# Copyright 2022 The thomaspinder Contributors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
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# See the License for the specific language governing permissions and
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# ==============================================================================
import beartype.typing as tp
from jaxtyping import Float
from paramax import AbstractUnwrappable
from gpjax.kernels.base import _val
from gpjax.kernels.computations import (
AbstractKernelComputation,
DenseKernelComputation,
)
from gpjax.kernels.stationary.base import StationaryKernel
from gpjax.kernels.stationary.utils import squared_distance
from gpjax.typing import (
Array,
ScalarArray,
ScalarFloat,
)
Lengthscale = tp.Union[Float[Array, "D"], ScalarArray]
LengthscaleCompatible = tp.Union[ScalarFloat, list[float], Lengthscale]
[docs]
class RationalQuadratic(StationaryKernel):
r"""The Rational Quadratic kernel.
Computes the covariance for pairs of inputs $(x, y)$ with lengthscale parameter
$\ell$, variance $\sigma^2$ and shape parameter $\alpha$.
$$
k(x,y)=\sigma^2\Bigg(1+\frac{\lVert x-y\rVert^2_2}{2\alpha\ell^2}\Bigg)^{-\alpha}
$$
As $\alpha \to \infty$ this recovers the :class:`~gpjax.kernels.RBF` kernel; it is
equivalently a scale mixture of RBF kernels with a Gamma-distributed inverse
squared lengthscale.
"""
name: str = "Rational Quadratic"
alpha: tp.Any
def __init__(
self,
active_dims: tp.Union[list[int], slice, None] = None,
lengthscale: tp.Union[LengthscaleCompatible, AbstractUnwrappable] = 1.0,
variance: tp.Union[ScalarFloat, AbstractUnwrappable] = 1.0,
alpha: tp.Union[ScalarFloat, AbstractUnwrappable] = 1.0,
n_dims: tp.Union[int, None] = None,
compute_engine: AbstractKernelComputation = DenseKernelComputation(),
):
"""Initializes the kernel.
Args:
active_dims: The indices of the input dimensions that the kernel operates on.
lengthscale: 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: the variance of the kernel σ.
alpha: the alpha parameter of the kernel α.
n_dims: The number of input dimensions. If `lengthscale` is an array, this
argument is ignored.
compute_engine: The computation engine that the kernel uses to compute the
covariance matrix.
"""
self.alpha = alpha
super().__init__(active_dims, lengthscale, variance, n_dims, compute_engine)
def __call__(self, x: Float[Array, " D"], y: Float[Array, " D"]) -> ScalarFloat:
x = self.slice_input(x) / _val(self.lengthscale)
y = self.slice_input(y) / _val(self.lengthscale)
alpha_val = _val(self.alpha)
K = _val(self.variance) * (1 + 0.5 * squared_distance(x, y) / alpha_val) ** (
-alpha_val
)
return K.squeeze()