Powered Exponential

gpjax.kernels.stationary.powered_exponential

PoweredExponential dataclass

Bases: AbstractKernel

The powered exponential family of kernels. This also equivalent to the symmetric generalized normal distribution.

See Diggle and Ribeiro (2007) - "Model-based Geostatistics". and https://en.wikipedia.org/wiki/Generalized_normal_distribution#Symmetric_version

compute_engine: AbstractKernelComputation = static_field(DenseKernelComputation()) class-attribute instance-attribute

active_dims: Optional[List[int]] = static_field(None) class-attribute instance-attribute

ndims property

spectral_density: Optional[tfd.Distribution] property

lengthscale: Union[ScalarFloat, Float[Array, ' D']] = param_field(jnp.array(1.0), bijector=tfb.Softplus()) class-attribute instance-attribute

variance: ScalarFloat = param_field(jnp.array(1.0), bijector=tfb.Softplus()) class-attribute instance-attribute

power: ScalarFloat = param_field(jnp.array(1.0), bijector=tfb.Sigmoid()) class-attribute instance-attribute

name: str = 'Powered Exponential' class-attribute instance-attribute

__init_subclass__(mutable: bool = False)

replace(**kwargs: Any) -> Self

Replace the values of the fields of the object.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the fields of the object.	{}

Returns

Module: with the fields replaced.

replace_meta(**kwargs: Any) -> Self

Replace the metadata of the fields.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the metadata of the fields of the object.	{}

Returns

Module: with the metadata of the fields replaced.

update_meta(**kwargs: Any) -> Self

Update the metadata of the fields. The metadata must already exist.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the fields of the object.	{}

Returns

Module: with the fields replaced.

replace_trainable(**kwargs: Dict[str, bool]) -> Self

Replace the trainability status of local nodes of the Module.

replace_bijector(**kwargs: Dict[str, tfb.Bijector]) -> Self

Replace the bijectors of local nodes of the Module.

constrain() -> Self

Transform model parameters to the constrained space according to their defined bijectors.

Returns

Module: transformed to the constrained space.

unconstrain() -> Self

Transform model parameters to the unconstrained space according to their defined bijectors.

Returns

Module: transformed to the unconstrained space.

stop_gradient() -> Self

Stop gradients flowing through the Module.

Returns

Module: with gradients stopped.

trainables() -> Self

cross_covariance(x: Num[Array, 'N D'], y: Num[Array, 'M D'])

gram(x: Num[Array, 'N D'])

slice_input(x: Float[Array, '... D']) -> Float[Array, '... Q']

Slice out the relevant columns of the input matrix.

Select the relevant columns of the supplied matrix to be used within the kernel's evaluation.

Parameters:

Name	Type	Description	Default
x	Float[Array, '... D']	The matrix or vector that is to be sliced.	required

Returns

Float[Array, "... Q"]: A sliced form of the input matrix.

__add__(other: Union[AbstractKernel, ScalarFloat]) -> AbstractKernel

Add two kernels together. Args: other (AbstractKernel): The kernel to be added to the current kernel.

Returns

AbstractKernel: A new kernel that is the sum of the two kernels.

__radd__(other: Union[AbstractKernel, ScalarFloat]) -> AbstractKernel

Add two kernels together. Args: other (AbstractKernel): The kernel to be added to the current kernel.

Returns

AbstractKernel: A new kernel that is the sum of the two kernels.

__mul__(other: Union[AbstractKernel, ScalarFloat]) -> AbstractKernel

Multiply two kernels together.

Parameters:

Name	Type	Description	Default
other	AbstractKernel	The kernel to be multiplied with the current kernel.	required

Returns

AbstractKernel: A new kernel that is the product of the two kernels.

__call__(x: Float[Array, ' D'], y: Float[Array, ' D']) -> ScalarFloat

Compute the Powered Exponential kernel between a pair of arrays.

Evaluate the kernel on a pair of inputs $(x, y)$ with length-scale parameter $\ell$, $\sigma$ and power $\kappa$.

$$k(x, y)=\sigma^2\exp\Bigg(-\Big(\frac{\lVert x-y\rVert^2}{\ell^2}\Big)^\kappa\Bigg)$$

Parameters:

Name	Type	Description	Default
x	Float[Array, ' D']	The left hand argument of the kernel function's call.	required
y	Float[Array, ' D']	The right hand argument of the kernel function's call	required

Returns

ScalarFloat: The value of $k(x, y)$.