Nonparametric Estimation of the Division Rate of a Size-Structured Population

Abstract

We consider the problem of estimating the division rate of a size-structured population in a nonparametric setting. The size of the system evolves according to a transport-fragmentation equation: each individual grows with a given transport rate, and splits into two offsprings of the same size, following a binary fragmentation process with unknown division rate that depends on its size. In contrast to a deterministic inverse problem approach, as in [23,4], we take in this paper the perspective of statistical inference: our data consists in a large sample of the size of individuals, when the evolution of the system is close to its time-asymptotic behavior, so that it can be related to the eigenproblem of the considered transport-fragmentation equation (see [22] for instance). By estimating statistically each term of the eigenvalue problem and by suitably inverting a certain linear operator (see [4]), we are able to construct a more realistic estimator of the division rate that achieves the same optimal error bound as in related deterministic inverse problems. Our procedure relies on kernel methods with automatic bandwidth selection. It is inspired by model selection and recent results of Goldenschluger and Lepski [13,14].

Estimating the division rate for the growth-fragmentation equation

“…Due to this small regularization, as shown by Figures 8, 10, the noise is filtered but not as much as we hoped first -especially for smaller x, that are farer from the departing point of the algorithm. Finally, the parameter α needs to stay in a confidence interval, selected, for a given growth rate g(x), from a range of simulations carried out for various plausible birth rates (see for instance Figures 5,7,9).…”

“…As in [4,7], we consider the problem of recovering the cell division rate B and the constant c from the a priori knowledge of the shape of the growth rate g(x) and the experimental measure of the asymptotic distribution N and exponential growth λ 0 . To model this, we suppose that we have two given measurements…”

“…To overcome this difficulty, two regularization methods were proposed in [2,3] for the particular case of division into two equal cells, i.e. when κ(x, y) = δ x=y/2 , a third method has also been proposed in [11], and a statistical treatment to estimate the derivative in [7]. Indeed, looking at the problem in terms of H = BN and not in terms of B makes it almost linear in H; almost, because λ 0 being also measured, the term λ 0 N can be viewed as quadratic.…”

“…For this case, we consider as input data the values of the solution N of the direct problem in which we add a multiplicative random noise uniformely distributed in [ −ε 2 , ε 2 ] (see [7] for a more precise statistical setting of noisy informations). The nonnegativity of the data is insured by the choice…”

Structured Models for Cell Populations: Direct and Inverse Problems

“…In one spatial dimension, the amount of data needed to estimate the division kernel k(·, ··) would be unrealistic in practical applications without further hypothesis. See [1][2][3] …”