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 ℓ. 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 σ.

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

  the power of the kernel κ.

• 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.