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ai-content-maker/.venv/Lib/site-packages/sklearn/gaussian_process/kernels.py

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"""
The :mod:`sklearn.gaussian_process.kernels` module implements a set of kernels that
can be combined by operators and used in Gaussian processes.
"""
# Kernels for Gaussian process regression and classification.
#
# The kernels in this module allow kernel-engineering, i.e., they can be
# combined via the "+" and "*" operators or be exponentiated with a scalar
# via "**". These sum and product expressions can also contain scalar values,
# which are automatically converted to a constant kernel.
#
# All kernels allow (analytic) gradient-based hyperparameter optimization.
# The space of hyperparameters can be specified by giving lower und upper
# boundaries for the value of each hyperparameter (the search space is thus
# rectangular). Instead of specifying bounds, hyperparameters can also be
# declared to be "fixed", which causes these hyperparameters to be excluded from
# optimization.
# Author: Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
# License: BSD 3 clause
# Note: this module is strongly inspired by the kernel module of the george
# package.
import math
import warnings
from abc import ABCMeta, abstractmethod
from collections import namedtuple
from inspect import signature
import numpy as np
from scipy.spatial.distance import cdist, pdist, squareform
from scipy.special import gamma, kv
from ..base import clone
from ..exceptions import ConvergenceWarning
from ..metrics.pairwise import pairwise_kernels
from ..utils.validation import _num_samples
def _check_length_scale(X, length_scale):
length_scale = np.squeeze(length_scale).astype(float)
if np.ndim(length_scale) > 1:
raise ValueError("length_scale cannot be of dimension greater than 1")
if np.ndim(length_scale) == 1 and X.shape[1] != length_scale.shape[0]:
raise ValueError(
"Anisotropic kernel must have the same number of "
"dimensions as data (%d!=%d)" % (length_scale.shape[0], X.shape[1])
)
return length_scale
class Hyperparameter(
namedtuple(
"Hyperparameter", ("name", "value_type", "bounds", "n_elements", "fixed")
)
):
"""A kernel hyperparameter's specification in form of a namedtuple.
.. versionadded:: 0.18
Attributes
----------
name : str
The name of the hyperparameter. Note that a kernel using a
hyperparameter with name "x" must have the attributes self.x and
self.x_bounds
value_type : str
The type of the hyperparameter. Currently, only "numeric"
hyperparameters are supported.
bounds : pair of floats >= 0 or "fixed"
The lower and upper bound on the parameter. If n_elements>1, a pair
of 1d array with n_elements each may be given alternatively. If
the string "fixed" is passed as bounds, the hyperparameter's value
cannot be changed.
n_elements : int, default=1
The number of elements of the hyperparameter value. Defaults to 1,
which corresponds to a scalar hyperparameter. n_elements > 1
corresponds to a hyperparameter which is vector-valued,
such as, e.g., anisotropic length-scales.
fixed : bool, default=None
Whether the value of this hyperparameter is fixed, i.e., cannot be
changed during hyperparameter tuning. If None is passed, the "fixed" is
derived based on the given bounds.
Examples
--------
>>> from sklearn.gaussian_process.kernels import ConstantKernel
>>> from sklearn.datasets import make_friedman2
>>> from sklearn.gaussian_process import GaussianProcessRegressor
>>> from sklearn.gaussian_process.kernels import Hyperparameter
>>> X, y = make_friedman2(n_samples=50, noise=0, random_state=0)
>>> kernel = ConstantKernel(constant_value=1.0,
... constant_value_bounds=(0.0, 10.0))
We can access each hyperparameter:
>>> for hyperparameter in kernel.hyperparameters:
... print(hyperparameter)
Hyperparameter(name='constant_value', value_type='numeric',
bounds=array([[ 0., 10.]]), n_elements=1, fixed=False)
>>> params = kernel.get_params()
>>> for key in sorted(params): print(f"{key} : {params[key]}")
constant_value : 1.0
constant_value_bounds : (0.0, 10.0)
"""
# A raw namedtuple is very memory efficient as it packs the attributes
# in a struct to get rid of the __dict__ of attributes in particular it
# does not copy the string for the keys on each instance.
# By deriving a namedtuple class just to introduce the __init__ method we
# would also reintroduce the __dict__ on the instance. By telling the
# Python interpreter that this subclass uses static __slots__ instead of
# dynamic attributes. Furthermore we don't need any additional slot in the
# subclass so we set __slots__ to the empty tuple.
__slots__ = ()
def __new__(cls, name, value_type, bounds, n_elements=1, fixed=None):
if not isinstance(bounds, str) or bounds != "fixed":
bounds = np.atleast_2d(bounds)
if n_elements > 1: # vector-valued parameter
if bounds.shape[0] == 1:
bounds = np.repeat(bounds, n_elements, 0)
elif bounds.shape[0] != n_elements:
raise ValueError(
"Bounds on %s should have either 1 or "
"%d dimensions. Given are %d"
% (name, n_elements, bounds.shape[0])
)
if fixed is None:
fixed = isinstance(bounds, str) and bounds == "fixed"
return super(Hyperparameter, cls).__new__(
cls, name, value_type, bounds, n_elements, fixed
)
# This is mainly a testing utility to check that two hyperparameters
# are equal.
def __eq__(self, other):
return (
self.name == other.name
and self.value_type == other.value_type
and np.all(self.bounds == other.bounds)
and self.n_elements == other.n_elements
and self.fixed == other.fixed
)
class Kernel(metaclass=ABCMeta):
"""Base class for all kernels.
.. versionadded:: 0.18
Examples
--------
>>> from sklearn.gaussian_process.kernels import Kernel, RBF
>>> import numpy as np
>>> class CustomKernel(Kernel):
... def __init__(self, length_scale=1.0):
... self.length_scale = length_scale
... def __call__(self, X, Y=None):
... if Y is None:
... Y = X
... return np.inner(X, X if Y is None else Y) ** 2
... def diag(self, X):
... return np.ones(X.shape[0])
... def is_stationary(self):
... return True
>>> kernel = CustomKernel(length_scale=2.0)
>>> X = np.array([[1, 2], [3, 4]])
>>> print(kernel(X))
[[ 25 121]
[121 625]]
"""
def get_params(self, deep=True):
"""Get parameters of this kernel.
Parameters
----------
deep : bool, default=True
If True, will return the parameters for this estimator and
contained subobjects that are estimators.
Returns
-------
params : dict
Parameter names mapped to their values.
"""
params = dict()
# introspect the constructor arguments to find the model parameters
# to represent
cls = self.__class__
init = getattr(cls.__init__, "deprecated_original", cls.__init__)
init_sign = signature(init)
args, varargs = [], []
for parameter in init_sign.parameters.values():
if parameter.kind != parameter.VAR_KEYWORD and parameter.name != "self":
args.append(parameter.name)
if parameter.kind == parameter.VAR_POSITIONAL:
varargs.append(parameter.name)
if len(varargs) != 0:
raise RuntimeError(
"scikit-learn kernels should always "
"specify their parameters in the signature"
" of their __init__ (no varargs)."
" %s doesn't follow this convention." % (cls,)
)
for arg in args:
params[arg] = getattr(self, arg)
return params
def set_params(self, **params):
"""Set the parameters of this kernel.
The method works on simple kernels as well as on nested kernels.
The latter have parameters of the form ``<component>__<parameter>``
so that it's possible to update each component of a nested object.
Returns
-------
self
"""
if not params:
# Simple optimisation to gain speed (inspect is slow)
return self
valid_params = self.get_params(deep=True)
for key, value in params.items():
split = key.split("__", 1)
if len(split) > 1:
# nested objects case
name, sub_name = split
if name not in valid_params:
raise ValueError(
"Invalid parameter %s for kernel %s. "
"Check the list of available parameters "
"with `kernel.get_params().keys()`." % (name, self)
)
sub_object = valid_params[name]
sub_object.set_params(**{sub_name: value})
else:
# simple objects case
if key not in valid_params:
raise ValueError(
"Invalid parameter %s for kernel %s. "
"Check the list of available parameters "
"with `kernel.get_params().keys()`."
% (key, self.__class__.__name__)
)
setattr(self, key, value)
return self
def clone_with_theta(self, theta):
"""Returns a clone of self with given hyperparameters theta.
Parameters
----------
theta : ndarray of shape (n_dims,)
The hyperparameters
"""
cloned = clone(self)
cloned.theta = theta
return cloned
@property
def n_dims(self):
"""Returns the number of non-fixed hyperparameters of the kernel."""
return self.theta.shape[0]
@property
def hyperparameters(self):
"""Returns a list of all hyperparameter specifications."""
r = [
getattr(self, attr)
for attr in dir(self)
if attr.startswith("hyperparameter_")
]
return r
@property
def theta(self):
"""Returns the (flattened, log-transformed) non-fixed hyperparameters.
Note that theta are typically the log-transformed values of the
kernel's hyperparameters as this representation of the search space
is more amenable for hyperparameter search, as hyperparameters like
length-scales naturally live on a log-scale.
