gpflow.kernels¶
gpflow.kernels.AnisotropicStationary¶

class
gpflow.kernels.
AnisotropicStationary
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.stationaries.Stationary
Base class for anisotropic stationary kernels, i.e. kernels that only depend on
d = x  x’
Derived classes should implement K_d(self, d): Returns the kernel evaluated on d, which is the pairwise difference matrix, scaled by the lengthscale parameter ℓ (i.e. [(X  X2ᵀ) / ℓ]). The last axis corresponds to the input dimension.
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
scaled_difference_matrix
(X[, X2])Returns [(X  X2ᵀ) / ℓ].
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_d
K_diag
scale
gpflow.kernels.ArcCosine¶

class
gpflow.kernels.
ArcCosine
(order=0, variance=1.0, weight_variances=1.0, bias_variance=1.0, *, active_dims=None, name=None)[source]¶ Bases:
gpflow.kernels.base.Kernel
The Arccosine family of kernels which mimics the computation in neural networks. The order parameter specifies the assumed activation function. The Multi Layer Perceptron (MLP) kernel is closely related to the ArcCosine kernel of order 0. The key reference is
@incollection{NIPS2009_3628, title = {Kernel Methods for Deep Learning}, author = {Youngmin Cho and Lawrence K. Saul}, booktitle = {Advances in Neural Information Processing Systems 22}, year = {2009}, url = {http://papers.nips.cc/paper/3628kernelmethodsfordeeplearning.pdf} }
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
 Parameters
order (
int
) –active_dims (
Union
[slice
,list
,None
]) –name (
Optional
[str
]) –

_J
(theta)[source]¶ Implements the order dependent family of functions defined in equations 4 to 7 in the reference paper.

__init__
(order=0, variance=1.0, weight_variances=1.0, bias_variance=1.0, *, active_dims=None, name=None)[source]¶  Parameters
order (
int
) – specifies the activation function of the neural network the function is a rectified monomial of the chosen ordervariance – the (initial) value for the variance parameter
weight_variances – the (initial) value for the weight_variances parameter, to induce ARD behaviour this must be initialised as an array the same length as the the number of active dimensions e.g. [1., 1., 1.]
bias_variance – the (initial) value for the bias_variance parameter defaults to 1.0
active_dims (
Union
[slice
,list
,None
]) – a slice or list specifying which columns of X are used
 Parameters
name (
Optional
[str
]) –

property
ard
¶ Whether ARD behaviour is active.
 Return type
bool
gpflow.kernels.Constant¶

class
gpflow.kernels.
Constant
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.statics.Static
The Constant (aka Bias) kernel. Functions drawn from a GP with this kernel are constant, i.e. f(x) = c, with c ~ N(0, σ^2). The kernel equation is
k(x, y) = σ²
where: σ² is the variance parameter.
 Attributes
 active_dims
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
gpflow.kernels.ChangePoints¶

class
gpflow.kernels.
ChangePoints
(kernels, locations, steepness=1.0, name=None)[source]¶ Bases:
gpflow.kernels.base.Combination
The ChangePoints kernel defines a fixed number of changepoints along a 1d input space where different kernels govern different parts of the space.
The kernel is by multiplication and addition of the base kernels with sigmoid functions (σ). A single changepoint kernel is defined as:
K₁(x, x’) * (1  σ(x)) * (1  σ(x’)) + K₂(x, x’) * σ(x) * σ(x’)
where K₁ is deactivated around the changepoint and K₂ is activated. The single changepoint version can be found in citet{lloyd2014}. Each sigmoid is a logistic function defined as:
σ(x) = 1 / (1 + exp{s(x  x₀)})
parameterized by location “x₀” and steepness “s”.
 @incollection{lloyd2014,
author = {Lloyd, James Robert et al}, title = {Automatic Construction and Naturallanguage Description of Nonparametric Regression Models}, booktitle = {Proceedings of the TwentyEighth AAAI Conference on Artificial Intelligence}, year = {2014}, url = {http://dl.acm.org/citation.cfm?id=2893873.2894066},
}
 Attributes
 active_dims
on_separate_dimensions
Checks whether the kernels in the combination act on disjoint subsets of dimensions.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
 Parameters
kernels (
List
[Kernel
]) –locations (
List
[float
]) –steepness (
Union
[float
,List
[float
]]) –name (
Optional
[str
]) –

