Kalman filter with impulse noised outliers: a robust sequential algorithm to filter data with a large number of outliers.

Impulse noised outliers are data points that differ significantly from other observations. They are generally removed from the data set through local regression or the Kalman filter algorithm. However, these methods, or their generalizations, are not well suited when the number of outliers is of the same order as the number of low-noise data (often called nominal measurement). In this article, we propose a new model for impulsed noise outliers. It is based on a hierarchical model and a simple linear Gaussian process as with the Kalman Filter. We present a fast forward-backward algorithm to filter and smooth sequential data and which also detects these outliers. We compare the robustness and efficiency of this algorithm with classical methods. Finally, we apply this method on a real data set from a Walk Over Weighing system admitting around 60 % of outliers. For this application, we further develop an (explicit) EM algorithm to calibrate some algorithm parameters.

From the distributions of times of interactions to preys and predators dynamical systems.

We consider a stochastic individual based model where each predator searches and then manipulates its prey or rests during random times. The time distributions may be non-exponential and density dependent. An age structure allows to describe these interactions and get a Markovian setting. The process is characterized by a measure-valued stochastic differential equation. We prove averaging results in this infinite dimensional setting and get the convergence of the slow-fast macroscopic prey predator process to a two dimensional dynamical system. We recover classical functional responses. We also get new forms arising in particular when births and deaths of predators are affected by the lack of food.

Long-time behavior and Darwinian optimality for an asymmetric size-structured branching process.

We study the long time behavior of an asymmetric size-structured measure-valued growth-fragmentation branching process that models the dynamics of a population of cells taking into account physiological and morphological asymmetry at division. We show that the process exhibits a Malthusian behavior; that is that the global population size grows exponentially fast and that the trait distribution of individuals converges to some stable distribution. The proof is based on a generalization of Lyapunov function techniques for non-conservative semi-groups. We then investigate the fluctuations of the growth rate with respect to the parameters guiding asymmetry. In particular, we exhibit that, under some special assumptions, symmetric division is sub-optimal in a Darwinian sense.

Gaussian approximations for chemostat models in finite and infinite dimensions.

In a chemostat, bacteria live in a growth container of constant volume in which liquid is injected continuously. Recently, Campillo and Fritsch introduced a mass-structured individual-based model to represent this dynamics and proved its convergence to a more classic partial differential equation. In this work, we are interested in the convergence of the fluctuation process. We consider this process in some Sobolev spaces and use central limit theorems on Hilbert space to prove its convergence in law to an infinite-dimensional Gaussian process. As a consequence, we obtain a two-dimensional Gaussian approximation of the Crump-Young model for which the long time behavior is relatively misunderstood. For this approximation, we derive the invariant distribution and the convergence to it. We also present numerical simulations illustrating our results.