RIVOIRARD Vincent

Hawkes processes are a form of self-exciting process that has been used in numerous applications, including neuroscience, seismology, and terrorism. While these self-exciting processes have a simple formulation, they can model incredibly complex phenomena. Traditionally Hawkes processes are a continuous-time process, however we enable these models to be applied to a wider range of problems by considering a discrete-time variant of Hawkes processes. We illustrate this through the novel coronavirus disease (COVID-19) as a substantive case study. While alternative models, such as compartmental and growth curve models, have been widely applied to the COVID-19 epidemic, the use of discrete-time Hawkes processes allows us to gain alternative insights. This paper evaluates the capability of discrete-time Hawkes processes by modelling daily mortality counts as distinct phases in the COVID-19 outbreak. We first consider the initial stage of exponential growth and the subsequent decline as preventative measures become effective. We then explore subsequent phases with more recent data. Various countries that have been adversely affected by the epidemic are considered, namely, Brazil, China, France, Germany, India, Italy, Spain, Sweden, the United Kingdom and the United States. These countries are all unique concerning the spread of the virus and their corresponding response measures. However, we find that this simple model is useful in accurately capturing the dynamics of the process, despite hidden interactions that are not directly modelled due to their complexity, and differences both within and between countries. The utility of this model is not confined to the current COVID-19 epidemic, rather this model could explain many other complex phenomena. It is of interest to have simple models that adequately describe these complex processes with unknown dynamics. As models become more complex, a simpler representation of the process can be desirable for the sake of parsimony.

The speed of information exchange in the Internet network is measured using latency: a time that measures the time elapsed between the sending of the first bit of information of a request and the reception of the first bit of information of the response. In this thesis realized in collaboration with Citrix, we are interested in the study and modeling of latency data in a context of Internet traffic optimization. Citrix collects data through two different channels, generating latency measures suspected to share common properties. In a first step, we address a distributional fitting problem where the co-variates and the responses are probability measures imaged from each other by a deterministic transport, and the observables are independent samples drawn according to these laws. We propose an estimator of this transport and show its convergence properties. We show that our estimator can be used to match the distributions of the latency measures generated by the two channels.In a second step we propose a modeling strategy to predict the process obtained by computing the moving median of the latency measures on regular partitions of the interval [0, T] with a mesh size D > 0. We show that the conditional mean of this process, which plays a major role in Internet traffic optimization, is correctly described by a Fourier series decomposition and that its conditional variance is organized in clusters that we model using an ARMA Seasonal-GARCH process, i.e., an ARMA-GARCH process with added deterministic seasonal terms. The predictive performance of this model is compared to the reference models used in the industry. A new measure of the amount of residual information not captured by the model based on a certain entropy criterion is introduced.We then address the problem of fault detection in the Internet network. We propose an algorithm for detecting changes in the distribution of a stream of latency data based on the comparison of two sliding windows using a certain weighted Wasserstein distance.Finally, we describe how to select the training data of predictive algorithms in order to reduce their size to limit the computational cost without impacting the accuracy.

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We consider a stochastic individual-based model in continuous time to describe a size-structured population for cell divisions. This model is motivated by the detection of cellular aging in biology. We address here the problem of nonparametric estimation of the kernel ruling the divisions based on the eigenvalue problem related to the asymptotic behavior in large population. This inverse problem involves a multiplicative deconvolution operator. Using Fourier technics we derive a nonparametric estimator whose consistency is studied. The main difficulty comes from the non-standard equations connecting the Fourier transforms of the kernel and the parameters of the model. A numerical study is carried out and we pay special attention to the derivation of bandwidths by using resampling.

This PhD thesis deals with variational inference and robustness in statistics and machine learning. Specifically, it focuses on the statistical properties of variational approximations and the design of efficient algorithms to compute them sequentially, and studies Maximum Mean Discrepancy based estimators as learning rules that are robust to model misspecification.In recent years, variational inference has been widely studied from a computational perspective, however, the literature has paid little attention to its theoretical properties until very recently. In this thesis, we study the consistency of variational approximations in various statistical models and the conditions that ensure their consistency. In particular, we address the case of mixture models and deep neural networks. We also justify from a theoretical point of view the use of the ELBO maximization strategy, a numerical criterion that is widely used in the VB community for model selection and whose effectiveness has already been confirmed in practice. In addition, Bayesian inference provides an attractive online learning framework for analyzing sequential data, and offers generalization guarantees that remain valid even under model misspecification and in the presence of adversaries. Unfortunately, exact Bayesian inference is rarely tractable in practice and approximation methods are usually employed, but do these methods preserve the generalization properties of Bayesian inference? In this thesis, we show that this is indeed the case for some variational inference (VI) algorithms. We propose new online tempered algorithms and derive generalization bounds. Our theoretical result relies on the convexity of the variational objective, but we argue that our result should be more general and present empirical evidence in support. Our work provides theoretical justifications for online algorithms that rely on approximate Bayesian methods.Another question of major interest in statistics that is addressed in this thesis is the design of a universal estimation procedure. This question is of major interest, especially because it leads to robust estimators, a topical issue in statistics and machine learning. We address the problem of universal estimation by using a distance minimization estimator based on Maximum Mean Discrepancy. We show that the estimator is robust to both dependence and the presence of outliers in the dataset. We also highlight the links that can exist with distance minimization estimators using the L2 distance. Finally, we present a theoretical study of the stochastic gradient descent algorithm used to compute the estimator, and we support our findings with numerical simulations. We also propose a Bayesian version of our estimator, which we study from both a theoretical and a computational point of view.