__init__
(kernels, locations, steepness=1.0, name=None)[source]¶  Parameters
kernels (
List
[Kernel
]) – list of kernels defining the different regimeslocations (
List
[float
]) – list of changepoint locations in the 1d input spacesteepness (
Union
[float
,List
[float
]]) – the steepness parameter(s) of the sigmoids, this can be common between them or decoupled
 Parameters
name (
Optional
[str
]) –
gpflow.kernels.Combination¶

class
gpflow.kernels.
Combination
(kernels, name=None)[source]¶ Bases:
gpflow.kernels.base.Kernel
Combine a list of kernels, e.g. by adding or multiplying (see inheriting classes).
The names of the kernels to be combined are generated from their class names.
 Attributes
 active_dims
on_separate_dimensions
Checks whether the kernels in the combination act on disjoint subsets of dimensions.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
 Parameters
kernels (
List
[Kernel
]) –name (
Optional
[str
]) –

property
on_separate_dimensions
¶ Checks whether the kernels in the combination act on disjoint subsets of dimensions. Currently, it is hard to asses whether two slice objects will overlap, so this will always return False.
 Returns
Boolean indicator.
gpflow.kernels.Convolutional¶

class
gpflow.kernels.
Convolutional
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.base.Kernel
Plain convolutional kernel as described in citet{vdw2017convgp}. Defines a GP f( ) that is constructed from a sum of responses of individual patches in an image:
f(x) = sum_p x^{[p]}
where x^{[p]} is the pth patch in the image.
 @incollection{vdw2017convgp,
title = {Convolutional Gaussian Processes}, author = {van der Wilk, Mark and Rasmussen, Carl Edward and Hensman, James}, booktitle = {Advances in Neural Information Processing Systems 30}, year = {2017}, url = {http://papers.nips.cc/paper/6877convolutionalgaussianprocesses.pdf}
}
 Attributes
 active_dims
 num_patches
 parameters
 patch_len
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
get_patches
(X)Extracts patches from the images X.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
gpflow.kernels.Coregion¶

class
gpflow.kernels.
Coregion
(output_dim, rank, *, active_dims=None, name=None)[source]¶ Bases:
gpflow.kernels.base.Kernel
A Coregionalization kernel. The inputs to this kernel are _integers_ (we cast them from floats as needed) which usually specify the outputs of a Coregionalization model.
The kernel function is an indexing of a positivedefinite matrix:
K(x, y) = B[x, y] .
To ensure that B is positivedefinite, it is specified by the two parameters of this kernel, W and kappa:
B = W Wᵀ + diag(kappa) .
We refer to the size of B as “output_dim x output_dim”, since this is the number of outputs in a coregionalization model. We refer to the number of columns on W as ‘rank’: it is the number of degrees of correlation between the outputs.
NB. There is a symmetry between the elements of W, which creates a local minimum at W=0. To avoid this, it is recommended to initialize the optimization (or MCMC chain) using a random W.
 Attributes
 active_dims
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
output_covariance
output_variance
 Parameters
output_dim (
int
) –rank (
int
) –active_dims (
Union
[slice
,list
,None
]) –name (
Optional
[str
]) –
gpflow.kernels.Cosine¶

class
gpflow.kernels.
Cosine
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.stationaries.AnisotropicStationary
The Cosine kernel. Functions drawn from a GP with this kernel are sinusoids (with a random phase). The kernel equation is
k(r) = σ² cos{2πd}
where: d is the sum of the perdimension differences between the input points, scaled by the lengthscale parameter ℓ (i.e. Σᵢ [(X  X2ᵀ) / ℓ]ᵢ), σ² is the variance parameter.
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
scaled_difference_matrix
(X[, X2])Returns [(X  X2ᵀ) / ℓ].
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_d
K_diag
scale
gpflow.kernels.Exponential¶

