## Module contents¶

Interface for a Gaussian process as used by quadrature models.

Implementations of this class can be used by Gaussian process based quadrature models such as WarpedBayesianQuadratureModel.

Note

When this class is initialized with data, use the raw evaluations Y of the integrand as this is a general Gaussian process class that can be used with various quadrature methods. Possible transformations of Y such as the ones applied in WarpedBayesianQuadratureModel will be performed automatically there.

Parameters

kern (QuadratureKernel) – An instance of a quadrature kernel.

property observation_noise_variance: float

The variance of the Gaussian observation noise.

Return type

float

predict_with_full_covariance(X_pred)

Predictive mean and full co-variance at the locations X_pred.

Parameters

X_pred (ndarray) – Points at which to predict, shape (n_points, input_dim)

Return type

Tuple[ndarray, ndarray]

Returns

Predictive mean, predictive full co-variance, shapes (num_points, 1) and (num_points, num_points).

solve_linear(z)

Solve the linear system $$Gx=z$$ for $$x$$.

$$G$$ is the Gram matrix $$G := K(X, X) + \sigma^2 I$$, of shape (num_dat, num_dat) and $$z$$ is a matrix of shape (num_dat, num_obs).

Parameters

z (ndarray) – A matrix of shape (num_dat, num_obs).

Return type

ndarray

Returns

The solution $$x$$ of the linear, shape (num_dat, num_obs).

graminv_residual()

The solution $$z$$ of the linear system

$(K_{XX} + \zeta^2 I) z = (Y - m(X))$

where $$X$$ and $$Y$$ are the available nodes and function evaluation, $$m(X)$$ is the predictive mean at $$X$$, and $$\zeta^2$$ the observation noise variance.

Return type

ndarray

Returns

The solution $$z$$ of the linear system, shape (num_dat, 1).

Evaluates the predictive gradients of mean and variance at X.

Parameters

X (ndarray) – Points at which the gradients are evaluated, shape (n_points, input_dim).

Return type

Tuple

Returns

The gradients of mean and variance, each of shape (n_points, input_dim).

Bases: object

Interface for a standard kernel k(x, x’) that in principle can be integrated.

K(x1, x2)

The kernel k(x1, x2) evaluated at x1 and x2.

Parameters
• x1 (ndarray) – First argument of the kernel, shape (n_points N, input_dim)

• x2 (ndarray) – Second argument of the kernel, shape (n_points M, input_dim)

Return type

ndarray

Returns

Kernel evaluated at x1, x2, shape (N, M).

dK_dx1(x1, x2)

Gradient of the kernel wrt x1 evaluated at pair x1, x2.

Parameters
• x1 (ndarray) – First argument of the kernel, shape (n_points N, input_dim)

• x2 (ndarray) – Second argument of the kernel, shape (n_points M, input_dim)

Return type

ndarray

Returns

The gradient of the kernel wrt x1 evaluated at (x1, x2), shape (input_dim, N, M)

dKdiag_dx(x)

The gradient of the diagonal of the kernel (the variance) v(x):=k(x, x) evaluated at x.

Parameters

x (ndarray) – The locations where the gradient is evaluated, shape (n_points, input_dim).

Return type

ndarray

Returns

The gradient of the diagonal of the kernel evaluated at x, shape (input_dim, n_points).

Interface for a Brownian motion kernel.

$k(x, x') = \sigma^2 \operatorname{min}(x, x')\quad\text{with}\quad x, x' \geq 0,$

where $$\sigma^2$$ is the variance property.

Note

Inherit from this class to wrap your standard Brownian motion kernel. The wrapped kernel can then be handed to a quadrature Brownian motion kernel that augments it with integrability.

property variance: float

The scale $$\sigma^2$$ of the kernel.

Return type

float

dK_dx1(x1, x2)

Gradient of the kernel wrt x1 evaluated at pair x1, x2.

Parameters
• x1 (ndarray) – First argument of the kernel, shape (n_points N, input_dim)

• x2 (ndarray) – Second argument of the kernel, shape (n_points M, input_dim)

Return type

ndarray

Returns

The gradient of the kernel wrt x1 evaluated at (x1, x2), shape (input_dim, N, M)

dKdiag_dx(x)

The gradient of the diagonal of the kernel (the variance) v(x):=k(x, x) evaluated at x.

Parameters

x (ndarray) – The locations where the gradient is evaluated, shape (n_points, input_dim).

Return type

ndarray

Returns

The gradient of the diagonal of the kernel evaluated at x, shape (input_dim, n_points).

Interface for an RBF kernel.

$k(x, x') = \sigma^2 e^{-\frac{1}{2}\sum_{i=1}^{d}r_i^2},$

where $$d$$ is the input dimensionality, $$r_i = \frac{x_i-x_i'}{\lambda_i}$$ is the scaled vector difference of dimension $$i$$, $$\lambda_i$$ is the $$i$$ th element of the lengthscales property and $$\sigma^2$$ is the variance property.

Note

Inherit from this class to wrap your standard RBF kernel. The wrapped kernel can then be handed to a quadrature RBF kernel that augments it with integrability.

property lengthscales: ndarray

The lengthscales $$\lambda$$ of the kernel.

Return type

ndarray

property variance: float

The scale $$\sigma^2$$ of the kernel.

Return type

float

dK_dx1(x1, x2)

Gradient of the kernel wrt x1 evaluated at pair x1, x2.