Returns
-------
theta : ndarray of shape (n_dims,)
The non-fixed, log-transformed hyperparameters of the kernel
"""
theta = []
params = self.get_params()
for hyperparameter in self.hyperparameters:
if not hyperparameter.fixed:
theta.append(params[hyperparameter.name])
if len(theta) > 0:
return np.log(np.hstack(theta))
else:
return np.array([])
@theta.setter
def theta(self, theta):
"""Sets the (flattened, log-transformed) non-fixed hyperparameters.
Parameters
----------
theta : ndarray of shape (n_dims,)
The non-fixed, log-transformed hyperparameters of the kernel
"""
params = self.get_params()
i = 0
for hyperparameter in self.hyperparameters:
if hyperparameter.fixed:
continue
if hyperparameter.n_elements > 1:
# vector-valued parameter
params[hyperparameter.name] = np.exp(
theta[i : i + hyperparameter.n_elements]
)
i += hyperparameter.n_elements
else:
params[hyperparameter.name] = np.exp(theta[i])
i += 1
if i != len(theta):
raise ValueError(
"theta has not the correct number of entries."
" Should be %d; given are %d" % (i, len(theta))
)
self.set_params(**params)
@property
def bounds(self):
"""Returns the log-transformed bounds on the theta.
Returns
-------
bounds : ndarray of shape (n_dims, 2)
The log-transformed bounds on the kernel's hyperparameters theta
"""
bounds = [
hyperparameter.bounds
for hyperparameter in self.hyperparameters
if not hyperparameter.fixed
]
if len(bounds) > 0:
return np.log(np.vstack(bounds))
else:
return np.array([])
def __add__(self, b):
if not isinstance(b, Kernel):
return Sum(self, ConstantKernel(b))
return Sum(self, b)
def __radd__(self, b):
if not isinstance(b, Kernel):
return Sum(ConstantKernel(b), self)
return Sum(b, self)
def __mul__(self, b):
if not isinstance(b, Kernel):
return Product(self, ConstantKernel(b))
return Product(self, b)
def __rmul__(self, b):
if not isinstance(b, Kernel):
return Product(ConstantKernel(b), self)
return Product(b, self)
def __pow__(self, b):
return Exponentiation(self, b)
def __eq__(self, b):
if type(self) != type(b):
return False
params_a = self.get_params()
params_b = b.get_params()
for key in set(list(params_a.keys()) + list(params_b.keys())):
if np.any(params_a.get(key, None) != params_b.get(key, None)):
return False
return True
def __repr__(self):
return "{0}({1})".format(
self.__class__.__name__, ", ".join(map("{0:.3g}".format, self.theta))
)
@abstractmethod
def __call__(self, X, Y=None, eval_gradient=False):
"""Evaluate the kernel."""
@abstractmethod
def diag(self, X):
"""Returns the diagonal of the kernel k(X, X).
The result of this method is identical to np.diag(self(X)); however,
it can be evaluated more efficiently since only the diagonal is
evaluated.
Parameters
----------
X : array-like of shape (n_samples,)
Left argument of the returned kernel k(X, Y)
Returns
-------
K_diag : ndarray of shape (n_samples_X,)
Diagonal of kernel k(X, X)
"""
@abstractmethod
def is_stationary(self):
"""Returns whether the kernel is stationary."""
@property
def requires_vector_input(self):
"""Returns whether the kernel is defined on fixed-length feature
vectors or generic objects. Defaults to True for backward
compatibility."""
return True
def _check_bounds_params(self):
"""Called after fitting to warn if bounds may have been too tight."""
list_close = np.isclose(self.bounds, np.atleast_2d(self.theta).T)
idx = 0
for hyp in self.hyperparameters:
if hyp.fixed:
continue
for dim in range(hyp.n_elements):
if list_close[idx, 0]:
warnings.warn(
"The optimal value found for "
"dimension %s of parameter %s is "
"close to the specified lower "
"bound %s. Decreasing the bound and"
" calling fit again may find a "
"better value." % (dim, hyp.name, hyp.bounds[dim][0]),
ConvergenceWarning,
)
elif list_close[idx, 1]:
warnings.warn(
"The optimal value found for "
"dimension %s of parameter %s is "
"close to the specified upper "
"bound %s. Increasing the bound and"
" calling fit again may find a "
"better value." % (dim, hyp.name, hyp.bounds[dim][1]),
ConvergenceWarning,
)
idx += 1
class NormalizedKernelMixin:
"""Mixin for kernels which are normalized: k(X, X)=1.
.. versionadded:: 0.18
"""
def diag(self, X):
"""Returns the diagonal of the kernel k(X, X).
The result of this method is identical to np.diag(self(X)); however,
it can be evaluated more efficiently since only the diagonal is
evaluated.
Parameters
----------
X : ndarray of shape (n_samples_X, n_features)
Left argument of the returned kernel k(X, Y)
Returns
-------
K_diag : ndarray of shape (n_samples_X,)
Diagonal of kernel k(X, X)
"""
return np.ones(X.shape[0])
class StationaryKernelMixin:
"""Mixin for kernels which are stationary: k(X, Y)= f(X-Y).
.. versionadded:: 0.18
"""
def is_stationary(self):
"""Returns whether the kernel is stationary."""
return True
class GenericKernelMixin:
"""Mixin for kernels which operate on generic objects such as variable-
length sequences, trees, and graphs.
.. versionadded:: 0.22
"""
@property
def requires_vector_input(self):
"""Whether the kernel works only on fixed-length feature vectors."""
return False
class CompoundKernel(Kernel):
"""Kernel which is composed of a set of other kernels.
.. versionadded:: 0.18
Parameters
----------
kernels : list of Kernels
The other kernels
Examples
--------
>>> from sklearn.gaussian_process.kernels import WhiteKernel
>>> from sklearn.gaussian_process.kernels import RBF
>>> from sklearn.gaussian_process.kernels import CompoundKernel
>>> kernel = CompoundKernel(
... [WhiteKernel(noise_level=3.0), RBF(length_scale=2.0)])
>>> print(kernel.bounds)
[[-11.51292546 11.51292546]
[-11.51292546 11.51292546]]
>>> print(kernel.n_dims)
2
>>> print(kernel.theta)
[1.09861229 0.69314718]
"""
def __init__(self, kernels):
self.kernels = kernels
def get_params(self, deep=True):
"""Get parameters of this kernel.
Parameters
----------
deep : bool, default=True
If True, will return the parameters for this estimator and
contained subobjects that are estimators.
Returns
-------
params : dict
Parameter names mapped to their values.
"""
return dict(kernels=self.kernels)
@property
def theta(self):
"""Returns the (flattened, log-transformed) non-fixed hyperparameters.
Note that theta are typically the log-transformed values of the
kernel's hyperparameters as this representation of the search space
is more amenable for hyperparameter search, as hyperparameters like
length-scales naturally live on a log-scale.
Returns
-------
theta : ndarray of shape (n_dims,)
The non-fixed, log-transformed hyperparameters of the kernel
"""
return np.hstack([kernel.theta for kernel in self.kernels])
@theta.setter
def theta(self, theta):
"""Sets the (flattened, log-transformed) non-fixed hyperparameters.
Parameters
----------
theta : array of shape (n_dims,)
The non-fixed, log-transformed hyperparameters of the kernel
"""
k_dims = self.k1.n_dims
for i, kernel in enumerate(self.kernels):
kernel.theta = theta[i * k_dims : (i + 1) * k_dims]
@property
def bounds(self):
"""Returns the log-transformed bounds on the theta.
Returns
-------
bounds : array of shape (n_dims, 2)
The log-transformed bounds on the kernel's hyperparameters theta
"""
return np.vstack([kernel.bounds for kernel in self.kernels])
def __call__(self, X, Y=None, eval_gradient=False):
"""Return the kernel k(X, Y) and optionally its gradient.
Note that this compound kernel returns the results of all simple kernel
stacked along an additional axis.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object, \
default=None
Left argument of the returned kernel k(X, Y)
Y : array-like of shape (n_samples_X, n_features) or list of object, \
default=None
Right argument of the returned kernel k(X, Y). If None, k(X, X)
is evaluated instead.
eval_gradient : bool, default=False
Determines whether the gradient with respect to the log of the
kernel hyperparameter is computed.
Returns
-------
K : ndarray of shape (n_samples_X, n_samples_Y, n_kernels)
Kernel k(X, Y)
K_gradient : ndarray of shape \
(n_samples_X, n_samples_X, n_dims, n_kernels), optional
The gradient of the kernel k(X, X) with respect to the log of the
hyperparameter of the kernel. Only returned when `eval_gradient`
is True.
"""
if eval_gradient:
K = []
K_grad = []
for kernel in self.kernels:
K_single, K_grad_single = kernel(X, Y, eval_gradient)
K.append(K_single)
K_grad.append(K_grad_single[..., np.newaxis])
return np.dstack(K), np.concatenate(K_grad, 3)
else:
return np.dstack([kernel(X, Y, eval_gradient) for kernel in self.kernels])
def __eq__(self, b):
if type(self) != type(b) or len(self.kernels) != len(b.kernels):
return False
return np.all(
[self.kernels[i] == b.kernels[i] for i in range(len(self.kernels))]
)
def is_stationary(self):
"""Returns whether the kernel is stationary."""
return np.all([kernel.is_stationary() for kernel in self.kernels])
@property
def requires_vector_input(self):
"""Returns whether the kernel is defined on discrete structures."""
return np.any([kernel.requires_vector_input for kernel in self.kernels])
def diag(self, X):
"""Returns the diagonal of the kernel k(X, X).
