"""Gaussian processes regression."""
# Authors: Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
# Modified by: Pete Green <p.l.green@liverpool.ac.uk>
# License: BSD 3 clause
import warnings
from operator import itemgetter
import numpy as np
from scipy.linalg import cholesky, cho_solve, solve_triangular
import scipy.optimize
from ..base import BaseEstimator, RegressorMixin, clone
from ..base import MultiOutputMixin
from .kernels import RBF, ConstantKernel as C
from ..preprocessing._data import _handle_zeros_in_scale
from ..utils import check_random_state
from ..utils.optimize import _check_optimize_result
GPR_CHOLESKY_LOWER = True
class GaussianProcessRegressor(MultiOutputMixin, RegressorMixin, BaseEstimator):
"""Gaussian process regression (GPR).
The implementation is based on Algorithm 2.1 of [1]_.
In addition to standard scikit-learn estimator API,
:class:`GaussianProcessRegressor`:
* allows prediction without prior fitting (based on the GP prior)
* provides an additional method `sample_y(X)`, which evaluates samples
drawn from the GPR (prior or posterior) at given inputs
* exposes a method `log_marginal_likelihood(theta)`, which can be used
externally for other ways of selecting hyperparameters, e.g., via
Markov chain Monte Carlo.
Read more in the :ref:`User Guide <gaussian_process>`.
.. versionadded:: 0.18
Parameters
----------
kernel : kernel instance, default=None
The kernel specifying the covariance function of the GP. If None is
passed, the kernel ``ConstantKernel(1.0, constant_value_bounds="fixed"
* RBF(1.0, length_scale_bounds="fixed")`` is used as default. Note that
the kernel hyperparameters are optimized during fitting unless the
bounds are marked as "fixed".
alpha : float or ndarray of shape (n_samples,), default=1e-10
Value added to the diagonal of the kernel matrix during fitting.
This can prevent a potential numerical issue during fitting, by
ensuring that the calculated values form a positive definite matrix.
It can also be interpreted as the variance of additional Gaussian
measurement noise on the training observations. Note that this is
different from using a `WhiteKernel`. If an array is passed, it must
have the same number of entries as the data used for fitting and is
used as datapoint-dependent noise level. Allowing to specify the
noise level directly as a parameter is mainly for convenience and
for consistency with :class:`~sklearn.linear_model.Ridge`.
optimizer : "fmin_l_bfgs_b" or callable, default="fmin_l_bfgs_b"
Can either be one of the internally supported optimizers for optimizing
the kernel's parameters, specified by a string, or an externally
defined optimizer passed as a callable. If a callable is passed, it
must have the signature::
def optimizer(obj_func, initial_theta, bounds):
# * 'obj_func': the objective function to be minimized, which
# takes the hyperparameters theta as a parameter and an
# optional flag eval_gradient, which determines if the
# gradient is returned additionally to the function value
# * 'initial_theta': the initial value for theta, which can be
# used by local optimizers
# * 'bounds': the bounds on the values of theta
....
# Returned are the best found hyperparameters theta and
# the corresponding value of the target function.
return theta_opt, func_min
Per default, the L-BFGS-B algorithm from `scipy.optimize.minimize`
is used. If None is passed, the kernel's parameters are kept fixed.
Available internal optimizers are: `{'fmin_l_bfgs_b'}`.
n_restarts_optimizer : int, default=0
The number of restarts of the optimizer for finding the kernel's
parameters which maximize the log-marginal likelihood. The first run
of the optimizer is performed from the kernel's initial parameters,
the remaining ones (if any) from thetas sampled log-uniform randomly
from the space of allowed theta-values. If greater than 0, all bounds
must be finite. Note that `n_restarts_optimizer == 0` implies that one
run is performed.
normalize_y : bool, default=False
Whether or not to normalized the target values `y` by removing the mean
and scaling to unit-variance. This is recommended for cases where
zero-mean, unit-variance priors are used. Note that, in this
implementation, the normalisation is reversed before the GP predictions
are reported.
.. versionchanged:: 0.23
copy_X_train : bool, default=True
If True, a persistent copy of the training data is stored in the
object. Otherwise, just a reference to the training data is stored,
which might cause predictions to change if the data is modified
externally.
random_state : int, RandomState instance or None, default=None
Determines random number generation used to initialize the centers.
Pass an int for reproducible results across multiple function calls.
See :term:`Glossary <random_state>`.
Attributes
----------
X_train_ : array-like of shape (n_samples, n_features) or list of object
Feature vectors or other representations of training data (also
required for prediction).
y_train_ : array-like of shape (n_samples,) or (n_samples, n_targets)
Target values in training data (also required for prediction).
kernel_ : kernel instance
The kernel used for prediction. The structure of the kernel is the
same as the one passed as parameter but with optimized hyperparameters.
