arXiv subjects

Sophie Hautphenne

Publications and source records attributed to Sophie Hautphenne.

At least 19 recordsLinked to original sources

arXiv2026

Computing the density of the Kesten-Stigum limit in supercritical Galton-Watson processes

This paper proposes a novel numerical method for computing the density of the limit random variable associated with a supercritical Galton-Watson process. This random variable captures the effect of early demographic fluctuations and determines the random amplitude of long-term exponential population growth. While the existence of a non-trivial limit is ensured by the Kesten-Stigum theorem, computing its density in a stable and efficient manner for arbitrary offspring laws remains a significant challenge. The proposed approach leverages a functional equation that characterizes the Laplace-Stieltjes transform of the limit distribution and combines it with a moment-matching method to obtain accurate approximations within a class of linear combinations of Laguerre polynomials with exponential damping. The effectiveness of the approach is validated on several examples in which the offspring generating function is a polynomial of bounded degree.

math.PR↗

arXiv2025

Consistent estimation in subcritical birth-and-death processes

We investigate parameter estimation in subcritical continuous-time birth-and-death processes with multiple births. We show that the classical maximum likelihood estimators for the model parameters, based on the continuous observation of a single non-extinct trajectory, are not consistent in the usual sense: conditional on survival up to time $t$, they converge as $t \to \infty$ to the corresponding quantities in the associated $Q$-process, namely the process conditioned to survive in the distant future. We develop the first $C$-consistent estimators in this setting, which converge to the true parameter values when conditioning on survival up to time $t$, and establish their asymptotic normality. The analysis relies on spine decompositions and coupling techniques.

math.ST↗

arXiv2025

Existence and non-existence of consistent estimators in supercritical controlled branching processes

We consider the problem of estimating the parameters of a supercritical controlled branching process consistently from a single observed trajectory of population size counts. Our goal is to establish which parameters can and cannot be consistently estimated. When a parameter can be consistently estimated, we derive an explicit expression for the estimator. We address these questions in three scenarios: when the distribution of the control function distribution is known, when it is unknown, and when progenitor numbers are observed alongside population size counts. Our results offer a theoretical justification for the common practice in population ecology of estimating demographic and environmental stochasticity using separate observation schemes.

math.PR↗

arXiv2025

A hybrid framework for compartmental models enabling simulation-based inference

Multi-scale systems often exhibit a combination of stochastic and deterministic dynamics. In compartmental models, low occupancy compartments tend to exhibit stochastic dynamics while high occupancy compartments tend to follow deterministic dynamics. Representing both dynamics with existing methods is challenging. Failing to account for stochasticity in small populations can produce ``atto-foxes'', for example in the Lotka-Volterra ordinary differential equation (ODE) model. This limitation becomes problematic when studying the extinction of species or the clearance of infection, but it can be overcome by using discrete stochastic models, such as continuous time Markov chains (CTMCs). Unfortunately, simulating CTMCs is impractical for many realistic models, where discrete events have very high frequencies. In this work, we develop a novel mathematical framework to couple continuous ODEs and discrete CTMCs: ``Jump-Switch-Flow'' (JSF). In this framework, compartments can reach extinct states (``absorbing states''), thereby resolving atto-fox-type problems. JSF has the desired behaviours of exact CTMC simulation, but is substantially computationally faster than existing alternatives, by at least one order of magnitude, and can even obtain constant scaling, irrespective of compartment occupancy. We demonstrate JSF's utility for simulation-based inference, particularly multi-scale problems, with several case-studies. In a simulation study, we demonstrate how JSF can enable a more nuanced analysis of the efficacy of public health interventions. We also carry out a novel analysis of longitudinal within-host data from SARS-CoV-2 infections to quantify the timing of viral clearance. In this work, we show how JSF offers a novel approach to compartmental model simulation.

q-bio.PE↗

arXiv2024

Consistent least squares estimation in population-size-dependent branching processes

We derive the first conditionally consistent estimators for a class of parametric Markov population models with logistic growth, which are suitable for modelling endangered populations in restricted habitats with a carrying capacity. We focus on discrete-time parametric population-size-dependent branching processes, for which we propose a new class of weighted least-squares estimators based on a single trajectory of population size counts. We establish the consistency and asymptotic normality of our estimators, conditional on non-extinction up to time $n$, as $n\to\infty$. Since Markov population models with a carrying capacity become extinct almost surely under general conditions, our proofs rely on arguments distinct from those in the existing literature. Our results are motivated by conservation biology, where endangered populations are often studied precisely because they are still alive, leading to an observation bias. Through simulated examples, we show that our conditionally consistent estimators generally reduce this bias for key quantities such as a habitat's carrying capacity. We apply our methodology to estimate the carrying capacity of the Chatham Island black robin, a population reduced to a single breeding female in the 1970's, which has since recovered but has yet to reach the island's carrying capacity.

