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White

White 

White(
    active_dims: list[int] | slice | None = None,
    variance: ScalarFloat | AbstractUnwrappable = 1.0,
    n_dims: int | None = None,
    compute_engine: AbstractKernelComputation = ConstantDiagonalKernelComputation(),
)

Bases: StationaryKernel

The White noise kernel.

Computes the covariance for pairs of inputs \((x, y)\) with variance \(\sigma^2\): $$ k(x, y) = \sigma^2 \delta(x-y) $$

Parameters:

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

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

  • variance (ScalarFloat | AbstractUnwrappable, default: 1.0 ) –

    the variance of the kernel Οƒ.

  • n_dims (int | None, default: None ) –

    The number of input dimensions.

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

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

spectral_density property 

spectral_density: MultivariateNormal | MultivariateStudentT

The normalised spectral measure \(p(\boldsymbol{\omega})\) of the kernel.

By Bochner's theorem, a stationary kernel is the Fourier transform of a finite measure. This property returns that measure normalised to a probability distribution over \(\mathbb{R}^D\), so that

\[ k(\boldsymbol{\tau}) = \sigma^2 \, \mathbb{E}_{p(\boldsymbol{\omega})} \big[e^{i \boldsymbol{\omega}^\top \boldsymbol{\tau}}\big]. \]

The measure depends on the lengthscale \(\ell\) (as an inverse scale) but not on the variance \(\sigma^2\): the variance is the measure's total mass, which normalisation divides out, and it re-enters as the explicit prefactor above. The unnormalised spectral density of Rasmussen & Williams (2006, Β§4.2.1) is recovered as \(S(\boldsymbol{\omega}) = \sigma^2 (2\pi)^D p(\boldsymbol{\omega})\) under the convention \(k(\boldsymbol{\tau}) = (2\pi)^{-D}\int S(\boldsymbol{\omega}) e^{i\boldsymbol{\omega}^\top\boldsymbol{\tau}}\,d\boldsymbol{\omega}\).

Returns:

  • MultivariateNormal | MultivariateStudentT –

    The spectral measure as a \(D\)-dimensional numpyro distribution.

cross_covariance 

cross_covariance(
    x: Num[Array, "N D"], y: Num[Array, "M D"]
) -> Float[Array, "N M"]

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: Num[Array, 'N D']) -> lx.AbstractLinearOperator

Compute the gram matrix of the kernel.

Parameters:

  • x (Num[Array, 'N D']) –

    the input matrix of shape (N, D).

Returns:

  • AbstractLinearOperator –

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

diagonal 

diagonal(x: Num[Array, 'N D']) -> lx.AbstractLinearOperator

Compute the diagonal of the gram matrix of the kernel.

Parameters:

  • x (Num[Array, 'N D']) –

    the input matrix of shape (N, D).

Returns:

  • AbstractLinearOperator –

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

slice_input 

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:

  • 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: Union[AbstractKernel, ScalarFloat],
) -> AbstractKernel

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: Union[AbstractKernel, ScalarFloat],
) -> AbstractKernel

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.