|quantitative analysis in biology||
PhD research: Statistics for unbalanced tree data. Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging.
Together with J.-F. Delmas, we have studied dependent data on a binary tree. Think of the descent of an initial individual, where each individual in one generation gives rise to two offspring, one of type 0 and one of type 1, in the next generation. Suppose we are interested in a particular characteristic of these individuals. We assume that this characteristic is stochastic and depends on the ancestors' only through the mother's. The dependency structure may be described by a transition probability P(x,dydz) which gives the probability that the couple of daughters' characteristics is around (y,z) given that the mother's characteristic is x. Note that y, the characteristic of the daughter of type 0, and z, that of the daughter of type 1, may be conditionally dependent given x, and their respective conditional ldistributions may differ (that is why we speak of unbalanced data tree). We then speak of bifurcating Markov chains. We have derived laws of large numbers and central limit theorems for such stochastic processes. We have applied these results to detect cellular aging in Escherichia Coli, using the data of the Laboratoire de Genetique moleculaire, evolutive et medicale (INSERM U571, Faculte de Medecine Necker, Paris) and a bifurcating autoregressive model.