We are interested in reconstructing the initial condition of a non-linear partial differential equation (PDE), namely the Fokker-Planck equation, from the observation of a Dyson Brownian motion at a given time t > 0. The Fokker-Planck equation describes the evolution of electrostatic repulsive particle systems, and can be seen as the large particle limit of correctly renormalized Dyson Brownian motions. The solution of the Fokker-Planck equation can be written as the free convolution of the initial condition and the semi-circular distribution. We propose a nonparametric estimator for the initial condition obtained by performing the free deconvolution via the subordination functions method. This statistical estimator is original as it involves the resolution of a fixed point equation, and a classical deconvolution by a Cauchy distribution. This is due to the fact that, in free probability, the analogue of the Fourier transform is the R-transform, related to the Cauchy transform. In past literature, there has been a focus on the estimation of the initial conditions of linear PDEs such as the heat equation, but to the best of our knowledge, this is the first time that the problem is tackled for a non-linear PDE. The convergence of the estimator is proved and the integrated mean square error is computed, providing rates of convergence similar to the ones known for non-parametric deconvolution methods. Finally, a simulation study illustrates the good performances of our estimator.

We focus on the estimation of the intensity of a Poisson process in the presence of a uniform noise. We propose a kernel-based procedure fully calibrated in theory and practice. We show that our adaptive estimator is optimal from the oracle and minimax points of view, and provide new lower bounds when the intensity belongs to a Sobolev ball. By developing the Goldenshluger-Lepski methodology in the case of deconvolution for Pois-son processes, we propose an optimal data-driven selection of the kernel's bandwidth, and we provide a heuristic framework to calibrate the estimator in practice. Our method is illustrated on the spatial repartition of replication origins along the human genome.

We observe the actions of a K subsample of N individuals, during a time interval of length t>0, for some large K≤N. We model the individuals' relationships by i.i.d. Bernoulli (p) random variables, where p∈(0,1] is an unknown parameter. The action rate of each individual depends on an unknown parameter μ>0 and on the sum of some function ϕ of the ages of the actions of the individuals that influence it. The function ϕ is unknown but we assume that it decays quickly. The goal of this thesis is to estimate the parameter p, which is the main feature of the interaction graph, in the asymptotic where population size N→∞, the observed population size K→∞, and in a long time t→∞. Let mt be the average number of actions per individual up to time t, which depends on all model parameters. In the subcritical case, where mt increases linearly, we construct an estimator of p with convergence rate 1K√+NmtK√+NKmt√. In the supercritical case, where mt increases exponentially fast, we construct an estimator of p with convergence rate 1K√+NmtK√. In a second step, we study the asymptotic normality of these estimators. In the subcritical case, the work is very technical but quite general, and we are led to study three possible regimes, depending on the dominant term in 1K√+NmtK√+NKmt√ at 0. In the supercritical case, we unfortunately assume some additional conditions and consider only one of the two possible regimes.

We first focus on the parsimonious Gaussian sequence model. An empirical Bayesian approach on the a priori Spike and Slab allows us to obtain the convergence at minimax speed of the second order moment a posteriori for Cauchy Slabs and we prove a suboptimality result for a Laplace Slab. A better choice of Slab allows us to obtain the exact constant. In the density estimation model, an a priori Polya tree such that the variables in the tree have a Spike and Slab distribution gives minimax and adaptive speed convergence for the sup norm of the a posteriori law and a nonparametric Bernstein-von Mises theorem.

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Kernel density estimation is a well known method involving a smoothing parameter (the bandwidth) that needs to be tuned by the user. Although this method has been widely used the bandwidth selection remains a challenging issue in terms of balancing algorithmic performance and statistical relevance. The purpose of this paper is to compare a recently developped bandwidth selection method for kernel density estimation to those which are commonly used by now (at least those which are implemented in the R-package). This new method is called Penalized Comparison to Overfitting (PCO). It has been proposed by some of the authors of this paper in a previous work devoted to its statistical relevance from a purely theoretical perspective. It is compared here to other usual bandwidth selection methods for univariate and also multivariate kernel density estimation on the basis of intensive simulation studies. In particular, cross-validation and plug-in criteria are numerically investigated and compared to PCO. The take home message is that PCO can outperform the classical methods without algorithmic additionnal cost.

