Branching Processes Markovian Branching Processes Markovian Binary Trees

Branching Processes Markovian Branching Processes Markovian Binary Trees

Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Branching processes Sophie Hautphenne University of Melbourne & EPFL Stochastic Modelling Meets Phylogenetics 16{18 November 2015 1 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Introduction Branching processes Branching processes are stochastic processes describing the dynamics of a population of individuals which reproduce and die independently, according to some specific probability distributions. Branching processes have numerous applications in population biology and phylogenetics 2 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Introduction Branching processes There are many types of branching processes : Discrete time (Galton-Watson branching processes) Continuous time, with exponential lifetime distributions (Markovian branching process), or general lifetime distributions (age-dependent, Bellman-Harris branching process) Single type, or multitype (with finitely or 1ly many types) Individuals reproduction rules may depend on the actual size of the population (population size-dependent branching process) Branching processes can undergo catastrophes or live in a random environment ... 3 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Outline 1 Introduction 2 Galton-Watson branching processes 3 Markovian branching processes 4 Markovian binary trees 4 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees 1 Introduction 2 Galton-Watson branching processes 3 Markovian branching processes 4 Markovian binary trees 5 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees The Galton-Watson branching process Time is discrete and represents successive generations Each individual has a unit lifetime, at the end of which it might give birth to one or more offsprings simultaneously The offspring distribution is described by a random variable ξ taking non-negative integer values with corresponding probabilities pk = P[ξ = k]; k ≥ 0: All individuals behave independently of each other 6 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees The Galton-Watson branching process A realisation of a GW process through 3 generations starting with a single individual at generation 0 : 0 1 2 3 Generation n 7 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees The Galton-Watson branching process The population size at generation n is denoted by Zn We have the branching process equation Zn−1 X Zn = ξi ; n ≥ 1; i=1 where ξ1; ξ2;::: are i.i.d. copies of ξ. The process fZn; n ≥ 0g is a discrete-time Markov chain on the state space f0; 1; 2; 3;:::g where state 0 is absorbing and all other states are transient. 8 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Population size distribution The probability generating function (p.g.f.) of ξ is 1 X P(s) := E[sξ] = P[ξ = k]sk ; s 2 [0; 1]: k=0 If Z0 = 1, then P(s) corresponds to the p.g.f of Z1. Let Pn(s) denote the p.g.f. of Zn, 1 X k Pn(s) := P[Zn = k]s : k=0 9 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Composition of probability generating functions Define the random variable SN as N X SN := Xi ; i=1 where Xi are i.i.d. with p.g.f. GX (s), and N is an independent + random variable taking values in Z , with p.g.f. GN (s). Then the p.g.f. of SN is given by GSN = GN (GX (s)): 10 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Population size distribution Recall that for all n ≥ 1, Zn−1 X Zn = ξi i=1 where ξ1; ξ2;::: are i.i.d. copies of ξ. Proposition Conditionally on Z0 = 1, the p.g.f. of Zn satisfies Pn(s) = Pn−1(P(s)) = P(P(::: P(s))) | {z } n = P(Pn−1(s)); n ≥ 1; with P1(s) = P(s). 11 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Examples of progeny distributions Binary case : 0 w.p. 1=3; ξ = 2 w.p. 2=3 X 1 2 ! P(s) = P[ξ = k]sk = + s2 3 3 k Geometric case : ξ ∼ Geom(p), p P[ξ = k] = (1 − p)k p ! P(s) = 1 − (1 − p)s Poisson case : ξ ∼ Poisson(λ), P[ξ = k] = e−λλk =k! ! P(s) = eλ(s−1) 12 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Mean progeny and criticality The mean progeny is m := E[ξ] = P0(1−) 2 (0; 1): Corollary The mean population size at generation n, conditional on Z0 = 1, is given by E 0 n Mn := [Zn j Z0 = 1] = Pn(1−) = m : m < 1 : subcritical case, Mn & 0 as n ! 1 m = 1 : critical case Mn = 1 for all n m > 1 : supercritical case, Mn % 1 as n ! 