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Powered Exponential

PoweredExponential 

PoweredExponential(active_dims=None, lengthscale=1.0, variance=1.0, power=1.0, n_dims=None, compute_engine=DenseKernelComputation())

Bases: StationaryKernel

The powered exponential family of kernels.

Computes the covariance for pairs of inputs \((x, y)\) with length-scale parameter \(\ell\), \(\sigma\) and power \(\kappa\). $$ k(x, y)=\sigma^2\exp\Bigg(-\Big(\frac{\lVert x-y\rVert^2}{\ell^2}\Big)^\kappa\Bigg) $$

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

Parameters:

  • active_dims (Union[list[int], slice, None], default: None ) –

    the indices of the input dimensions that the kernel operates on.

  • lengthscale (Union[LengthscaleCompatible, Variable[Lengthscale]], default: 1.0 ) –

    the lengthscale(s) of the kernel β„“. If a scalar or an array of length 1, the kernel is isotropic, meaning that the same lengthscale is used for all input dimensions. If an array with length > 1, the kernel is anisotropic, meaning that a different lengthscale is used for each input.

  • variance (Union[ScalarFloat, Variable[ScalarArray]], default: 1.0 ) –

    the variance of the kernel Οƒ.

  • power (Union[ScalarFloat, Variable[ScalarArray]], default: 1.0 ) –

    the power of the kernel ΞΊ.

  • n_dims (Union[int, None], default: None ) –

    the number of input dimensions. If lengthscale is an array, this argument is ignored.

  • compute_engine (AbstractKernelComputation, default: DenseKernelComputation() ) –

    the computation engine that the kernel uses to compute the covariance matrix.

spectral_density property 

spectral_density

The spectral density of the kernel.

Returns:

  • Normal | StudentT –

    Callable[[Float[Array, "D"]], Float[Array, "D"]]: The spectral density function.

cross_covariance 

cross_covariance(x, y)

Compute the cross-covariance matrix of the kernel.

Parameters:

  • x (Num[Array, 'N D']) –

    the first input matrix of shape (N, D).

  • y (Num[Array, 'M D']) –

    the second input matrix of shape (M, D).

Returns:

  • Float[Array, 'N M'] –

    The cross-covariance matrix of the kernel of shape (N, M).

gram 

gram(x)

Compute the gram matrix of the kernel.

Parameters:

  • x (Num[Array, 'N D']) –

    the input matrix of shape (N, D).

Returns:

  • LinearOperator –

    The gram matrix of the kernel of shape (N, N).

diagonal 

diagonal(x)

Compute the diagonal of the gram matrix of the kernel.

Parameters:

  • x (Num[Array, 'N D']) –

    the input matrix of shape (N, D).

Returns:

  • Float[Array, ' N'] –

    The diagonal of the gram matrix of the kernel of shape (N,).

slice_input 

slice_input(x)

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:

  • x (Float[Array, '... D']) –

    the matrix or vector that is to be sliced.

Returns:

  • Float[Array, '... Q'] –

    The sliced form of the input matrix.

__add__ 

__add__(other)

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

Returns:

  • AbstractKernel ( AbstractKernel ) –

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

__mul__ 

__mul__(other)

Multiply two kernels together.

Parameters:

  • other (AbstractKernel) –

    The kernel to be multiplied with the current kernel.

Returns:

  • AbstractKernel ( AbstractKernel ) –

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