Statistical estimation of a growth-fragmentation model observed on a genealogical tree

Abstract

We raise the issue of estimating the division rate for a growing and dividing population modelled by a piecewise deterministic Markov branching tree. Such models have broad applications, ranging from TCP/IP window size protocol to bacterial growth. Here, the individuals split into two offsprings at a division rate B(x) that depends on their size x, whereas their size grow exponentially in time, at a rate that exhibits variability. The mean empirical measure of the model satisfies a growth-fragmentation type equation, and we bridge the deterministic and probabilistic viewpoints. We then construct a nonparametric estimator of the division rate B(x) based on the observation of the population over different sampling schemes of size n on the genealogical tree. Our estimator nearly achieves the rate n −s/(2s+1) in squared-loss error asymptotically, generalizing and improving on the rate n −s/(2s+3) obtained in [13,15] through indirect observation schemes. Our method is consistently tested numerically and implemented on Escherichia coli data, which demonstrates its major interest for practical applications.

Adaptive estimation for bifurcating Markov chains

“…These include binary growth-fragmentation processes, where we subsequently estimate adaptively the splitting rate of a size-dependent model, thus extending previous results of Doumic et al [26] and bifurcating autoregressive processes, where we complete previous studies of Bitseki Penda et al [12] and Bitseki Penda and Olivier [13].…”

“…We consider in Section 4.1 the growth-fragmentation model as studied in Doumic et al [26], where we estimate the sizedependent splitting rate of the model as a function of the invariant measure of an associated BMC in Theorem 11. This enables us to extend the recent results of Doumic et al in several directions: adaptive estimation, extension of the smoothness classes and the loss functions considered, and also a proof of a minimax lower bound.…”

“…In continuous time, BMC encode certain piecewise deterministic Markov processes on trees that serve as the stochastic realisation of growth-fragmentation models (see e.g. Doumic et al [26], Robert et al [38] for modelling cell division in Escherichia coli and the references therein).…”

Statistical inference for partial differential equations

“…Theorem 1. Let us consider the algorithm (8) and the function u K defined by (10). The following convergence result holds:…”

“…The construction of B is based on the key representation formula proved in [10] B(y) = y 2 ν B (y/2)…”