, or approximation, of such smoothed distributions, and to analyse in a common unifying framework different schemes to reach this goal. Through a cohesive and generic exposition of the scientific literature we offer several novel extensions allowing to approximate joint smoothing distribution in the most general case with a cost growing linearly with the number of particles.

math.ST↗

arXiv2008

Forgetting of the initial distribution for non-ergodic Hidden Markov Chains

In this paper, the forgetting of the initial distribution for a non-ergodic Hidden Markov Models (HMM) is studied. A new set of conditions is proposed to establish the forgetting property of the filter, which significantly extends all the existing results. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using generic models of non-ergodic HMM and extend all the results known so far.

math.PR↗

arXiv2008

Adaptive methods for sequential importance sampling with application to state space models

In this paper we discuss new adaptive proposal strategies for sequential Monte Carlo algorithms--also known as particle filters--relying on criteria evaluating the quality of the proposed particles. The choice of the proposal distribution is a major concern and can dramatically influence the quality of the estimates. Thus, we show how the long-used coefficient of variation of the weights can be used for estimating the chi-square distance between the target and instrumental distributions of the auxiliary particle filter. As a by-product of this analysis we obtain an auxiliary adjustment multiplier weight type for which this chi-square distance is minimal. Moreover, we establish an empirical estimate of linear complexity of the Kullback-Leibler divergence between the involved distributions. Guided by these results, we discuss adaptive designing of the particle filter proposal distribution and illustrate the methods on a numerical example.

stat.CO↗

arXiv2008

A Wavelet Whittle estimator of the memory parameter of a non-stationary Gaussian time series

We consider a time series $X=\{X_k, k\in\mathbb{Z}\}$ with memory parameter $d\in\mathbb{R}$. This time series is either stationary or can be made stationary after differencing a finite number of times. We study the "Local Whittle Wavelet Estimator" of the memory parameter $d$. This is a wavelet-based semiparametric pseudo-likelihood maximum method estimator. The estimator may depend on a given finite range of scales or on a range which becomes infinite with the sample size. We show that the estimator is consistent and rate optimal if $X$ is a linear process and is asymptotically normal if $X$ is Gaussian.

math.ST↗

arXiv2008

On Upper-Confidence Bound Policies for Non-Stationary Bandit Problems

Multi-armed bandit problems are considered as a paradigm of the trade-off between exploring the environment to find profitable actions and exploiting what is already known. In the stationary case, the distributions of the rewards do not change in time, Upper-Confidence Bound (UCB) policies have been shown to be rate optimal. A challenging variant of the MABP is the non-stationary bandit problem where the gambler must decide which arm to play while facing the possibility of a changing environment. In this paper, we consider the situation where the distributions of rewards remain constant over epochs and change at unknown time instants. We analyze two algorithms: the discounted UCB and the sliding-window UCB. We establish for these two algorithms an upper-bound for the expected regret by upper-bounding the expectation of the number of times a suboptimal arm is played. For that purpose, we derive a Hoeffding type inequality for self normalized deviations with a random number of summands. We establish a lower-bound for the regret in presence of abrupt changes in the arms reward distributions. We show that the discounted UCB and the sliding-window UCB both match the lower-bound up to a logarithmic factor.

math.ST↗

arXiv2008

Testing for Homogeneity with Kernel Fisher Discriminant Analysis

We propose to investigate test statistics for testing homogeneity in reproducing kernel Hilbert spaces. Asymptotic null distributions under null hypothesis are derived, and consistency against fixed and local alternatives is assessed. Finally, experimental evidence of the performance of the proposed approach on both artificial data and a speaker verification task is provided.

stat.ML↗

arXiv2008

The ODE method for stability of skip-free Markov chains with applications to MCMC

Fluid limit techniques have become a central tool to analyze queueing networks over the last decade, with applications to performance analysis, simulation and optimization. In this paper, some of these techniques are extended to a general class of skip-free Markov chains. As in the case of queueing models, a fluid approximation is obtained by scaling time, space and the initial condition by a large constant. The resulting fluid limit is the solution of an ordinary differential equation (ODE) in ``most'' of the state space. Stability and finer ergodic properties for the stochastic model then follow from stability of the set of fluid limits. Moreover, similarly to the queueing context where fluid models are routinely used to design control policies, the structure of the limiting ODE in this general setting provides an understanding of the dynamics of the Markov chain. These results are illustrated through application to Markov chain Monte Carlo methods.

math.PR↗

arXiv2008

Sequential Monte Carlo smoothing with application to parameter estimation in non-linear state space models