The result of this method is identical to `np.diag(self(X))`; however,
it can be evaluated more efficiently since only the diagonal is
evaluated.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object
Argument to the kernel.
Returns
-------
K_diag : ndarray of shape (n_samples_X, n_kernels)
Diagonal of kernel k(X, X)
"""
return np.vstack([kernel.diag(X) for kernel in self.kernels]).T
class KernelOperator(Kernel):
"""Base class for all kernel operators.
.. versionadded:: 0.18
"""
def __init__(self, k1, k2):
self.k1 = k1
self.k2 = k2
def get_params(self, deep=True):
"""Get parameters of this kernel.
Parameters
----------
deep : bool, default=True
If True, will return the parameters for this estimator and
contained subobjects that are estimators.
Returns
-------
params : dict
Parameter names mapped to their values.
"""
params = dict(k1=self.k1, k2=self.k2)
if deep:
deep_items = self.k1.get_params().items()
params.update(("k1__" + k, val) for k, val in deep_items)
deep_items = self.k2.get_params().items()
params.update(("k2__" + k, val) for k, val in deep_items)
return params
@property
def hyperparameters(self):
"""Returns a list of all hyperparameter."""
r = [
Hyperparameter(
"k1__" + hyperparameter.name,
hyperparameter.value_type,
hyperparameter.bounds,
hyperparameter.n_elements,
)
for hyperparameter in self.k1.hyperparameters
]
for hyperparameter in self.k2.hyperparameters:
r.append(
Hyperparameter(
"k2__" + hyperparameter.name,
hyperparameter.value_type,
hyperparameter.bounds,
hyperparameter.n_elements,
)
)
return r
@property
def theta(self):
"""Returns the (flattened, log-transformed) non-fixed hyperparameters.
Note that theta are typically the log-transformed values of the
kernel's hyperparameters as this representation of the search space
is more amenable for hyperparameter search, as hyperparameters like
length-scales naturally live on a log-scale.
Returns
-------
theta : ndarray of shape (n_dims,)
The non-fixed, log-transformed hyperparameters of the kernel
"""
return np.append(self.k1.theta, self.k2.theta)
@theta.setter
def theta(self, theta):
"""Sets the (flattened, log-transformed) non-fixed hyperparameters.
Parameters
----------
theta : ndarray of shape (n_dims,)
The non-fixed, log-transformed hyperparameters of the kernel
"""
k1_dims = self.k1.n_dims
self.k1.theta = theta[:k1_dims]
self.k2.theta = theta[k1_dims:]
@property
def bounds(self):
"""Returns the log-transformed bounds on the theta.
Returns
-------
bounds : ndarray of shape (n_dims, 2)
The log-transformed bounds on the kernel's hyperparameters theta
"""
if self.k1.bounds.size == 0:
return self.k2.bounds
if self.k2.bounds.size == 0:
return self.k1.bounds
return np.vstack((self.k1.bounds, self.k2.bounds))
def __eq__(self, b):
if type(self) != type(b):
return False
return (self.k1 == b.k1 and self.k2 == b.k2) or (
self.k1 == b.k2 and self.k2 == b.k1
)
def is_stationary(self):
"""Returns whether the kernel is stationary."""
return self.k1.is_stationary() and self.k2.is_stationary()
@property
def requires_vector_input(self):
"""Returns whether the kernel is stationary."""
return self.k1.requires_vector_input or self.k2.requires_vector_input
class Sum(KernelOperator):
"""The `Sum` kernel takes two kernels :math:`k_1` and :math:`k_2`
and combines them via
.. math::
k_{sum}(X, Y) = k_1(X, Y) + k_2(X, Y)
Note that the `__add__` magic method is overridden, so
`Sum(RBF(), RBF())` is equivalent to using the + operator
with `RBF() + RBF()`.
Read more in the :ref:`User Guide <gp_kernels>`.
.. versionadded:: 0.18
Parameters
----------
k1 : Kernel
The first base-kernel of the sum-kernel
k2 : Kernel
The second base-kernel of the sum-kernel
Examples
--------
>>> from sklearn.datasets import make_friedman2
>>> from sklearn.gaussian_process import GaussianProcessRegressor
>>> from sklearn.gaussian_process.kernels import RBF, Sum, ConstantKernel
>>> X, y = make_friedman2(n_samples=500, noise=0, random_state=0)
>>> kernel = Sum(ConstantKernel(2), RBF())
>>> gpr = GaussianProcessRegressor(kernel=kernel,
... random_state=0).fit(X, y)
>>> gpr.score(X, y)
1.0
>>> kernel
1.41**2 + RBF(length_scale=1)
"""
def __call__(self, X, Y=None, eval_gradient=False):
"""Return the kernel k(X, Y) and optionally its gradient.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object
Left argument of the returned kernel k(X, Y)
Y : array-like of shape (n_samples_X, n_features) or list of object,\
default=None
Right argument of the returned kernel k(X, Y). If None, k(X, X)
is evaluated instead.
eval_gradient : bool, default=False
Determines whether the gradient with respect to the log of
the kernel hyperparameter is computed.
Returns
-------
K : ndarray of shape (n_samples_X, n_samples_Y)
Kernel k(X, Y)
K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims),\
optional
The gradient of the kernel k(X, X) with respect to the log of the
hyperparameter of the kernel. Only returned when `eval_gradient`
is True.
"""
if eval_gradient:
K1, K1_gradient = self.k1(X, Y, eval_gradient=True)
K2, K2_gradient = self.k2(X, Y, eval_gradient=True)
return K1 + K2, np.dstack((K1_gradient, K2_gradient))
else:
return self.k1(X, Y) + self.k2(X, Y)
def diag(self, X):
"""Returns the diagonal of the kernel k(X, X).
The result of this method is identical to `np.diag(self(X))`; however,
it can be evaluated more efficiently since only the diagonal is
evaluated.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object
Argument to the kernel.
Returns
-------
K_diag : ndarray of shape (n_samples_X,)
Diagonal of kernel k(X, X)
"""
return self.k1.diag(X) + self.k2.diag(X)
def __repr__(self):
return "{0} + {1}".format(self.k1, self.k2)
class Product(KernelOperator):
"""The `Product` kernel takes two kernels :math:`k_1` and :math:`k_2`
and combines them via
.. math::
k_{prod}(X, Y) = k_1(X, Y) * k_2(X, Y)
Note that the `__mul__` magic method is overridden, so
`Product(RBF(), RBF())` is equivalent to using the * operator
with `RBF() * RBF()`.
Read more in the :ref:`User Guide <gp_kernels>`.
.. versionadded:: 0.18
Parameters
----------
k1 : Kernel
The first base-kernel of the product-kernel
k2 : Kernel
The second base-kernel of the product-kernel
Examples
--------
>>> from sklearn.datasets import make_friedman2
>>> from sklearn.gaussian_process import GaussianProcessRegressor
>>> from sklearn.gaussian_process.kernels import (RBF, Product,
... ConstantKernel)
>>> X, y = make_friedman2(n_samples=500, noise=0, random_state=0)
>>> kernel = Product(ConstantKernel(2), RBF())
>>> gpr = GaussianProcessRegressor(kernel=kernel,
... random_state=0).fit(X, y)
>>> gpr.score(X, y)
1.0
>>> kernel
1.41**2 * RBF(length_scale=1)
"""
def __call__(self, X, Y=None, eval_gradient=False):
"""Return the kernel k(X, Y) and optionally its gradient.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object
Left argument of the returned kernel k(X, Y)
Y : array-like of shape (n_samples_Y, n_features) or list of object,\
default=None
Right argument of the returned kernel k(X, Y). If None, k(X, X)
is evaluated instead.
eval_gradient : bool, default=False
Determines whether the gradient with respect to the log of
the kernel hyperparameter is computed.
Returns
-------
K : ndarray of shape (n_samples_X, n_samples_Y)
Kernel k(X, Y)
K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims), \
optional
The gradient of the kernel k(X, X) with respect to the log of the
hyperparameter of the kernel. Only returned when `eval_gradient`
is True.
"""
if eval_gradient:
K1, K1_gradient = self.k1(X, Y, eval_gradient=True)
K2, K2_gradient = self.k2(X, Y, eval_gradient=True)
return K1 * K2, np.dstack(
(K1_gradient * K2[:, :, np.newaxis], K2_gradient * K1[:, :, np.newaxis])
)
else:
return self.k1(X, Y) * self.k2(X, Y)
def diag(self, X):
"""Returns the diagonal of the kernel k(X, X).
The result of this method is identical to np.diag(self(X)); however,
it can be evaluated more efficiently since only the diagonal is
evaluated.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object
Argument to the kernel.
Returns
-------
K_diag : ndarray of shape (n_samples_X,)
Diagonal of kernel k(X, X)
"""
return self.k1.diag(X) * self.k2.diag(X)
def __repr__(self):
return "{0} * {1}".format(self.k1, self.k2)
class Exponentiation(Kernel):
"""The Exponentiation kernel takes one base kernel and a scalar parameter
:math:`p` and combines them via
.. math::
k_{exp}(X, Y) = k(X, Y) ^p
Note that the `__pow__` magic method is overridden, so
`Exponentiation(RBF(), 2)` is equivalent to using the ** operator
with `RBF() ** 2`.
