Priors

IWP([dim::Integer=1,] num_derivatives::Integer)

$q$-times integrated Wiener process.

This is the recommended prior! The IWP is the most common prior choice in the probabilistic ODE solver literature (see the references) and is the default choice for the solvers in ProbNumDiffEq.jl. It is also the prior that has the most efficient implementation.

The IWP can be created without specifying the dimension of the Wiener process, in which case it will be one-dimensional. The ODE solver then assumes that each dimension of the ODE solution should be modeled with the same prior. This is typically the preferred usage.

In math

The IWP is a Gauss–Markov process, which we model with a state representation

\[\begin{aligned} Y(t) = \left[ Y^{(0)}(t), Y^{(1)}(t), \dots Y^{(q)}(t) \right], \end{aligned}\]

defined as the solution of the stochastic differential equation

\[\begin{aligned} \text{d} Y^{(i)}(t) &= Y^{(i+1)}(t) \ \text{d}t, \qquad i = 0, \dots, q-1 \\ \text{d} Y^{(q)}(t) &= \Gamma \ \text{d}W(t). \end{aligned}\]

Then, $Y^{(0)}(t)$ is the $q$-times integrated Wiener process (IWP) and $Y^{(i)}(t)$ is the $i$-th derivative of the IWP, for $i = 1, \dots, q$.

Examples

julia> solve(prob, EK1(prior=IWP(2)))

IOUP([dim::Integer=1,]
     num_derivatives::Integer,
     rate_parameter::Union{Number,AbstractVector,AbstractMatrix})

Integrated Ornstein–Uhlenbeck process.

This prior is mostly used in the context of Probabilistic Exponential Integrators to include the linear part of a semi-linear ODE in the prior, so it is used in the ExpEK and the RosenbrockExpEK.

In math

The IOUP is a Gauss–Markov process, which we model with a state representation

\[\begin{aligned} Y(t) = \left[ Y^{(0)}(t), Y^{(1)}(t), \dots Y^{(q)}(t) \right], \end{aligned}\]

defined as the solution of the stochastic differential equation

\[\begin{aligned} \text{d} Y^{(i)}(t) &= Y^{(i+1)}(t) \ \text{d}t, \qquad i = 0, \dots, q-1 \\ \text{d} Y^{(q)}(t) &= L Y^{(q)}(t) \ \text{d}t + \Gamma \ \text{d}W(t), \end{aligned}\]

where $L$ is called the rate_parameter. Then, $Y^{(0)}(t)$ is the $q$-times integrated Ornstein–Uhlenbeck process (IOUP) and $Y^{(i)}(t)$ is the $i$-th derivative of the IOUP, for $i = 1, \dots, q$.

Examples

julia> solve(prob, EK1(prior=IOUP(2, -1)))

Matern([dim::Integer=1,]
       num_derivatives::Integer,
       lengthscale::Number)

Matern process.

The class of Matern processes is well-known in the Gaussian process literature, and they also have a corresponding SDE representation similarly to the IWP and the IOUP. See also [6] for more details.

In math

A Matern process is a Gauss–Markov process, which we model with a state representation

\[\begin{aligned} Y(t) = \left[ Y^{(0)}(t), Y^{(1)}(t), \dots Y^{(q)}(t) \right], \end{aligned}\]

defined as the solution of the stochastic differential equation

\[\begin{aligned} \text{d} Y^{(i)}(t) &= Y^{(i+1)}(t) \ \text{d}t, \qquad i = 0, \dots, q-1 \\ \text{d} Y^{(q)}(t) &= - \sum_{j=0}^q \left( \begin{pmatrix} q+1 \\ j \end{pmatrix} \left( \frac{\sqrt{2q - 1}}{l} \right)^{q-j} Y^{(j)}(t) \right) \ \text{d}t + \Gamma \ \text{d}W(t). \end{aligned}\]

where $l$ is called the lengthscale parameter. Then, $Y^{(0)}(t)$ is a Matern process and $Y^{(i)}(t)$ is the $i$-th derivative of this process, for $i = 1, \dots, q$.

Examples

julia> solve(prob, EK1(prior=Matern(2, 1)))

AbstractGaussMarkovProcess{elType}

Abstract type for Gauss-Markov processes.

Gauss-Markov processes are solutions to linear time-invariant stochastic differential equations (SDEs). Here we assume SDEs of the form

\[\begin{aligned} dX_t &= F X_t dt + L dW_t \\ X_0 &= \mathcal{N} \left( X_0; \mu_0, \Sigma_0 \right) \end{aligned}\]

where $X_t$ is the state, $W_t$ is a Wiener process, and $F$ and $L$ are matrices.

Currently, ProbNumDiffEq.jl makes many assumptions about the structure of the SDEs that it can solve. In particular, it assumes that the state vector $X_t$ contains a range of dervatives, and that the Wiener process only enters the highest one. It also assumes a certain ordering of dimensions and derivatives. This is not a limitation of the underlying mathematics, but rather a limitation of the current implementation. In the future, we hope to remove these limitations.

LTISDE(F::AbstractMatrix, L::AbstractMatrix)

Linear time-invariant stochastic differential equation.

A LTI-SDE is a stochastic differential equation of the form

\[dX_t = F X_t dt + L dW_t\]

where $X_t$ is the state, $W_t$ is a Wiener process, and $F$ and $L$ are matrices. This LTISDE object holds the matrices $F$ and $L$. It also provides some functionality to discretize the SDE via a matrix-fraction decomposition. See: discretize(::LTISDE, ::Real).

dim(p::AbstractGaussMarkovProcess)

Return the dimension of the process.

This is not the dimension of the "state" that is used to efficiently model the prior process as a state-space model, but it is the dimension of the process itself that we aim to model.

See AbstractGaussMarkovProcess for more details on Gauss-Markov processes in ProbNumDiffEq.

num_derivatives(p::AbstractGaussMarkovProcess)

Return the number of derivatives that are represented by the processes state.

See AbstractGaussMarkovProcess for more details on Gauss-Markov processes in ProbNumDiffEq.

to_sde(p::AbstractGaussMarkovProcess)

Convert the prior to the corresponding SDE.

Gauss-Markov processes are solutions to linear time-invariant stochastic differential equations (SDEs) of the form

\[\begin{aligned} dX_t &= F X_t dt + L dW_t \\ X_0 &= \mathcal{N} \left( X_0; \mu_0, \Sigma_0 \right) \end{aligned}\]

where $X_t$ is the state, $W_t$ is a Wiener process, and $F$ and $L$ are matrices. This function returns the corresponding SDE, i.e. the matrices $F$ and $L$, as a LTISDE.

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

Compute the transition matrices of the process for a given step size.

discretize(p::LTISDE, step_size::Real)

Compute the transition matrices of the SDE solution for a given step size.

initial_distribution(p::AbstractGaussMarkovProcess)

Return the initial distribution of the process.

Currently this is always a Gaussian distribution with zero mean and unit variance, unless explicitly overwitten (e.g. for Matern processes to have the stationary distribution). This implementation is likely to change in the future to allow for more flexibility.

marginalize(process::AbstractGaussMarkovProcess, times)

Compute the marginal distributions of the process at the given time points.

This function computes the marginal distributions of the process at the given times. It does so by discretizing the process with the given step sizes (using ProbNumDiffEq.discretize), and then computing the marginal distributions of the resulting Gaussian distributions.

See also: sample.

sample(process::AbstractGaussMarkovProcess, times, N=1)

Samples from the Gauss-Markov process on the given time grid.

See also: marginalize.