# ornstein_uhlenbeck¶

probnum.problems.zoo.filtsmooth.ornstein_uhlenbeck(rng, measurement_variance=0.1, driftspeed=0.21, process_diffusion=0.5, time_grid=None, initrv=None, forward_implementation='classic', backward_implementation='classic')[source]

Filtering/smoothing setup based on an Ornstein Uhlenbeck process.

A linear, time-invariant state space model for the dynamics of a time-invariant Ornstein-Uhlenbeck process. See e.g. Example 10.19 in Särkkä et. al, 2019. 1 Here, we formulate a continuous-discrete state space model:

$\begin{split}d x(t) &= \lambda x(t) d t + L d w(t) \\ y_n &= x(t_n) + r_n\end{split}$

for a drift constant $$\lambda$$ and a driving Wiener process $$w(t)$$. $$r_n \sim \mathcal{N}(0, R)$$ is Gaussian distributed measurement noise with covariance matrix $$R$$. Note that the linear, time-invariant dynamics have an equivalent discretization.

Parameters
Returns

• regression_problemTimeSeriesRegressionProblem object with time points and noisy observations.

• info – Dictionary containing additional information like the prior process.

References

1

Särkkä, Simo, and Solin, Arno. Applied Stochastic Differential Equations. Cambridge University Press, 2019