Example 20: relaxed Gaussian process

Script: examples/example20_regp.py

Purpose

The script demonstrates relaxed Gaussian-process (reGP) remodeling for a threshold-oriented target. Selected observations are allowed to move inside a relaxation interval, and the relaxed values are optimized together with covariance parameters before prediction. See Petit et al. [6] for the method.

What is computed

• an initial GP model and posterior predictions.

• membership of observations in the relaxation interval.

• relaxed observation values and updated covariance parameters.

• posterior predictions from the relaxed model.

• a selected relaxation threshold based on a tCRPS criterion.

• a plot comparing observations, relaxed observations, and predictions.

Main objects

• gpmpcontrib.regp.regp.predict

• gpmpcontrib.regp.regp.remodel

• gpmpcontrib.regp.regp.select_optimal_threshold_above_t0

Outputs

Run python examples/example20_regp.py from the repository root to execute the example. Regenerate the static figure with cd docs && python make_example_results.py.

Black dashed curve: reference function. Red curve: initial REML posterior mean. Blue and green curves: reGP posterior means obtained after relaxing observations above the corresponding horizontal thresholds. Green points: relaxed observation values used by the selected-threshold reGP model.

Source excerpt

    Visualize the results using gp.plot (a matplotlib wrapper).

    Parameters
    ----------
    xt : numpy.ndarray
        Target x values
    zt : numpy.ndarray
        Target z values
    xi : numpy.ndarray
        Input x values
    zi : numpy.ndarray
        Input z values
    zpm : numpy.ndarray
        Posterior mean
    zpv : numpy.ndarray
        Posterior variance
    """
    if fig is None:
        fig = gp.plot.Figure(isinteractive=True)
    fig.plot(xt, zt, "k", linewidth=1, linestyle=(0, (5, 5)))
    fig.plotdata(xi, zi)
    fig.plotgp(xt, zpm, zpv, colorscheme="hue", rgb_hue=rgb_hue)
    fig.xlabel("$x$")
    fig.ylabel("$z$")