# 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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# ==============================================================================
import beartype.typing as tp
import jax.numpy as jnp
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 euclidean_distance
from gpjax.typing import (
Array,
ScalarArray,
ScalarFloat,
)
Lengthscale = tp.Union[Float[Array, "D"], ScalarArray]
LengthscaleCompatible = tp.Union[ScalarFloat, list[float], Lengthscale]
[docs]
class PoweredExponential(StationaryKernel):
r"""The powered exponential family of kernels.
Computes the covariance for pairs of inputs $(x, y)$ with length-scale parameter
$\ell$, variance $\sigma^2$ and power $\kappa$.
$$
k(x, y)=\sigma^2\exp\Bigg(-\Big(\frac{\lVert x-y\rVert_2}{\ell}\Big)^\kappa\Bigg)
$$
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
"""
name: str = "Powered Exponential"
power: 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,
power: 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.
power: the power of the kernel κ.
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.power = power
super().__init__(active_dims, lengthscale, variance, n_dims, compute_engine)
def __call__(
self, x: Float[Array, " D"], y: Float[Array, " D"]
) -> Float[Array, ""]:
x = self.slice_input(x) / _val(self.lengthscale)
y = self.slice_input(y) / _val(self.lengthscale)
power_val = _val(self.power)
K = _val(self.variance) * jnp.exp(-(euclidean_distance(x, y) ** power_val))
return K.squeeze()