Examples¶

The repository contains runnable scripts in the examples/ directory. The pages below are rendered from a selected subset of those scripts. Each page states the objective, shows at least one plot, and then includes the full script so that the example can be copied or run directly.

Example coverage¶

The selected examples cover the main GPmp operations:

1D interpolation gives the minimal sequence: model construction, covariance-parameter selection, prediction, and plotting.
2D interpolation, Higher-dimensional interpolation, and Higher-dimensional interpolation with Matérn regularity selection show the same operations when the input dimension is larger than one.
ML / REML / REMAP parameter selection compares ML, REML, and REMAP on one data set.
Dataloader-based parameter selection passes observations through Dataset and DataLoader objects during parameter selection.
Posterior parameter sampling uses reduced iteration counts in the documentation build. The code block shows the sampler arguments that control longer runs.

Notation used below¶

The example pages use the same names as the scripts. Observation points are stored in xi and written mathematically as \(x_i\). Prediction points are stored in xt and written as \(x_t\).

The latent process is denoted by \(Z\). A typical model is

\[Z \sim \mathcal{GP}(m, k_\theta),\]

where \(m\) is the mean function and \(k_\theta\) is the covariance kernel. In the fixed-regularity Matérn examples, the covariance parameter vector follows the sigma2_rho layout

\[\theta = \mathrm{covparam} = \left(\log(\sigma^2), -\log(\rho_0), \ldots, -\log(\rho_{d-1})\right).\]

The matern example uses the sigma2_nu_rho layout

\[\theta = \left(\log(\sigma^2), \log(\nu), -\log(\rho_0), \ldots, -\log(\rho_{d-1})\right).\]

The covariance kernel is written with a lowercase symbol \(k_\theta\). Covariance matrices and blocks are written with uppercase symbols. For example, \(K_{ii}\) is the matrix with entries \(k_\theta(x_i^a, x_i^b)\), and \(K_{it}\) contains the covariances between observation points and prediction points.

Random variables use uppercase letters. Thus \(Z_i = Z(x_i)\) and \(Z_t = Z(x_t)\) are random variables. The arrays zi and zt store realizations, denoted by lowercase \(z_i\) and \(z_t\).

Selected examples¶

Topic	Page	What it illustrates
Matérn covariance functions	Plotting Matérn covariances	Compare half-integer and `nu`-parameterized Matérn kernels.
1D interpolation	1D interpolation	Build a noise-free GP interpolator, select covariance parameters, and plot the posterior.
2D interpolation	2D interpolation	Build an anisotropic Matérn GP and inspect reference, prediction, error, and uncertainty fields.
Higher-dimensional interpolation	Higher-dimensional interpolation	Use leave-one-out diagnostics when spatial plotting is no longer practical.
Matérn regularity selection	Higher-dimensional interpolation with Matérn regularity selection	Select the Matérn regularity together with variance and lengthscales.
Custom covariance	Custom covariance	Define and use a covariance callable with the GPmp model interface.
Noisy observations	Noisy observations	Model noisy observations while predicting the latent process.
Conditional sample paths	Conditional sample paths	Generate posterior sample paths in the noise-free setting.
Noisy conditional sample paths	Noisy conditional sample paths	Generate conditional paths when observations have heteroscedastic noise.
ML / REML / REMAP comparison	ML / REML / REMAP parameter selection	Compare parameter-selection criteria on the same one-dimensional setup.
Posterior parameter sampling	Posterior parameter sampling	Start from REMAP selection and explore covariance-parameter uncertainty.
Dataloader-based selection	Dataloader-based parameter selection	Use `Dataset` and `DataLoader` objects for batched parameter selection.

Running examples locally¶

Run an example from the repository root, for instance:

python examples/gpmp_example02_1d_interpolation.py

When writing new examples, prefer the same structure: define data generation, define the model, select covariance parameters, run prediction or diagnostics, and keep plotting in small functions.

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