Kernels

Kernels specific to state-space GPs (e.g. TruncatedPeriodic) and the to_sde dispatcher.

TruncatedPeriodic

TruncatedPeriodic(
    *,
    active_dims=None,
    lengthscale=1.0,
    variance=1.0,
    period=1.0,
    truncation_order=6,
    n_dims=None,
    compute_engine=None,
)

Bases: StationaryKernel

Truncated-Fourier approximation of the periodic kernel.

See Solin & Särkkä (2014). The kernel is the truncated cosine series

k(τ) = σ² · (Ĩ_0(c) + 2 Σ_{k=1}^{K} Ĩ_k(c) cos(2π k τ / period))

where Ĩ_k(c) = e^{-c} I_k(c) is the scaled modified Bessel function and c = 1/(4ℓ²). Acceptance of a 1-D temporal input is enforced by _validate_temporal_kernel at to_sde time, not here.

Example:

    >>> from gpjax.state_space import TruncatedPeriodic
    >>> kernel = TruncatedPeriodic(
    ...     lengthscale=1.0, variance=1.0, period=1.0, truncation_order=6,
    ... )
    >>> kernel.truncation_order
    6

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, §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.

to_sde

to_sde(kernel) -> LinearSDE

Convert a kernel to its state-space (linear SDE) representation.

Default handler raises NotImplementedError listing supported kernels.

Example:

    >>> import gpjax as gpx
    >>> from gpjax.state_space import to_sde
    >>> sde = to_sde(gpx.kernels.Matern32(lengthscale=1.0, variance=1.0))
    >>> sde.__class__.__name__
    'Matern32SDE'