Read more in the :ref:`User Guide <gp_kernels>`.
.. versionadded:: 0.18
Parameters
----------
kernel : Kernel
The base kernel
exponent : float
The exponent for the base kernel
Examples
--------
>>> from sklearn.datasets import make_friedman2
>>> from sklearn.gaussian_process import GaussianProcessRegressor
>>> from sklearn.gaussian_process.kernels import (RationalQuadratic,
... Exponentiation)
>>> X, y = make_friedman2(n_samples=500, noise=0, random_state=0)
>>> kernel = Exponentiation(RationalQuadratic(), exponent=2)
>>> gpr = GaussianProcessRegressor(kernel=kernel, alpha=5,
... random_state=0).fit(X, y)
>>> gpr.score(X, y)
0.419...
>>> gpr.predict(X[:1,:], return_std=True)
(array([635.5...]), array([0.559...]))
"""
def __init__(self, kernel, exponent):
self.kernel = kernel
self.exponent = exponent
def get_params(self, deep=True):
"""Get parameters of this kernel.
Parameters
----------
deep : bool, default=True
If True, will return the parameters for this estimator and
contained subobjects that are estimators.
Returns
-------
params : dict
Parameter names mapped to their values.
"""
params = dict(kernel=self.kernel, exponent=self.exponent)
if deep:
deep_items = self.kernel.get_params().items()
params.update(("kernel__" + k, val) for k, val in deep_items)
return params
@property
def hyperparameters(self):
"""Returns a list of all hyperparameter."""
r = []
for hyperparameter in self.kernel.hyperparameters:
r.append(
Hyperparameter(
"kernel__" + hyperparameter.name,
hyperparameter.value_type,
hyperparameter.bounds,
hyperparameter.n_elements,
)
)
return r
@property
def theta(self):
"""Returns the (flattened, log-transformed) non-fixed hyperparameters.
Note that theta are typically the log-transformed values of the
kernel's hyperparameters as this representation of the search space
is more amenable for hyperparameter search, as hyperparameters like
length-scales naturally live on a log-scale.
Returns
-------
theta : ndarray of shape (n_dims,)
The non-fixed, log-transformed hyperparameters of the kernel
"""
return self.kernel.theta
@theta.setter
def theta(self, theta):
"""Sets the (flattened, log-transformed) non-fixed hyperparameters.
Parameters
----------
theta : ndarray of shape (n_dims,)
The non-fixed, log-transformed hyperparameters of the kernel
"""
self.kernel.theta = theta
@property
def bounds(self):
"""Returns the log-transformed bounds on the theta.
Returns
-------
bounds : ndarray of shape (n_dims, 2)
The log-transformed bounds on the kernel's hyperparameters theta
"""
return self.kernel.bounds
def __eq__(self, b):
if type(self) != type(b):
return False
return self.kernel == b.kernel and self.exponent == b.exponent
def __call__(self, X, Y=None, eval_gradient=False):
"""Return the kernel k(X, Y) and optionally its gradient.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object
Left argument of the returned kernel k(X, Y)
Y : array-like of shape (n_samples_Y, n_features) or list of object,\
default=None
Right argument of the returned kernel k(X, Y). If None, k(X, X)
is evaluated instead.
eval_gradient : bool, default=False
Determines whether the gradient with respect to the log of
the kernel hyperparameter is computed.
Returns
-------
K : ndarray of shape (n_samples_X, n_samples_Y)
Kernel k(X, Y)
K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims),\
optional
The gradient of the kernel k(X, X) with respect to the log of the
hyperparameter of the kernel. Only returned when `eval_gradient`
is True.
"""
if eval_gradient:
K, K_gradient = self.kernel(X, Y, eval_gradient=True)
K_gradient *= self.exponent * K[:, :, np.newaxis] ** (self.exponent - 1)
return K**self.exponent, K_gradient
else:
K = self.kernel(X, Y, eval_gradient=False)
return K**self.exponent
def diag(self, X):
"""Returns the diagonal of the kernel k(X, X).
The result of this method is identical to np.diag(self(X)); however,
it can be evaluated more efficiently since only the diagonal is
evaluated.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object
Argument to the kernel.
Returns
-------
K_diag : ndarray of shape (n_samples_X,)
Diagonal of kernel k(X, X)
"""
return self.kernel.diag(X) ** self.exponent
def __repr__(self):
return "{0} ** {1}".format(self.kernel, self.exponent)
def is_stationary(self):
"""Returns whether the kernel is stationary."""
return self.kernel.is_stationary()
@property
def requires_vector_input(self):
"""Returns whether the kernel is defined on discrete structures."""
return self.kernel.requires_vector_input
class ConstantKernel(StationaryKernelMixin, GenericKernelMixin, Kernel):
"""Constant kernel.
Can be used as part of a product-kernel where it scales the magnitude of
the other factor (kernel) or as part of a sum-kernel, where it modifies
the mean of the Gaussian process.
.. math::
k(x_1, x_2) = constant\\_value \\;\\forall\\; x_1, x_2
Adding a constant kernel is equivalent to adding a constant::
kernel = RBF() + ConstantKernel(constant_value=2)
is the same as::
kernel = RBF() + 2
Read more in the :ref:`User Guide <gp_kernels>`.
.. versionadded:: 0.18
Parameters
----------
constant_value : float, default=1.0
The constant value which defines the covariance:
k(x_1, x_2) = constant_value
constant_value_bounds : pair of floats >= 0 or "fixed", default=(1e-5, 1e5)
The lower and upper bound on `constant_value`.
If set to "fixed", `constant_value` cannot be changed during
hyperparameter tuning.
Examples
--------
>>> from sklearn.datasets import make_friedman2
>>> from sklearn.gaussian_process import GaussianProcessRegressor
>>> from sklearn.gaussian_process.kernels import RBF, ConstantKernel
>>> X, y = make_friedman2(n_samples=500, noise=0, random_state=0)
>>> kernel = RBF() + ConstantKernel(constant_value=2)
>>> gpr = GaussianProcessRegressor(kernel=kernel, alpha=5,
... random_state=0).fit(X, y)
>>> gpr.score(X, y)
0.3696...
>>> gpr.predict(X[:1,:], return_std=True)
(array([606.1...]), array([0.24...]))
"""
def __init__(self, constant_value=1.0, constant_value_bounds=(1e-5, 1e5)):
self.constant_value = constant_value
self.constant_value_bounds = constant_value_bounds
@property
def hyperparameter_constant_value(self):
return Hyperparameter("constant_value", "numeric", self.constant_value_bounds)
def __call__(self, X, Y=None, eval_gradient=False):
"""Return the kernel k(X, Y) and optionally its gradient.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object
Left argument of the returned kernel k(X, Y)
Y : array-like of shape (n_samples_X, n_features) or list of object, \
default=None
Right argument of the returned kernel k(X, Y). If None, k(X, X)
is evaluated instead.
eval_gradient : bool, default=False
Determines whether the gradient with respect to the log of
the kernel hyperparameter is computed.
Only supported when Y is None.
Returns
-------
K : ndarray of shape (n_samples_X, n_samples_Y)
Kernel k(X, Y)
K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims), \
optional
The gradient of the kernel k(X, X) with respect to the log of the
hyperparameter of the kernel. Only returned when eval_gradient
is True.
"""
if Y is None:
Y = X
elif eval_gradient:
raise ValueError("Gradient can only be evaluated when Y is None.")
K = np.full(
(_num_samples(X), _num_samples(Y)),
self.constant_value,
dtype=np.array(self.constant_value).dtype,
)
if eval_gradient:
if not self.hyperparameter_constant_value.fixed:
return (
K,
np.full(
(_num_samples(X), _num_samples(X), 1),
self.constant_value,
dtype=np.array(self.constant_value).dtype,
),
)
else:
return K, np.empty((_num_samples(X), _num_samples(X), 0))
else:
return K
def diag(self, X):
"""Returns the diagonal of the kernel k(X, X).
The result of this method is identical to np.diag(self(X)); however,
it can be evaluated more efficiently since only the diagonal is
evaluated.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object
Argument to the kernel.
Returns
-------
K_diag : ndarray of shape (n_samples_X,)
Diagonal of kernel k(X, X)
"""
return np.full(
_num_samples(X),
self.constant_value,
dtype=np.array(self.constant_value).dtype,
)
def __repr__(self):
return "{0:.3g}**2".format(np.sqrt(self.constant_value))
class WhiteKernel(StationaryKernelMixin, GenericKernelMixin, Kernel):
"""White kernel.
The main use-case of this kernel is as part of a sum-kernel where it
explains the noise of the signal as independently and identically
normally-distributed. The parameter noise_level equals the variance of this
noise.
.. math::
k(x_1, x_2) = noise\\_level \\text{ if } x_i == x_j \\text{ else } 0
Read more in the :ref:`User Guide <gp_kernels>`.
.. versionadded:: 0.18
Parameters
----------
noise_level : float, default=1.0
Parameter controlling the noise level (variance)
noise_level_bounds : pair of floats >= 0 or "fixed", default=(1e-5, 1e5)
The lower and upper bound on 'noise_level'.
