Fitzhugh-Nagumo Bayesian Parameter Estimation Benchmarks
Vaibhav Dixit, Chris Rackauckas
using DiffEqBayes, BenchmarkTools
using OrdinaryDiffEq, RecursiveArrayTools, Distributions, ParameterizedFunctions, CmdStan, DynamicHMC using Plots
gr(fmt=:png)
Plots.GRBackend()
Defining the problem.
The FitzHugh-Nagumo model is a simplified version of Hodgkin-Huxley model and is used to describe an excitable system (e.g. neuron).
fitz = @ode_def FitzhughNagumo begin dv = v - v^3/3 -w + l dw = τinv*(v + a - b*w) end a b τinv l
(::Main.WeaveSandBox0.FitzhughNagumo{Main.WeaveSandBox0.var"###Parameterize
dDiffEqFunction#496",Main.WeaveSandBox0.var"###ParameterizedTGradFunction#4
97",Main.WeaveSandBox0.var"###ParameterizedJacobianFunction#498",Nothing,No
thing,ModelingToolkit.ODESystem}) (generic function with 1 method)
prob_ode_fitzhughnagumo = ODEProblem(fitz,[1.0,1.0],(0.0,10.0),[0.7,0.8,1/12.5,0.5]) sol = solve(prob_ode_fitzhughnagumo, Tsit5())
retcode: Success
Interpolation: specialized 4th order "free" interpolation
t: 14-element Array{Float64,1}:
0.0
0.15079562872319327
0.6663735500745417
1.4549121831880751
2.6341751496828474
3.7872864628874394
5.149282290423124
6.764810407399299
7.606020974182365
8.324334146165869
9.040772814596577
9.552575705603262
9.985208121599765
10.0
u: 14-element Array{Array{Float64,1},1}:
[1.0, 1.0]
[1.0242787914016627, 1.0109527801835287]
[1.0925382825360388, 1.0495725586393927]
[1.147894455050522, 1.1102123746508352]
[1.134543873591793, 1.1975474781177977]
[1.0432761941043434, 1.2718688798460578]
[0.8446920007269357, 1.3381007267503957]
[0.3135440377028956, 1.3689380033842313]
[-0.4098348685955019, 1.342759540998098]
[-1.4082544459528368, 1.2706202503513042]
[-1.909783303000839, 1.1563318788556225]
[-1.9618464536295719, 1.0688710996087507]
[-1.9544223037206336, 0.9966722929830949]
[-1.9538629866249133, 0.9942458205399927]
Data is genereated by adding noise to the solution obtained above.
t = collect(range(1,stop=10,length=10)) sig = 0.20 data = convert(Array, VectorOfArray([(sol(t[i]) + sig*randn(2)) for i in 1:length(t)]))
2×10 Array{Float64,2}:
1.36325 1.08641 1.03527 0.87398 … -1.0363 -1.67129 -1.68434
1.14972 1.21111 1.14868 0.819744 1.1275 0.94216 0.801884
Plot of the data and the solution.
scatter(t, data[1,:]) scatter!(t, data[2,:]) plot!(sol)
Priors for the parameters which will be passed for the Bayesian Inference
priors = [Truncated(Normal(1.0,0.5),0,1.5),Truncated(Normal(1.0,0.5),0,1.5),Truncated(Normal(0.0,0.5),0.0,0.5),Truncated(Normal(0.5,0.5),0,1)]
4-element Array{Truncated{Normal{Float64},Continuous,Float64},1}:
Truncated(Normal{Float64}(μ=1.0, σ=0.5), range=(0.0, 1.5))
Truncated(Normal{Float64}(μ=1.0, σ=0.5), range=(0.0, 1.5))
Truncated(Normal{Float64}(μ=0.0, σ=0.5), range=(0.0, 0.5))
Truncated(Normal{Float64}(μ=0.5, σ=0.5), range=(0.0, 1.0))
Benchmarks
Stan.jl backend
@btime bayesian_result_stan = stan_inference(prob_ode_fitzhughnagumo,t,data,priors;num_samples = 10_000,printsummary=false)
File /Users/vaibhav/DiffEqBenchmarks.jl/tmp/parameter_estimation_model.stan
will be updated.