This paper concerns the use of sequential Monte Carlo methods (SMC) for smoothing in general state space models. A well-known problem when applying the standard SMC technique in the smoothing mode is that the resampling mechanism introduces degeneracy of the approximation in the path space. However, when performing maximum likelihood estimation via the EM algorithm, all functionals involved are of additive form for a large subclass of models. To cope with the problem in this case, a modification of the standard method (based on a technique proposed by Kitagawa and Sato) is suggested. Our algorithm relies on forgetting properties of the filtering dynamics and the quality of the estimates produced is investigated, both theoretically and via simulations.

math.ST↗

arXiv2008

Estimators of Long-Memory: Fourier versus Wavelets

There have been a number of papers written on semi-parametric estimation methods of the long-memory exponent of a time series, some applied, others theoretical. Some using Fourier methods, others using a wavelet-based technique. In this paper, we compare the Fourier and wavelet approaches to the local regression method and to the local Whittle method. We provide an overview of these methods, describe what has been done, indicate the available results and the conditions under which they hold. We discuss their relative strengths and weaknesses both from a practical and a theoretical perspective. We also include a simulation-based comparison. The software written to support this work is available on demand and we illustrate its use at the end of the paper.

math.ST↗

arXiv2007

Frequency estimation based on the cumulated Lomb-Scargle periodogram

We consider the problem of estimating the period of an unknown periodic function observed in additive noise sampled at irregularly spaced time instants in a semiparametric setting. To solve this problem, we propose a novel estimator based on the cumulated Lomb-Scargle periodogram. We prove that this estimator is consistent, asymptotically Gaussian and we provide an explicit expression of the asymptotic variance. Some Monte-Carlo experiments are then presented to support our claims.

math.ST↗

arXiv2007

Log-average periodogram estimator of the memory parameter

This paper introduces a semiparametric regression estimator of the memory parameter for long-memory time series process. It is based on the regression in a neighborhood of the zero-frequency of the periodogram averaged over epochs. The proposed estimator is theoretically justified and empirical Monte Carlo investigation gives evidence that the method is very promising to estimate the long-memory parameter.

math.ST↗

arXiv2007

Forgetting of the initial condition for the filter in general state-space hidden Markov chain: a coupling approach

We give simple conditions that ensure exponential forgetting of the initial conditions of the filter for general state-space hidden Markov chain. The proofs are based on the coupling argument applied to the posterior Markov kernels. These results are useful both for filtering hidden Markov models using approximation methods (e.g., particle filters) and for proving asymptotic properties of estimators. The results are general enough to cover models like the Gaussian state space model, without using the special structure that permits the application of the Kalman filter.

math.ST↗

arXiv2007

On the auxiliary particle filter

In this article we study asymptotic properties of weighted samples produced by the auxiliary particle filter (APF) proposed by pitt and shephard (1999). Besides establishing a central limit theorem (CLT) for smoothed particle estimates, we also derive bounds on the Lp error and bias of the same for a finite particle sample size. By examining the recursive formula for the asymptotic variance of the CLT we identify first-stage importance weights for which the increase of asymptotic variance at a single iteration of the algorithm is minimal. In the light of these findings, we discuss and demonstrate on several examples how the APF algorithm can be improved.

math.ST↗

arXiv2007

Subgeometric ergodicity of Markov chains

The goal of this paper is to give a short and self contained proof of general bounds for subgeometric rates of convergence, under practical conditions. The main result whose proof, based on coupling, provides an intuitive understanding of the results of Nummelin and Tuominen (1983) and Tuominen and Tweedie (1994). To obtain practical rates, a very general drift condition, recently introduced in Douc et al (2004) is used.

math.ST↗

arXiv2007

Forgetting of the initial distribution for Hidden Markov Models

The forgetting of the initial distribution for discrete Hidden Markov Models (HMM) is addressed: a new set of conditions is proposed, to establish the forgetting property of the filter, at a polynomial and geometric rate. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using different HMM of interest: the dynamic tobit model, the non-linear state space model and the stochastic volatility model.

math.ST↗

arXiv2006

On the spectral density of the wavelet coefficients of long memory time series with application to the log-regression estimation of the memory parameter

In the recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semi-parametric asymptotic theory, comparable to the one developed for Fourier methods, is still missing. In this contribution, we adapt the classical semi-parametric framework introduced by Robinson and his co-authors for estimating the memory parameter of a (possibly) non-stationary process. As an application, we obtain minimax upper bounds for the log-scale regression estimator of the memory parameter for a Gaussian process and we derive an explicit expression of its variance.

math.ST↗