The toxicity and efficacy of more than 30 anticancer agents presents very high variations, depend-

ing on the dosing time. Therefore the biologists studying the circadian rhythm require a very precise

method for estimating the Periodic Components (PC) vector of chronobiological signals. Moreover,

in recent developments not only the dominant period or the PC vector present a crucial interest, but

also their stability or variability. In cancer treatment experiments the recorded signals corresponding

to different phases of treatment are short, from seven days for the synchronization segment to two or

three days for the after treatment segment. When studying the stability of the dominant period we have

to consider very short length signals relative to the prior knowledge of the dominant period, placed in

the circadian domain. The classical approaches, based on Fourier Transform (FT) methods are ineffi-

cient (i.e. lack of precision) considering the particularities of the data (i.e. the short length). Another

particularity of the signals considered in such experiments is the level of noise: such signals are very

noisy and establishing the periodic components that are associated with the biological phenomena and

distinguish them from the ones associated with the noise is a difficult task. In this thesis we propose

a new method for the estimation of the PC vector of biomedical signals, using the biological prior

informations and considering a model that accounts for the noise.

The experiments developed in the cancer treatment context are recording signals expressing a lim-

ited number of periods. This is a prior information that can be translated as the sparsity of the PC

vector. The proposed method considers the PC vector estimation as an Inverse Problem (IP) using

the general Bayesian inference in order to infer all the unknowns of our model, i.e. the PC vector

but also the hyperparameters. The sparsity prior information is modelled using a sparsity enforcing

prior law. In this thesis we propose a Student-t distribution, viewed as the marginal distribution of

a bivariate Normal - Inverse Gamma distribution. In fact, when the equality between the shape and

scale parameters corresponding to the Inverse Gamma distribution is not imposed, the marginal of the

Normal-Inverse Gamma distribution is a generalization of the Student-t distribution. We build a general

Infinite Gaussian Scale Mixture (IGSM) hierarchical model where we also assign prior distributions for

the hyperparameters. The expression of the joint posterior law of the unknown PC vector and the hy-

perparameters is obtained via the Bayes rule and then the unknowns are estimated via Joint Maximum

A Posteriori (JMAP) or Posterior Mean (PM). For the PM estimator, the expression of the posterior

distribution is approximated by a separable one, via Variational Bayesian Approximation (VBA), us-

ing the Kullback-Leibler (KL) divergence. Two possibilities are considered: an approximation with

partially separable distributions and an approximation with a fully separable one. The algorithms are

presented in detail and are compared with the ones corresponding to the Gaussian model. We examine

the practical convergency of the algorithms and give simulation results to compare their performances.

Finally we show simulation results on synthetic and real data in cancer treatment applications. The

real data considered in this thesis examines the rest-activity patterns and gene expressions of KI/KI

Per2::luc mouse, aged 10 weeks, singly housed in RT-BIO.

Keywords: Periodic Components (PC) vector estimation, Sparsity enforcing, Bayesian parameter

estimation, Variational Bayesian Approximation (VBA), Kullback-Leibler (KL) divergence, Infinite

Gaussian Scale Mixture (IGSM), Normal - Inverse Gamma, Inverse problem, Joint Maximum A Pos-

teriori (JMAP), Posterior Mean (PM), Chronobiology, Circadian rhythm, Cancer treatment.

__Composition du jury__

M. Ali MOHAMMAD-DJAFARI, Directeur de recherche CNRS, L2S, Gif-sur-Yvette, Directeur de thèse

M. Francis LÉVI, Professeur des Universités, University of Warwick, Angleterre, Co-directeur de thèse

M. Jean-François GIOVANELLI, Professeur des Universités, IMS, Bordeaux, Rapporteur

M. Ercan Engin KURUOGLU, Chercheur sénior CNRS, ISTI, Italie, Rapporteur

M. Alexandre RENAUX, Maître de conférences, Paris-Sud, Orsay, Examinateur

M. Michel KIEFFER, Professeur des Universités, Paris-Sud, Orsay, Examinateur