If set to "fixed", 'noise_level' cannot be changed during
hyperparameter tuning.
Examples
--------
>>> from sklearn.datasets import make_friedman2
>>> from sklearn.gaussian_process import GaussianProcessRegressor
>>> from sklearn.gaussian_process.kernels import DotProduct, WhiteKernel
>>> X, y = make_friedman2(n_samples=500, noise=0, random_state=0)
>>> kernel = DotProduct() + WhiteKernel(noise_level=0.5)
>>> gpr = GaussianProcessRegressor(kernel=kernel,
... random_state=0).fit(X, y)
>>> gpr.score(X, y)
0.3680...
>>> gpr.predict(X[:2,:], return_std=True)
(array([653.0..., 592.1... ]), array([316.6..., 316.6...]))
"""
def __init__(self, noise_level=1.0, noise_level_bounds=(1e-5, 1e5)):
self.noise_level = noise_level
self.noise_level_bounds = noise_level_bounds
@property
def hyperparameter_noise_level(self):
return Hyperparameter("noise_level", "numeric", self.noise_level_bounds)
def __call__(self, X, Y=None, eval_gradient=False):
"""Return the kernel k(X, Y) and optionally its gradient.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object
Left argument of the returned kernel k(X, Y)
Y : array-like of shape (n_samples_X, n_features) or list of object,\
default=None
Right argument of the returned kernel k(X, Y). If None, k(X, X)
is evaluated instead.
eval_gradient : bool, default=False
Determines whether the gradient with respect to the log of
the kernel hyperparameter is computed.
Only supported when Y is None.
Returns
-------
K : ndarray of shape (n_samples_X, n_samples_Y)
Kernel k(X, Y)
K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims),\
optional
The gradient of the kernel k(X, X) with respect to the log of the
hyperparameter of the kernel. Only returned when eval_gradient
is True.
"""
if Y is not None and eval_gradient:
raise ValueError("Gradient can only be evaluated when Y is None.")
if Y is None:
K = self.noise_level * np.eye(_num_samples(X))
if eval_gradient:
if not self.hyperparameter_noise_level.fixed:
return (
K,
self.noise_level * np.eye(_num_samples(X))[:, :, np.newaxis],
)
else:
return K, np.empty((_num_samples(X), _num_samples(X), 0))
else:
return K
else:
return np.zeros((_num_samples(X), _num_samples(Y)))
def diag(self, X):
"""Returns the diagonal of the kernel k(X, X).
The result of this method is identical to np.diag(self(X)); however,
it can be evaluated more efficiently since only the diagonal is
evaluated.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object
Argument to the kernel.
Returns
-------
K_diag : ndarray of shape (n_samples_X,)
Diagonal of kernel k(X, X)
"""
return np.full(
_num_samples(X), self.noise_level, dtype=np.array(self.noise_level).dtype
)
def __repr__(self):
return "{0}(noise_level={1:.3g})".format(
self.__class__.__name__, self.noise_level
)
class RBF(StationaryKernelMixin, NormalizedKernelMixin, Kernel):
"""Radial basis function kernel (aka squared-exponential kernel).
The RBF kernel is a stationary kernel. It is also known as the
"squared exponential" kernel. It is parameterized by a length scale
parameter :math:`l>0`, which can either be a scalar (isotropic variant
of the kernel) or a vector with the same number of dimensions as the inputs
X (anisotropic variant of the kernel). The kernel is given by:
.. math::
k(x_i, x_j) = \\exp\\left(- \\frac{d(x_i, x_j)^2}{2l^2} \\right)
where :math:`l` is the length scale of the kernel and
:math:`d(\\cdot,\\cdot)` is the Euclidean distance.
For advice on how to set the length scale parameter, see e.g. [1]_.
This kernel is infinitely differentiable, which implies that GPs with this
kernel as covariance function have mean square derivatives of all orders,
and are thus very smooth.
See [2]_, Chapter 4, Section 4.2, for further details of the RBF kernel.
Read more in the :ref:`User Guide <gp_kernels>`.
.. versionadded:: 0.18
Parameters
----------
length_scale : float or ndarray of shape (n_features,), default=1.0
The length scale of the kernel. If a float, an isotropic kernel is
used. If an array, an anisotropic kernel is used where each dimension
of l defines the length-scale of the respective feature dimension.
length_scale_bounds : pair of floats >= 0 or "fixed", default=(1e-5, 1e5)
The lower and upper bound on 'length_scale'.
If set to "fixed", 'length_scale' cannot be changed during
hyperparameter tuning.
References
----------
.. [1] `David Duvenaud (2014). "The Kernel Cookbook:
Advice on Covariance functions".
<https://www.cs.toronto.edu/~duvenaud/cookbook/>`_
.. [2] `Carl Edward Rasmussen, Christopher K. I. Williams (2006).
"Gaussian Processes for Machine Learning". The MIT Press.
<http://www.gaussianprocess.org/gpml/>`_
Examples
--------
>>> from sklearn.datasets import load_iris
>>> from sklearn.gaussian_process import GaussianProcessClassifier
>>> from sklearn.gaussian_process.kernels import RBF
>>> X, y = load_iris(return_X_y=True)
>>> kernel = 1.0 * RBF(1.0)
>>> gpc = GaussianProcessClassifier(kernel=kernel,
... random_state=0).fit(X, y)
>>> gpc.score(X, y)
0.9866...
>>> gpc.predict_proba(X[:2,:])
array([[0.8354..., 0.03228..., 0.1322...],
[0.7906..., 0.0652..., 0.1441...]])
"""
def __init__(self, length_scale=1.0, length_scale_bounds=(1e-5, 1e5)):
self.length_scale = length_scale
self.length_scale_bounds = length_scale_bounds
@property
def anisotropic(self):
return np.iterable(self.length_scale) and len(self.length_scale) > 1
@property
def hyperparameter_length_scale(self):
if self.anisotropic:
return Hyperparameter(
"length_scale",
"numeric",
self.length_scale_bounds,
len(self.length_scale),
)
return Hyperparameter("length_scale", "numeric", self.length_scale_bounds)
def __call__(self, X, Y=None, eval_gradient=False):
"""Return the kernel k(X, Y) and optionally its gradient.
Parameters
----------
X : ndarray of shape (n_samples_X, n_features)
Left argument of the returned kernel k(X, Y)
Y : ndarray of shape (n_samples_Y, n_features), default=None
Right argument of the returned kernel k(X, Y). If None, k(X, X)
if evaluated instead.
eval_gradient : bool, default=False
Determines whether the gradient with respect to the log of
the kernel hyperparameter is computed.
Only supported when Y is None.
Returns
-------
K : ndarray of shape (n_samples_X, n_samples_Y)
Kernel k(X, Y)
K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims), \
optional
The gradient of the kernel k(X, X) with respect to the log of the
hyperparameter of the kernel. Only returned when `eval_gradient`
is True.
"""
X = np.atleast_2d(X)
length_scale = _check_length_scale(X, self.length_scale)
if Y is None:
dists = pdist(X / length_scale, metric="sqeuclidean")
K = np.exp(-0.5 * dists)
# convert from upper-triangular matrix to square matrix
K = squareform(K)
np.fill_diagonal(K, 1)
else:
if eval_gradient:
raise ValueError("Gradient can only be evaluated when Y is None.")
dists = cdist(X / length_scale, Y / length_scale, metric="sqeuclidean")
K = np.exp(-0.5 * dists)
if eval_gradient:
if self.hyperparameter_length_scale.fixed:
# Hyperparameter l kept fixed
return K, np.empty((X.shape[0], X.shape[0], 0))
elif not self.anisotropic or length_scale.shape[0] == 1:
K_gradient = (K * squareform(dists))[:, :, np.newaxis]
return K, K_gradient
elif self.anisotropic:
# We need to recompute the pairwise dimension-wise distances
K_gradient = (X[:, np.newaxis, :] - X[np.newaxis, :, :]) ** 2 / (
length_scale**2
)
K_gradient *= K[..., np.newaxis]
return K, K_gradient
else:
return K
def __repr__(self):
if self.anisotropic:
return "{0}(length_scale=[{1}])".format(
self.__class__.__name__,
", ".join(map("{0:.3g}".format, self.length_scale)),
)
else: # isotropic
return "{0}(length_scale={1:.3g})".format(
self.__class__.__name__, np.ravel(self.length_scale)[0]
)
class Matern(RBF):
"""Matern kernel.
The class of Matern kernels is a generalization of the :class:`RBF`.
It has an additional parameter :math:`\\nu` which controls the
smoothness of the resulting function. The smaller :math:`\\nu`,
the less smooth the approximated function is.
As :math:`\\nu\\rightarrow\\infty`, the kernel becomes equivalent to
the :class:`RBF` kernel. When :math:`\\nu = 1/2`, the Matérn kernel
becomes identical to the absolute exponential kernel.
Important intermediate values are
:math:`\\nu=1.5` (once differentiable functions)
and :math:`\\nu=2.5` (twice differentiable functions).
