Kernels#

fvgp.kernels.squared_exponential_kernel(distance, length)[source]#

Function for the squared exponential kernel. kernel = np.exp(-(distance ** 2) / (2.0 * (length ** 2)))

Parameters:

  • distance (scalar or np.ndarray) – Distance between a set of points.

  • length (scalar) – The length scale hyperparameters.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.squared_exponential_kernel_robust(distance, phi)[source]#

Function for the squared exponential kernel (robust version) kernel = np.exp(-(distance ** 2) * (phi ** 2))

Parameters:

  • distance (scalar or np.ndarray) – Distance between a set of points.

  • phi (scalar) – The length scale hyperparameters.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.exponential_kernel(distance, length)[source]#

Function for the exponential kernel. kernel = np.exp(-(distance) / (length))

Parameters:

  • distance (scalar or np.ndarray) – Distance between a set of points.

  • length (scalar) – The length scale hyperparameters.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.exponential_kernel_robust(distance, phi)[source]#

Function for the exponential kernel (robust version) kernel = np.exp(-(distance) * (phi**2))

Parameters:

  • distance (scalar or np.ndarray) – Distance between a set of points.

  • phi (scalar) – The length scale hyperparameters.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.matern_kernel_diff1(distance, length)[source]#

Function for the Matern kernel, order of differentiability = 1. kernel = (1.0 + ((np.sqrt(3.0) * distance) / (length))) * np.exp(

-(np.sqrt(3.0) * distance) / length

Parameters:

  • distance (scalar or np.ndarray) – Distance between a set of points.

  • length (scalar) – The length scale hyperparameters.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.matern_kernel_diff1_grad(distance, dist_der)[source]#

Function for the Matern kernel derivative, order of differentiability = 1. kernel = (1.0 + ((np.sqrt(3.0) * distance) / (length))) * np.exp(

-(np.sqrt(3.0) * distance) / length

Parameters:

  • distance (scalar or np.ndarray) – Distance between a set of points.

  • dist_der (scalar or np.ndarray) – The derivative of the distance matrix. We assume here that the distance is a function of the hyperparameters.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.matern_kernel_diff1_robust(distance, phi)[source]#

Function for the Matern kernel, order of differentiability = 1, robust version. kernel = (1.0 + ((np.sqrt(3.0) * distance) * (phi**2))) * np.exp(

-(np.sqrt(3.0) * distance) * (phi**2))

Parameters:

  • distance (scalar or np.ndarray) – Distance between a set of points.

  • phi (scalar) – The length scale hyperparameters.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.matern_kernel_diff2(distance, length)[source]#

Function for the Matern kernel, order of differentiability = 2. kernel = (

1.0 + ((np.sqrt(5.0) * distance) / (length)) + ((5.0 * distance ** 2) / (3.0 * length ** 2))

) * np.exp(-(np.sqrt(5.0) * distance) / length)

Parameters:

  • distance (scalar or np.ndarray) – Distance between a set of points.

  • length (scalar) – The length scale hyperparameters.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.matern_kernel_diff2_robust(distance, phi)[source]#

Function for the Matern kernel, order of differentiability = 2, robust version. kernel = (

1.0 + ((np.sqrt(5.0) * distance) * (phi**2)) + ((5.0 * distance ** 2) * (3.0 * phi ** 4))

) * np.exp(-(np.sqrt(5.0) * distance) * (phi**2))

Parameters:

  • distance (scalar or np.ndarray) – Distance between a set of points.

  • phi (scalar) – The length scale hyperparameters.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.sparse_kernel(distance, radius)[source]#

Function for a compactly supported kernel.

Parameters:

  • distance (scalar or np.ndarray) – Distance between a set of points.

  • radius (scalar) – Radius of support.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.periodic_kernel(distance, length, p)[source]#

Function for a periodic kernel. kernel = np.exp(-(2.0/length**2)*(np.sin(np.pi*distance/p)**2))

Parameters:

  • distance (scalar or np.ndarray) – Distance between a set of points.

  • length (scalar) – Length scale.

  • p (scalar) – Period.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.linear_kernel(x1, x2, hp1, hp2, hp3)[source]#

Function for a linear kernel. kernel = hp1 + (hp2*(x1-hp3)*(x2-hp3))

Parameters:

  • x1 (float) – Point 1.

  • x2 (float) – Point 2.

  • hp1 (float) – Hyperparameter.

  • hp2 (float) – Hyperparameter.

  • hp3 (float) – Hyperparameter.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.dot_product_kernel(x1, x2, hp, matrix)[source]#

Function for a dot-product kernel. kernel = hp + x1.T @ matrix @ x2

Parameters:

  • x1 (np.ndarray) – Point 1.

  • x2 (np.ndarray) – Point 2.

  • hp (float) – Offset hyperparameter.