We consider the problem of estimating conditional densities in moderately high dimensions. Much more informative than regression functions, conditional densities are of major interest in recent methods, especially in the Bayesian framework (study of the posterior distribution, search for its modes.). After recalling the problems related to high-dimensional estimation in the introduction, the next two chapters develop two methods that tackle the scourge of dimension by requiring: to be computationally efficient thanks to a gluttonous iterative procedure, to detect the relevant variables under a parsimony assumption, and to converge at a near-optimal minimax speed. More precisely, both methods consider kernel estimators well adapted to the estimation of conditional densities and select a point multivariate window by revisiting the gluttonous RODEO (Re- gularisation Of Derivative Expectation Operator) algorithm. The first method has ini- tialization problems and additional logarithmic factors in the speed of convergence, the second method solves these problems, while adding regularity adaptation. In the penultimate chapter, we discuss the calibration and numerical performance of these two procedures, before giving some comments and perspectives in the last chapter.

We consider a stochastic individual-based model in continuous time to describe a size-structured population for cell divisions. This model is motivated by the detection of cellular aging in biology. We address here the problem of nonparametric estimation of the kernel ruling the divisions based on the eigenvalue problem related to the asymptotic behavior in large population. This inverse problem involves a multiplicative deconvolution operator. Using Fourier technics we derive a nonparametric estimator whose consistency is studied. The main difficulty comes from the non-standard equations connecting the Fourier transforms of the kernel and the parameters of the model. A numerical study is carried out and we pay special attention to the derivation of bandwidths by using resampling.

This paper studies the estimation of the conditional density f (x, ·) of Y i given X i = x, from the observation of an i.i.d. sample (X i , Y i) ∈ R d , i = 1,. . , n. We assume that f depends only on r unknown components with typically r d. We provide an adaptive fully-nonparametric strategy based on kernel rules to estimate f. To select the bandwidth of our kernel rule, we propose a new fast iterative algorithm inspired by the Rodeo algorithm (Wasserman and Lafferty (2006)) to detect the sparsity structure of f. More precisely, in the minimax setting, our pointwise estimator, which is adaptive to both the regularity and the sparsity, achieves the quasi-optimal rate of convergence. Its computational complexity is only O(dn log n).

This thesis deals with the subject of linear regression in different frameworks, notably related to learning. The first two chapters present the context of the work, its contributions and the mathematical tools used. The third chapter is devoted to the construction of an optimal regularization function, allowing for example to improve on the theoretical level the regularization of the LASSO estimator. The fourth chapter presents, in the field of convex sequential optimization, accelerations of a recent and promising algorithm, MetaGrad, and a conversion from a so-called "deterministic sequential" framework to a so-called "stochastic batch" framework for this algorithm. The fifth chapter focuses on successive interval forecasts, based on the aggregation of predictors, without intermediate feedback or stochastic modeling. Finally, the sixth chapter applies several aggregation methods to a petroleum dataset, resulting in short-term point forecasts and long-term forecast intervals.

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We provide conditions on the statistical model and the prior probability law to derive contraction rates of posterior distributions corresponding to data-dependent priors in an empirical Bayes approach for selecting prior hyper-parameter values. We aim at giving conditions in the same spirit as those in the seminal article of Ghosal and van der Vaart [23]. We then apply the result to specific statistical settings: density estimation using Dirichlet process mixtures of Gaussian densities with base measure depending on data-driven chosen hyper-parameter values and intensity function estimation of counting processes obeying the Aalen model. In the former setting, we also derive recovery rates for the related inverse problem of density deconvolution. In the latter, a simulation study for inhomogeneous Poisson processes illustrates the results.

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This paper studies nonparametric estimation of parameters of multivariate Hawkes processes. We consider the Bayesian setting and derive posterior concentration rates. First rates are derived for L1-metrics for stochastic intensities of the Hawkes process. We then deduce rates for the L1-norm of interactions functions of the process. Our results are exemplified by using priors based on piecewise constant functions, with regular or random partitions and priors based on mixtures of Betas distributions. Numerical illustrations are then proposed with in mind applications for inferring functional connec-tivity graphs of neurons.

