MarginalDivergence.jl

A fast, Bayesian method for Practical Identifiability in differential equation and other models. Implemented in Julia.

The methodology is described in the following paper:

Case, Young, & Hébert-Dufresne, "Accurately summarizing an outbreak using epidemiological models takes time." Royal Society Open Science. 2023. doi: 10.1098/rsos.230634

Installation

Using Pkg mode, simply

add https://github.com/brendandaisy/MarginalDivergence.jl

to create a static installation or use dev to clone a version you can edit.

Check the package is loaded by typing st in Pkg model. To test everything is working:

test MarginalDivergence

To update to the latest version, simply

update MarginalDivergence

Dependencies

Currently, getting full use of this package requires use of the unregistered package ConditionalTransform.jl, which is used to get samples from the conditional distribution $P(\theta \mid u)$ for arbitrary function of the model parameters $u=f(\theta)$. That package is here.

Examples

using MarginalDivergence
using Distributions, MonteCarloMeasurements
using Parameters

Define the latent process

There are two classes of interfaces: models inheriting from GenericModel must simply define a solve method, while models inheriting from ODEModel and DDEModel requires implementing at least timespan initial_values parameters and de_func, which point to the relavent componenets of DifferentialEquations.jl's implementation. The solve method as well as some other helpers are then inherited.

Here we implement a simple ODEModel, the SIS model. First, we define the differential equations:

function sis!(dx, x, p, t)
    I = x[1]
    β, α = p
    dx[1] = β * I * (1-I) - α * I
end

sis! (generic function with 1 method)

These should be implemented in the same way as in DifferentialEquations.jl. Now we implement our model object, which stores the parameters, timespan, etc, as well as complete the ODEModel interface.

@with_kw struct SISModel{T<:Real} <: ODEModel{T}
    start::T = 0.
    stop::T = 20.
    I₀::Param{T} = 0.01
    β::Param{T} = 0.3
    α::Param{T} = 0.1
end

MarginalDivergence.timespan(m::SISModel) = (m.start, m.stop)
MarginalDivergence.initial_values(m::SISModel) = [m.I₀]
MarginalDivergence.parameters(m::SISModel) = [m.β, m.α]
MarginalDivergence.de_func(::SISModel) = sis!

And we're all set. Define an instance of the model with uncertain $\beta$ and $\alpha$, seeing data only up to $T=10$.

mod = SISModel(β=0..1, α=0.1..0.5)
sol = solve(mod; saveat=0:10, save_idxs=1).u

11-element Vector{Particles{Float64, 2000}}:
 0.01
 0.0127 ± 0.0039
 0.0175 ± 0.011
 0.0258 ± 0.023
 0.0393 ± 0.044
 0.0594 ± 0.077
 0.0858 ± 0.12
 0.116 ± 0.16
 0.146 ± 0.2
 0.174 ± 0.23
 0.198 ± 0.25

To compute the RMD of $\beta$, we also simulate the model holding $\beta$ fixed to the "true" value.

θtrue = (β=0.7, α=0.2)
sol_restr = solve(mod, (β=0.7,); saveat=0:10, save_idxs=1).u

11-element Vector{Particles{Float64, 2000}}:
 0.01
 0.0149 ± 0.0017
 0.0224 ± 0.0051
 0.0337 ± 0.011
 0.0508 ± 0.022
 0.0758 ± 0.039
 0.111 ± 0.065
 0.156 ± 0.098
 0.208 ± 0.13
 0.265 ± 0.17
 0.319 ± 0.2

Finally, we need to define the observation process, here using the provided Gaussian noise model, and get the Distribution of the data.

obs_mod = NConstVar(0.01)

sol_true = solve(mod, θtrue; saveat=0:10, save_idxs=1).u
ytrue = observe_dist(obs_mod, sol_true)

11-element Vector{Normal{Float64}}:
 Normal{Float64}(μ=0.01, σ=0.1)
 Normal{Float64}(μ=0.016338818919136543, σ=0.1)
 Normal{Float64}(μ=0.02654426937177269, σ=0.1)
 Normal{Float64}(μ=0.04273385817344045, σ=0.1)
 Normal{Float64}(μ=0.06782393946706626, σ=0.1)
 Normal{Float64}(μ=0.10533481065432604, σ=0.1)
 Normal{Float64}(μ=0.15850378577161595, σ=0.1)
 Normal{Float64}(μ=0.22844561737900243, σ=0.1)
 Normal{Float64}(μ=0.31192175591246146, σ=0.1)
 Normal{Float64}(μ=0.40075126698866187, σ=0.1)
 Normal{Float64}(μ=0.48437783675404145, σ=0.1)

And we get the RMD, approximating with 100 outer product steps (in practice, $N$ and $M$ should probably be much larger)

marginal_divergence(ytrue, sol_restr, sol, obs_mod; N=100)

0.8384965917102682

Additional examples can be found in the test folder.