math.ST↗

arXiv2024

Approximate Bayesian computation for Markovian binary trees in phylogenetics

Phylogenetic trees describe the relationships between species in the evolutionary process, and provide information about the rates of diversification. To understand the mechanisms behind macroevolution, we consider a class of multitype branching processes called Markovian binary trees (MBTs). MBTs allow for trait-based variation in diversification rates, and provide a flexible and realistic probabilistic model for phylogenetic trees. We develop an approximate Bayesian computation (ABC) scheme to infer the rates of MBT parameters by exploiting the information in the shapes of phylogenetic trees. We evaluate the accuracy of this inference method using simulation studies, and find that our method is able to detect variation in the diversification rates, with accuracy comparable to, and generally better than, likelihood-based methods. In an application to a real-life phylogeny of squamata, we reinforce conclusions drawn from earlier studies, in particular supporting the existence of ovi-/viviparity transitions in both directions. Our method demonstrates the potential for more complex models of evolution to be employed in phylogenetic inference, in conjunction with likelihood-free schemes.

q-bio.PE↗

arXiv2023

Linking Population-Size-Dependent and Controlled Branching Processes

Population-size dependent branching processes (PSDBP) and controlled branching processes (CBP) are two classes of branching processes widely used to model biological populations that exhibit logistic growth. In this paper we develop connections between the two, with the ultimate goal of determining when a population is more appropriately modelled with a PSDBP or a CBP. In particular, we state conditions for the existence of equivalent PSDBPs and CBPs, we then consider the subclass of CBPs with deterministic control functions (DCBPs), stating a necessary and sufficient condition for DCBP-PSDBP equivalence. Finally, we derive an upper bound on the total variation distance between non-equivalent DCBPs and PSDBPs with matching first and second moments and equal initial population size, and show that under certain conditions this bound tends to zero as the initial population size becomes large.

math.PR↗

arXiv2021

A fluid approach to total-progeny-dependent birth-and-death processes

We introduce a class of branching processes in which the reproduction or lifetime distribution at a given time depends on the total cumulative number of individuals who have been born in the population until that time. We focus on a continuous-time version of these processes, called total-progeny-dependent birth-and-death processes, and study some of their properties through the analysis of their fluid (deterministic) approximation. These properties include the maximum population size, the total progeny size at extinction, the time to reach the maximum population size, and the time until extinction. As the fluid approach does not allow us to approximate the time until extinction directly, we propose several methods to complement this approach. We also use the fluid approach to study the behaviour of the processes as we increase the magnitude of the individual birth rate.

math.PR↗

arXiv2021

Birth-and-death Processes in Python: The BirDePy Package

Birth-and-death processes (BDPs) form a class of continuous-time Markov chains that are particularly suited to describing the changes in the size of a population over time. Population-size-dependent BDPs (PSDBDPs) allow the rate at which a population grows to depend on the current population size. The main purpose of our new Python package BirDePy is to provide easy-to-use functions that allow the parameters of discretely-observed PSDBDPs to be estimated. The package can also be used to estimate parameters of continuously-observed PSDBDPs, simulate sample paths, approximate transition probabilities, and generate forecasts. We describe in detail several methods which have been incorporated into BirDePy to achieve each of these tasks. The usage and effectiveness of the package is demonstrated through a variety of examples of PSDBDPs, as well as case studies involving annual population count data of two endangered bird species.

stat.ME↗

arXiv2021

Markovian Transition Counting Processes: An Alternative to Markov Modulated Poisson Processes

Stochastic models for performance analysis, optimization and control of queues hinge on a multitude of alternatives for input point processes. In case of bursty traffic, one very popular model is the \textit{Markov Modulated Poisson Process} (MMPP), however it is not the only option. Here, we introduce an alternative that we call \textit{Markovian transition counting process} (MTCP). The latter is a point process counting the number of transitions of a finite continuous-time Markov chain. For a given MTCP one can establish an MMPP with the same first and second moments of counts. In this paper, we show the other direction by establishing a duality in terms of first and second moments of counts between MTCPs and a rich class of MMPPs which we refer to as slow MMPPs (modulation is slower than the events). Such a duality confirms the applicability of the MTCP as an alternative to the MMPP which is superior when it comes to moment matching and finding the important measures of the inter-event process. We illustrate the use of such equivalence in a simple queueing example, showing that the MTCP is a comparable and competitive model for performance analysis.

math.PR↗

arXiv2020

Extinction probabilities in branching processes with countably many types: a general framework

We consider Galton-Watson branching processes with countable typeset $\mathcal{X}$. We study the vectors ${\bf q}(A)=(q_x(A))_{x\in\mathcal{X}}$ recording the conditional probabilities of extinction in subsets of types $A\subseteq \mathcal{X}$, given that the type of the initial individual is $x$. We first investigate the location of the vectors ${\bf q}(A)$ in the set of fixed points of the progeny generating vector and prove that $q_x(\{x\})$ is larger than or equal to the $x$th entry of any fixed point, whenever it is different from 1. Next, we present equivalent conditions for $q_x(A)< q_x (B)$ for any initial type $x$ and $A,B\subseteq \mathcal{X}$. Finally, we develop a general framework to characterise all \emph{distinct} extinction probability vectors, and thereby to determine whether there are finitely many, countably many, or uncountably many distinct vectors. We illustrate our results with examples, and conclude with open questions.