File /Users/vaibhav/DiffEqBenchmarks.jl/tmp/parameter_estimation_model.stan
will be updated.
File /Users/vaibhav/DiffEqBenchmarks.jl/tmp/parameter_estimation_model.stan
will be updated.
File /Users/vaibhav/DiffEqBenchmarks.jl/tmp/parameter_estimation_model.stan
will be updated.
181.327 s (1361525 allocations: 57.16 MiB)
DiffEqBayes.StanModel{Stanmodel,Int64,Array{Float64,3},Array{String,1}}( n
ame = "parameter_estimation_model"
nchains = 1
num_samples = 10000
num_warmup = 1000
thin = 1
monitors = String[]
model_file = "parameter_estimation_model.stan"
data_file = "parameter_estimation_model_1.data.R"
output = Output()
file = "parameter_estimation_model_samples_1.csv"
diagnostics_file = ""
refresh = 100
pdir = "/Users/vaibhav/DiffEqBenchmarks.jl"
tmpdir = "/Users/vaibhav/DiffEqBenchmarks.jl/tmp"
output_format = :array
method = Sample()
num_samples = 10000
num_warmup = 1000
save_warmup = false
thin = 1
algorithm = HMC()
engine = NUTS()
max_depth = 10
metric = CmdStan.diag_e
stepsize = 1.0
stepsize_jitter = 1.0
adapt = Adapt()
gamma = 0.05
delta = 0.8
kappa = 0.75
t0 = 10.0
init_buffer = 75
term_buffer = 50
window = 25
, 0, [2.41684 0.98964 … 0.114672 0.496116; 1.34539 0.958575 … 0.0707656 0.4
58518; … ; -3.03367 0.942425 … 0.0719484 0.414652; 1.69935 0.937137 … 0.079
2119 0.474801], ["lp__", "accept_stat__", "stepsize__", "treedepth__", "n_l
eapfrog__", "divergent__", "energy__", "sigma1.1", "sigma1.2", "theta1", "t
heta2", "theta3", "theta4", "theta.1", "theta.2", "theta.3", "theta.4"])
Turing.jl backend
@btime bayesian_result_turing = turing_inference(prob_ode_fitzhughnagumo,Tsit5(),t,data,priors;num_samples = 10_000)
61.372 s (488812759 allocations: 36.82 GiB)
Object of type Chains, with data of type 9000×17×1 Array{Float64,3}
Iterations = 1:9000
Thinning interval = 1
Chains = 1
Samples per chain = 9000
internals = acceptance_rate, hamiltonian_energy, hamiltonian_energy
_error, is_accept, log_density, lp, max_hamiltonian_energy_error, n_steps,
nom_step_size, numerical_error, step_size, tree_depth
parameters = theta[1], theta[2], theta[3], theta[4], σ[1]
2-element Array{MCMCChains.ChainDataFrame,1}
Summary Statistics
parameters mean std naive_se mcse ess r_hat
────────── ────── ────── ──────── ────── ───────── ──────
theta[1] 0.8469 0.3148 0.0033 0.0058 2712.1944 0.9999
theta[2] 1.0511 0.2642 0.0028 0.0050 3078.2793 0.9999
theta[3] 0.1148 0.0467 0.0005 0.0006 4168.2588 1.0000
theta[4] 0.5130 0.0810 0.0009 0.0012 3859.9413 1.0000
σ[1] 0.2955 0.0599 0.0006 0.0010 3968.2518 1.0000
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
────────── ────── ────── ────── ────── ──────
theta[1] 0.1940 0.6283 0.8636 1.0866 1.4014
theta[2] 0.4934 0.8709 1.0721 1.2595 1.4718
theta[3] 0.0341 0.0825 0.1116 0.1426 0.2184
theta[4] 0.3671 0.4571 0.5082 0.5623 0.6895
σ[1] 0.2046 0.2530 0.2863 0.3271 0.4384
Conclusion
FitzHugh-Ngumo is a standard problem for parameter estimation studies. In the FitzHugh-Nagumo model the parameters to be estimated were [0.7,0.8,0.08,0.5]. dynamichmc_inference has issues with the model and hence was excluded from this benchmark.