class
gpflow.kernels.
Exponential
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.stationaries.IsotropicStationary
The Exponential kernel. It is equivalent to a Matern12 kernel with doubled lengthscales.
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
scaled_squared_euclid_dist
(X[, X2])Returns ‖(X  X2ᵀ) / ℓ‖², i.e. the squared L₂norm.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
K_r
K_r2
scale
gpflow.kernels.IndependentLatent¶

class
gpflow.kernels.
IndependentLatent
(active_dims=None, name=None)[source]¶ Bases:
gpflow.kernels.multioutput.kernels.MultioutputKernel
Base class for multioutput kernels that are constructed from independent latent Gaussian processes.
It should always be possible to specify inducing variables for such kernels that give a blockdiagonal Kuu, which can be represented as a [L, M, M] tensor. A reasonable (but not optimal) inference procedure can be specified by placing the inducing points in the latent processes and simply computing Kuu [L, M, M] and Kuf [N, P, M, L] and using fallback_independent_latent_ conditional(). This can be specified by using Fallback{SeparateShared} IndependentInducingVariables.
 Attributes
 active_dims
latent_kernels
The underlying kernels in the multioutput kernel
num_latent_gps
The number of latent GPs in the multioutput kernel
 parameters
 trainable_parameters
Methods
K
(X[, X2, full_output_cov])Returns the correlation of f(X) and f(X2), where f(.) can be multidimensional.
K_diag
(X[, full_output_cov])Returns the correlation of f(X) and f(X), where f(.) can be multidimensional.
__call__
(X[, X2, full_cov, full_output_cov, …])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
Kgg
 Parameters
active_dims (
Union
[slice
,list
,None
]) –name (
Optional
[str
]) –
gpflow.kernels.IsotropicStationary¶

class
gpflow.kernels.
IsotropicStationary
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.stationaries.Stationary
Base class for isotropic stationary kernels, i.e. kernels that only depend on
r = ‖x  x’‖
Derived classes should implement one of:
K_r2(self, r2): Returns the kernel evaluated on r² (r2), which is the squared scaled Euclidean distance Should operate elementwise on r2.
K_r(self, r): Returns the kernel evaluated on r, which is the scaled Euclidean distance. Should operate elementwise on r.
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
scaled_squared_euclid_dist
(X[, X2])Returns ‖(X  X2ᵀ) / ℓ‖², i.e. the squared L₂norm.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
K_r2
scale
gpflow.kernels.Kernel¶

class
gpflow.kernels.
Kernel
(active_dims=None, name=None)[source]¶ Bases:
gpflow.base.Module
The basic kernel class. Handles active dims.
 Attributes
 active_dims
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
 Parameters
active_dims (
Union
[slice
,list
,None
]) –name (
Optional
[str
]) –

__init__
(active_dims=None, name=None)[source]¶  Parameters
active_dims (
Union
[slice
,list
,None
]) – active dimensions, either a slice or list of indices into the columns of X.name (
Optional
[str
]) – optional kernel name.

_validate_ard_active_dims
(ard_parameter)[source]¶ Validate that ARD parameter matches the number of active_dims (provided active_dims has been specified as an array).

on_separate_dims
(other)[source]¶ Checks if the dimensions, over which the kernels are specified, overlap. Returns True if they are defined on different/separate dimensions and False otherwise.

slice
(X, X2=None)[source]¶ Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
 Parameters
X (
Tensor
) – Input 1 [N, D].X2 (
Optional
[Tensor
]) – Input 2 [M, D], can be None.
 Returns
Sliced X, X2, [N, I], I  input dimension.

