Example: Blowfly Model
Likelihood-free inference for the blow-fly model was introduced by Simon N. Wood. We model here the discrete time stochastic dynamics of the size $N$ of an adult blowfly population as given in section 1.2.3 of the supplementary information.
where $eₜ$ and $ϵₜ$ are independent Gamma random deviates with mean 1 and variance $σp²$ and $σd²$, respectively.
using Distributions, StatsBase, LikelihoodfreeInference
Base.@kwdef struct BlowFlyModel
burnin::Int = 50
T::Int = 1000
end
function (m::BlowFlyModel)(P, N₀, σd, σp, τ, δ)
p1 = Gamma(1/σp^2, σp^2)
p2 = Gamma(1/σd^2, σd^2)
T = m.T + m.burnin + τ
N = fill(180., T)
for t in τ+1:T-1
N[t+1] = P * N[t-τ] * exp(-N[t-τ]/N₀)*rand(p1) + N[t]*exp(-δ*rand(p2))
end
N[end-m.T+1:end]
end
Let us plot four realizations from this model with the same parameters.
using StatsPlots
gr()
m = BlowFlyModel()
plot([plot(m(29, 260, .6, .3, 7, .2),
xlabel = "t", ylabel = "N", legend = false) for _ in 1:4]...,
layout = (2, 2))
To compare different realizations we will use histogram summary statistics. In the literature one finds also other summary statistics for this data.
summary_statistics(N) = fit(Histogram, N, 140:16:16140).weights
summary_statistics (generic function with 1 method)
We will use a normal prior on log-transformed parameters.
function parameter(logparams)
lP, lN₀, lσd, lσp, lτ, lδ = logparams
(P = round(exp(2 + 2lP)),
N₀ = round(exp(4 + .5lN₀)),
σd = exp(-.5 + lσd),
σp = exp(-.5 + lσp),
τ = round(Int, max(1, min(500, exp(2 + lτ)))),
δ = exp(-1 + .4lδ))
end
(m::BlowFlyModel)(logparams) = m(parameter(logparams)...)
target(m::BlowFlyModel) = [(log(29) - 2)/2,
(log(260) - 4)*2,
log(.6) + .5,
log(.3) + .5,
log(7) - 2,
(log(.2) + 1)/.4]
lower(m::BlowFlyModel) = fill(-5., 6)
upper(m::BlowFlyModel) = fill(5., 6)
prior = TruncatedMultivariateNormal(zeros(6), ones(6),
lower = lower(m), upper = upper(m))
TruncatedMultivariateNormal{Distributions.MvNormal{Float64,PDMats.PDiagMat{Float64,Array{Float64,1}},Array{Float64,1}},Float64}(
mvnormal: DiagNormal(
dim: 6
μ: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
Σ: [1.0 0.0 … 0.0 0.0; 0.0 1.0 … 0.0 0.0; … ; 0.0 0.0 … 1.0 0.0; 0.0 0.0 … 0.0 1.0]
)
lower: [-5.0, -5.0, -5.0, -5.0, -5.0, -5.0]
upper: [5.0, 5.0, 5.0, 5.0, 5.0, 5.0]
)
Let us now generate some target data.
model = BlowFlyModel()
x0 = target(model)
data = summary_statistics(model(x0))
1000-element Array{Int64,1}:
0
0
0
0
0
0
2
2
3
3
⋮
0
0
0
0
0
0
0
0
0
Adaptive SMC
smc = AdaptiveSMC(prior = prior)
result = run!(smc, x -> summary_statistics(model(x)), data,
maxfevals = 2*10^5, verbose = false)
using PrettyTables
pretty_table([[keys(parameter(zeros(6)))...] quantile(smc, .05) median(smc) mean(smc) x0 quantile(smc, .95)],
["names", "5%", "median", "mean", "actual", "95%"],
formatter = ft_printf("%10.3f"))
┌ Warning: The API of formatters has changed in PrettyTables v0.9.
│ The keyword is now called `formatters` instead of `formatter` and it is a tuple
│ of functions instead of a dictionary.
