Covariance functions

gpyreg.covariance_functions

Each covariance function is implemented as a subclass of AbstractKernel, which defines their basic interface. An instance of one of these classes is passed to gpyreg.GP at initialization and defines the type of covariance function for that Gaussian process.

AbstractKernel

class gpyreg.covariance_functions.AbstractKernel

Abstract base class for covariance kernels.

abstractmethod compute(hyp: ndarray, X: ndarray, X_star: ndarray = None, compute_diag: bool = False, compute_grad: bool = False)

Compute the covariance matrix for given training points and test points.

Parameters:

hypndarray, shape (cov_N,)

A 1D array of hyperparameters, where cov_N is the number of hyperparameters.

Xndarray, shape (N, D)

A 2D array where each row is a training point.

X_starndarray, shape (M, D), optional

A 2D array where each row is a test point. If this is not given, the self-covariance matrix is being computed.

compute_diagbool, defaults to False

Whether to only compute the diagonal of the self-covariance matrix.

compute_gradbool, defaults to False

Whether to compute the gradient with respect to the hyperparameters.

Returns:

Kndarray

The covariance matrix which is by default of shape (N, N). If compute_diag = True the shape is (N, 1).

dKndarray, shape (N, N, cov_N), optional

The gradient of the covariance matrix with respect to the hyperparameters.

Raises:

ValueError

Raised when hyp has not the expected number of hyperparameters.

ValueError

Raised when hyp is not an 1D array but of higher dimension.

ValueError

Raised when compute_diag and compute_grad are both True: the gradient is available for the full covariance matrix only.

ValueError

Raised when the kernel has periods and their number differs from the number of columns of X.

get_bounds_info(X: ndarray, y: ndarray)

Return information on the lower, upper, plausible lower and plausible upper bounds of the hyperparameters of this covariance function.

Parameters:

Xndarray, shape (N, D)

A 2D array where each row is a test point.

yndarray, shape (N, 1)

A 2D array where each row is a test target.

Returns:

cov_bound_info: dict

A dictionary containing the bound info with the following elements:

LBnp.ndarray, shape (cov_N, 1)

The lower bounds of the hyperparameters.

UBnp.ndarray, shape (cov_N, 1)

The upper bounds of the hyperparameters.

PLBnp.ndarray, shape (cov_N, 1)

The plausible lower bounds of the hyperparameters.

PUBnp.ndarray, shape (cov_N, 1)

The plausible upper bounds of the hyperparameters.

x0np.ndarray, shape (cov_N, 1)

The plausible starting point.

where cov_N is the number of hyperparameters.

hyperparameter_count(D: int)

Return the number of hyperparameters this covariance function has.

Parameters:

Dint

The dimensionality of the kernel.

Returns:

countint

The number of hyperparameters.

hyperparameter_info(D: int)

Return information on the names of hyperparameters for setting them in other parts of the program.

Parameters:

Dint

The dimensionality of the kernel.

Returns:

hyper_infoarray_like

A list of tuples of hyperparameter names and their number, in the order they are in the hyperparameter array.

periods = None

The period of each input dimension, np.inf for a dimension that is not periodic, or None for a kernel that is not periodic along any dimension.

RationalQuadraticARD

class gpyreg.covariance_functions.RationalQuadraticARD(periods: ndarray = None)

Bases: AbstractKernel

Rational Quadratic ARD kernel.

Parameters:

periodsarray_like, shape (D,), optional

The period of each input dimension along which the kernel is periodic, and np.inf for each dimension along which it is not. Along a dimension of period p, the squared difference delta**2 of two coordinates is replaced by the squared chord (p / pi)**2 * sin(pi * delta / p)**2, the squared distance between the two points mapped onto a circle of circumference p. It is zero at every multiple of the period and equals delta**2 to second order in delta, so that the length scale keeps the units of the input. The periods are fixed constants, not hyperparameters. None, the default, or periods that are all infinite give the non-periodic kernel. A period that is NaN, zero or negative, or periods that are not a one-dimensional array, raise ValueError.

compute(hyp: ndarray, X: ndarray, X_star: ndarray = None, compute_diag: bool = False, compute_grad: bool = False)

Compute the covariance matrix for given training points and test points.

Parameters:

hypndarray, shape (cov_N,)

A 1D array of hyperparameters, where cov_N is the number of hyperparameters.

Xndarray, shape (N, D)

A 2D array where each row is a training point.

X_starndarray, shape (M, D), optional

A 2D array where each row is a test point. If this is not given, the self-covariance matrix is being computed.

compute_diagbool, defaults to False

Whether to only compute the diagonal of the self-covariance matrix.

compute_gradbool, defaults to False

Whether to compute the gradient with respect to the hyperparameters.

Returns:

Kndarray

The covariance matrix which is by default of shape (N, N). If compute_diag = True the shape is (N, 1).

dKndarray, shape (N, N, cov_N), optional

The gradient of the covariance matrix with respect to the hyperparameters.

Raises:

ValueError

Raised when hyp has not the expected number of hyperparameters.

ValueError

Raised when hyp is not an 1D array but of higher dimension.

ValueError

Raised when compute_diag and compute_grad are both True: the gradient is available for the full covariance matrix only.

ValueError

Raised when the kernel has periods and their number differs from the number of columns of X.

get_bounds_info(X: ndarray, y: ndarray)

Return information on the lower, upper, plausible lower and plausible upper bounds of the hyperparameters of this covariance function.