slice_cov
(cov)[source]¶ Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices. This requires slicing the rows and columns. This will also turn flattened diagonal matrices into a tensor of full diagonal matrices.
 Parameters
cov (
Tensor
) – Tensor of covariance matrices, [N, D, D] or [N, D]. Return type
Tensor
 Returns
[N, I, I].
gpflow.kernels.Linear¶

class
gpflow.kernels.
Linear
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.base.Kernel
The linear kernel. Functions drawn from a GP with this kernel are linear, i.e. f(x) = cx. The kernel equation is
k(x, y) = σ²xy
where σ² is the variance parameter.
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag

__init__
(variance=1.0, active_dims=None)[source]¶  Parameters
variance – the (initial) value for the variance parameter(s), to induce ARD behaviour this must be initialised as an array the same length as the the number of active dimensions e.g. [1., 1., 1.]
active_dims – a slice or list specifying which columns of X are used

property
ard
¶ Whether ARD behaviour is active.
 Return type
bool
gpflow.kernels.LinearCoregionalization¶

class
gpflow.kernels.
LinearCoregionalization
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.multioutput.kernels.IndependentLatent
,gpflow.kernels.base.Combination
Linear mixing of the latent GPs to form the output.
 Attributes
 active_dims
latent_kernels
The underlying kernels in the multioutput kernel
num_latent_gps
The number of latent GPs in the multioutput kernel
on_separate_dimensions
Checks whether the kernels in the combination act on disjoint subsets of dimensions.
 parameters
 trainable_parameters
Methods
K
(X[, X2, full_output_cov])Returns the correlation of f(X) and f(X2), where f(.) can be multidimensional.
K_diag
(X[, full_output_cov])Returns the correlation of f(X) and f(X), where f(.) can be multidimensional.
__call__
(X[, X2, full_cov, full_output_cov, …])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
Kgg

K
(X, X2=None, full_output_cov=True)[source]¶ Returns the correlation of f(X) and f(X2), where f(.) can be multidimensional. :param X: data matrix, [N1, D] :param X2: data matrix, [N2, D] :param full_output_cov: calculate correlation between outputs. :return: cov[f(X), f(X2)] with shape  [N1, P, N2, P] if full_output_cov = True  [P, N1, N2] if full_output_cov = False

K_diag
(X, full_output_cov=True)[source]¶ Returns the correlation of f(X) and f(X), where f(.) can be multidimensional. :param X: data matrix, [N, D] :param full_output_cov: calculate correlation between outputs. :return: var[f(X)] with shape  [N, P, N, P] if full_output_cov = True  [N, P] if full_output_cov = False

property
latent_kernels
¶ The underlying kernels in the multioutput kernel

property
num_latent_gps
¶ The number of latent GPs in the multioutput kernel
gpflow.kernels.Matern12¶

class
gpflow.kernels.
Matern12
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.stationaries.IsotropicStationary
The Matern 1/2 kernel. Functions drawn from a GP with this kernel are not differentiable anywhere. The kernel equation is
k(r) = σ² exp{r}
where: r is the Euclidean distance between the input points, scaled by the lengthscales parameter ℓ. σ² is the variance parameter
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
scaled_squared_euclid_dist
(X[, X2])Returns ‖(X  X2ᵀ) / ℓ‖², i.e. the squared L₂norm.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
K_r
K_r2
scale
gpflow.kernels.Matern32¶

class
gpflow.kernels.
Matern32
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.stationaries.IsotropicStationary
The Matern 3/2 kernel. Functions drawn from a GP with this kernel are once differentiable. The kernel equation is
k(r) = σ² (1 + √3r) exp{√3 r}
where: r is the Euclidean distance between the input points, scaled by the lengthscales parameter ℓ, σ² is the variance parameter.
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
scaled_squared_euclid_dist
(X[, X2])Returns ‖(X  X2ᵀ) / ℓ‖², i.e. the squared L₂norm.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
K_r
K_r2
scale
gpflow.kernels.Matern52¶