  • matrix (np.ndarray) – PSD matrix defining the inner product.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.polynomial_kernel(x1, x2, p)[source]#

Function for a polynomial kernel. kernel = (1.0+x1.T @ x2)**p

Parameters:

  • x1 (np.ndarray) – Point 1.

  • x2 (np.ndarray) – Point 2.

  • p (float) – Power hyperparameter.

Returns:

Kernel output

Return type:

same as distance parameter.

fvgp.kernels.wendland_kernel(d)[source]#

Function for the Wendland kernel with a given distance matrix. The Wendland kernel is compactly supported, leading to sparse covariance matrices.

Parameters:

d (np.ndarray) – Distance matrix.

Returns:

Covariance matrix

Return type:

np.ndarray

fvgp.kernels.wendland_anisotropic(x1, x2, hyperparameters)[source]#

Function for the Wendland kernel. The Wendland kernel is compactly supported, leading to sparse covariance matrices.

Parameters:

  • x1 (np.ndarray) – Numpy array of shape (U x D).

  • x2 (np.ndarray) – Numpy array of shape (V x D).

  • hyperparameters (np.ndarray) – Array of hyperparameters. For this kernel we need D + 1 hyperparameters.

Returns:

Covariance matrix

Return type:

np.ndarray

fvgp.kernels.non_stat_kernel(x1, x2, x0, w, l)[source]#

Non-stationary kernel. kernel = g(x1) g(x2)

Parameters:

  • x1 (np.ndarray) – Numpy array of shape (U x D).

  • x2 (np.ndarray) – Numpy array of shape (V x D).

  • x0 (np.ndarray) – Numpy array of the basis function locations.

  • w (np.ndarray) – 1d np.ndarray of weights. len(w) = len(x0).

  • l (float) – Width measure of the basis functions.

Returns:

Covariance matrix

Return type:

np.ndarray

fvgp.kernels.non_stat_kernel_gradient(x1, x2, x0, w, l)[source]#

Non-stationary kernel gradient. kernel = g(x1) g(x2)

Parameters:

  • x1 (np.ndarray) – Numpy array of shape (U x D).

  • x2 (np.ndarray) – Numpy array of shape (V x D).

  • x0 (np.ndarray) – Numpy array of the basis function locations.

  • w (np.ndarray) – 1d np.ndarray of weights. len(w) = len(x0).

  • l (float) – Width measure of the basis functions.

Returns:

Covariance matrix

Return type:

np.ndarray

fvgp.kernels.get_distance_matrix(x1, x2)[source]#

Function to calculate the pairwise distance matrix of points in x1 and x2.

Parameters:

  • x1 (np.ndarray) – Numpy array of shape (U x D).

  • x2 (np.ndarray) – Numpy array of shape (V x D).

Returns:

distance matrix

Return type:

np.ndarray

fvgp.kernels.get_anisotropic_distance_matrix(x1, x2, hps)[source]#

Function to calculate the pairwise axial-anisotropic distance matrix of points in x1 and x2.

Parameters:

  • x1 (np.ndarray) – Numpy array of shape (U x D).

  • x2 (np.ndarray) – Numpy array of shape (V x D).

  • hps (np.ndarray) – 1d array of values. The diagonal of the metric tensor describing the axial anisotropy.

Returns:

distance matrix

Return type:

np.ndarray

fvgp.kernels.wendland_anisotropic_gp2Scale_cpu(x1, x2, hps)[source]#

Function for the anisotropic Wendland kernel computed on the CPU. The Wendland kernel is compactly supported, leading to sparse covariance matrices.

Parameters:

  • x1 (np.ndarray) – Numpy array of shape (U x D).

  • x2 (np.ndarray) – Numpy array of shape (V x D).

  • hps (np.ndarray) – Array of hyperparameters. For this kernel we need D + 1 hyperparameters.

Returns:

Covariance matrix

Return type:

np.ndarray

fvgp.kernels.wendland_anisotropic_gp2Scale_gpu(x1, x2, hps)[source]#

Function for the anisotropic Wendland kernel computed on the GPU. Needs pytorch. The Wendland kernel is compactly supported, leading to sparse covariance matrices.

Parameters:

  • x1 (np.ndarray) – Numpy array of shape (U x D).

  • x2 (np.ndarray) – Numpy array of shape (V x D).

  • hps (np.ndarray) – Array of hyperparameters. For this kernel we need D + 1 hyperparameters.

Returns:

Covariance matrix

Return type:

np.ndarray

fvgp.kernels.wasserstein_1d(a, b)[source]#

The 1d Wasserstein distance.

Parameters:

  • a (np.ndarray) – 1d Numpy array. Input distribution.

  • b (np.ndarray) – 1d Numpy array. Input distribution.

Returns:

Wasserstein distance

Return type:

float