The kernel is given by:
.. math::
k(x_i, x_j) = \\frac{1}{\\Gamma(\\nu)2^{\\nu-1}}\\Bigg(
\\frac{\\sqrt{2\\nu}}{l} d(x_i , x_j )
\\Bigg)^\\nu K_\\nu\\Bigg(
\\frac{\\sqrt{2\\nu}}{l} d(x_i , x_j )\\Bigg)
where :math:`d(\\cdot,\\cdot)` is the Euclidean distance,
:math:`K_{\\nu}(\\cdot)` is a modified Bessel function and
:math:`\\Gamma(\\cdot)` is the gamma function.
See [1]_, Chapter 4, Section 4.2, for details regarding the different
variants of the Matern kernel.
Read more in the :ref:`User Guide <gp_kernels>`.
.. versionadded:: 0.18
Parameters
----------
length_scale : float or ndarray of shape (n_features,), default=1.0
The length scale of the kernel. If a float, an isotropic kernel is
used. If an array, an anisotropic kernel is used where each dimension
of l defines the length-scale of the respective feature dimension.
length_scale_bounds : pair of floats >= 0 or "fixed", default=(1e-5, 1e5)
The lower and upper bound on 'length_scale'.
If set to "fixed", 'length_scale' cannot be changed during
hyperparameter tuning.
nu : float, default=1.5
The parameter nu controlling the smoothness of the learned function.
The smaller nu, the less smooth the approximated function is.
For nu=inf, the kernel becomes equivalent to the RBF kernel and for
nu=0.5 to the absolute exponential kernel. Important intermediate
values are nu=1.5 (once differentiable functions) and nu=2.5
(twice differentiable functions). Note that values of nu not in
[0.5, 1.5, 2.5, inf] incur a considerably higher computational cost
(appr. 10 times higher) since they require to evaluate the modified
Bessel function. Furthermore, in contrast to l, nu is kept fixed to
its initial value and not optimized.
References
----------
.. [1] `Carl Edward Rasmussen, Christopher K. I. Williams (2006).
"Gaussian Processes for Machine Learning". The MIT Press.
<http://www.gaussianprocess.org/gpml/>`_
Examples
--------
>>> from sklearn.datasets import load_iris
>>> from sklearn.gaussian_process import GaussianProcessClassifier
>>> from sklearn.gaussian_process.kernels import Matern
>>> X, y = load_iris(return_X_y=True)
>>> kernel = 1.0 * Matern(length_scale=1.0, nu=1.5)
>>> gpc = GaussianProcessClassifier(kernel=kernel,
... random_state=0).fit(X, y)
>>> gpc.score(X, y)
0.9866...
>>> gpc.predict_proba(X[:2,:])
array([[0.8513..., 0.0368..., 0.1117...],
[0.8086..., 0.0693..., 0.1220...]])
"""
def __init__(self, length_scale=1.0, length_scale_bounds=(1e-5, 1e5), nu=1.5):
super().__init__(length_scale, length_scale_bounds)
self.nu = nu
def __call__(self, X, Y=None, eval_gradient=False):
"""Return the kernel k(X, Y) and optionally its gradient.
Parameters
----------
X : ndarray of shape (n_samples_X, n_features)
Left argument of the returned kernel k(X, Y)
Y : ndarray of shape (n_samples_Y, n_features), default=None
Right argument of the returned kernel k(X, Y). If None, k(X, X)
if evaluated instead.
eval_gradient : bool, default=False
Determines whether the gradient with respect to the log of
the kernel hyperparameter is computed.
Only supported when Y is None.
Returns
-------
K : ndarray of shape (n_samples_X, n_samples_Y)
Kernel k(X, Y)
K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims), \
optional
The gradient of the kernel k(X, X) with respect to the log of the
hyperparameter of the kernel. Only returned when `eval_gradient`
is True.
"""
X = np.atleast_2d(X)
length_scale = _check_length_scale(X, self.length_scale)
if Y is None:
dists = pdist(X / length_scale, metric="euclidean")
else:
if eval_gradient:
raise ValueError("Gradient can only be evaluated when Y is None.")
dists = cdist(X / length_scale, Y / length_scale, metric="euclidean")
if self.nu == 0.5:
K = np.exp(-dists)
elif self.nu == 1.5:
K = dists * math.sqrt(3)
K = (1.0 + K) * np.exp(-K)
elif self.nu == 2.5:
K = dists * math.sqrt(5)
K = (1.0 + K + K**2 / 3.0) * np.exp(-K)
elif self.nu == np.inf:
K = np.exp(-(dists**2) / 2.0)
else: # general case; expensive to evaluate
K = dists
K[K == 0.0] += np.finfo(float).eps # strict zeros result in nan
tmp = math.sqrt(2 * self.nu) * K
K.fill((2 ** (1.0 - self.nu)) / gamma(self.nu))
K *= tmp**self.nu
K *= kv(self.nu, tmp)
if Y is None:
# convert from upper-triangular matrix to square matrix
K = squareform(K)
np.fill_diagonal(K, 1)
if eval_gradient:
if self.hyperparameter_length_scale.fixed:
# Hyperparameter l kept fixed
K_gradient = np.empty((X.shape[0], X.shape[0], 0))
return K, K_gradient
# We need to recompute the pairwise dimension-wise distances
if self.anisotropic:
D = (X[:, np.newaxis, :] - X[np.newaxis, :, :]) ** 2 / (
length_scale**2
)
else:
D = squareform(dists**2)[:, :, np.newaxis]
if self.nu == 0.5:
denominator = np.sqrt(D.sum(axis=2))[:, :, np.newaxis]
divide_result = np.zeros_like(D)
np.divide(
D,
denominator,
out=divide_result,
where=denominator != 0,
)
K_gradient = K[..., np.newaxis] * divide_result
elif self.nu == 1.5:
K_gradient = 3 * D * np.exp(-np.sqrt(3 * D.sum(-1)))[..., np.newaxis]
elif self.nu == 2.5:
tmp = np.sqrt(5 * D.sum(-1))[..., np.newaxis]
K_gradient = 5.0 / 3.0 * D * (tmp + 1) * np.exp(-tmp)
elif self.nu == np.inf:
K_gradient = D * K[..., np.newaxis]
else:
# approximate gradient numerically
def f(theta): # helper function
return self.clone_with_theta(theta)(X, Y)
return K, _approx_fprime(self.theta, f, 1e-10)
if not self.anisotropic:
return K, K_gradient[:, :].sum(-1)[:, :, np.newaxis]
else:
return K, K_gradient
else:
return K
def __repr__(self):
if self.anisotropic:
return "{0}(length_scale=[{1}], nu={2:.3g})".format(
self.__class__.__name__,
", ".join(map("{0:.3g}".format, self.length_scale)),
self.nu,
)
else:
return "{0}(length_scale={1:.3g}, nu={2:.3g})".format(
self.__class__.__name__, np.ravel(self.length_scale)[0], self.nu
)
class RationalQuadratic(StationaryKernelMixin, NormalizedKernelMixin, Kernel):
"""Rational Quadratic kernel.
The RationalQuadratic kernel can be seen as a scale mixture (an infinite
sum) of RBF kernels with different characteristic length scales. It is
parameterized by a length scale parameter :math:`l>0` and a scale
mixture parameter :math:`\\alpha>0`. Only the isotropic variant
where length_scale :math:`l` is a scalar is supported at the moment.
The kernel is given by:
.. math::
k(x_i, x_j) = \\left(
1 + \\frac{d(x_i, x_j)^2 }{ 2\\alpha l^2}\\right)^{-\\alpha}
where :math:`\\alpha` is the scale mixture parameter, :math:`l` is
the length scale of the kernel and :math:`d(\\cdot,\\cdot)` is the
Euclidean distance.
For advice on how to set the parameters, see e.g. [1]_.
Read more in the :ref:`User Guide <gp_kernels>`.
.. versionadded:: 0.18
Parameters
----------
length_scale : float > 0, default=1.0
The length scale of the kernel.
alpha : float > 0, default=1.0
Scale mixture parameter
length_scale_bounds : pair of floats >= 0 or "fixed", default=(1e-5, 1e5)
The lower and upper bound on 'length_scale'.
If set to "fixed", 'length_scale' cannot be changed during
hyperparameter tuning.
alpha_bounds : pair of floats >= 0 or "fixed", default=(1e-5, 1e5)
The lower and upper bound on 'alpha'.
If set to "fixed", 'alpha' cannot be changed during
hyperparameter tuning.
References
----------
.. [1] `David Duvenaud (2014). "The Kernel Cookbook:
Advice on Covariance functions".
<https://www.cs.toronto.edu/~duvenaud/cookbook/>`_
Examples
--------
>>> from sklearn.datasets import load_iris
>>> from sklearn.gaussian_process import GaussianProcessClassifier
>>> from sklearn.gaussian_process.kernels import RationalQuadratic
>>> X, y = load_iris(return_X_y=True)
>>> kernel = RationalQuadratic(length_scale=1.0, alpha=1.5)
>>> gpc = GaussianProcessClassifier(kernel=kernel,
... random_state=0).fit(X, y)
>>> gpc.score(X, y)
0.9733...
>>> gpc.predict_proba(X[:2,:])
array([[0.8881..., 0.0566..., 0.05518...],
[0.8678..., 0.0707... , 0.0614...]])