This thesis deals with dependence problems between stochastic processes in continuous time. In a first part, new copulas are established to model the dependence between two Brownian movements and to control the distribution of their difference. It is shown that the class of admissible copulas for Brownians contains asymmetric copulas. With these copulas, the survival function of the difference of the two Brownians is higher in its positive part than with a Gaussian dependence. The results are applied to the joint modeling of electricity prices and other energy commodities. In a second part, we consider a discretely observed stochastic process defined by the sum of a continuous semi-martingale and a compound Poisson process with mean reversion. An estimation procedure for the mean-reverting parameter is proposed when the mean-reverting parameter is large in a high frequency finite horizon statistical framework. In a third part, we consider a doubly stochastic Poisson process whose stochastic intensity is a function of a continuous semi-martingale. To estimate this function, a local polynomial estimator is used and a window selection method is proposed leading to an oracle inequality. A test is proposed to determine if the intensity function belongs to a certain parametric family. With these results, the dependence between the intensity of electricity price peaks and exogenous factors such as wind generation is modeled.

In the multidimensional setting, we consider the errors-in-variables model. We aim at estimating the unknown nonparametric multivariate regression function with errors in the covariates. We devise an adaptive estimator based on projection kernels on wavelets and a deconvolution operator. We propose an automatic and fully data driven procedure to select the wavelet level resolution. We obtain an oracle inequality and optimal rates of convergence over anisotropic Hölder classes. Our theoretical results are illustrated by some simulations.

In this thesis, we address two topics, high dimensional clustering on the one hand and mixture density estimation on the other hand. The first chapter is an introduction to clustering. We present different methods and we focus on one of the main models of our work which is the Gaussian mixture. We also discuss the problems inherent to high dimensional estimation and the difficulty of estimating the number of clusters. We briefly explain here the concepts discussed in this manuscript. Let's consider a mixed distribution of K Gaussians in R^p. One of the common approaches to estimate the parameters of the mixture is to use the maximum likelihood estimator. Since this problem is not convex, the convergence of classical methods cannot be guaranteed. However, by exploiting the biconvexity of the negative log-likelihood, we can use the iterative procedure 'Expectation-Maximization' (EM). Unfortunately, this method is not well suited to address the challenges posed by high dimensionality. Moreover, this method requires to know the number of clusters. Chapter 2 presents three methods that we have developed to try to solve the problems described above. The work presented here has not been extensively researched for various reasons. The first method, 'graphical lasso on Gaussian mixtures', consists in estimating the inverse matrices of the covariance matrices under the assumption that they are parsimonious. We adapt the graphical lasso method of [Friedman et al., 2007] on a component in the case of a mixture and we experimentally evaluate this method. The other two methods address the problem of estimating the number of clusters in the mixture. The first one is a penalized estimation of the posterior probability matrix whose component (i,j) is the probability that the i-th observation is in the j-th cluster. Unfortunately, this method proved to be too expensive in complexity. Finally, the second method considered consists in penalizing the weight vector in order to make it parsimonious. This method shows promising results. In Chapter 3, we study the maximum likelihood estimator of a density of n observations i.e. under the hypothesis that it is well approximated by a mixture of several given densities. We are interested in the performance of the estimator with respect to the Kullback-Leibler loss. We establish risk bounds in the form of exact oracle inequalities, either in probability or in expectation. We show through these bounds that, in the case of the convex aggregation problem, the maximum likelihood estimator reaches the speed (log K)/n)^{1/2}, which is optimal to within one logarithmic term, when the number of components is larger than n^{1/2}. More importantly, under the additional assumption that the Gram matrix of the dictionary components satisfies the compatibility condition, the obtained oracle inequalities give the optimal speed in the parsimonious scenario. In other words, if the weight vector is (almost) D-parcimonious, we obtain a speed (Dlog K)/n. In addition to these oracle inequalities, we introduce the notion of (almost)-D-parcimonious aggregation and establish the corresponding lower bounds for this type of aggregation. Finally, in Chapter 4, we propose an algorithm that performs the Kullback-Leibler aggregation of dictionary components as studied in Chapter 3. We compare its performance with different methods. We then propose a method to build the density dictionary and study it numerically. This thesis was carried out within the framework of a CIFRE agreement with the company ARTEFACT.

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Estimator selection has become a crucial issue in non parametric estimation. Two widely used methods are penalized empirical risk minimization (such as penalized log-likelihood estimation) or pairwise comparison (such as Lepski's method). Our aim in this paper is twofold. First we explain some general ideas about the calibration issue of estimator selection methods. We review some known results, putting the emphasis on the concept of minimal penalty which is helpful to design data-driven selection criteria. Secondly we present a new method for bandwidth selection within the framework of kernel density density estimation which is in some sense intermediate between these two main methods mentioned above. We provide some theoretical results which lead to some fully data-driven selection strategy.

