Gauss-Markov Processes

IWP(wiener_process_dimension::Integer, num_derivatives::Integer)

Integrated Wiener process.

By itself it does not have much utility right now, but together with discretize it provides discrete transition matrices that are useful for defining discrete state-space models.

IOUP(wiener_process_dimension::Integer, num_derivatives::Integer, drift_speed::Real)

Integrated Ornstein-Uhlenbeck process.

Matern(wiener_process_dimension::Integer, num_derivatives::Integer, lengthscale::Real)

Matern process. Typically, they are defined with a half-integer parameter $ν$. To get the corresponding process here, set num_derivatives = ν+1/2.

discretize(p::AbstractGaussMarkovProcess, dt::Real)

Discretize the Gauss-Markov process.

Computes the discrete transition matrices for a time step of size dt.

discretize(sde::LTISDE, dt::Real)

Compute the discrete transition via the matrix fraction decomposition.

See also: matrix_fraction_decomposition

projectionmatrix(process::AbstractGaussMarkovProcess, derivative::Integer)

Compute the projection matrix that maps the state to the specified derivative.

preconditioner(p::AbstractGaussMarkovProcess, dt::Real)

Computes a preconditioner, which is helpful for numerically stable implementation.

This corresponds to the preconditioner as described by [1], developed for integrated Wiener process priors (IWP), but it might also be useful for other priors.

[1] Krämer & Hennig, "Stable Implementation of Probabilistic ODE Solvers", 2020.

matrix_fraction_decomposition(drift::AbstractMatrix, dispersion::AbstractVecOrMat, dt::Real)

preconditioned_discretize(p::IWP)

Preconditioned discretization of the integrated Wiener process.

For the IWP, the preconditioned discretization is independend of the current step size and known in closed form. This is very helpful for numerically stable implementations and high efficiency.

LTISDE(drift::AbstractMatrix, dispersion::AbstractVecOrMat)

Linear time-invariant SDE

\[\begin{aligned} dx = F x dt + L dβ \end{aligned}\]

with drift F and dispersion L, and diffusion σ.

to_1d_sde(p::AbstractGaussMarkovProcess)

Build a LTISDE that corresponds to a single dimension of the given Gauss-Markov process.

Just working on a single dimension leads to more efficient code, so that the full matrices can be built at the very end (e.g. after computing the one-dimensional transition matrices via matrix fraction decomposition) with a Kronecker product.