arzwa/birthdeathprocesses.jl

(general) birth death processes

(General) birth death processes

This is a module with some functionalities related to (general) birth-death processes. A birth-death process X(t) is a continuous-time Markov process on the non-negative integers where the only allowed transitions are the birth of a new 'particle' and 'death' of an extant particle. The transition probabilities are defined as P(X(t+Δt) = i+1|X(t)=i) = λᵢΔt + o(Δt) and P(X(t+Δt)=i-1|X(t)=i) = μᵢΔt + o(Δt). An important special case is where λᵢ = iλ and μᵢ = iμ, in which case the process is called a linear birth-death process.

Currently the module contains functions for working with linear and general birth-death processes. For the latter, the approach of Crawford & Suchard (2012) is used to compute the Laplace transform of the transition probabilities, using continued fractions, which is numerically inverted using InverseLaplace.jl.

The methods for general BDPs are not yet very efficiently implemented.

Example

using BirthDeathProcesses

Linear BDP

lbdp = LinearBDP(0.3, 0.2)
tp(lbdp, 0.2, 6, 9)  # transition probability

0.0059196166724168565

Transient distribution for the linear BDP for t=2 and X₀=2

tbdp = lbdp(2., 2)

BirthDeathProcesses.TransientLinearBDP{Float64}(λ=0.3, μ=0.2, a=2, t=2.0, α=0.26607578096471346, β=0.3991136714470701)

This acts like a distribution from Distributions.jl

pdf.(tbdp, 0:10)  # probabilities to end in states 0,1,...,10

11-element Array{Float64,1}:
 0.07079632121598217
 0.23468151522042963
 0.2881500371272334
 0.1926264156371718
 0.1078597561048541
 0.05541281292948958
 0.02705085606317386
 0.012765930525135248
 0.005881137338864456
 0.00266097756552092
 0.0011872485532721843

The probability generating function and a sampler are implemented for the linear BDP

pgf(tbdp, 0.)  # the extinction probability == pgf evaluated at 0
rand(tbdp)

2

General BDP

There are two implementations, GeneralBDP is a fast implementation which uses a bound on the continued fraction (default at 100 terms) and uses Arrays as data structures:

λ = (k)->0.3k^2*exp(-1.2*k)
μ = (k)->0.3k
gbdp = GeneralBDP(λ, μ)
tp(gbdp, 1.0, 19, 27)

1.0442021641887911e-84

If one exceeds the depth of the BDP, an error is raised

try tp(gbdp, 1.0, 101, 101); catch ex @show ex; end

AssertionError("(101, 101) exceeds BDP depth (100)")

So we should increase the BDP depth

gbdp = GeneralBDP(λ, μ, 200)
@time tp(gbdp, 1.0, 101, 102)

6.611263593795198e-63

There is another implementation using Lazy lists, which is more elegant but vastly less efficient

import BirthDeathProcesses: LazyGeneralBDP
gbdp = LazyGeneralBDP(λ, μ)
@time tp(gbdp, 1.0, 101, 102)

6.1535547441041e-63

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