Parameters:

Xndarray, shape (N, D)

A 2D array where each row is a test point.

yndarray, shape (N, 1)

A 2D array where each row is a test target.

Returns:

cov_bound_info: dict

A dictionary containing the bound info with the following elements:

LBnp.ndarray, shape (cov_N, 1)

The lower bounds of the hyperparameters.

UBnp.ndarray, shape (cov_N, 1)

The upper bounds of the hyperparameters.

PLBnp.ndarray, shape (cov_N, 1)

The plausible lower bounds of the hyperparameters.

PUBnp.ndarray, shape (cov_N, 1)

The plausible upper bounds of the hyperparameters.

x0np.ndarray, shape (cov_N, 1)

The plausible starting point.

where cov_N is the number of hyperparameters.

hyperparameter_count(D: int)

Return the number of hyperparameters this covariance function has.

Parameters:

Dint

The dimensionality of the kernel.

Returns:

countint

The number of hyperparameters.

hyperparameter_info(D: int)

Return information on the names of hyperparameters for setting them in other parts of the program.

Parameters:

Dint

The dimensionality of the kernel.

Returns:

hyper_infoarray_like

A list of tuples of hyperparameter names and their number, in the order they are in the hyperparameter array.

Matern

class gpyreg.covariance_functions.Matern(degree: int, periods: ndarray = None)

Bases: AbstractKernel

Matern kernel.

Parameters:

degree{1, 3, 5}

The degree of the Matern kernel.

Currently the only supported degrees are 1, 3, 5, and if some other degree is provided a ValueError exception is raised.

periodsarray_like, shape (D,), optional

The period of each input dimension along which the kernel is periodic, and np.inf for each dimension along which it is not. Along a dimension of period p, the squared difference delta**2 of two coordinates is replaced by the squared chord (p / pi)**2 * sin(pi * delta / p)**2, the squared distance between the two points mapped onto a circle of circumference p. It is zero at every multiple of the period and equals delta**2 to second order in delta, so that the length scale keeps the units of the input. The periods are fixed constants, not hyperparameters. None, the default, or periods that are all infinite give the non-periodic kernel. A period that is NaN, zero or negative, or periods that are not a one-dimensional array, raise ValueError.

compute(hyp: ndarray, X: ndarray, X_star: ndarray = None, compute_diag: bool = False, compute_grad: bool = False)

Compute the covariance matrix for given training points and test points.

Parameters:

hypndarray, shape (cov_N,)

A 1D array of hyperparameters, where cov_N is the number of hyperparameters.

Xndarray, shape (N, D)

A 2D array where each row is a training point.

X_starndarray, shape (M, D), optional

A 2D array where each row is a test point. If this is not given, the self-covariance matrix is being computed.

compute_diagbool, defaults to False

Whether to only compute the diagonal of the self-covariance matrix.

compute_gradbool, defaults to False

Whether to compute the gradient with respect to the hyperparameters.

Returns:

Kndarray

The covariance matrix which is by default of shape (N, N). If compute_diag = True the shape is (N, 1).

dKndarray, shape (N, N, cov_N), optional

The gradient of the covariance matrix with respect to the hyperparameters.

Raises:

ValueError

Raised when hyp has not the expected number of hyperparameters.

ValueError

Raised when hyp is not an 1D array but of higher dimension.

ValueError

Raised when compute_diag and compute_grad are both True: the gradient is available for the full covariance matrix only.

ValueError

Raised when the kernel has periods and their number differs from the number of columns of X.

SquaredExponential

class gpyreg.covariance_functions.SquaredExponential(periods: ndarray = None)

Bases: AbstractKernel

Squared exponential kernel.

Parameters:

periodsarray_like, shape (D,), optional

The period of each input dimension along which the kernel is periodic, and np.inf for each dimension along which it is not. Along a dimension of period p, the squared difference delta**2 of two coordinates is replaced by the squared chord (p / pi)**2 * sin(pi * delta / p)**2, the squared distance between the two points mapped onto a circle of circumference p. It is zero at every multiple of the period and equals delta**2 to second order in delta, so that the length scale keeps the units of the input. The periods are fixed constants, not hyperparameters. None, the default, or periods that are all infinite give the non-periodic kernel. A period that is NaN, zero or negative, or periods that are not a one-dimensional array, raise ValueError.

compute(hyp: ndarray, X: ndarray, X_star: ndarray = None, compute_diag: bool = False, compute_grad: bool = False)

Compute the covariance matrix for given training points and test points.

Parameters:

hypndarray, shape (cov_N,)

A 1D array of hyperparameters, where cov_N is the number of hyperparameters.

Xndarray, shape (N, D)

A 2D array where each row is a training point.

X_starndarray, shape (M, D), optional

A 2D array where each row is a test point. If this is not given, the self-covariance matrix is being computed.

compute_diagbool, defaults to False

Whether to only compute the diagonal of the self-covariance matrix.

compute_gradbool, defaults to False

Whether to compute the gradient with respect to the hyperparameters.

Returns:

Kndarray

The covariance matrix which is by default of shape (N, N). If compute_diag = True the shape is (N, 1).

dKndarray, shape (N, N, cov_N), optional

The gradient of the covariance matrix with respect to the hyperparameters.

Raises:

ValueError

Raised when hyp has not the expected number of hyperparameters.

ValueError

Raised when hyp is not an 1D array but of higher dimension.

ValueError

Raised when compute_diag and compute_grad are both True: the gradient is available for the full covariance matrix only.

ValueError

Raised when the kernel has periods and their number differs from the number of columns of X.