## Abstract

The toxicity and efficacy of more than 30 anticancer agents present very high variations, depending on the dosing time. Therefore, the biologists studying the circadian rhythm require a very precise method for estimating the periodic component (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 7 days for the synchronization segment to 2 or 3 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 inefficient (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 distinguishing them from the ones associated with the noise are difficult tasks. In this paper, 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 cancer treatment context are recording signals expressing a limited 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 the unknown of our model, i.e. the PC vector but also the hyperparameters (i.e the variances). The sparsity prior information is modeled using a sparsity enforcing prior law. In this paper, we propose a Student’s *t* distribution, viewed as the marginal distribution of a bivariate normal-inverse gamma distribution. We build a general infinite Gaussian scale mixture (IGSM) hierarchical model where we assign prior distributions also for the hyperparameters. The expression of the joint posterior law of the unknown PC vector and hyperparameters is obtained via 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), using the Kullback-Leibler (KL) divergence. For the PM estimation, two possibilities are considered: an approximation with a partially separable distribution and an approximation with a fully separable one. Both resulting algorithms corresponding to the PM estimation and the one corresponding to the JMAP estimation are iterative algorithms. The algorithms are presented in detail and are compared with the ones corresponding to the Gaussian model. We examine the 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 paper examines the rest-activity patterns of KI/KI Per2::luc mouse, aged 10 weeks, singly housed in RealTime Biolumicorder (RT-BIO).