Rational Quadratic

RationalQuadratic

RationalQuadratic(
    active_dims: Union[list[int], slice, None] = None,
    lengthscale: Union[
        LengthscaleCompatible, Variable[Lengthscale]
    ] = 1.0,
    variance: Union[
        ScalarFloat, Variable[ScalarArray]
    ] = 1.0,
    alpha: Union[ScalarFloat, Variable[ScalarArray]] = 1.0,
    n_dims: Union[int, None] = None,
    compute_engine: AbstractKernelComputation = DenseKernelComputation(),
)

Bases: StationaryKernel

The Rational Quadratic kernel.

Computes the covariance for pairs of inputs $(x, y)$ with lengthscale parameter $\ell$ and variance $\sigma^2$. $$k(x,y)=\sigma^2\exp\Bigg(1+\frac{\lVert x-y\rVert^2_2}{2\alpha\ell^2}\Bigg)$$

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.

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

  the alpha parameter of the kernel α.

• 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: Normal | StudentT

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: 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']) -> LinearOperator

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: Num[Array, 'N D']) -> LinearOperator

Compute the diagonal of the gram matrix of the kernel.

Parameters:

• x (Num[Array, 'N D']) –

  the input matrix of shape (N, D).

Returns:

• LinearOperator –

  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.