class
gpflow.kernels.
Matern52
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.stationaries.IsotropicStationary
The Matern 5/2 kernel. Functions drawn from a GP with this kernel are twice differentiable. The kernel equation is
k(r) = σ² (1 + √5r + 5/3r²) exp{√5 r}
where: r is the Euclidean distance between the input points, scaled by the lengthscales parameter ℓ, σ² is the variance parameter.
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
scaled_squared_euclid_dist
(X[, X2])Returns ‖(X  X2ᵀ) / ℓ‖², i.e. the squared L₂norm.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
K_r
K_r2
scale
gpflow.kernels.MultioutputKernel¶

class
gpflow.kernels.
MultioutputKernel
(active_dims=None, name=None)[source]¶ Bases:
gpflow.kernels.base.Kernel
Multi Output Kernel class. This kernel can represent correlation between outputs of different datapoints. Therefore, subclasses of Mok should implement K which returns:  [N, P, N, P] if full_output_cov = True  [P, N, N] if full_output_cov = False and K_diag returns:  [N, P, P] if full_output_cov = True  [N, P] if full_output_cov = False The full_output_cov argument holds whether the kernel should calculate the covariance between the outputs. In case there is no correlation but full_output_cov is set to True the covariance matrix will be filled with zeros until the appropriate size is reached.
 Attributes
 active_dims
latent_kernels
The underlying kernels in the multioutput kernel
num_latent_gps
The number of latent GPs in the multioutput kernel
 parameters
 trainable_parameters
Methods
K
(X[, X2, full_output_cov])Returns the correlation of f(X) and f(X2), where f(.) can be multidimensional.
K_diag
(X[, full_output_cov])Returns the correlation of f(X) and f(X), where f(.) can be multidimensional.
__call__
(X[, X2, full_cov, full_output_cov, …])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
 Parameters
active_dims (
Union
[slice
,list
,None
]) –name (
Optional
[str
]) –

abstract
K
(X, X2=None, full_output_cov=True)[source]¶ Returns the correlation of f(X) and f(X2), where f(.) can be multidimensional. :param X: data matrix, [N1, D] :param X2: data matrix, [N2, D] :param full_output_cov: calculate correlation between outputs. :return: cov[f(X), f(X2)] with shape  [N1, P, N2, P] if full_output_cov = True  [P, N1, N2] if full_output_cov = False

abstract
K_diag
(X, full_output_cov=True)[source]¶ Returns the correlation of f(X) and f(X), where f(.) can be multidimensional. :param X: data matrix, [N, D] :param full_output_cov: calculate correlation between outputs. :return: var[f(X)] with shape  [N, P, N, P] if full_output_cov = True  [N, P] if full_output_cov = False

abstract property
latent_kernels
¶ The underlying kernels in the multioutput kernel

abstract property
num_latent_gps
¶ The number of latent GPs in the multioutput kernel
gpflow.kernels.Periodic¶

class
gpflow.kernels.
Periodic
(base_kernel, period=1.0)[source]¶ Bases:
gpflow.kernels.base.Kernel
The periodic family of kernels. Can be used to wrap any Stationary kernel to transform it into a periodic version. The canonical form (based on the SquaredExponential kernel) can be found in Equation (47) of
D.J.C.MacKay. Introduction to Gaussian processes. In C.M.Bishop, editor, Neural Networks and Machine Learning, pages 133–165. Springer, 1998.
The derivation can be achieved by mapping the original inputs through the transformation u = (cos(x), sin(x)).
For the SquaredExponential base kernel, the result can be expressed as:
k(r) = σ² exp{ 0.5 sin²(π r / γ) / ℓ²}
where: r is the Euclidean distance between the input points ℓ is the lengthscales parameter, σ² is the variance parameter, γ is the period parameter.
 NOTE: usually we have a factor of 4 instead of 0.5 in front but this
is absorbed into the lengthscales hyperparameter.
 NOTE: periodic kernel uses active_dims of a base kernel, therefore
the constructor doesn’t have it as an argument.
 Attributes
 active_dims
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
 Parameters
base_kernel (
IsotropicStationary
) –period (
Union
[float
,List
[float
]]) –