│
│ If you are using a predefined formatter, then you only need to replace the
│ keyword `formatter` by `formatters`. Otherwise, you will need to rewrite your
│ formmaters.
│
│ For now, the old API is wrapped in a function that converts it to
│ the new one.
│
│ For more information, see the documentation.
└ @ PrettyTables ~/.julia/packages/PrettyTables/BRTPU/src/print.jl:727
┌───────┬────────────┬────────────┬────────────┬────────────┬────────────┐
│ names │ 5% │ median │ mean │ actual │ 95% │
├───────┼────────────┼────────────┼────────────┼────────────┼────────────┤
│ P │ -2.008 │ 0.301 │ -0.041 │ 0.684 │ 1.920 │
│ N₀ │ -1.770 │ -0.109 │ -0.063 │ 3.121 │ 1.792 │
│ σd │ -1.672 │ -0.053 │ -0.101 │ -0.011 │ 1.429 │
│ σp │ -1.599 │ -0.069 │ -0.004 │ -0.704 │ 1.974 │
│ τ │ -1.310 │ 0.283 │ 0.234 │ -0.054 │ 1.659 │
│ δ │ -1.592 │ 0.059 │ 0.061 │ -1.524 │ 1.723 │
└───────┴────────────┴────────────┴────────────┴────────────┴────────────┘
histogram(smc)
corrplot(smc)
KernelABC
k = KernelABC(prior = prior, delta = 1e-1, K = 10^3, kernel = Kernel())
result = run!(k, x -> summary_statistics(model(x)), data)
pretty_table([[keys(parameter(zeros(6)))...] quantile(k, .05) median(k) mean(k) x0 quantile(k, .95)],
["names", "5%", "median", "mean", "actual", "95%"],
formatter = ft_printf("%10.3f"))
┌ Warning: The API of formatters has changed in PrettyTables v0.9.
│ The keyword is now called `formatters` instead of `formatter` and it is a tuple
│ of functions instead of a dictionary.
│
│ If you are using a predefined formatter, then you only need to replace the
│ keyword `formatter` by `formatters`. Otherwise, you will need to rewrite your
│ formmaters.
│
│ For now, the old API is wrapped in a function that converts it to
│ the new one.
│
│ For more information, see the documentation.
└ @ PrettyTables ~/.julia/packages/PrettyTables/BRTPU/src/print.jl:727
┌───────┬────────────┬────────────┬────────────┬────────────┬────────────┐
│ names │ 5% │ median │ mean │ actual │ 95% │
├───────┼────────────┼────────────┼────────────┼────────────┼────────────┤
│ P │ -1.827 │ 0.406 │ 0.129 │ 0.684 │ 1.962 │
│ N₀ │ -1.712 │ 0.064 │ 0.056 │ 3.121 │ 1.922 │
│ σd │ -1.741 │ -0.026 │ 0.010 │ -0.011 │ 1.721 │
│ σp │ -1.665 │ 0.088 │ 0.005 │ -0.704 │ 1.639 │
│ τ │ -1.416 │ 0.202 │ 0.114 │ -0.054 │ 1.660 │
│ δ │ -1.531 │ 0.110 │ 0.076 │ -1.524 │ 1.747 │
└───────┴────────────┴────────────┴────────────┴────────────┴────────────┘
histogram(k)
Kernel Recursive ABC (with callback)
k = KernelRecursiveABC(prior = prior,
K = 100,
delta = 1e-3,
kernel = Kernel(bandwidth = Bandwidth(heuristic = MedianHeuristic(2^3))),
kernelx = Kernel());
We will use a callback here to show how the estimated parameters evolves.
using LinearAlgebra
res_krabc = run!(k, x -> summary_statistics(model(x)), data,
maxfevals = 1300,
verbose = true,
callback = () -> @show norm(k.theta - x0)/norm(x0))
(x = [0.6856825795339864, 0.46820286945349177, -0.09602277695882587, 0.0076403068295181975, -0.03486446901293221, 0.07808071052524887],)
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