Parameters
• x1 (ndarray) – First argument of the kernel, shape (n_points N, input_dim)

• x2 (ndarray) – Second argument of the kernel, shape (n_points M, input_dim)

Return type

ndarray

Returns

The gradient of the kernel wrt x1 evaluated at (x1, x2), shape (input_dim, N, M)

dKdiag_dx(x)

The gradient of the diagonal of the kernel (the variance) v(x):=k(x, x) evaluated at x.

Parameters

x (ndarray) – The locations where the gradient is evaluated, shape (n_points, input_dim).

Return type

ndarray

Returns

The gradient of the diagonal of the kernel evaluated at x, shape (input_dim, n_points).

Interface for a Brownian product kernel.

The product kernel is of the form $$k(x, x') = \sigma^2 \prod_{i=1}^d k_i(x, x')$$ where

$k_i(x, x') = \operatorname{min}(x_i-c, x_i'-c)\quad\text{with}\quad x_i, x_i' \geq c,$

$$d$$ is the input dimensionality, $$\sigma^2$$ is the variance property and $$c$$ is the offset property.

Make sure to encode only a single variance parameter, and not one for each individual $$k_i$$.

Note

Inherit from this class to wrap your standard product Brownian kernel. The wrapped kernel can then be handed to a quadrature product Brownian kernel that augments it with integrability.

property variance: float

The scale $$\sigma^2$$ of the kernel.

Return type

float

property offset: float

The offset $$c$$ of the kernel.

Return type

float

Interface for a Matern12 (a.k.a. Exponential) product kernel.

The product kernel is of the form $$k(x, x') = \sigma^2 \prod_{i=1}^d k_i(x, x')$$ where

$k_i(x, x') = e^{-r_i}.$

$$d$$ is the input dimensionality, $$r_i:=\frac{|x_i - x'_i|}{\lambda_i}$$, $$\sigma^2$$ is the variance property and $$\lambda_i$$ is the $$i$$ th element of the lengthscales property.

Make sure to encode only a single variance parameter, and not one for each individual $$k_i$$.

Note

Inherit from this class to wrap your standard product Matern12 kernel. The wrapped kernel can then be handed to a quadrature product Matern12 kernel that augments it with integrability.

property nu: float

The smoothness parameter of the kernel.

Return type

float

property lengthscales: ndarray

The lengthscales $$\lambda$$ of the kernel.

Return type

ndarray

property variance: float

The scale $$\sigma^2$$ of the kernel.

Return type

float

dKdiag_dx(x)

The gradient of the diagonal of the kernel (the variance) v(x):=k(x, x) evaluated at x.

Parameters

x (ndarray) – The locations where the gradient is evaluated, shape (n_points, input_dim).

Return type

ndarray

Returns

The gradient of the diagonal of the kernel evaluated at x, shape (input_dim, n_points).

Interface for a Matern32 product kernel.

The product kernel is of the form $$k(x, x') = \sigma^2 \prod_{i=1}^d k_i(x, x')$$ where

$k_i(x, x') = (1 + \sqrt{3}r_i ) e^{-\sqrt{3} r_i}.$

$$d$$ is the input dimensionality, $$r_i:=\frac{|x_i - x'_i|}{\lambda_i}$$, $$\sigma^2$$ is the variance property and $$\lambda_i$$ is the $$i$$ th element of the lengthscales property.

Make sure to encode only a single variance parameter, and not one for each individual $$k_i$$.

Note

Inherit from this class to wrap your standard product Matern32 kernel. The wrapped kernel can then be handed to a quadrature product Matern32 kernel that augments it with integrability.

property nu: float

The smoothness parameter of the kernel.

Return type

float

property lengthscales: ndarray

The lengthscales $$\lambda$$ of the kernel.

Return type

ndarray

property variance: float

The scale $$\sigma^2$$ of the kernel.

Return type

float

dKdiag_dx(x)

The gradient of the diagonal of the kernel (the variance) v(x):=k(x, x) evaluated at x.

Parameters

x (ndarray) – The locations where the gradient is evaluated, shape (n_points, input_dim).

Return type

ndarray

Returns

The gradient of the diagonal of the kernel evaluated at x, shape (input_dim, n_points).

Interface for a Matern52 product kernel.

The product kernel is of the form $$k(x, x') = \sigma^2 \prod_{i=1}^d k_i(x, x')$$ where

$k_i(x, x') = (1 + \sqrt{5} r_i + \frac{5}{3} r_i^2) \exp(- \sqrt{5} r_i).$

$$d$$ is the input dimensionality, $$r_i:=\frac{|x_i - x'_i|}{\lambda_i}$$, $$\sigma^2$$ is the variance property and $$\lambda_i$$ is the $$i$$ th element of the lengthscales property.

Make sure to encode only a single variance parameter, and not one for each individual $$k_i$$.

Note

Inherit from this class to wrap your standard product Matern52 kernel. The wrapped kernel can then be handed to a quadrature product Matern52 kernel that augments it with integrability.

property nu: float

The smoothness parameter of the kernel.

Return type

float

property lengthscales: ndarray

The lengthscales $$\lambda$$ of the kernel.

Return type

ndarray

property variance: float

The scale $$\sigma^2$$ of the kernel.

Return type

float

dKdiag_dx(x)

The gradient of the diagonal of the kernel (the variance) v(x):=k(x, x) evaluated at x.

Parameters

x (ndarray) – The locations where the gradient is evaluated, shape (n_points, input_dim).

Return type

ndarray

Returns

The gradient of the diagonal of the kernel evaluated at x, shape (input_dim, n_points).