Processus de branchement | EPFL Graph Search

En théorie des probabilités, un processus de branchement est un processus stochastique formé par une collection de variables aléatoires. Les variables aléatoires d'un processus stochastique sont indexées par les nombres entiers naturels. Les processus de branchement ont été développés en premier lieu pour décrire une population dans laquelle chaque individu de la génération n produit un nombre aléatoire d'individus dans la génération n+1. Par exemple, une population de bactéries dans laquelle chaque individu, indépendamment des autres, produit un nombre aléatoire de descendants à chaque pas de temps. Les processus de branchement peuvent aussi être utilisés pour modéliser d'autres systèmes avec des dynamiques similaires, par exemple la propagation de noms de familles dans une généalogie ou de neutrons dans un réacteur nucléaire. Formulation mathématique La formulation la plus commune d'un processus de branchement est le processus de Galt

Statistical Inference for Partially Observed Stochastic Epidemics

Andrea Kraus

This work is concerned with the estimation of the spreading potential of the disease in the initial stages of an epidemic. A speedy and accurate estimation is important for determining whether or not interventions are necessary to prevent a major outbreak. At the same time, the information available in the early stages is scarce and data collection imperfect. We consider an epidemic in a large susceptible population, and address the estimation based on temporally aggregated counts of new cases that are subject to unknown random under-reporting. We allow for an influence of the detection process on the evolution of the epidemic. While the proportion of infectious individuals in the population is small, the role of chance in the spread of the disease may be substantial. Therefore, stochastic epidemic models are applied. As these are difficult to analyse, the time evolution of the number of infectious individuals is approximated by branching processes. We study the estimation in a partially observed Galton–Watson process and in a partially observed linear birth and death process; and in each case focus on the parameter characterising the growth of the process. We aim at estimators that perform well in the asymptotic sense where a single trajectory is observed over a long period of time, and study the asymptotics conditionally on the eventual explosion of the process. The partially observed Galton–Watson process has been recently proposed in the literature as a model for the initial stages of an epidemic. Its probabilistic structure has been explored and estimation has been partially addressed, in that consistent estimators have been constructed. However, the estimation-related uncertainty has not been evaluated. We address this issue here by constructing estimators that are motivated from the asymptotic dependence structure of the process. We show that they are consistent and asymptotically normal, consistently estimate their asymptotic variances, and construct asymptotic confidence intervals. In addition, we evaluate their finite-sample performance in a simulation study and their practical performance on real data. The observation mechanism in the partially observed Galton–Watson process is inherently discrete. To allow for continuous-time dynamics, we incorporate partial observation in the linear birth and death process. In particular, we propose a model where the birth process is completely unobserved, while a random proportion of the death process is observed at discrete time points. We study the estimation in this model. Motivated by its counting process structure, we arrive at consistent and asymptotically normal estimators, consistently estimate their asymptotic variances, and construct asymptotic confidence intervals. We also evaluate the finite-sample and practical performance of the estimators in a simulation study and on real data.

EPFL2013

Critical values in an anisotropic percolation on $\mathbb{Z}^2$

Hao Xue

In this thesis, we consider an anisotropic finite-range bond percolation model on

\mathbb{Z}^2

. On each horizontal layer

\{(x,i):x\in\mathbb{Z}\}

for

i\in\mathbb{Z}

, we have edges

\langle(x,i),(y,i)\rangle

for

1\leq|x-y|\leq N

with

N\in\mathbb{N}

. There are also vertical edges connecting two nearest neighbor vertices on distinct layers

\langle(x,i),(x,i+1)\rangle

for

x,i\in\mathbb{Z}

. On this graph, we consider the following anisotropic percolation model: horizontal edges are open with probability

\lambda/(2N)

with

\lambda\geq 1

, while vertical edges are open with probability

\epsilon

to be suitably tuned as

N

grows to infinity. This question is motivated by a result on the analogous layered ferromagnetic Ising model at mean field critical temperature. We first deal with the critical case when

\lambda=1

. If

\epsilon=\kappa N^{-2/5}

, we see a phase transition in

\kappa

: positive and finite constants

C_1,C_2

exist so that there is no percolation if

\kappaC_2

. The derivation of the scaling limit is inspired by works on the long range contact process. The proof relies on the analysis of the scaling limit of the critical branching random walk that dominates the growth process restricted to each horizontal layer and a careful analysis of the true horizontal growth process, which is interesting by itself. A renormalization argument is used for the percolative regime. We then deal with the supercritical case when

\lambda>1

. If

\epsilon=e^{-\kappa N}

, we can also see a phase transition in

\kappa

. The horizontal and vertical edges can be discovered through subordinate process in each regime. The proof is based on the analysis of supercritical branching random walk but several levels of attritions are introduced to make sure the independent structure. The comparison between our original percolation and the percolation on the inhomogeneous square lattice is used in the renormalization scheme.

EPFL2021

Lyapunov Exponents for Branching Processes in a Random Environment: The Effect of Information

Sophie Myriam Hautphenne

We consider multitype branching processes evolving in a Markovian random environment. To determine whether or not the branching process becomes extinct almost surely is akin to computing the maximal Lyapunov exponent of a sequence of random matrices, which is a notoriously difficult problem. We define Markov chains associated to the branching process, and we construct bounds for the Lyapunov exponent. The bounds are obtained by adding or by removing information: to add information results in a lower bound, to remove information results in an upper bound, and we show that adding less information improves the lower bound. We give a few illustrative examples and we observe that the upper bound is generally more accurate than the lower bounds.

Springer Verlag2016

Critical values in an anisotropic percolation on $\mathbb{Z}^2$

Hao Xue

In this thesis, we consider an anisotropic finite-range bond percolation model on

\mathbb{Z}^2

. On each horizontal layer

\{(x,i):x\in\mathbb{Z}\}

for

i\in\mathbb{Z}

, we have edges

\langle(x,i),(y,i)\rangle

for

1\leq|x-y|\leq N

with

N\in\mathbb{N}

. There are also vertical edges connecting two nearest neighbor vertices on distinct layers

\langle(x,i),(x,i+1)\rangle

for

x,i\in\mathbb{Z}

. On this graph, we consider the following anisotropic percolation model: horizontal edges are open with probability

\lambda/(2N)

with

\lambda\geq 1

, while vertical edges are open with probability

\epsilon

to be suitably tuned as

N

grows to infinity. This question is motivated by a result on the analogous layered ferromagnetic Ising model at mean field critical temperature. We first deal with the critical case when

\lambda=1

. If

\epsilon=\kappa N^{-2/5}

, we see a phase transition in

\kappa

: positive and finite constants

C_1,C_2

exist so that there is no percolation if

\kappaC_2

\lambda>1

. If

\epsilon=e^{-\kappa N}

, we can also see a phase transition in

\kappa

EPFL2021

Statistical Inference for Partially Observed Stochastic Epidemics

Andrea Kraus

EPFL2013