Patricia Reynaud-Bouret

Adaptive estimation for Hawkes processes; application to genome analysis

The aim of this paper is to provide a new method for the detection of either favored or avoided distances between genomic events along DNA sequences. These events are modeled by a Hawkes’ process. The biological problem is actually complex enough to need a non asymptotic penalized model selection approach. We provide a theoretical penalty that satisfies an oracle inequality even for quite complex families of models. The consecutive theoretical estimator is

Lasso and probabilistic inequalities for multivariate point processes

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 to be estimated by linear combinations of a fixed dictionary. To select coefficients, we propose an adaptive

Exponential Inequalities, with Constants, for U-statistics of Order Two

\ell_1

Nonparametric Estimation of the Division Rate of a Size-Structured Population

-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 activities inference, we finally lead 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 with respect to the problem of supports recovery. 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.

Adaptive tests of homogeneity for a Poisson process

A martingale proof of a sharp exponential inequality (with constants) is given for U-statistics of order two as well as for double integrals of Poisson processes.

Microscopic approach of a time elapsed neural model

We consider the problem of estimating thedivision rate of a size-structured population in anonparametric setting. The size of the system evolvesaccording to a transport-fragmentation equation: eachindividual grows with a given transport rate, and splitsinto two offsprings of the same size, followinga binary fragmentation process with unknown division ratethat depends on its size. In contrast to a deterministicinverse problem approach, we take in this talk theperspective of statistical inference: our data consists ina large sample of the size of individuals, when theevolution of the system is close to its time-asymptoticbehavior, so that it can be related to the eigenproblem ofthe considered transport-fragmentation equation. Byestimating statistically each term of the eigenvalueproblem and by suitably inverting a certain linearoperator, we are able to construct a more realisticestimator of the division rate that achieves the sameoptimal error bound as in related deterministic inverseproblems. Our procedure relies on kernel methods withautomatic bandwidth selection.

Bootstrap and permutation tests of independence for point processes

We propose to test the homogeneity of a Poisson process observed on a finite interval. In this framework, we first provide lower bounds for the uniform separation rates in

The two-sample problem for Poisson processes: adaptive tests with a non-asymptotic wild bootstrap approach

\mathbb{L}^2

Concentration for Norms of Infinitely Divisible Vectors With Independent Components

norm over classical Besov bodies and weak Besov bodies. Surprisingly, the obtained lower bounds over weak Besov bodies coincide with the minimax estimation rates over such classes. Then we construct non asymptotic and nonparametric testing procedures that are adaptive in the sense that they achieve, up to a possible logarithmic factor, the optimal uniform separation rates over various Besov bodies simultaneously. These procedures are based on model selection and thresholding methods. We finally complete our theoretical study with a Monte Carlo evaluation of the power of our tests under various alternatives.

Multiple tests based on a gaussian approximation of the unitary events method with delayed coincidence count

The spike trains are the main components of the information processing in the brain. To model spike trains several point processes have been investigated in the literature. And more macroscopic approaches have also been studied, using partial differential equation models. The main aim of the present article is to build a bridge between several point processes models (Poisson, Wold, Hawkes) that have been proved to statistically fit real spike trains data and age-structured partial differential equations as introduced by Pakdaman, Perthame and Salort.