ou: Ornstein-Uhlenbeck Process Model

In ngme2: Linear Latent Non-Gaussian Models with Flexible Distributions

Ornstein-Uhlenbeck Process Model

Description

Implements the exact discrete-time representation of the continuous Ornstein-Uhlenbeck (OU) process using a matrix formulation \mathbf{K}\mathbf{X} = \boldsymbol{\delta}.

Usage

ou(mesh = NULL, theta = 1)

Arguments

mesh	numerical vector or 'inla.mesh.1d' object giving the ordered time locations. Must be strictly increasing.
theta	positive mean-reversion rate (stiffness parameter). Internally stored on the log scale so optimization remains unconstrained.

Details

The OU process is defined by the stochastic differential equation (SDE):

dX_t = -\theta X_t dt + \sigma dW_t

where \theta > 0 is the mean-reversion rate (stiffness) and \sigma is the volatility (diffusion coefficient).

## Exact Discretization

Unlike Euler-Maruyama approximations, the OU process has an exact solution between time points t_i and t_{i+1} with step size \Delta t_i:

X_{i+1} = X_i e^{-\theta \Delta t_i} + \epsilon_i

This is an AR(1) form with autoregressive coefficient:

\rho_i = e^{-\theta \Delta t_i}

and Gaussian noise:

\epsilon_i \sim \mathcal{N}\left(0, \frac{\sigma^2}{2\theta}(1 - \rho_i^2)\right)

## Matrix Representation

The precision matrix \mathbf{K} for the linear system \mathbf{K}\mathbf{X} = \boldsymbol{\delta} is constructed as:

\mathbf{K} = \begin{bmatrix} \sqrt{1-\rho_1^2} & 0 & 0 & \cdots & 0 \\ -\rho_1 & 1 & 0 & \cdots & 0 \\ 0 & -\rho_2 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & -\rho_{n-1} & 1 \end{bmatrix}

where \rho_i = \exp(-\theta \Delta t_i) with \Delta t_i = t_{i+1} - t_i.

**Why the \sqrt{1-\rho_1^2} term?** The first row ensures that X_1 is drawn from the stationary distribution \mathcal{N}(0, \sigma^2/(2\theta)). This scaling factor matches the variance of the first component of \boldsymbol{\delta} with the transition noise variance in subsequent steps.

## Non-uniform Mesh

For non-uniform meshes, each \rho_i is computed locally based on the varying time step \Delta t_i, allowing the model to naturally handle irregularly spaced observations.

## The h vector

The h vector represents integration weights for the mass matrix. For the OU model, it is set as:

h = [\Delta t_1, \Delta t_2, \ldots, \Delta t_{n-1}, \Delta t_{n-1}]

where the last element is duplicated. This ensures proper weighting in the finite element formulation.

## The \boldsymbol{\delta} vector

In the equation \mathbf{K}\mathbf{X} = \boldsymbol{\delta}, the vector \boldsymbol{\delta} contains independent identically distributed (i.i.d.) random variables with unit variance. The precision matrix \mathbf{K} is constructed such that the resulting process \mathbf{X} has the correct OU covariance structure.

Value

An 'ngme_operator' object containing:

• 'K': the precision matrix

• 'h': integration weights

• 'theta_K': log-transformed theta for unconstrained optimization

• 'mesh': the mesh object

References

Uhlenbeck, G. E., & Ornstein, L. S. (1930). On the theory of the Brownian motion. Physical review, 36(5), 823.

Examples

# Uniform mesh
mesh_uniform <- seq(0, 10, by = 0.5)
ou_uniform <- ou(mesh = mesh_uniform, theta = 0.5)
print(ou_uniform)

# Non-uniform mesh
mesh_nonuniform <- c(0, 1, 2.5, 3.5, 5, 8, 11)
ou_nonunif <- ou(mesh = mesh_nonuniform, theta = 0.8)
print(ou_nonunif)

ngme2 documentation built on May 20, 2026, 9:10 a.m.