__init__
(base_kernel, period=1.0)[source]¶  Parameters
base_kernel (
IsotropicStationary
) – the base kernel to make periodic; must inherit from Stationary Note that active_dims should be specified in the base kernel.period (
Union
[float
,List
[float
]]) – the period; to induce a different period per active dimension this must be initialized with an array the same length as the number of active dimensions e.g. [1., 1., 1.]
gpflow.kernels.Polynomial¶

class
gpflow.kernels.
Polynomial
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.linears.Linear
The Polynomial kernel. Functions drawn from a GP with this kernel are polynomials of degree d. The kernel equation is
k(x, y) = (σ²xy + γ)ᵈ
where: σ² is the variance parameter, γ is the offset parameter, d is the degree parameter.
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag

__init__
(degree=3.0, variance=1.0, offset=1.0, active_dims=None)[source]¶  Parameters
degree – the degree of the polynomial
variance – the (initial) value for the variance parameter(s), to induce ARD behaviour this must be initialised as an array the same length as the the number of active dimensions e.g. [1., 1., 1.]
offset – the offset of the polynomial
active_dims – a slice or list specifying which columns of X are used
gpflow.kernels.Product¶

class
gpflow.kernels.
Product
(kernels, name=None)[source]¶ Bases:
gpflow.kernels.base.ReducingCombination
 Attributes
 active_dims
on_separate_dimensions
Checks whether the kernels in the combination act on disjoint subsets of dimensions.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
 Parameters
kernels (
List
[Kernel
]) –name (
Optional
[str
]) –
gpflow.kernels.SquaredExponential¶

class
gpflow.kernels.
SquaredExponential
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.stationaries.IsotropicStationary
The radial basis function (RBF) or squared exponential kernel. The kernel equation is
k(r) = σ² exp{½ r²}
where: r is the Euclidean distance between the input points, scaled by the lengthscales parameter ℓ. σ² is the variance parameter
Functions drawn from a GP with this kernel are infinitely differentiable!
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
scaled_squared_euclid_dist
(X[, X2])Returns ‖(X  X2ᵀ) / ℓ‖², i.e. the squared L₂norm.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
K_r2
scale
gpflow.kernels.RationalQuadratic¶

class
gpflow.kernels.
RationalQuadratic
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.stationaries.IsotropicStationary
Rational Quadratic kernel,
k(r) = σ² (1 + r² / 2αℓ²)^(α)
σ² : variance ℓ : lengthscales α : alpha, determines relative weighting of smallscale and largescale fluctuations
For α → ∞, the RQ kernel becomes equivalent to the squared exponential.
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
scaled_squared_euclid_dist
(X[, X2])Returns ‖(X  X2ᵀ) / ℓ‖², i.e. the squared L₂norm.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
K_r2
scale
gpflow.kernels.SeparateIndependent¶

class
gpflow.kernels.
SeparateIndependent
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.multioutput.kernels.MultioutputKernel
,gpflow.kernels.base.Combination
Separate: we use different kernel for each output latent
Independent: Latents are uncorrelated a priori.
 Attributes
 active_dims
latent_kernels
The underlying kernels in the multioutput kernel
num_latent_gps
The number of latent GPs in the multioutput kernel
on_separate_dimensions
Checks whether the kernels in the combination act on disjoint subsets of dimensions.
 parameters
 trainable_parameters
Methods
K
(X[, X2, full_output_cov])Returns the correlation of f(X) and f(X2), where f(.) can be multidimensional.
K_diag
(X[, full_output_cov])Returns the correlation of f(X) and f(X), where f(.) can be multidimensional.
__call__
(X[, X2, full_cov, full_output_cov, …])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.

