glearn.Covariance.cross_covariance

Covariance.cross_covariance(test_points)

Compute the cross-covariance between training and test points.

Parameters:

training_pointsnumpy.ndarray

An array of the size \(n^{\ast} \times d\) representing the coordinates of \(n^{\ast}\) test points. Each row of the array is the coordinates of a point \(\boldsymbol{x} = (x_1, \dots, x_d)\).

Returns:

S_star_starnumpy.ndarray

The covariance array \(\boldsymbol{\Sigma}^{\ast}\) of the size \(n \times n^{\ast}\) where \(n\) and \(n^{\ast}\) are respectively the number of training and test points.

See also

glearn.Covariance.get_matrix

glearn.Covariance.cross_covariance

Notes

Cross-Covariance:

Suppose the training points \(\{ \boldsymbol{x}_i \}_{i=1}^{n}\) that were given at the time of creation of the covariance object. Given a set of test points \(\{ \boldsymbol{x}^{\ast}_i \}_{i=1}^{n^{\ast}}\), this function generates the \(n \times n^{\ast}\) cross-covariance \(\boldsymbol{\Sigma}^{\ast \ast}\) where each element \(\Sigma^{\ast}_{ij}\) of the matrix is the cross covariance between the points in the \(i\)-th training point and \(j\)-th test point, namely,

\[\Sigma^{\ast}_{ij} = \mathrm{cov}( \boldsymbol{x}_i, \boldsymbol{x}^{\ast}_j).\]

Specifying Hyperparameters:

The cross-covariance matrix \(\boldsymbol{\Sigma}^{\ast}\) depends on a set of hyperparameters as it is defined by

\[\boldsymbol{\Sigma}^{\ast}(\boldsymbol{\alpha}, \sigma, \varsigma) = \sigma^2 \mathbf{K}^{\ast}(\boldsymbol{\alpha}) + \varsigma^2 \mathbf{I}.\]

In the above, \(\mathbf{I}\) is the identity matrix and \(\mathbf{K}^{\ast}\) is the cross-correlation matrix that depends on a set of scale hyperparameters \(\boldsymbol{\alpha}=(\alpha_1, \dots, \alpha_d)\).

Note

Before using glearn.Covariance.cross_covariance(), the hyperparameters \(\sigma\), \(\varsigma\), and \(\boldsymbol{\alpha}\) of the covariance object should be defined. These hyperparameters can be either defined at the time of instantiation of glearn.Covariance, or to be set by

• glearn.Covariance.set_sigmas() to set \(\sigma\) and \(\varsigma\).

• glearn.Covariance.set_scale() to set \(\boldsymbol{\alpha}\).

Summary of Covariance Functions:

Suppose \(\{ \boldsymbol{x}_i \}_{i=1}^{n}\) and \(\{ \boldsymbol{x}^{\ast}_i \}_{i=1}^{n^{\ast}}\) are respectively training and test points. Three covariance matrices can be generated:

• glearn.Covariance.get_matrix() returns the auto-covariance between training points by the \(n \times n\) matrix \(\boldsymbol{\Sigma}\) with the components

  \[\Sigma_{ij} = \mathrm{cov}(\boldsymbol{x}_i, \boldsymbol{x}_j).\]

• glearn.Covariance.cross_covariance() returns the cross-covariance between the training points and test points by the \(n \times n^{\ast}\) matrix \(\boldsymbol{\Sigma}^{\ast}\) with the components

  \[\Sigma_{ij}^{\ast} = \mathrm{cov}(\boldsymbol{x}_i, \boldsymbol{x}^{\ast}_j).\]

• glearn.Covariance.auto_covariance() returns the cross-covariance between the test points by the \(n^{\ast} \times n^{\ast}\) matrix \(\boldsymbol{\Sigma}^{\ast \ast}\) with the components

  \[\Sigma_{ij}^{\ast \ast} = \mathrm{cov}( \boldsymbol{x}^{\ast}_i, \boldsymbol{x}^{\ast}_j).\]

Examples

Create a covariance matrix based on a set of sample data with four points in \(d=2\) dimensional space.

>>> # Generate a set of points
>>> from glearn.sample_data import generate_points
>>> x = generate_points(num_points=4, dimension=2)

>>> # Create a covariance object
>>> from glearn import Covariance
>>> cov = Covariance(x, sigma=2.0, sigma0=3.0, scale=[1.0, 2.0])

Now, create a set of test points \(\boldsymbol{x}^{\ast}\), and compute the auto-covariance between the test points.

>>> # Generate a random set of points
>>> x_star = generate_points(num_points=2, dimension=2, seed=42)

>>> # Auto-covariance between test points
>>> cov.cross_covariance(x_star)
array([[3.24126331, 3.30048921],
       [2.94735574, 3.50537082],
       [3.40813768, 2.93601147],
       [3.7310863 , 2.87895123]])