math.PR↗

arXiv2020

Parameter estimation in branching processes with almost sure extinction

We consider population-size-dependent branching processes (PSDBPs) which eventually become extinct with probability one. For these processes, we derive maximum likelihood estimators for the mean number of offspring born to individuals when the current population size is $z\geq 1$. As is standard in branching process theory, an asymptotic analysis of the estimators requires us to condition on non-extinction up to a finite generation $n$ and let $n\to\infty$; however, because the processes become extinct with probability one, we are able to demonstrate that our estimators do not satisfy the classical consistency property ($C$-consistency). This leads us to define the concept of $Q$-consistency, and we prove that our estimators are $Q$-consistent and asymptotically normal. To investigate the circumstances in which a $C$-consistent estimator is preferable to a $Q$-consistent estimator, we then provide two $C$-consistent estimators for subcritical Galton-Watson branching processes. Our results rely on a combination of linear operator theory, coupling arguments, and martingale methods.

math.ST↗

arXiv2020

A low-rank technique for computing the quasi-stationary distribution of subcritical Galton-Watson processes

We present a new algorithm for computing the quasi-stationary distribution of subcritical Galton--Watson branching processes. This algorithm is based on a particular discretization of a well-known functional equation that characterizes the quasi-stationary distribution of these processes. We provide a theoretical analysis of the approximate low-rank structure that stems from this discretization, and we extend the procedure to multitype branching processes. We use numerical examples to demonstrate that our algorithm is both more accurate and more efficient than other approaches.

math.NA↗

arXiv2019

The probabilities of extinction in a branching random walk on a strip

We consider a class of multitype Galton-Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, the probability $\boldsymbol{q}(A)$ of extinction in subsets of types $A\subseteq \mathcal{X}_d$ may differ from the global extinction probability $\boldsymbol{q}$ and the partial extinction probability $\tilde{\boldsymbol{q}}$. After deriving partial and global extinction criteria, we develop conditions for $\boldsymbol{q}<\boldsymbol{q}(A)<\tilde{\boldsymbol{q}}$. We then present an iterative method to compute the vector $\boldsymbol{q}(A)$ for any set $A$. Finally, we investigate the location of the vectors $\boldsymbol{q}(A)$ in the set of fixed points of the progeny generating vector.

math.PR↗

arXiv2018

Parameter estimation for discretely-observed linear birth-and-death processes

Birth-and-death processes are widely used to model the development of biological populations. Although they are relatively simple models, their parameters can be challenging to estimate, because the likelihood can become numerically unstable when data arise from the most common sampling schemes, such as annual population censuses. Simple estimators may be based on an embedded Galton-Watson process, but this presupposes that the observation times are equi-spaced. We estimate the birth, death, and growth rates of a linear birth-and-death process whose population size is periodically observed via an embedded Galton-Watson process, and by maximizing a saddlepoint approximation to the likelihood. We show that a Gaussian approximation to the saddlepoint-based likelihood connects the two approaches, we establish consistency and asymptotic normality of quasi-likelihood estimators, compare our estimators on some numerical examples, and apply our results to census data for two endangered bird populations and the H1N1 influenza pandemic.

math.ST↗

arXiv2018

Extinction in lower Hessenberg branching processes with countably many types

We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton-Watson processes with typeset $\mathcal{X}=\{0,1,2,\dots\}$, in which individuals of type $i$ may give birth to offspring of type $j\leq i+1$ only. For this class of processes, we study the set $S$ of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector $\boldsymbol{q}$ and whose maximum is the partial extinction probability vector $\boldsymbol{\tilde{q}}$. In the case where $\boldsymbol{\tilde{q}}=\boldsymbol{1}$, we derive a global extinction criterion which holds under second moment conditions, and when $\boldsymbol{\tilde{q}}<\boldsymbol{1}$ we develop necessary and sufficient conditions for $\boldsymbol{q}=\boldsymbol{\tilde{q}}$.

math.PR↗

arXiv2017

A pathwise iterative approach to the extinction of branching processes with countably many types

We consider the extinction events of Galton-Watson processes with countably infinitely many types. In particular, we construct truncated and augmented Galton-Watson processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some sufficient conditions, the corresponding sequence of extinction probability vectors converges to the global extinction probability vector of the Galton-Watson processes with countably infinitely many types. This gives rise to a number of iterative methods for the computation of the global extinction probability vector.

math.PR↗

arXiv2017

Fitting Markovian binary trees using global and individual demographic data

We consider a class of branching processes called Markovian binary trees, in which the individuals lifetime and reproduction epochs are modeled using a transient Markovian arrival process (TMAP). We estimate the parameters of the TMAP based on population data containing information on age-specific fertility and mortality rates. Depending on the degree of detail of the available data, a weighted non-linear regression method or a maximum likelihood method is applied. We discuss the optimal choice of the number of phases in the TMAP, and we provide confidence intervals for the model outputs. The results are illustrated using real data on human and bird populations.

stat.AP↗