arXiv subjects

Bernard Delyon

Publications and source records attributed to Bernard Delyon.

16 recordsLinked to original sources

arXiv2024

Stochastic mirror descent for nonparametric adaptive importance sampling

This paper addresses the problem of approximating an unknown probability distribution with density $f$ -- which can only be evaluated up to an unknown scaling factor -- with the help of a sequential algorithm that produces at each iteration $n\geq 1$ an estimated density $q_n$.The proposed method optimizes the Kullback-Leibler divergence using a mirror descent (MD) algorithm directly on the space of density functions, while a stochastic approximation technique helps to manage between algorithm complexity and variability. One of the key innovations of this work is the theoretical guarantee that is provided for an algorithm with a fixed MD learning rate $η\in (0,1 )$. The main result is that the sequence $q_n$ converges almost surely to the target density $f$ uniformly on compact sets. Through numerical experiments, we show that fixing the learning rate $η\in (0,1 )$ significantly improves the algorithm's performance, particularly in the context of multi-modal target distributions where a small value of $η$ allows to increase the chance of finding all modes. Additionally, we propose a particle subsampling method to enhance computational efficiency and compare our method against other approaches through numerical experiments.

math.ST↗

arXiv2020

Safe and adaptive importance sampling: a mixture approach

This paper investigates adaptive importance sampling algorithms for which the policy, the sequence of distributions used to generate the particles, is a mixture distribution between a flexible kernel density estimate (based on the previous particles), and a "safe" heavy-tailed density. When the share of samples generated according to the safe density goes to zero but not too quickly, two results are established: (i) uniform convergence rates are derived for the policy toward the target density; (ii) a central limit theorem is obtained for the resulting integral estimates. The fact that the asymptotic variance is the same as the variance of an "oracle" procedure with variance-optimal policy, illustrates the benefits of the approach. In addition, a subsampling step (among the particles) can be conducted before constructing the kernel estimate in order to decrease the computational effort without altering the performance of the method. The practical behavior of the algorithms is illustrated in a simulation study.

math.ST↗

arXiv2018

Asymptotic optimality of adaptive importance sampling

Adaptive importance sampling (AIS) uses past samples to update the \textit{sampling policy} $q_t$ at each stage $t$. Each stage $t$ is formed with two steps : (i) to explore the space with $n_t$ points according to $q_t$ and (ii) to exploit the current amount of information to update the sampling policy. The very fundamental question raised in this paper concerns the behavior of empirical sums based on AIS. Without making any assumption on the allocation policy $n_t$, the theory developed involves no restriction on the split of computational resources between the explore (i) and the exploit (ii) step. It is shown that AIS is asymptotically optimal : the asymptotic behavior of AIS is the same as some "oracle" strategy that knows the targeted sampling policy from the beginning. From a practical perspective, weighted AIS is introduced, a new method that allows to forget poor samples from early stages.

math.ST↗

arXiv2018

On the Asymptotic Normality of Adaptive Multilevel Splitting

Adaptive Multilevel Splitting (AMS for short) is a generic Monte Carlo method for Markov processes that simulates rare events and estimates associated probabilities. Despite its practical efficiency, there are almost no theoretical results on the convergence of this algorithm. The purpose of this paper is to prove both consistency and asymptotic normality results in a general setting. This is done by associating to the original Markov process a level-indexed process, also called a stochastic wave, and by showing that AMS can then be seen as a Fleming-Viot type particle system. This being done, we can finally apply general results on Fleming-Viot particle systems that we have recently obtained.

math.PR↗

arXiv2018

Mixing properties and central limit theorem for associated point processes

Positively (resp. negatively) associated point processes are a class of point processes that induce attraction (resp. inhibition) between the points. As an important example, determinantal point processes (DPPs) are negatively associated. We prove $α$-mixing properties for associated spatial point processes by controlling their $α$-coefficients in terms of the first two intensity functions. A central limit theorem for functionals of associated point processes is deduced, using both the association and the $α$-mixing properties. We discuss in detail the case of DPPs, for which we obtain the limiting distribution of sums, over subsets of close enough points of the process, of any bounded function of the DPP. As an application, we get the asymptotic properties of the parametric two-step estimator of some inhomogeneous DPPs.

