fit_ou: Inference about the parameters of an Ornstein-Uhlenbeck...

In mixediffusion: Mixed-Effects Diffusion Models with General Drift

Inference about the parameters of an Ornstein–Uhlenbeck process

Description

Implements the EM algorithm to perform inference on the parameters of an Ornstein–Uhlenbeck process with mixed drift effects.

Usage

fit_ou(df,
  mu = "at^1",
  tol = 1e-4,
  max_iter = 100,
  theta = NULL,
  M = 100,
  verbose = TRUE,
  mu_cond = NULL,
  n_mcmc = 1000,
  burnin = 500)

Arguments

df	Data frame with the observed data. It must include the columns t (time), unit (unit identifier) and Y (the observed trajectory).
mu	Functional form of the drift. Supported drifts include "at^p", "exp(at)", "cos(at)" and "sin(at)". The default is "at^1".
tol	Convergence tolerance for the EM algorithm. The algorithm stops when the maximum absolute difference between the parameter estimates at two consecutive EM iterations is smaller than tol.
max_iter	Maximum number of EM iterations.
theta	Optional named vector of initial parameter values. If NULL, default initial values are used.
M	Number of Monte Carlo samples used to approximate the conditional expectations in the E-step.
verbose	Logical indicating whether to print EM iteration progress.
mu_cond	Optional user-supplied function defining the conditional mean of the process. If NULL, it is constructed automatically from mu.
n_mcmc	Number of MCMC iterations used in the E-step to sample the random effects.
burnin	Number of initial MCMC iterations discarded.

Details

The model is a one-dimensional Ornstein–Uhlenbeck diffusion defined by

dY_{kt} = \lambda\{\mu(t, a_k) - Y_{kt}\}\,dt + \sigma\,dW_{kt},

where \mu(t, a_k) is a user-specified drift function depending on a unit-specific random effect a_k \sim \mathcal{N}(\mu_a, \sigma_a^2).

The parameter \lambda controls the strength of mean reversion toward the time-dependent mean.

For discretely observed trajectories, the conditional mean is given by

E\{Y_{kt_i} \mid Y_{kt_{i-1}}, a_k\} = Y_{kt_{i-1}} e^{-\lambda \Delta t_i} + \lambda \int_{t_{i-1}}^{t_i} e^{-\lambda (t_i - s)} \mu(s,a_k)\,ds.

The function mu_cond represents this conditional mean. If not provided, it is constructed automatically from the drift specification supplied through mu using closed-form expressions.

Value

A named numeric vector containing the estimated model parameters:

mu_a	Estimated mean of the random effects distribution.
sigma2_a	Estimated variance of the random effects distribution.
sigma2	Estimated diffusion variance of the process.
lambda	Estimated mean reversion parameter.

Examples

library(mixediffusion)
data(datasim03)
plot_paths(df = datasim03)
fit <- fit_ou(df = datasim03, mu = "at^1",
              verbose = FALSE, max_iter = 2)
fit

mixediffusion documentation built on March 20, 2026, 5:10 p.m.