stationaries.py
# Copyright 2017-2020 The GPflow Contributors. All Rights Reserved.
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# Licensed under the Apache License, Version 2.0 (the "License");
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# http://www.apache.org/licenses/LICENSE-2.0
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import numpy as np
import tensorflow as tf
from ..base import Parameter
from ..utilities import positive
from ..utilities.ops import difference_matrix, square_distance
from .base import Kernel
class Stationary(Kernel):
"""
Base class for kernels that are stationary, that is, they only depend on
d = x - x'
This class handles 'ard' behaviour, which stands for 'Automatic Relevance
Determination'. This means that the kernel has one lengthscale per
dimension, otherwise the kernel is isotropic (has a single lengthscale).
"""
def __init__(self, variance=1.0, lengthscales=1.0, **kwargs):
"""
:param variance: the (initial) value for the variance parameter.
:param lengthscales: the (initial) value for the lengthscale
parameter(s), to induce ARD behaviour this must be initialised as
an array the same length as the the number of active dimensions
e.g. [1., 1., 1.]. If only a single value is passed, this value
is used as the lengthscale of each dimension.
:param kwargs: accepts `name` and `active_dims`, which is a list or
slice of indices which controls which columns of X are used (by
default, all columns are used).
"""
for kwarg in kwargs:
if kwarg not in {"name", "active_dims"}:
raise TypeError(f"Unknown keyword argument: {kwarg}")
super().__init__(**kwargs)
self.variance = Parameter(variance, transform=positive())
self.lengthscales = Parameter(lengthscales, transform=positive())
self._validate_ard_active_dims(self.lengthscales)
@property
def ard(self) -> bool:
"""
Whether ARD behaviour is active.
"""
return self.lengthscales.shape.ndims > 0
def scale(self, X):
X_scaled = X / self.lengthscales if X is not None else X
return X_scaled
def K_diag(self, X):
return tf.fill(tf.shape(X)[:-1], tf.squeeze(self.variance))
class IsotropicStationary(Stationary):
"""
Base class for isotropic stationary kernels, i.e. kernels that only
depend on
r = ‖x - x'‖
Derived classes should implement one of:
K_r2(self, r2): Returns the kernel evaluated on r² (r2), which is the
squared scaled Euclidean distance Should operate element-wise on r2.
K_r(self, r): Returns the kernel evaluated on r, which is the scaled
Euclidean distance. Should operate element-wise on r.
"""
def K(self, X, X2=None):
r2 = self.scaled_squared_euclid_dist(X, X2)
return self.K_r2(r2)
def K_r2(self, r2):
if hasattr(self, "K_r"):
# Clipping around the (single) float precision which is ~1e-45.
r = tf.sqrt(tf.maximum(r2, 1e-36))
return self.K_r(r) # pylint: disable=no-member
raise NotImplementedError
def scaled_squared_euclid_dist(self, X, X2=None):
"""
Returns ‖(X - X2ᵀ) / ℓ‖², i.e. the squared L₂-norm.
"""
return square_distance(self.scale(X), self.scale(X2))
class AnisotropicStationary(Stationary):
"""
Base class for anisotropic stationary kernels, i.e. kernels that only
depend on
d = x - x'
Derived classes should implement K_d(self, d): Returns the kernel evaluated
on d, which is the pairwise difference matrix, scaled by the lengthscale
parameter ℓ (i.e. [(X - X2ᵀ) / ℓ]). The last axis corresponds to the
input dimension.
"""
def K(self, X, X2=None):
return self.K_d(self.scaled_difference_matrix(X, X2))
def scaled_difference_matrix(self, X, X2=None):
"""
Returns [(X - X2ᵀ) / ℓ]. If X has shape [..., N, D] and
X2 has shape [..., M, D], the output will have shape [..., N, M, D].
"""
return difference_matrix(self.scale(X), self.scale(X2))
def K_d(self, d):
raise NotImplementedError
class SquaredExponential(IsotropicStationary):
"""
The radial basis function (RBF) or squared exponential kernel. The kernel equation is
k(r) = σ² exp{-½ r²}
where:
r is the Euclidean distance between the input points, scaled by the lengthscales parameter ℓ.
σ² is the variance parameter
Functions drawn from a GP with this kernel are infinitely differentiable!
"""
def K_r2(self, r2):
return self.variance * tf.exp(-0.5 * r2)
class RationalQuadratic(IsotropicStationary):
"""
Rational Quadratic kernel,
k(r) = σ² (1 + r² / 2αℓ²)^(-α)
σ² : variance
ℓ : lengthscales
α : alpha, determines relative weighting of small-scale and large-scale fluctuations
For α → ∞, the RQ kernel becomes equivalent to the squared exponential.
"""
def __init__(self, variance=1.0, lengthscales=1.0, alpha=1.0, active_dims=None):
super().__init__(variance=variance, lengthscales=lengthscales, active_dims=active_dims)
self.alpha = Parameter(alpha, transform=positive())
def K_r2(self, r2):
return self.variance * (1 + 0.5 * r2 / self.alpha) ** (-self.alpha)
class Exponential(IsotropicStationary):
"""
The Exponential kernel. It is equivalent to a Matern12 kernel with doubled lengthscales.
"""
def K_r(self, r):
return self.variance * tf.exp(-0.5 * r)
class Matern12(IsotropicStationary):
"""
The Matern 1/2 kernel. Functions drawn from a GP with this kernel are not
differentiable anywhere. The kernel equation is
k(r) = σ² exp{-r}
where:
r is the Euclidean distance between the input points, scaled by the lengthscales parameter ℓ.
σ² is the variance parameter
"""
def K_r(self, r):
return self.variance * tf.exp(-r)
class Matern32(IsotropicStationary):
"""
The Matern 3/2 kernel. Functions drawn from a GP with this kernel are once
differentiable. The kernel equation is
k(r) = σ² (1 + √3r) exp{-√3 r}
where:
r is the Euclidean distance between the input points, scaled by the lengthscales parameter ℓ,
σ² is the variance parameter.
"""
def K_r(self, r):
sqrt3 = np.sqrt(3.0)
return self.variance * (1.0 + sqrt3 * r) * tf.exp(-sqrt3 * r)
class Matern52(IsotropicStationary):
"""
The Matern 5/2 kernel. Functions drawn from a GP with this kernel are twice
differentiable. The kernel equation is
k(r) = σ² (1 + √5r + 5/3r²) exp{-√5 r}
where:
r is the Euclidean distance between the input points, scaled by the lengthscales parameter ℓ,
σ² is the variance parameter.
"""
def K_r(self, r):
sqrt5 = np.sqrt(5.0)
return self.variance * (1.0 + sqrt5 * r + 5.0 / 3.0 * tf.square(r)) * tf.exp(-sqrt5 * r)
class Cosine(AnisotropicStationary):
"""
The Cosine kernel. Functions drawn from a GP with this kernel are sinusoids
(with a random phase). The kernel equation is
k(r) = σ² cos{2πd}
where:
d is the sum of the per-dimension differences between the input points, scaled by the
lengthscale parameter ℓ (i.e. Σᵢ [(X - X2ᵀ) / ℓ]ᵢ),
σ² is the variance parameter.
"""
def K_d(self, d):
d = tf.reduce_sum(d, axis=-1)
return self.variance * tf.cos(2 * np.pi * d)