using DiffEqBenchmarks DiffEqBenchmarks.bench_footer(WEAVE_ARGS[:folder],WEAVE_ARGS[:file])
Appendix
These benchmarks are a part of the DiffEqBenchmarks.jl repository, found at: https://github.com/JuliaDiffEq/DiffEqBenchmarks.jl
To locally run this tutorial, do the following commands:
using DiffEqBenchmarks
DiffEqBenchmarks.weave_file("ParameterEstimation","DiffEqBayesFitzHughNagumo.jmd")Computer Information:
Julia Version 1.4.0
Commit b8e9a9ecc6 (2020-03-21 16:36 UTC)
Platform Info:
OS: macOS (x86_64-apple-darwin18.6.0)
CPU: Intel(R) Core(TM) i7-6700HQ CPU @ 2.60GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-8.0.1 (ORCJIT, skylake)
Package Information:
Status: `/Users/vaibhav/DiffEqBenchmarks.jl/Project.toml`
[28f2ccd6-bb30-5033-b560-165f7b14dc2f] ApproxFun 0.11.10
[a134a8b2-14d6-55f6-9291-3336d3ab0209] BlackBoxOptim 0.5.0
[a93c6f00-e57d-5684-b7b6-d8193f3e46c0] DataFrames 0.20.2
[2b5f629d-d688-5b77-993f-72d75c75574e] DiffEqBase 6.25.2
[ebbdde9d-f333-5424-9be2-dbf1e9acfb5e] DiffEqBayes 2.9.1
[eb300fae-53e8-50a0-950c-e21f52c2b7e0] DiffEqBiological 4.2.0
[f3b72e0c-5b89-59e1-b016-84e28bfd966d] DiffEqDevTools 2.18.0
[c894b116-72e5-5b58-be3c-e6d8d4ac2b12] DiffEqJump 6.5.0
[1130ab10-4a5a-5621-a13d-e4788d82bd4c] DiffEqParamEstim 1.13.0
[a077e3f3-b75c-5d7f-a0c6-6bc4c8ec64a9] DiffEqProblemLibrary 4.6.4
[ef61062a-5684-51dc-bb67-a0fcdec5c97d] DiffEqUncertainty 1.4.1
[0c46a032-eb83-5123-abaf-570d42b7fbaa] DifferentialEquations 6.12.0
[7073ff75-c697-5162-941a-fcdaad2a7d2a] IJulia 1.21.1
[7f56f5a3-f504-529b-bc02-0b1fe5e64312] LSODA 0.6.1
[76087f3c-5699-56af-9a33-bf431cd00edd] NLopt 0.5.1
[c030b06c-0b6d-57c2-b091-7029874bd033] ODE 2.6.0
[54ca160b-1b9f-5127-a996-1867f4bc2a2c] ODEInterface 0.4.6
[09606e27-ecf5-54fc-bb29-004bd9f985bf] ODEInterfaceDiffEq 3.6.0
[1dea7af3-3e70-54e6-95c3-0bf5283fa5ed] OrdinaryDiffEq 5.32.2
[2dcacdae-9679-587a-88bb-8b444fb7085b] ParallelDataTransfer 0.5.0
[65888b18-ceab-5e60-b2b9-181511a3b968] ParameterizedFunctions 5.0.3
[91a5bcdd-55d7-5caf-9e0b-520d859cae80] Plots 0.29.9
[b4db0fb7-de2a-5028-82bf-5021f5cfa881] ReactionNetworkImporters 0.1.5
[f2c3362d-daeb-58d1-803e-2bc74f2840b4] RecursiveFactorization 0.1.0
[9672c7b4-1e72-59bd-8a11-6ac3964bc41f] SteadyStateDiffEq 1.5.0
[c3572dad-4567-51f8-b174-8c6c989267f4] Sundials 3.9.0
[a759f4b9-e2f1-59dc-863e-4aeb61b1ea8f] TimerOutputs 0.5.3
[44d3d7a6-8a23-5bf8-98c5-b353f8df5ec9] Weave 0.9.4
[b77e0a4c-d291-57a0-90e8-8db25a27a240] InteractiveUtils
[d6f4376e-aef5-505a-96c1-9c027394607a] Markdown
[44cfe95a-1eb2-52ea-b672-e2afdf69b78f] Pkg
[9a3f8284-a2c9-5f02-9a11-845980a1fd5c] Random