math.ST↗

arXiv2017

Integral estimation based on Markovian design

Suppose that a mobile sensor describes a Markovian trajectory in the ambient space. At each time the sensor measures an attribute of interest, e.g., the temperature. Using only the location history of the sensor and the associated measurements, the aim is to estimate the average value of the attribute over the space. In contrast to classical probabilistic integration methods, e.g., Monte Carlo, the proposed approach does not require any knowledge on the distribution of the sensor trajectory. Probabilistic bounds on the convergence rates of the estimator are established. These rates are better than the traditional "root n"-rate, where n is the sample size, attached to other probabilistic integration methods. For finite sample sizes, the good behaviour of the procedure is demonstrated through simulations and an application to the evaluation of the average temperature of oceans is considered.

math.ST↗

arXiv2017

A Central Limit Theorem for Fleming-Viot Particle Systems with Soft Killing

The distribution of a Markov process with killing, conditioned to be still alive at a given time, can be approximated by a Fleming-Viot type particle system. In such a system, each particle is simulated independently according to the law of the underlying Markov process, and branches onto another particle at each killing time. The consistency of this method in the large population limit was the subject of several recent articles. In the present paper, we go one step forward and prove a central limit theorem for the law of the Fleming-Viot particle system at a given time under two conditions: a "soft killing" assumption and a boundedness condition involving the "carré du champ" operator of the underlying Markov process.

math.PR↗

arXiv2017

A Central Limit Theorem for Fleming-Viot Particle Systems with Hard Killing

Fleming-Viot type particle systems represent a classical way to approximate the distribution of a Markov process with killing, given that it is still alive at a final deterministic time. In this context, each particle evolves independently according to the law of the underlying Markov process until its killing, and then branches instantaneously on another randomly chosen particle. While the consistency of this algorithm in the large population limit has been recently studied in several articles, our purpose here is to prove Central Limit Theorems under very general assumptions. For this, we only suppose that the particle system does not explode in finite time, and that the jump and killing times have atomless distributions. In particular, this includes the case of elliptic diffusions with hard killing.

math.PR↗

arXiv2016

Integral approximation by kernel smoothing

Let $(X_1,\ldots,X_n)$ be an i.i.d. sequence of random variables in $\mathbb{R}^d$, $d\geq 1$. We show that, for any function $φ:\mathbb{R}^d\rightarrow\mathbb{R}$, under regularity conditions, \[n^ {1/2}\Biggl(n^{-1}\sum_{i=1}^n\frac{φ(X_i)}{\widehat{f}^(X_i)}- \int φ(x)\,dx\Biggr)\stackrel{\mathbb{P}}{\longrightarrow}0,\] where $\widehat{f}$ is the classical kernel estimator of the density of $X_1$. This result is striking because it speeds up traditional rates, in root $n$, derived from the central limit theorem when $\widehat{f}=f$. Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions $φ$ and $f$, the dimension $d$ and the bandwidth of the kernel estimator $\widehat{f}$. Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of $φ$. As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations.

math.ST↗

arXiv2015

Dependent time changed processes with applications to nonlinear ocean waves

Many records in environmental sciences exhibit asymmetric trajectories and there is a need for simple and tractable models which can reproduce such features. In this paper we explore an approach based on applying both a time change and a marginal transformation on Gaussian processes. The main originality of the proposed model is that the time change depends on the observed trajectory. We first show that the proposed model is stationary and ergodic and provide an explicit characterization of the stationary distribution. This result is then used to build both parametric and non-parametric estimates of the time change function whereas the estimation of the marginal transformation is based on up-crossings. Simulation results are provided to assess the quality of the estimates. The model is applied to shallow water wave data and it is shown that the fitted model is able to reproduce important statistics of the data such as its spectrum and marginal distribution which are important quantities for practical applications. An important benefit of the proposed model is its ability to reproduce the observed asymmetries between the crest and the troughs and between the front and the back of the waves by accelerating the chronometer in the crests and in the front of the waves.