"""
def __init__(
self,
length_scale=1.0,
alpha=1.0,
length_scale_bounds=(1e-5, 1e5),
alpha_bounds=(1e-5, 1e5),
):
self.length_scale = length_scale
self.alpha = alpha
self.length_scale_bounds = length_scale_bounds
self.alpha_bounds = alpha_bounds
@property
def hyperparameter_length_scale(self):
return Hyperparameter("length_scale", "numeric", self.length_scale_bounds)
@property
def hyperparameter_alpha(self):
return Hyperparameter("alpha", "numeric", self.alpha_bounds)
def __call__(self, X, Y=None, eval_gradient=False):
"""Return the kernel k(X, Y) and optionally its gradient.
Parameters
----------
X : ndarray of shape (n_samples_X, n_features)
Left argument of the returned kernel k(X, Y)
Y : ndarray of shape (n_samples_Y, n_features), default=None
Right argument of the returned kernel k(X, Y). If None, k(X, X)
if evaluated instead.
eval_gradient : bool, default=False
Determines whether the gradient with respect to the log of
the kernel hyperparameter is computed.
Only supported when Y is None.
Returns
-------
K : ndarray of shape (n_samples_X, n_samples_Y)
Kernel k(X, Y)
K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims)
The gradient of the kernel k(X, X) with respect to the log of the
hyperparameter of the kernel. Only returned when eval_gradient
is True.
"""
if len(np.atleast_1d(self.length_scale)) > 1:
raise AttributeError(
"RationalQuadratic kernel only supports isotropic version, "
"please use a single scalar for length_scale"
)
X = np.atleast_2d(X)
if Y is None:
dists = squareform(pdist(X, metric="sqeuclidean"))
tmp = dists / (2 * self.alpha * self.length_scale**2)
base = 1 + tmp
K = base**-self.alpha
np.fill_diagonal(K, 1)
else:
if eval_gradient:
raise ValueError("Gradient can only be evaluated when Y is None.")
dists = cdist(X, Y, metric="sqeuclidean")
K = (1 + dists / (2 * self.alpha * self.length_scale**2)) ** -self.alpha
if eval_gradient:
# gradient with respect to length_scale
if not self.hyperparameter_length_scale.fixed:
length_scale_gradient = dists * K / (self.length_scale**2 * base)
length_scale_gradient = length_scale_gradient[:, :, np.newaxis]
else: # l is kept fixed
length_scale_gradient = np.empty((K.shape[0], K.shape[1], 0))
# gradient with respect to alpha
if not self.hyperparameter_alpha.fixed:
alpha_gradient = K * (
-self.alpha * np.log(base)
+ dists / (2 * self.length_scale**2 * base)
)
alpha_gradient = alpha_gradient[:, :, np.newaxis]
else: # alpha is kept fixed
alpha_gradient = np.empty((K.shape[0], K.shape[1], 0))
return K, np.dstack((alpha_gradient, length_scale_gradient))
else:
return K
def __repr__(self):
return "{0}(alpha={1:.3g}, length_scale={2:.3g})".format(
self.__class__.__name__, self.alpha, self.length_scale
)
class ExpSineSquared(StationaryKernelMixin, NormalizedKernelMixin, Kernel):
r"""Exp-Sine-Squared kernel (aka periodic kernel).
The ExpSineSquared kernel allows one to model functions which repeat
themselves exactly. It is parameterized by a length scale
parameter :math:`l>0` and a periodicity parameter :math:`p>0`.
Only the isotropic variant where :math:`l` is a scalar is
supported at the moment. The kernel is given by:
.. math::
k(x_i, x_j) = \text{exp}\left(-
\frac{ 2\sin^2(\pi d(x_i, x_j)/p) }{ l^ 2} \right)
where :math:`l` is the length scale of the kernel, :math:`p` the
periodicity of the kernel and :math:`d(\cdot,\cdot)` is the
Euclidean distance.
Read more in the :ref:`User Guide <gp_kernels>`.
.. versionadded:: 0.18
Parameters
----------
length_scale : float > 0, default=1.0
The length scale of the kernel.
periodicity : float > 0, default=1.0
The periodicity of the kernel.
length_scale_bounds : pair of floats >= 0 or "fixed", default=(1e-5, 1e5)
The lower and upper bound on 'length_scale'.
If set to "fixed", 'length_scale' cannot be changed during
hyperparameter tuning.
periodicity_bounds : pair of floats >= 0 or "fixed", default=(1e-5, 1e5)
The lower and upper bound on 'periodicity'.
If set to "fixed", 'periodicity' cannot be changed during
hyperparameter tuning.
Examples
--------
>>> from sklearn.datasets import make_friedman2
>>> from sklearn.gaussian_process import GaussianProcessRegressor
>>> from sklearn.gaussian_process.kernels import ExpSineSquared
>>> X, y = make_friedman2(n_samples=50, noise=0, random_state=0)
>>> kernel = ExpSineSquared(length_scale=1, periodicity=1)
>>> gpr = GaussianProcessRegressor(kernel=kernel, alpha=5,
... random_state=0).fit(X, y)
>>> gpr.score(X, y)
0.0144...
>>> gpr.predict(X[:2,:], return_std=True)
(array([425.6..., 457.5...]), array([0.3894..., 0.3467...]))
"""
def __init__(
self,
length_scale=1.0,
periodicity=1.0,
length_scale_bounds=(1e-5, 1e5),
periodicity_bounds=(1e-5, 1e5),
):
self.length_scale = length_scale
self.periodicity = periodicity
self.length_scale_bounds = length_scale_bounds
self.periodicity_bounds = periodicity_bounds
@property
def hyperparameter_length_scale(self):
"""Returns the length scale"""
return Hyperparameter("length_scale", "numeric", self.length_scale_bounds)
@property
def hyperparameter_periodicity(self):
return Hyperparameter("periodicity", "numeric", self.periodicity_bounds)
def __call__(self, X, Y=None, eval_gradient=False):
"""Return the kernel k(X, Y) and optionally its gradient.
Parameters
----------
X : ndarray of shape (n_samples_X, n_features)
Left argument of the returned kernel k(X, Y)
Y : ndarray of shape (n_samples_Y, n_features), default=None
Right argument of the returned kernel k(X, Y). If None, k(X, X)
if evaluated instead.
eval_gradient : bool, default=False
Determines whether the gradient with respect to the log of
the kernel hyperparameter is computed.
Only supported when Y is None.
Returns
-------
K : ndarray of shape (n_samples_X, n_samples_Y)
Kernel k(X, Y)
K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims), \
optional
The gradient of the kernel k(X, X) with respect to the log of the
hyperparameter of the kernel. Only returned when `eval_gradient`
is True.
"""
X = np.atleast_2d(X)
if Y is None:
dists = squareform(pdist(X, metric="euclidean"))
arg = np.pi * dists / self.periodicity
sin_of_arg = np.sin(arg)
K = np.exp(-2 * (sin_of_arg / self.length_scale) ** 2)
else:
if eval_gradient:
raise ValueError("Gradient can only be evaluated when Y is None.")
dists = cdist(X, Y, metric="euclidean")
K = np.exp(
-2 * (np.sin(np.pi / self.periodicity * dists) / self.length_scale) ** 2
)
if eval_gradient:
cos_of_arg = np.cos(arg)
# gradient with respect to length_scale
if not self.hyperparameter_length_scale.fixed:
length_scale_gradient = 4 / self.length_scale**2 * sin_of_arg**2 * K
length_scale_gradient = length_scale_gradient[:, :, np.newaxis]
else: # length_scale is kept fixed
length_scale_gradient = np.empty((K.shape[0], K.shape[1], 0))
# gradient with respect to p
if not self.hyperparameter_periodicity.fixed:
periodicity_gradient = (
4 * arg / self.length_scale**2 * cos_of_arg * sin_of_arg * K
)
periodicity_gradient = periodicity_gradient[:, :, np.newaxis]
else: # p is kept fixed
periodicity_gradient = np.empty((K.shape[0], K.shape[1], 0))
return K, np.dstack((length_scale_gradient, periodicity_gradient))
else:
return K
def __repr__(self):
return "{0}(length_scale={1:.3g}, periodicity={2:.3g})".format(
self.__class__.__name__, self.length_scale, self.periodicity
)
class DotProduct(Kernel):
r"""Dot-Product kernel.
The DotProduct kernel is non-stationary and can be obtained from linear
regression by putting :math:`N(0, 1)` priors on the coefficients
of :math:`x_d (d = 1, . . . , D)` and a prior of :math:`N(0, \sigma_0^2)`
on the bias. The DotProduct kernel is invariant to a rotation of
the coordinates about the origin, but not translations.
It is parameterized by a parameter sigma_0 :math:`\sigma`
which controls the inhomogenity of the kernel. For :math:`\sigma_0^2 =0`,
the kernel is called the homogeneous linear kernel, otherwise
it is inhomogeneous. The kernel is given by
.. math::
k(x_i, x_j) = \sigma_0 ^ 2 + x_i \cdot x_j
The DotProduct kernel is commonly combined with exponentiation.
See [1]_, Chapter 4, Section 4.2, for further details regarding the
DotProduct kernel.
Read more in the :ref:`User Guide <gp_kernels>`.