We provide general conditions to derive posterior concentration rates for Aalen counting processes. The conditions are designed to resemble those proposed in the literature for the problem of density estimation, so that existing results on density estimation can be adapted to the present setting. We apply the general theorem to some prior models including Dirichlet process mixtures of uniform densities to estimate monotone non-increasing intensities and log-splines.

Background: Statistical models that predict neuron spike occurrence from the earlier spiking activity of the whole recorded network are promising tools to reconstruct functional connectivity graphs. Some of the previously used methods were in the general statistical framework of the multivariate Hawkes processes but they often required huge amount of data, prior knowledge about the recorded network, and may generate non stationary models that could not be directly used in simulation. New Method: Here, we present a method, based on least-square estimators and LASSO penalty criteria, optimizing Hawkes models that can be used for simulation. Results: Challenging our method to multiple Integrate and Fire models of neuron networks demonstrated that it eciently detects both excitatory and inhibitory connections. The few errors that occasionally occurred with complex networks including common inputs, weak and chained connections, could easily be discarded based on objective criteria. Conclusions: The present method is robust, stable, applicable with an experimentally realistic amount of data, and does not require any prior knowledge of the studied network. Therefore, it can be used on a personal computer as a turn-key procedure to infer connectivity graphs and generate simulation models from simultaneous spike train recordings.

The goal of this thesis is to show that the arsenal of new optimization methods allows us to solve difficult estimation problems based on event models.These dated events are ordered chronologically and therefore cannot be considered as independent.This simple fact justifies the use of a particular mathematical tool called point process to learn a certain structure from these events. The first is the point process behind the Cox proportional hazards model: its conditional strength allows to define the hazard ratio, a fundamental quantity in the survival analysis literature.The Cox regression model relates the time to the occurrence of an event, called a failure, to the covariates of an individual.This model can be reformulated using the point process framework. The second is the Hawkes process which models the impact of past events on the probability of future events.The multivariate case allows to encode a notion of causality between the different dimensions considered.This theme is divided into three parts.The first part is concerned with a new optimization algorithm that we have developed.It allows to estimate the parameter vector of the Cox regression when the number of observations is very large.Our algorithm is based on the SVRG (Stochastic Variance Reduced Gradient) algorithm and uses an MCMC (Monte Carlo Marker Model) method.We have proved convergence speeds for our algorithm and have shown its numerical performance on simulated and real-world data sets.The second part shows that causality in the Hawkes sense can be reduced to a minimum. The second part shows that the causality in the Hawkes sense can be estimated in a non-parametric way thanks to the integrated cumulants of the multivariate point process.We have developed two methods for estimating the integrals of the kernels of the Hawkes process, without making any assumption on the shape of these kernels. Our methods are faster and more robust, with respect to the shape of the kernels, compared to the state of the art. We have demonstrated the statistical consistency of the first method, and have shown that the second one can be applied to a convex optimization problem.The last part highlights the order book dynamics using the first non-parametric estimation method introduced in the previous part.We have used data from the EUREX futures market, defined new order book models (e.g., the order book of the same day), and developed a new method for the estimation of the order book.We have used data from the EUREX futures market, developed new order book models (based on the previous work of Bacry et al.) and applied the estimation method on these point processes.The results obtained are very satisfactory and consistent with an economic analysis.Such a work proves that the method we have developed allows to extract a structure from data as complex as those from high-frequency finance.

In this paper we consider the problem of estimating $f$, the conditional density of $Y$ given $X$, by using an independent sample distributed as $(X,Y)$ in the multivariate setting. We consider the estimation of $f(x,.)$ where $x$ is a fixed point. We define two different procedures of estimation, the first one using kernel rules, the second one inspired from projection methods. Both adapted estimators are tuned by using the Goldenshluger and Lepski methodology. After deriving lower bounds, we show that these procedures satisfy oracle inequalities and are optimal from the minimax point of view on anisotropic Hölder balls. Furthermore, our results allow us to measure precisely the influence of $\mathrm{f}_X(x)$ on rates of convergence, where $\mathrm{f}_X$ is the density of $X$. Finally, some simulations illustrate the good behavior of our tuned estimates in practice.

In this thesis we focus on low rank matrix completion methods and study some related problems. A first set of results aims at extending the existing statistical guarantees for completion models with additive sub-Gaussian noise to more general distributions. In particular, we consider multinational distributions and distributions belonging to the exponential family. For the latter, we prove the optimality (in the minimax sense) to within one logarithmic factor of the trace norm penalty estimators. A second set of results concerns the conditional gradient algorithm which is notably used to compute the previous estimators. In particular, we consider two conditional gradient algorithms in the context of stochastic optimization. We give the conditions under which these algorithms reach the performance of projected gradient algorithms.