1 13 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Extinction probability Let qn = P[Zn = 0] = Pn(0) be the probability that the nth generation is empty. If Z0 = 1, then q0 = 0 and Pn(s) = P(Pn−1(s)) ) qn = P(qn−1); n ≥ 1: The probability of extinction of the branching process is q = lim P[Zn = 0] = P[ lim Zn = 0] ) q = P(q) n!1 n!1 Theorem The extinction probability q is the minimal nonnegative solution of the fixed point equation s = P(s): 14 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Computation of q In practice, when s = P(s) cannot be solved explicitly, q is obtained as the limit of the sequence qn computed using the functional iteration qn = P(qn−1) with q0 = 0. Examples : Binary case : s = (1=3) + (2=3)s2 ! q = 1=2 p Geometric case : s = ! q = min(p=(1 − p); 1), 1 − (1 − p)s i.e. 1 if p ≥ 1=2 q = p=(1 − p) if p < 1=2 Poisson case : s = eλ(s−1) ! the functional iteration is useful here ! 15 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees A closer look at the progeny generating function P(s) is an increasing, convex function such that P(1) = 1 ! P(s) has at most two fixed points in [0; 1] : P(s) P(s) 6 6 1 1 r r p0 p0 r - - q = 1sq 1 s m = P0(1) ≤ 1 m = P0(1) > 1 ! P(s) has a fixed point q < 1 if and only if m > 1 16 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Extinction criterion Proposition (Extinction criterion) q < 1 , m > 1 Recall in the supercritical case, m > 1 ) q < 1 in the critical case, m = 1 ) q = 1 in the subcritical case, m > 1 ) q = 1 17 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Dichotomy : extinction or explosion ! Theorem Regardless the value of m, any state k 6= 0 is transient, so that lim P[Zn = k] = 0 for any k = 1; 2;:::. n!1 Moreover, P[Zn ! 0] + P[Zn ! 1] = 1: | {z } | {z } q 1−q In the supercritical case, conditionally on non-extinction, limn!1 Zn = +1 a.s. Can we tell more about the growth rate of Zn ? 18 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Limiting behaviour in the supercritical case 1 < m < 1 n Let Wn = Zn=m . fWn; n ≥ 0g is a martingale with E[Wn] = 1 for all n, so it converges a.s. to a nonnegative random variable W , Zn n W := lim ! Zn ∼ W m n!1 mn Theorem (Kesten-Stigum) Either P[W = 0] = q or P[W = 0] = 1. The following are equivalent : (i) E[ξ log+ ξ] < 1 (ii) P[W = 0] = q 1 (iii) Wn converges in mean (L ) (iv) E[W ] = 1. 19 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Quasi-stationary distribution in the subcritical case m < 1 Theorem (Yaglom) For each j = 1; 2;:::; lim P[Zn = j j Zn 6= 0] = bj n!1 P exists, and j bj = 1. P k Moreover, the p.g.f. G(s) = k bk s satisfies the equation G(P(s)) = m G(s) + 1 − m: In addition, the vector b = (b1; b2;:::) satisfies bQ = m b; where Q is the truncated probability transition matrix of the GW restricted to the transient states. 20 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees The critical case m = 1 Theorem (Kesten-Ney-Spitzer) Assume σ2 :=Var(ξ) < 1. Then we have Kolmogorov's estimate 2 lim n P[Zn 6= 0] = : n!1 σ2 Yaglom's universal limit law 2 lim P[Zn=n ≥ x j Zn 6= 0] = exp(−2x/σ ); x > 0: n!1 21 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Multitype Galton-Watson branching process Suppose now there are r > 1 types of individuals, each type having its own reproduction law. Example with r = 2 : 0 1 2 3 Generation n 22 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Multitype Galton-Watson branching process Population size vector : Zn = (Zn1; Zn2;:::; Znr ); n 2 N, where Zni : # of individuals of type i at the nth generation r fZng is an r-dimensional Markov process with state space N and an absorbing state 0 = (0; 0;:::; 0)T Progeny distribution : pij : j = (j1; j2;:::; jr ), where pij = probability that a type i gives birth to j1 children of type1, j2 children of type2,. , jr children of type r: 23 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Multitype Galton-Watson branching processes Progeny generating vector P(s) = (P1(s); P2(s);:::; Pr (s)), where Pi (s) is the progeny generating function of an individual of type i r X j X Y jk Pi (s) = pij s = pij sk ; si 2 [0; 1] j2Nr j2Nr k=1 Mean progeny matrix M with elements @Pi (s) Mij = @sj s=1 = expected number of direct offsprings of type j born to a parent of type i Irreducible branching process ≡ M is irreducible 24 Introduction Galton-Watson branching processes Markovian branching processes Markovian binary trees Extinction probability As usual, we assume that the process starts with a single individual > Extinction probability vector q = (q1; q2;:::; qr ) , with h i qi = P lim jZnj = 0 '0 = i n!1 h i q = P lim jZnj = 0 '0 ; n!1 where '0 is the type of the first individual in generation 0.

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    86 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us