Séminaire Probabilités et Statistique

Bivariate signals appear in a broad range of applications: polarized waveforms in seismology and optics, current velocities in oceanography, etc. Formally, bivariate signals are 2D vector time series. Existing approaches for bivariate signal processing do not provide a straightforward description of the signal in terms of its polarization properties. For this purpose we introduce a new and generic framework based on a tailored quaternion Fourier transform.
This new framework re-establishes a clear interpretability in terms of polarization attributes of usual quantities such as spectral densities, linear filters, etc.
In this talk, I will introduce the main features of this approach, with the focus on second-order stationary random bivariate signals. I will discuss spectral analysis, linear filtering and some original decompositions of bivariate signals. Synthetic data will illustrate the usefulness of the proposed framework.

Pendant le séminaire, nous introduirons une formule d’intégration par partie non linéaire qui peut être vu comme une généralisation stochastique du lemme de Alekseev-Gröbner.

La preuve est basée sur le calcule de Malliavin et sur l’expression de certains intégrales stochastiques anticipatifs comme intégrales de Skorohod.

La formule que l’on présente induit une théorie de perturbations, i.e. une façon d’estimer, en terme de caractéristiques locales, l’erreur globale entre la solution exacte d’une équation différentielle stochastique et un processus d’Itô quelconque.

Si le temps le permet, nous parlerons des différences par rapport au résultat de perturbation établi précédemment par M. Hutzenthaler et A. Jentzen, et des applications comme la dérivation des taux de convergence en moyenne quadratique des schémas d’approximations pour ED(P)S.

(Travail en collaboration avec A. Hudde, M. Hutzenthaler, et A. Jentzen)

Integrating the increasing number of available multi-omics cancer data remains one of the main challenges to improve our understanding of cancer. Our approach is based on AMARETTO, an algorithm that integrates DNA methylation, DNA copy number and gene expression data to identify cancer driver genes and associates them to modules of co-expressed genes. We then propose a pancancer version of AMARETTO by connecting all modules in pancancer communities. This leads to the identification of major oncogenic pathways and master regulators involved in different cancers.

The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite time ruin probability in a standard risk model is greater or equal to the corresponding ruin probability evaluated in the associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T.
In this talk, we will consider the diffusion approximations of both the standard risk model and the associated risk model. We will prove that the associated model, when conveniently renormalized, converges in distribution to a gaussian process satisfying a simple SDE with explicit coefficients. We will then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model, which is well known. We will then conclude that the De Vylder and Goovaerts conjecture holds true for these diffusion limits.
This is a joint work with Stefan Ankirchner (University of Jena) and Christophette Blanchet-Scalliet (Ecole Centrale de Lyon and ICJ).

In a quasicrystal, the arrangement of the atoms is highly ordered (as
in an ordinary crystal) but non-periodic (unlike in a crystal). There
are various mathematical challenges in connection with quasicrystals.
From the point of view of statistical mechanics, the major open
problem is to provide a mathematical explanation of the formation and
stability of quasicrystals in presence of thermal fluctuations. In
this talk, I will present a (toy) lattice gas model with finite-range
interactions that has stable quasicrystal phases at positive
temperature (i.e., Gibbs measures supported at perturbations of
non-periodic tilings). The construction is based on old results on
cellular automata and tilings, in particular, a method of simulating
one cellular automaton with another that is resilient against noise,
and the existence of aperiodic sets of Wang tiles that are
deterministic in one direction.

The distribution of a Markov process with killing, conditioned to be
still alive at a given time, can be approximated by a Fleming-Viot
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 purpose of this
talk is to present a central limit theorem for the law of the
Fleming-Viot particle system at a given time in the large population
limit. We will illustrate this result on an application in molecular
dynamics. This is a joint work with Frédéric Cérou, Bernard Delyon and
Mathias Rousset.

In the majority of solid cancers, secondary tumors (metastases) and associated complications are the main cause of death. In order to define the optimal therapeutic strategy for a given patient, one of the major current challenges is to estimate, at diagnosis, the burden of invisible metastases and how they will respond to treatments. In this talk, I will present research efforts towards the establishment of a predictive computational tool of metastatic development, with a particular emphasis on the assessment of mathematical models to empirical data (both experimental and clinical). I will first present the model’s framework, which is based on a physiologically-structured partial differential equation for the time dynamics of a population of metastases, combined to a nonlinear mixed-effects model for statistical representation of the distribution of the parameters in the population. Then, I will show results about the descriptive power of the model on data from clinically relevant ortho-surgical animal models of metastasis (breast and kidney tumors), with recent findings about differential effect of therapies between primary and secondary tumors. The talk will further be devoted to the translation of this modeling approach toward the clinical reality. Using clinical imaging data of brain metastasis from non-small cell lung cancer, several biological processes will be investigated to establish a minimal and biologically realistic model able to describe the data. Integration of this model into a biostatistical approach for individualized prediction of the model’s parameters from data only available at diagnosis will also be discussed. Together, these results represent a step forward towards the integration of mathematical modeling as a predictive tool for personalized medicine in oncology.

Nous présentons plusieurs algorithmes d’exploration de graphes aléatoires, markoviens dans le sens o๠leur implémentation est simultanée à  la construction-même du graphe par le modèle de configuration. Pour différents modèles, par des approximations fluides des processus markoviens sous-jacents, nous obtenons des estimations en grand graphe de (i) la taille de la famille indépendante maximale, avec des applications au protocole de télécommunication CSMA; (ii) la dynamique d’une épidémie de type SIR sur un réseau hétérogène et (iii) la taille d’un couplage maximal sur un grand graphe aléatoire, éventuellement orienté.

I will talk about the problem of detection of a change in mean in a Gaussian vector. A bump is a stepwise change within an interval of a given length but unknown location. We consider the problem of heterogeneous bump detection when the change occurs in mean and in variance of the observed vector and the detection of a bump for dependent observations. Minimax detectability conditions will be presented.
Joint work with A. Munk, M. Pohlmann and F. Werner.

Population dynamics of pathogens invading new territories continues to be of primary concern for both biologists and mathematicians. Extensive researches are mainly carried out throughout mathematical modeling to reconstruct the past dynamics of the alien species.We present a mechanistic-statistical approach that allows us to date and localize the invasion of an alien species and describe other epidemiological parameters as per example, the diffusion, the reproduction and the mortality parameter. The used approach is based on (i) a coupled reaction-diffusion-absorption sub-model that describes the dynamics of the epidemics in a heterogeneous domain and (ii) a stochastic sub-model that represents the observation process. Then, we will jointly estimate the initial conditions (date and site) and the epidemiological parameters using a Bayesian framework through an adaptive multiple importance sampling algorithm. We will show the results obtained in this framework on the basis of abundant post-introduction data gathered to draw up a surveillance plan on the expansion of Xylella fastidiosa, a phytopathogenic bacterium detected in South Corsica in 2015. Nevertheless, this approach could be applied to other post-emergent species in order to endorse a fast reaction.