.. versionadded:: 0.18
Parameters
----------
sigma_0 : float >= 0, default=1.0
Parameter controlling the inhomogenity of the kernel. If sigma_0=0,
the kernel is homogeneous.
sigma_0_bounds : pair of floats >= 0 or "fixed", default=(1e-5, 1e5)
The lower and upper bound on 'sigma_0'.
If set to "fixed", 'sigma_0' cannot be changed during
hyperparameter tuning.
References
----------
.. [1] `Carl Edward Rasmussen, Christopher K. I. Williams (2006).
"Gaussian Processes for Machine Learning". The MIT Press.
<http://www.gaussianprocess.org/gpml/>`_
Examples
--------
>>> from sklearn.datasets import make_friedman2
>>> from sklearn.gaussian_process import GaussianProcessRegressor
>>> from sklearn.gaussian_process.kernels import DotProduct, WhiteKernel
>>> X, y = make_friedman2(n_samples=500, noise=0, random_state=0)
>>> kernel = DotProduct() + WhiteKernel()
>>> gpr = GaussianProcessRegressor(kernel=kernel,
... random_state=0).fit(X, y)
>>> gpr.score(X, y)
0.3680...
>>> gpr.predict(X[:2,:], return_std=True)
(array([653.0..., 592.1...]), array([316.6..., 316.6...]))
"""
def __init__(self, sigma_0=1.0, sigma_0_bounds=(1e-5, 1e5)):
self.sigma_0 = sigma_0
self.sigma_0_bounds = sigma_0_bounds
@property
def hyperparameter_sigma_0(self):
return Hyperparameter("sigma_0", "numeric", self.sigma_0_bounds)
def __call__(self, X, Y=None, eval_gradient=False):
"""Return the kernel k(X, Y) and optionally its gradient.
Parameters
----------
X : ndarray of shape (n_samples_X, n_features)
Left argument of the returned kernel k(X, Y)
Y : ndarray of shape (n_samples_Y, n_features), default=None
Right argument of the returned kernel k(X, Y). If None, k(X, X)
if evaluated instead.
eval_gradient : bool, default=False
Determines whether the gradient with respect to the log of
the kernel hyperparameter is computed.
Only supported when Y is None.
Returns
-------
K : ndarray of shape (n_samples_X, n_samples_Y)
Kernel k(X, Y)
K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims),\
optional
The gradient of the kernel k(X, X) with respect to the log of the
hyperparameter of the kernel. Only returned when `eval_gradient`
is True.
"""
X = np.atleast_2d(X)
if Y is None:
K = np.inner(X, X) + self.sigma_0**2
else:
if eval_gradient:
raise ValueError("Gradient can only be evaluated when Y is None.")
K = np.inner(X, Y) + self.sigma_0**2
if eval_gradient:
if not self.hyperparameter_sigma_0.fixed:
K_gradient = np.empty((K.shape[0], K.shape[1], 1))
K_gradient[..., 0] = 2 * self.sigma_0**2
return K, K_gradient
else:
return K, np.empty((X.shape[0], X.shape[0], 0))
else:
return K
def diag(self, X):
"""Returns the diagonal of the kernel k(X, X).
The result of this method is identical to np.diag(self(X)); however,
it can be evaluated more efficiently since only the diagonal is
evaluated.
Parameters
----------
X : ndarray of shape (n_samples_X, n_features)
Left argument of the returned kernel k(X, Y).
Returns
-------
K_diag : ndarray of shape (n_samples_X,)
Diagonal of kernel k(X, X).
"""
return np.einsum("ij,ij->i", X, X) + self.sigma_0**2
def is_stationary(self):
"""Returns whether the kernel is stationary."""
return False
def __repr__(self):
return "{0}(sigma_0={1:.3g})".format(self.__class__.__name__, self.sigma_0)
# adapted from scipy/optimize/optimize.py for functions with 2d output
def _approx_fprime(xk, f, epsilon, args=()):
f0 = f(*((xk,) + args))
grad = np.zeros((f0.shape[0], f0.shape[1], len(xk)), float)
ei = np.zeros((len(xk),), float)
for k in range(len(xk)):
ei[k] = 1.0
d = epsilon * ei
grad[:, :, k] = (f(*((xk + d,) + args)) - f0) / d[k]
ei[k] = 0.0
return grad
class PairwiseKernel(Kernel):
"""Wrapper for kernels in sklearn.metrics.pairwise.
A thin wrapper around the functionality of the kernels in
sklearn.metrics.pairwise.
Note: Evaluation of eval_gradient is not analytic but numeric and all
kernels support only isotropic distances. The parameter gamma is
considered to be a hyperparameter and may be optimized. The other
kernel parameters are set directly at initialization and are kept
fixed.
.. versionadded:: 0.18
Parameters
----------
gamma : float, default=1.0
Parameter gamma of the pairwise kernel specified by metric. It should
be positive.
gamma_bounds : pair of floats >= 0 or "fixed", default=(1e-5, 1e5)
The lower and upper bound on 'gamma'.
If set to "fixed", 'gamma' cannot be changed during
hyperparameter tuning.
metric : {"linear", "additive_chi2", "chi2", "poly", "polynomial", \
"rbf", "laplacian", "sigmoid", "cosine"} or callable, \
default="linear"
The metric to use when calculating kernel between instances in a
feature array. If metric is a string, it must be one of the metrics
in pairwise.PAIRWISE_KERNEL_FUNCTIONS.
If metric is "precomputed", X is assumed to be a kernel matrix.
Alternatively, if metric is a callable function, it is called on each
pair of instances (rows) and the resulting value recorded. The callable
should take two arrays from X as input and return a value indicating
the distance between them.
pairwise_kernels_kwargs : dict, default=None
All entries of this dict (if any) are passed as keyword arguments to
the pairwise kernel function.
Examples
--------
>>> from sklearn.datasets import load_iris
>>> from sklearn.gaussian_process import GaussianProcessClassifier
>>> from sklearn.gaussian_process.kernels import PairwiseKernel
>>> X, y = load_iris(return_X_y=True)
>>> kernel = PairwiseKernel(metric='rbf')
>>> gpc = GaussianProcessClassifier(kernel=kernel,
... random_state=0).fit(X, y)
>>> gpc.score(X, y)
0.9733...
>>> gpc.predict_proba(X[:2,:])
array([[0.8880..., 0.05663..., 0.05532...],
[0.8676..., 0.07073..., 0.06165...]])
"""
def __init__(
self,
gamma=1.0,
gamma_bounds=(1e-5, 1e5),
metric="linear",
pairwise_kernels_kwargs=None,
):
self.gamma = gamma
self.gamma_bounds = gamma_bounds
self.metric = metric
self.pairwise_kernels_kwargs = pairwise_kernels_kwargs
@property
def hyperparameter_gamma(self):
return Hyperparameter("gamma", "numeric", self.gamma_bounds)
def __call__(self, X, Y=None, eval_gradient=False):
"""Return the kernel k(X, Y) and optionally its gradient.
Parameters
----------
X : ndarray of shape (n_samples_X, n_features)
Left argument of the returned kernel k(X, Y)
Y : ndarray of shape (n_samples_Y, n_features), default=None
Right argument of the returned kernel k(X, Y). If None, k(X, X)
if evaluated instead.
eval_gradient : bool, default=False
Determines whether the gradient with respect to the log of
the kernel hyperparameter is computed.
Only supported when Y is None.
Returns
-------
K : ndarray of shape (n_samples_X, n_samples_Y)
Kernel k(X, Y)
K_gradient : ndarray of shape (n_samples_X, n_samples_X, n_dims),\
optional
The gradient of the kernel k(X, X) with respect to the log of the
hyperparameter of the kernel. Only returned when `eval_gradient`
is True.
"""
pairwise_kernels_kwargs = self.pairwise_kernels_kwargs
if self.pairwise_kernels_kwargs is None:
pairwise_kernels_kwargs = {}
X = np.atleast_2d(X)
K = pairwise_kernels(
X,
Y,
metric=self.metric,
gamma=self.gamma,
filter_params=True,
**pairwise_kernels_kwargs,
)
if eval_gradient:
if self.hyperparameter_gamma.fixed:
return K, np.empty((X.shape[0], X.shape[0], 0))
else:
# approximate gradient numerically
def f(gamma): # helper function
return pairwise_kernels(
X,
Y,
metric=self.metric,
gamma=np.exp(gamma),
filter_params=True,
**pairwise_kernels_kwargs,
)
return K, _approx_fprime(self.theta, f, 1e-10)
else:
return K
def diag(self, X):
"""Returns the diagonal of the kernel k(X, X).
The result of this method is identical to np.diag(self(X)); however,
it can be evaluated more efficiently since only the diagonal is
evaluated.
Parameters
----------
X : ndarray of shape (n_samples_X, n_features)
Left argument of the returned kernel k(X, Y)
Returns
-------
K_diag : ndarray of shape (n_samples_X,)
Diagonal of kernel k(X, X)
"""
# We have to fall back to slow way of computing diagonal
return np.apply_along_axis(self, 1, X).ravel()
def is_stationary(self):
"""Returns whether the kernel is stationary."""
return self.metric in ["rbf"]
def __repr__(self):
return "{0}(gamma={1}, metric={2})".format(
self.__class__.__name__, self.gamma, self.metric
)