High dimensional Poisson regression has become a standard framework for the analysis of massive counts datasets. In this work we estimate the intensity function of the Poisson regression model by using a dictionary approach, which generalizes the classical basis approach , combined with a Lasso or a group-Lasso procedure. Selection depends on penalty weights that need to be calibrated. Standard methodologies developed in the Gaussian framework can not be directly applied to Poisson models due to heteroscedasticity. Here we provide data-driven weights for the Lasso and the group-Lasso derived from concentration inequalities adapted to the Poisson case. We show that the associated Lasso and group-Lasso procedures satisfy fast and slow oracle inequalities. Simulations are used to assess the empirical performance of our procedure, and an original application to the analysis of Next Generation Sequencing data is provided.

This thesis is divided into two independent parts. In the first part, we consider a stochastic individual-centered model in continuous time describing a population structured by size. The population is represented by a point measure evolving according to a deterministic piecewise random process. We study here the non-parametric estimation of the kernel governing the splits, under two different observation schemes. First, in the case where we obtain the entire tree of splits, we construct a kernel estimator with data-dependent adaptive window selection. We obtain an oracle inequality and optimal exponential convergence speeds. Second, in the case where the splitting tree is not completely observed, we show that the renormalized microscopic process describing the evolution of the population converges to the weak solution of a partial differential equation. We propose an estimator of the division kernel using Fourier techniques. We show the consistency of the estimator. In the second part, we consider the non-parametric regression model with errors on the variables in the multidimensional context. Our objective is to estimate the unknown multivariate regression function. We propose an adaptive estimator based on projection kernels founded on a multi-index wavelet basis and a deconvolution operator. The resolution level of the wavelets is obtained by the Goldenshluger-Lepski method. We obtain an oracle inequality and optimal convergence speeds on anisotropic Hölder spaces.

The distances between DNA Transcription Regulatory Elements (TRE) provide important clues to their dependencies and function within the gene regulation process. However, the locations of those TREs as well as their cross distances between occurrences are stochastic, in part due to the inherent limitations of Next Generation Sequencing methods used to localize them, in part due to biology itself. This paper describes a novel approach to analyzing these locations and their cross distances even at long range via a Poisson random convolution. The resulting deconvolution problem is ill-posed, and sparsity regularization is used to offset this challenge. Unlike previous work on sparse Poisson inverse problems, this paper adopts a weighted LASSO estimator with data-dependent weights calculated using concentration inequalities that account for the Poisson noise. This method exhibits better squared error performance than the classical (unweighted) LASSO both in theoretical performance bounds and in simulation studies, and can easily be computed using off-the-shelf LASSO solvers.

This thesis is based on the estimation of a density, considered from a non-parametric and non-asymptotic point of view. It deals with the problem of the selection of a kernel estimation method. The latter is a generalization of, among others, the problem of model selection and window selection. We study classical procedures, by penalization and resampling (in particular V-fold cross-validation), which evaluate the quality of a method by estimating its risk. We propose, thanks to concentration inequalities, a method to optimally calibrate the penalty to select a linear estimator and prove oracle inequalities and adaptation properties for these procedures. Moreover, a new resampled procedure, based on the comparison between estimators by robust tests, is proposed as an alternative to procedures based on the principle of unbiased risk estimation. A second objective is the comparison of all these procedures from a theoretical point of view and the analysis of the role of the V-parameter for the V-fold penalties. We validate the theoretical results by simulation studies.

Nonparametric Bayesian estimation for multidimensional Hawkes processes. SMAI Congress 2015.

Due to its low computational cost, Lasso is an attractive regularization method for high-dimensional statistical settings. In this paper, we consider multivariate counting processes depending on an unknown function parameter to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive $\ell_1$-penalization methodology, where data-driven weights of the penalty are derived from new Bernstein type inequalities for martingales. Oracle inequalities are established under assumptions on the Gram matrix of the dictionary. Non-asymptotic probabilistic results for multivariate Hawkes processes are proven, which allows us to check these assumptions by considering general dictionaries based on histograms, Fourier or wavelet bases. Motivated by problems of neuronal activity inference, we finally carry out a simulation study for multivariate Hawkes processes and compare our methodology with the {\it adaptive Lasso procedure} proposed by Zou in \cite{Zou}. We observe an excellent behavior of our procedure. We rely on theoretical aspects for the essential question of tuning our methodology. Unlike adaptive Lasso of \cite{Zou}, our tuning procedure is proven to be robust with respect to all the parameters of the problem, revealing its potential for concrete purposes, in particular in neuroscience.

