arXiv subjects

Adélaïde Olivier

Publications and source records attributed to Adélaïde Olivier.

5 recordsLinked to original sources

arXiv2019

Estimating the division rate from indirect measurements of single cells

Is it possible to estimate the dependence of a growing and dividing population on a given trait in the case where this trait is not directly accessible by experimental measurements, but making use of measurements of another variable? This article adresses this general question for a very recent and popular model describing bacterial growth, the so-called incremental or adder model. In this model, the division rate depends on the increment of size between birth and division, whereas the most accessible trait is the size itself. We prove that estimating the division 10 rate from size measurements is possible, we state a reconstruction formula in a deterministic and then in a statistical setting, and solve numerically the problem on simulated and experimental data. Though this represents a severely ill-posed inverse problem, our numerical results prove to be satisfactory.

math.ST↗

arXiv2017

How does variability in cells aging and growth rates influence the malthus parameter?

The aim of this study is to compare the growth speed of different cell populations measured by their Malthus parameter. We focus on both the age-structured and size-structured equations. A first population (of reference) is composed of cells all aging or growing at the same rate $\bar v$. A second population (with variability) is composed of cells each aging or growing at a rate $v$ drawn according to a non-degenerated distribution $ρ$ with mean $\bar v$. In a first part, analytical answers -- based on the study of an eigenproblem -- are provided for the age-structured model. In a second part, numerical answers -- based on stochastic simulations -- are derived for the size-structured model. It appears numerically that the population with variability proliferates more slowly than the population of reference (for experimentally plausible division rates). The decrease in the Malthus parameter we measure, around 2% for distributions $ρ$ with realistic coefficients of variations around 15-20\%, is determinant since it controls the {\it exponential} growth of the whole population.

math.PR↗

arXiv2016

Autoregressive Functions Estimation in Nonlinear Bifurcating Autoregressive Models

Bifurcating autoregressive processes, which can be seen as an adaptation of au-toregressive processes for a binary tree structure, have been extensively studied during the last decade in a parametric context. In this work we do not specify any a priori form for the two autoregressive functions and we use nonparametric techniques. We investigate both nonasymp-totic and asymptotic behavior of the Nadaraya-Watson type estimators of the autoregressive functions. We build our estimators observing the process on a finite subtree denoted by Tn, up to the depth n. Estimators achieve the classical rate |Tn| --$β$/(2$β$+1) in quadratic loss over H{ö}lder classes of smoothness. We prove almost sure convergence, asymptotic normality giving the bias expression when choosing the optimal bandwidth and a moderate deviations principle. Our proofs rely on specific techniques used to study bifurcating Markov chains. Finally, we address the question of asymmetry and develop an asymptotic test for the equality of the two autoregressive functions.

math.ST↗

arXiv2015

Nonparametric estimation of the division rate of an age dependent branching process

We study the nonparametric estimation of the branching rate $B(x)$ of a supercritical Bellman-Harris population: a particle with age $x$ has a random lifetime governed by $B(x)$; at its death time, it gives rise to $k \geq 2$ children with lifetimes governed by the same division rate and so on. We observe in continuous time the process over $[0,T]$. Asymptotics are taken as $T \rightarrow \infty$; the data are stochastically dependent and one has to face simultaneously censoring, bias selection and non-ancillarity of the number of observations. In this setting, under appropriate ergodicity properties, we construct a kernel-based estimator of $B(x)$ that achieves the rate of convergence $\exp(-λ_B \fracβ{2β+1}T)$, where $λ_B$ is the Malthus parameter and $β>0$ is the smoothness of the function $B(x)$ in a vicinity of $x$. We prove that this rate is optimal in a minimax sense and we relate it explicitly to classical nonparametric models such as density estimation observed on an appropriate (parameter dependent) scale. We also shed some light on the fact that estimation with kernel estimators based on data alive at time $T$ only is not sufficient to obtain optimal rates of convergence, a phenomenon which is specific to nonparametric estimation and that has been observed in other related growth-fragmentation models.

stat.OT↗

arXiv2015

Adaptive estimation for bifurcating Markov chains

In a first part, we prove Bernstein-type deviation inequalities for bifurcating Markov chains (BMC) under a geometric ergodicity assumption, completing former results of Guyon and Bitseki Penda, Djellout and Guillin. These preliminary results are the key ingredient to implement nonparametric wavelet thresholding estimation procedures: in a second part, we construct nonparametric estimators of the transition density of a BMC, of its mean transition density and of the corresponding invariant density, and show smoothness adaptation over various multivariate Besov classes under $L^p$-loss error, for $1 \leq p < \infty$. We prove that our estimators are (nearly) optimal in a minimax sense. As an application, we obtain new results for the estimation of the splitting size-dependent rate of growth-fragmentation models and we extend the statistical study of bifurcating autoregressive processes.

math.ST↗