covariance_functions

Methods for the covariance_functions module in hetgpy

suite of covariance functions in hetGPy

Note that most of these are not called by the user, but these functions provide the covariance kernels (and their gradients)

cov_Gaussian(X1, X2=None, theta=None)

Euclidean distances, scaled by theta

Notes

For an alternative implementation: https://stackoverflow.com/questions/27948363/numpy-broadcast-to-perform-euclidean-distance-vectorized threeSums = np.sum(np.square(A)[:,np.newaxis,:], axis=2) - 2 * A.dot(B.T) + np.sum(np.square(B), axis=1) np.exp(-threeSums/theta)

cov_gen(X1, X2=None, theta=None, type=None)

Correlation function of selected type, supporting both isotropic and product forms

Parameters:

• X1 (ndarray) – matrix of design locations, one point per row

• X2 (ndarray) – matrix of design locations if correlation is calculated between X1 and X2 (otherwise calculated between X1 and itself)

• theta (np.array or scalar) – vector of lengthscale parameters (either of size one if isotropic or of size d if anisotropic)

• type (str) – one of “Gaussian”, “Matern5_2”, “Matern3_2”

Return type:

matrix of covariances between design locations

Examples

>>> from hetgpy.covariance_functions import cov_gen
>>> import numpy as np
>>> X = np.random.default_rng(42).integers(low=1,high=20,size=(50,2))
>>> K = cov_gen(X1=X,X2=X,theta=np.array([1,2]),type="Gaussian")

Notes

Definition of univariate correlation function and hyperparameters:

• “Gaussian”: \(c(x, y) = \exp(-(x-y)^2/\theta)\)

• “Matern5_2”: \(c(x, y) = (1+\sqrt{5}/\theta * |x-y\ + 5/(3*\theta^2)(x-y)^2) * \exp(-\sqrt{5}*|x-y|/\theta)\)

• “Matern3_2”: \(c(x, y) = (1+\sqrt{3}/\theta * |x-y|) * \exp(-\sqrt{3}*|x-y|/\theta)\)

Multivariate correlations are product of univariate ones.

partial_d_C_Gaussian_dX_i_j(X1, theta, i1, i2)

Derivative with respect to X[i,j]. Useful for pseudo inputs, to be multiplied by the covariance matrix

Parameters:

• i1 (int) – row

• i2 (int) – column

partial_d_C_Gaussian_dtheta_k(X1, theta)

Partial derivative of the covariance matrix with respect to theta[k] (to be multiplied by the covariance matrix)

partial_d_C_Matern3_2_dX_i_j(X1, theta, i1, i2)

Derivative with respect to X[i,j]. Useful for pseudo inputs, to be multiplied by the covariance matrix

Parameters:

• i1 (int) – row

• i2 (int) – column

partial_d_C_Matern3_2_dtheta_k(X1, theta)

Partial derivative of the covariance matrix with respect to theta[k] (to be multiplied by the covariance matrix)

partial_d_C_Matern5_2_dX_i_j(X1, theta, i1, i2)

Derivative with respect to X[i,j]. Useful for pseudo inputs, to be multiplied by the covariance matrix

Parameters:

• i1 (int) – row

• i2 (int) – column

partial_d_C_Matern5_2_dtheta_k(X1, theta)

Partial derivative of the covariance matrix with respect to theta[k] (to be multiplied by the covariance matrix)

partial_d_Cg_Gaussian_d_k_theta_g(X1, theta, k_theta_g)

Partial derivative of the covariance matrix of the noise process with respect to k_theta_g (to be multiplied by the covariance matrix)

partial_d_Cg_Matern3_2_d_k_theta_g(X1, theta, k_theta_g)

Partial derivative of the covariance matrix of the noise process with respect to k_theta_g (to be multiplied by the covariance matrix)

partial_d_Cg_Matern5_2_d_k_theta_g(X1, theta, k_theta_g)

Partial derivative of the covariance matrix of the noise process with respect to k_theta_g (to be multiplied by the covariance matrix)

partial_d_k_Gaussian_dX_i_j(X1, X2, theta, i1, i2)

Derivative with respect to X[i,j]. Useful for pseudo inputs, to be multiplied by the covariance matrix

Parameters:

• i1 (int) – row

• i2 (int) – column

partial_d_k_Gaussian_dtheta_k(X1, X2, theta)

Partial derivative of the covariance vector with respect to theta[k] (to be multiplied by the covariance vector)

partial_d_k_Matern3_2_dX_i_j(X1, X2, theta, i1, i2)

Derivative with respect to X[i,j]. Useful for pseudo inputs, to be multiplied by the covariance matrix

Parameters:

• i1 (int) – row

• i2 (int) – column

• theta (int) – lengthscales

partial_d_k_Matern3_2_dtheta_k(X1, X2, theta)

Partial derivative of the covariance vector with respect to theta[k] (to be multiplied by the covariance vector)

partial_d_k_Matern5_2_dX_i_j(X1, X2, theta, i1, i2)

Derivative with respect to X[i,j]. Useful for pseudo inputs, to be multiplied by the covariance matrix

Parameters: i1: int

row

i2: int

column

theta: ndarray

lengthscales

partial_d_k_Matern5_2_dtheta_k(X1, X2, theta)

Partial derivative of the covariance vector with respect to theta[k] (to be multiplied by the covariance vector)

partial_d_kg_Gaussian_d_k_theta_g(X1, X2, theta, k_theta_g)

Partial derivative of the covariance vector of the noise process with respect to k_theta_g (to be multiplied by the covariance vector)

partial_d_kg_Matern3_2_d_k_theta_g(X1, X2, theta, k_theta_g)

Partial derivative of the covariance vector of the noise process with respect to k_theta_g (to be multiplied by the covariance vector)

partial_d_kg_Matern5_2_d_k_theta_g(X1, X2, theta, k_theta_g)

Partial derivative of the covariance vector of the noise process with respect to k_theta_g (to be multiplied by the covariance vector)