Sparse linear inverse problems appear in a variety of settings, but often the noise contaminating observations cannot accurately be described as bounded by or arising from a Gaussian distribution. Poisson observations in particular are a characteristic feature of several real-world applications. Previous work on sparse Poisson inverse problems encountered several limiting technical hurdles. This paper describes a novel alternative analysis approach for sparse Poisson inverse problems that (a) sidesteps the technical challenges present in previous work, (b) admits estimators that can readily be computed using off-the-shelf LASSO algorithms, and (c) hints at a general weighted LASSO framework for broad classes of problems. At the heart of this new approach lies a weighted LASSO estimator for which data-dependent weights are based on Pois-son concentration inequalities. Unlike previous analyses of the weighted LASSO, the proposed analysis depends on conditions which can be checked or shown to hold in general settings with high probability.

Multidimensional Hawkes processes are used to model neuronal action potentials. The estimation of intensity functions allows to understand the interaction structure of neurons. The non-parametric estimation of these functions has been proposed by LASSO type methods in a frequentist framework. We are interested in their non-parametric estimation in a Bayesian framework. For this purpose, we implement algorithms of the Sequential Monte Carlo Sampler type, particularly adapted to these point processes. Multidimensional Hawkes processes are used to modelise multivariate neuron spike data.

Non parametric Bayesian estimation for Hawkes processes. International Society for Bayesian Analysis World Meeting, ISBA 2014.

Research on non-parametric Bayesian methods has grown considerably over the last twenty years, especially since the development of simulation algorithms that allow them to be put into practice. It is therefore necessary to understand, from a theoretical point of view, the behavior of these methods. This thesis presents different contributions to the analysis of the frequentist properties of non-parametric Bayesian methods. Although being placed in an asymptotic framework may seem restrictive at first sight, it nevertheless allows us to understand the functioning of Bayesian procedures in extremely complex models. In particular, it allows us to detect the aspects of the a priori that are particularly influential on the inference. Many general results have been obtained in this framework, but as the models become more and more complex and realistic, they deviate from the classical assumptions and are no longer covered by the existing theory. In addition to the intrinsic interest of studying a specific model that does not satisfy the classical assumptions, this also allows for a better understanding of the mechanisms that govern the operation of non-parametric Bayesian methods.

In this paper we provide general conditions to check on the model and the prior to derive posterior concentration rates for data-dependent priors (or empirical Bayes approaches). We aim at providing conditions that are close to the conditions provided in the seminal paper by Ghosal & van der Vaart (2007). We then apply the general theorem to two different settings: the estimation of a density using Dirichlet process mixtures of Gaussian random variables with base measure depending on some empirical quantities and the estimation of the intensity of a counting process under the Aalen model. A simulation study for inhomogeneous Poisson processes also illustrates our results. In the former case we also derive some results on the estimation of the mixing density and on the deconvolution problem. In the latter, we provide a general theorem on posterior concentration rates for counting processes with Aalen multiplicative intensity with priors not depending on the data.

When dealing with classical spike train analysis, the practitioner often performs goodness-of-fit tests to test whether the observed process is a Poisson process, for instance, or if it obeys another type of probabilistic model. In doing so, there is a fundamental plug-in step, where the parameters of the supposed underlying model are estimated. The aim of this article is to show that plug-in has sometimes very undesirable effects. We propose a new method based on subsampling to deal with those plug-in issues in the case of the Kolmogorov- Smirnov test of uniformity. The method relies on the plug-in of good estimates of the underlying model, that have to be consistent with a controlled rate of convergence. Some non parametric estimates satisfying those constraints in the Poisson or in the Hawkes framework are highlighted. Moreover they share adaptive properties that are useful from a practical point of view. We show the performance of those methods on simulated data. We also provide a complete analysis with these tools on single unit activity recorded on a monkey during a sensory-motor task.

When dealing with classical spike train analysis, the practitioner often per-forms goodness-of-fit tests to test whether the observed process is a Poisson process, for instance, or if it obeys another type of probabilistic model (Yana et al. in Bio-phys. In doing so, there is a fundamental plug-in step, where the parameters of the supposed underlying model are estimated. The aim of this article is to show that plug-in has sometimes very un-desirable effects. We propose a new method based on subsampling to deal with those plug-in issues in the case of the Kolmogorov–Smirnov test of uniformity. The method relies on the plug-in of good estimates of the underlying model that have to be consis-tent with a controlled rate of convergence. Some nonparametric estimates satisfying those constraints in the Poisson or in the Hawkes framework are highlighted. More-over, they share adaptive properties that are useful from a practical point of view. We show the performance of those methods on simulated data. We also provide a com-plete analysis with these tools on single unit activity recorded on a monkey during a sensory-motor task. Electronic supplementary material The online version of this article (doi:10.1186/2190-8567-4-3) contains supplementary material.

