# Multivariate OU parameter estimation

I am interested in estimating the parameters M,D\in\mathbb{R}^{n\times n}, D\succeq0 of an OU process

dX_t = M(X_t-\mu)d_t + DdW_t

from observed data at frequency \Delta t.

Is there a generalized likelihood inference procedure in the SDE DiffEq package that is similar to the one used for DiffEqs (Generalized Likelihood Inference · DiffEqParamEstim.jl) but with a single dataset (i.e., if we only have observed one “run” of the SDE)?

Here is my approach so far (following Parameter Estimation for Stochastic Differential Equations and Ensembles · DiffEqParamEstim.jl), but it doesn’t seem to work, as the optimization procedure just returns the initial point:

using DifferentialEquations
using DiffEqParamEstim
using LinearAlgebra
using RecursiveArrayTools
using Optim
using PyPlot
fig, ax = subplots()

# parameters
M = [-1.0; 0.0; -1.0]
D = [1.0; 0.1; 1.0]
μ = [5.0; 10.0]
p = [vec(M); vec(D); vec(μ)]
dt = 0.001
tspan = (0.0, 1.0)

# drift and diffusion
f(du, u, p, t) = begin
M = reshape([p[1:2]; p[2:3]], (2, 2))
μ = p[7:8]
du .= M * (u - μ)
end
g(du, u, p, t) = begin
D = reshape([p[4:5]; p[5:6]], (2, 2))
du .= D
end

Set up the problem

# problem
prob = SDEProblem(f, g, μ, tspan, p, noise_rate_prototype=zeros(2, 2))
sol = solve(prob, EM(), dt=dt)
ax.cla()
ax.plot(sol.u)

Set up the estimation

# data
t = collect(range(0, stop=1, length=200))
function generate_data(t)
sol = solve(prob, EM(), dt=dt)
randomized = VectorOfArray([(sol(t[i]) + 0.01randn(2)) for i in 1:length(t)])
data = convert(Array, randomized)
end
ntraj = 200 # for now, doing more trajectories than a single run
aggregate_data = convert(Array, VectorOfArray([generate_data(t) for i in 1:ntraj]))

# estimation
p0 = [0.4vec([-1.0; 0.1; -1.0]); 0.4vec([1.0; 0.1; 1.0]); 0.4vec([1.0; 1.0])]
monte_prob = EnsembleProblem(prob)
# obj = build_loss_objective(monte_prob, SOSRA2(), L2Loss(t, aggregate_data),
#   maxiters=10000, verbose=false, trajectories=ntraj,
obj = build_loss_objective(monte_prob, EM(), dt=dt, L2Loss(t, aggregate_data),
maxiters=10000, verbose=false, trajectories=ntraj,
result = Optim.optimize(obj, p0, Optim.BFGS())

Show the result

pstar = Optim.minimizer(result)
hcat(p0, pstar, p)
julia> hcat(p0, pstar, p)
8×3 Matrix{Float64}:
-0.4   -0.39996    -1.0
0.04   0.0400358   0.0
-0.4   -0.400066   -1.0
0.4    0.40028     1.0
0.04   0.0400151   0.1
0.4    0.40048     1.0
0.4    0.399804    5.0
0.4    0.400003   10.0
1 Like

Maybe you can have it easy: I longer time ago I implemented the transition density of a OU process in Bridge.jl. So you can just have a likelihood and use a sampler. Let’s generate some toy data:

using Bridge
using LinearAlgebra
M = -I + 0.1*randn(5, 5)
@assert all(real.(eigvals(M)) .< 0) # otherwise explosion
D = randn(5, 5)
β = zeros(5)
P = LinPro(M, β, D)
x0 = zeros(5)
tt = collect(0:0.4:100.0)

# sorry some fixes for out of date code
Bridge.cholupper(a) = cholesky(Symmetric(a)).U
Bridge.rand(P::Bridge.Gaussian{<:Vector}) = P.μ + Bridge.cholupper(P.Σ)'*randn(eltype(P.μ), length(P.μ))

function generate(tt, x0, P)
x = copy(x0)
yy = [x]
for i in eachindex(tt)[2:end]
x = rand(transitionprob(tt[i-1], x, tt[i], P))
push!(yy, x)
end
tt, yy
end
tt, yy = generate(tt, x0, P)

Now we can evaluate the log-likelihood for any M and D:

function loglikelihood(tt, yy, P)
ll = 0.0
for i in eachindex(tt)[2:end]
ll += lp(tt[i-1], yy[i-1], tt[i], yy[i], P)
end
ll
end
loglikelihood(tt, yy, LinPro(M, β, D))
3 Likes

Indeed if it’s just an OU, using Bridge.jl in a way that’s specialized directly for OU will be much more efficient than using something for general stochastic differential equations. That’s definitely the right way to go.

1 Like