K
(X, X2=None, full_output_cov=True)[source]¶ Returns the correlation of f(X) and f(X2), where f(.) can be multidimensional. :param X: data matrix, [N1, D] :param X2: data matrix, [N2, D] :param full_output_cov: calculate correlation between outputs. :return: cov[f(X), f(X2)] with shape  [N1, P, N2, P] if full_output_cov = True  [P, N1, N2] if full_output_cov = False

K_diag
(X, full_output_cov=False)[source]¶ Returns the correlation of f(X) and f(X), where f(.) can be multidimensional. :param X: data matrix, [N, D] :param full_output_cov: calculate correlation between outputs. :return: var[f(X)] with shape  [N, P, N, P] if full_output_cov = True  [N, P] if full_output_cov = False

property
latent_kernels
¶ The underlying kernels in the multioutput kernel

property
num_latent_gps
¶ The number of latent GPs in the multioutput kernel
gpflow.kernels.Static¶

class
gpflow.kernels.
Static
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.base.Kernel
Kernels who don’t depend on the value of the inputs are ‘Static’. The only parameter is a variance, σ².
 Attributes
 active_dims
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
gpflow.kernels.Stationary¶

class
gpflow.kernels.
Stationary
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.base.Kernel
Base class for kernels that are stationary, that is, they only depend on
d = x  x’
This class handles ‘ard’ behaviour, which stands for ‘Automatic Relevance Determination’. This means that the kernel has one lengthscale per dimension, otherwise the kernel is isotropic (has a single lengthscale).
 Attributes
 active_dims
ard
Whether ARD behaviour is active.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
scale

__init__
(variance=1.0, lengthscales=1.0, **kwargs)[source]¶  Parameters
variance – the (initial) value for the variance parameter.
lengthscales – the (initial) value for the lengthscale parameter(s), to induce ARD behaviour this must be initialised as an array the same length as the the number of active dimensions e.g. [1., 1., 1.]. If only a single value is passed, this value is used as the lengthscale of each dimension.
kwargs – accepts name and active_dims, which is a list or slice of indices which controls which columns of X are used (by default, all columns are used).

property
ard
¶ Whether ARD behaviour is active.
 Return type
bool
gpflow.kernels.Sum¶

class
gpflow.kernels.
Sum
(kernels, name=None)[source]¶ Bases:
gpflow.kernels.base.ReducingCombination
 Attributes
 active_dims
on_separate_dimensions
Checks whether the kernels in the combination act on disjoint subsets of dimensions.
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
 Parameters
kernels (
List
[Kernel
]) –name (
Optional
[str
]) –
gpflow.kernels.White¶

class
gpflow.kernels.
White
(*args: Any, **kwargs: Any)[source]¶ Bases:
gpflow.kernels.statics.Static
The White kernel: this kernel produces ‘white noise’. The kernel equation is
k(x_n, x_m) = δ(n, m) σ²
where: δ(.,.) is the Kronecker delta, σ² is the variance parameter.
 Attributes
 active_dims
 parameters
 trainable_parameters
Methods
__call__
(X[, X2, full_cov, presliced])Call self as a function.
on_separate_dims
(other)Checks if the dimensions, over which the kernels are specified, overlap.
slice
(X[, X2])Slice the correct dimensions for use in the kernel, as indicated by self.active_dims.
slice_cov
(cov)Slice the correct dimensions for use in the kernel, as indicated by self.active_dims for covariance matrices.
K
K_diag
gpflow.kernels.base¶
Kernels form a core component of GPflow models and allow prior information to be encoded about a latent function of interest. The effect of choosing different kernels, and how it is possible to combine multiple kernels is shown in the “Using kernels in GPflow” notebook.
Broadcasting over leading dimensions: kernel.K(X1, X2) returns the kernel evaluated on every pair in X1 and X2. E.g. if X1 has shape [S1, N1, D] and X2 has shape [S2, N2, D], kernel.K(X1, X2) will return a tensor of shape [S1, N1, S2, N2]. Similarly, kernel.K(X1, X1) returns a tensor of shape [S1, N1, S1, N1]. In contrast, the return shape of kernel.K(X1) is [S1, N1, N1]. (Without leading dimensions, the behaviour of kernel.K(X, None) is identical to kernel.K(X, X).)