Merge pull request #305 from SciML/param_estim_doc_tutorial

Simple Parameter Estimation Tutorial

Author: TorkelE

docs/make.jl

@@ -49,7 +49,8 @@ makedocs(
             "tutorials/advanced.md",
             "tutorials/generated_systems.md",
             "tutorials/advanced_examples.md",
-            "tutorials/bifurcation_diagram.md"
+            "tutorials/bifurcation_diagram.md",
+            "tutorials/parameter_estimation.md"
         ],
         "API" => Any[
             "api/catalyst_api.md"

docs/src/assets/parameter_estimation_plot1.svg

@@ -0,0 +1,106 @@
+# Parameter Estimation
+The parameters of a model, generated by Catalyst, can be estimated using various packages available in the Julia ecosystem. Refer [here](https://diffeq.sciml.ai/stable/analysis/parameter_estimation/) for more extensive information. Below follows a quick tutorial of how [DiffEqFlux](https://diffeqflux.sciml.ai/dev/) can be used to fit a parameter set to data.
+
+First, we fetch the required packages.
+```julia
+using OrdinaryDiffEq
+using DiffEqFlux, Flux
+using Catalyst
+```
+
+Next, we declare our model. For our example, we will use the Brusselator, a simple oscillator.
+```julia
+brusselator = @reaction_network begin
+    A, ∅ → X
+    1, 2X + Y → 3X
+    B, X → Y
+    1, X → ∅
+end A B
+p_real = [1., 2.]
+```
+
+We simulate our model, and from the simulation generate sampled data points (with added noise), to which we will attempt to fit a parameter et.
+```julia
+u0 = [1.0, 1.0]
+tspan = (0.0, 30.0)
+
+sample_times = range(tspan[1],stop=tspan[2],length=100)
+prob = ODEProblem(brusselator, u0, tspan, p_real)
+sol_real = solve(prob, Rosenbrock23(), tstops=sample_times)
+
+sample_vals = [sol_real.u[findfirst(sol_real.t .>= ts)][var] * (1+(0.1rand()-0.05)) for var in 1:2, ts in sample_times];
+```
+
+We can plot the real solution, as well as the noisy samples.
+```julia
+using Plots
+plot(sol_real,size=(1200,400),label="",framestyle=:box,lw=3,color=[:darkblue :darkred])
+plot!(sample_times,sample_vals',seriestype=:scatter,color=[:blue :red],label="")
+```
+![parameter_estimation_plot1](../assets/parameter_estimation_plot1.svg)
+
+Next, we create an optimisation function. For a given initial estimate of the parameter values, p, this function will fit parameter values to our data samples. However, it will only do so on the interval [0,tend].
+```julia
+function optimise_p(p_init,tend)
+    function loss(p)
+        sol = solve(remake(prob,tspan=(0.,tend),p=p), Rosenbrock23(), tstops=sample_times)
+        vals = hcat(map(ts -> sol.u[findfirst(sol.t .>= ts)], sample_times[1:findlast(sample_times .<= tend)])...)    
+        loss = sum(abs2, vals .- sample_vals[:,1:size(vals)[2]])   
+        return loss, sol
+    end
+    return DiffEqFlux.sciml_train(loss,p_init,ADAM(0.1),maxiters = 100)
+end
+```
+
+Next, we will fit a parameter set to the data on the interval [0,10].
+```julia
+p_estimate = optimise_p([5.,5.],10.).minimizer
+```
+
+We can compare this to the real solution, as well as the sample data
+```julia
+sol_estimate = solve(remake(prob,tspan=(0.,10.),p=p_estimate), Rosenbrock23())
+plot(sol_real,size=(1200,400),color=[:blue :red],framestyle=:box,lw=3,label=["X real" "Y real"],linealpha=0.2)
+plot!(sample_times,sample_vals',seriestype=:scatter,color=[:blue :red],label=["Samples of X" "Samples of Y"],alpha=0.4)
+plot!(sol_estimate,color=[:darkblue :darkred], linestyle=:dash,lw=3,label=["X estimated" "Y estimated"],xlimit=tspan)
+```
+![parameter_estimation_plot2](../assets/parameter_estimation_plot2.svg)
+
+Next, we use this parameter estimation as the input to the next iteration of our fitting process, this time on the interval [0,20].
+```julia
+p_estimate = optimise_p(p_estimate,20.).minimizer
+
+sol_estimate = solve(remake(prob,tspan=(0.,20.),p=p_estimate), Rosenbrock23())
+plot(sol_real,size=(1200,400),color=[:blue :red],framestyle=:box,lw=3,label=["X real" "Y real"],linealpha=0.2)
+plot!(sample_times,sample_vals',seriestype=:scatter,color=[:blue :red],label=["Samples of X" "Samples of Y"],alpha=0.4)
+plot!(sol_estimate,color=[:darkblue :darkred], linestyle=:dash,lw=3,label=["X estimated" "Y estimated"],xlimit=tspan)
+```
+![parameter_estimation_plot3](../assets/parameter_estimation_plot3.svg)
+
+Finally, we use this estimate as the input to fit a parameter set on the full interval of sampled data.
+```julia
+p_estimate = optimise_p(p_estimate,30.).minimizer
+
+sol_estimate = solve(remake(prob,tspan=(0.,30.),p=p_estimate), Rosenbrock23())
+plot(sol_real,size=(1200,400),color=[:blue :red],framestyle=:box,lw=3,label=["X real" "Y real"],linealpha=0.2)
+plot!(sample_times,sample_vals',seriestype=:scatter,color=[:blue :red],label=["Samples of X" "Samples of Y"],alpha=0.4)
+plot!(sol_estimate,color=[:darkblue :darkred], linestyle=:dash,lw=3,label=["X estimated" "Y estimated"],xlimit=tspan)
+```
+![parameter_estimation_plot4](../assets/parameter_estimation_plot4.svg)
+
+The final parameter set becomes `[0.9996559014056948, 2.005632696191224]` (the real one was `[1.0, 2.0]`).
+
+
+### Why we fit the parameters in iterations.
+The reason we chose to fit the model on a smaller interval to begin with, and then extend the interval, is to avoid getting stuck in a local minimum. Here specifically, we chose our initial interval to be smaller than a full cycle of the oscillation. If we had chosen to fit a parameter set on the full interval immediately we would have received an inferior solution.
+```julia
+p_estimate = optimise_p([5.,5.],30.).minimizer
+
+sol_estimate = solve(remake(prob,tspan=(0.,30.),p=p_estimate), Rosenbrock23())
+plot(sol_real,size=(1200,400),color=[:blue :red],framestyle=:box,lw=3,label=["X real" "Y real"],linealpha=0.2)
+plot!(sample_times,sample_vals',seriestype=:scatter,color=[:blue :red],label=["Samples of X" "Samples of Y"],alpha=0.4)
+plot!(sol_estimate,color=[:darkblue :darkred], linestyle=:dash,lw=3,label=["X estimated" "Y estimated"],xlimit=tspan)
+```
+![parameter_estimation_plot5](../assets/parameter_estimation_plot5.svg)
+
+