stat.ME↗

arXiv2015

Investigation of asymmetry in E. coli growth rate

The data we analyze derives from the observation of numerous cells of the bacterium Escherichia coli (E. coli) growing and dividing. Single cells grow and divide to give birth to two daughter cells, that in turn grow and divide. Thus, a colony of cells from a single ancestor is structured as a binary genealogical tree. At each node the measured data is the growth rate of the bacterium. In this paper, we study two different data sets. One set corresponds to small complete trees, whereas the other one corresponds to long specific sub-trees. Our aim is to compare both sets. This paper is accessible to post graduate students and readers with advanced knowledge in statistics.

stat.AP↗

arXiv2013

On the acceleration of some empirical means with application to nonparametric regression

Let $(X_1,\ldots ,X_n)$ be an i.i.d. sequence of random variables in $\R^d$, $d\geq 1$, for some function $φ:\R^d\r \R$, under regularity conditions, we show that \begin{align*} n^{1/2} \left(n^{-1} \sum_{i=1}^n \frac{φ(X_i)}{\w f^{(i)}(X_i)}-\int_{} φ(x)dx \right) \overset¶{\lr} 0, \end{align*} where $\w f^{(i)}$ is the classical leave-one-out kernel estimator of the density of $X_1$. This result is striking because it speeds up traditional rates, in root $n$, derived from the central limit theorem when $\w f^{(i)}=f$. As a consequence, it improves the classical Monte Carlo procedure for integral approximation. The paper mainly addressed with theoretical issues related to the later result (rates of convergence, bandwidth choice, regularity of $φ$) but also interests some statistical applications dealing with random design regression. In particular, we provide the asymptotic normality of the estimation of the linear functionals of a regression function on which the only requirement is the Hölder regularity. This leads us to a new version of the \textit{average derivative estimator} introduced by Härdle and Stoker in \cite{hardle1989} which allows for \textit{dimension reduction} by estimating the \textit{index space} of a regression.

math.ST↗

arXiv2013

Bootstrap Testing of the Rank of a Matrix via Least Squared Constrained Estimation

In order to test if an unknown matrix has a given rank (null hypothesis), we consider the family of statistics that are minimum squared distances between an estimator and the manifold of fixed-rank matrix. Under the null hypothesis, every statistic of this family converges to a weighted chi-squared distribution. In this paper, we introduce the constrained bootstrap to build bootstrap estimate of the law under the null hypothesis of such statistics. As a result, the constrained bootstrap is employed to estimate the quantile for testing the rank. We provide the consistency of the procedure and the simulations shed light one the accuracy of the constrained bootstrap with respect to the traditional asymptotic comparison. More generally, the results are extended to test if an unknown parameter belongs to a sub-manifold locally smooth. Finally, the constrained bootstrap is easy to compute, it handles a large family of tests and it works under mild assumptions.

math.ST↗

arXiv2011

Test function: A new approach for covering the central subspace

In this paper we offer a complete methodology for sufficient dimension reduction called the test function (TF). TF provides a new family of methods for the estimation of the central subspace (CS) based on the introduction of a nonlinear transformation of the response. Theoretical background of TF is developed under weaker conditions than the existing methods. By considering order 1 and 2 conditional moments of the predictor given the response, we divide TF in two classes. In each class we provide conditions that guarantee an exhaustive estimation of the CS. Besides, the optimal members are calculated via the minimization of the asymptotic mean squared error deriving from the distance between the CS and its estimate. This leads us to two plug-in methods which are evaluated with several simulations.

math.ST↗

arXiv2006

Simulation of conditioned diffusions

In this paper, we propose some algorithms for the simulation of the distribution of certain diffusions conditioned on terminal point. We prove that the conditional distribution is absolutely continuous with respect to the distribution of another diffusion which is easy for simulation, and the formula for the density is given explicitly.

math.ST↗

arXiv2005

Convex domains and K-spectral sets

Let $Ω$ be an open convex domain of the complex plane. We study constants K such that $Ω$ is K-spectral or complete K-spectral for each continuous linear Hilbert space operator with numerical range included in $Ω$. Several approaches are discussed.

math.FA↗