function ornstein_uhlenbeck

deeptime.data.ornstein_uhlenbeck(h=0.001, n_steps=500)

The one-dimensional Ornstein-Uhlenbeck process. It is given by the stochastic differential equation

\[dX_t = -\alpha X_t dt + \sqrt{2\beta^{-1}}dW_t\]

with parameters \(\alpha=1\) and \(\beta=4\).

(Source code, png, hires.png, pdf)

Parameters:

• h (float, default = 1e-3) – Integration step size. The implementation uses an Euler-Maruyama integrator.

• n_steps (int, default = 500) – Number of integration steps between each evaluation. That means the default lag time is h*n_steps=10.

Returns:

system – The system.

Return type:

TimeIndependentSystem

Examples

The model possesses the capability to simulate trajectories as well as be evaluated at test points:

>>> import numpy as np
>>> import deeptime as dt

First, set up the model (which internally already creates the integrator).

>>> model = dt.data.ornstein_uhlenbeck(h=1e-3, n_steps=100)  # create model instance

Now, a trajectory can be generated:

>>> traj = model.trajectory(np.array([[-1.]]), 1000, seed=42, n_jobs=1)  # simulate trajectory
>>> assert traj.shape == (1000, 1)  # 1000 evaluations from initial condition [0, 0]

Or, alternatively the model can be evaluated at test points (mapping forward using the dynamical system):

>>> test_points = np.random.uniform(.5, 1.5, (100, 1))  # 100 test points
>>> evaluations = model(test_points, seed=53, n_jobs=1)
>>> assert evaluations.shape == (100, 1)