In this paper we provide general conditions to check on the model and the prior to derive posterior concentration rates for data-dependent priors (or empirical Bayes approaches). We aim at providing conditions that are close to the conditions provided in the seminal paper by \citet{ghosal:vdv:07}. We then apply the general theorem to two different settings: the estimation of a density using Dirichlet process mixtures of Gaussian random variables with base measure depending on some empirical quantities and the estimation of the intensity of a counting process under the Aalen model. A simulation study for inhomogeneous Poisson processes also illustrates our results. In the former case we also derive some results on the estimation of the mixing density and on the deconvolution problem. In the latter, we provide a general theorem on posterior concentration rates for counting processes with Aalen multiplicative intensity with priors not depending on the data. In this supplementary file, we present the Gibbs algorithm used in the numerical example.

We use Hawkes processes as models for spike trains analysis. A new Lasso method designed for general multivariate counting processes enables us to estimate the functional connectivity graph between the different recorded neurons.

We consider the problem of estimating a density of probability from indirect data in the spherical convolution model. We aim at building an estimate of the unknown density as a linear combination of functions of an overcomplete dictionary. The procedure is devised through a well-calibrated l1-penalized criterion. The spherical deconvolution setting has been barely studied so far, and the two main approches to this problem, namely the SVD and the hard thresholding ones considered only one basis at a time. The dictionary approach allows to combine various bases and thus enhances estimates sparsity. We provide an oracle inequality under global coherence assumptions. Moreover, the calibrated procedure that we put forward gives very satisfying results in the numerical study when compared with other procedures.

Parametric nonlinear mixed effects models (NLMEs) are now widely used in biometrical studies, especially in pharmacokinetics research and HIV dynamics models, due to, among other aspects, the computational advances achieved during the last years. However, this kind of models may not be flexible enough for complex longitudinal data analysis. Semiparametric NLMEs (SNMMs) have been proposed as an extension of NLMEs. These models are a good compro-mise and retain nice features of both parametric and non-parametric models resulting in more flexible models than standard parametric NLMEs. However, SNMMs are com-plex models for which estimation still remains a challenge. Previous estimation procedures are based on a combination of log-likelihood approximation methods for parametric es-timation and smoothing splines techniques for nonparamet-ric estimation. In this work, we propose new estimation strate-gies in SNMMs. On the one hand, we use the Stochastic Approximation version of EM algorithm (SAEM) to obtain exact ML and REML estimates of the fixed effects and vari-Ana Arribas-Gil is supported by projects MTM2010-17323 and ECO2011-25706, Spain. Karine Bertin is supported by projects FONDECYT 1090285 and ECOS/CONICYT C10E03 2010, Chile. Cristian Meza is supported by project FONDECYT 11090024, Chile.

In neuroscience, the main object of study is the spike train because it is considered as the main vector of information transmission of brain activity. Throughout the different studies, several models for spike trains have been proposed, more for biological than mathematical reasons. We propose here statistical procedures to test the various models.

We use Hawkes processes as models for spike trains analysis. A new Lasso method designed for general multivariate counting processes enables us to estimate the functional connectivity graph between the different recorded neurons.

We consider the problem of estimating a density of probability from indirect data in the spherical convolution model. We aim at building an estimate of the unknown density as a linear combination of functions of an overcomplete dictionary. The procedure is devised through a well-calibrated l1-penalized criterion. The spherical deconvolution setting has been barely studied so far, and the two main approches to this problem, namely the SVD and the hard thresholding ones considered only one basis at a time. The dictionary approach allows to combine various bases and thus enhances estimates sparsity. We provide an oracle inequality under global coherence assumptions. Moreover, the calibrated procedure that we put forward gives very satisfying results in the numerical study when compared with other procedures.

In the context of a wavelet analysis, we are interested in the statistical study of a particular class of Lorentz spaces: the weak Besov spaces which appear naturally in the context of the maxiset theory. With "white Gaussian noise" assumptions, we show, thanks to Bayesian techniques, that the minimax velocities of the strong and weak Besov spaces are the same. The worst-case distributions that we show for each weak Besov space are constructed from Pareto laws and differ from those of the strong Besov spaces. Using simulations of these distributions, we construct visual representations of "typical enemies". Finally, we exploit these distributions to build a minimax estimation procedure, of the "thresholding" type, called ParetoThresh, which we study from a practical point of view. In a second step, we place ourselves under the heteroskedastic white Gaussian noise model and under the maxiset approach, we establish the suboptimality of linear estimators compared to adaptive thresholding procedures. Then, we investigate the best way to model the sparse character of a sequence through a Bayesian approach. To this end, we study the maxima of classical Bayesian estimators - median, mean - associated with a model built on heavy-tailed densities. The maximal spaces for these estimators are Lorentz spaces, and coincide with those associated with thresholding estimators. We extend this result in a natural way by obtaining a necessary and sufficient condition on the parameters of the model so that the a priori law is almost certainly concentrated on a specific Lorentz space.

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