L_ : array-like of shape (n_samples, n_samples)
Lower-triangular Cholesky decomposition of the kernel in ``X_train_``.
alpha_ : array-like of shape (n_samples,)
Dual coefficients of training data points in kernel space.
log_marginal_likelihood_value_ : float
The log-marginal-likelihood of ``self.kernel_.theta``.
n_features_in_ : int
Number of features seen during :term:`fit`.
.. versionadded:: 0.24
feature_names_in_ : ndarray of shape (`n_features_in_`,)
Names of features seen during :term:`fit`. Defined only when `X`
has feature names that are all strings.
.. versionadded:: 1.0
See Also
--------
GaussianProcessClassifier : Gaussian process classification (GPC)
based on Laplace approximation.
References
----------
.. [1] `Rasmussen, Carl Edward.
"Gaussian processes in machine learning."
Summer school on machine learning. Springer, Berlin, Heidelberg, 2003
<http://www.gaussianprocess.org/gpml/chapters/RW.pdf>`_.
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,
kernel=None,
*,
alpha=1e-10,
optimizer="fmin_l_bfgs_b",
n_restarts_optimizer=0,
normalize_y=False,
copy_X_train=True,
random_state=None,
):
self.kernel = kernel
self.alpha = alpha
self.optimizer = optimizer
self.n_restarts_optimizer = n_restarts_optimizer
self.normalize_y = normalize_y
self.copy_X_train = copy_X_train
self.random_state = random_state
def fit(self, X, y):
"""Fit Gaussian process regression model.
Parameters
----------
X : array-like of shape (n_samples, n_features) or list of object
Feature vectors or other representations of training data.
y : array-like of shape (n_samples,) or (n_samples, n_targets)
Target values.
Returns
-------
self : object
GaussianProcessRegressor class instance.
"""
if self.kernel is None: # Use an RBF kernel as default
self.kernel_ = C(1.0, constant_value_bounds="fixed") * RBF(
1.0, length_scale_bounds="fixed"
)
else:
self.kernel_ = clone(self.kernel)
self._rng = check_random_state(self.random_state)
if self.kernel_.requires_vector_input:
dtype, ensure_2d = "numeric", True
else:
dtype, ensure_2d = None, False
X, y = self._validate_data(
X,
y,
multi_output=True,
y_numeric=True,
ensure_2d=ensure_2d,
dtype=dtype,
)
# Normalize target value
if self.normalize_y:
self._y_train_mean = np.mean(y, axis=0)
self._y_train_std = _handle_zeros_in_scale(np.std(y, axis=0), copy=False)
# Remove mean and make unit variance
y = (y - self._y_train_mean) / self._y_train_std
else:
self._y_train_mean = np.zeros(1)
self._y_train_std = 1
if np.iterable(self.alpha) and self.alpha.shape[0] != y.shape[0]:
if self.alpha.shape[0] == 1:
self.alpha = self.alpha[0]
else:
raise ValueError(
"alpha must be a scalar or an array with same number of "
f"entries as y. ({self.alpha.shape[0]} != {y.shape[0]})"
)
self.X_train_ = np.copy(X) if self.copy_X_train else X
self.y_train_ = np.copy(y) if self.copy_X_train else y
if self.optimizer is not None and self.kernel_.n_dims > 0:
# Choose hyperparameters based on maximizing the log-marginal
# likelihood (potentially starting from several initial values)
def obj_func(theta, eval_gradient=True):
if eval_gradient:
lml, grad = self.log_marginal_likelihood(
theta, eval_gradient=True, clone_kernel=False
)
return -lml, -grad
else:
return -self.log_marginal_likelihood(theta, clone_kernel=False)
# First optimize starting from theta specified in kernel
optima = [
(
self._constrained_optimization(
obj_func, self.kernel_.theta, self.kernel_.bounds
)
)
]
# Additional runs are performed from log-uniform chosen initial
# theta
if self.n_restarts_optimizer > 0:
if not np.isfinite(self.kernel_.bounds).all():
raise ValueError(
"Multiple optimizer restarts (n_restarts_optimizer>0) "
"requires that all bounds are finite."
)
bounds = self.kernel_.bounds
for iteration in range(self.n_restarts_optimizer):
theta_initial = self._rng.uniform(bounds[:, 0], bounds[:, 1])
optima.append(
self._constrained_optimization(obj_func, theta_initial, bounds)
)
# Select result from run with minimal (negative) log-marginal
# likelihood
lml_values = list(map(itemgetter(1), optima))
self.kernel_.theta = optima[np.argmin(lml_values)][0]
self.kernel_._check_bounds_params()
self.log_marginal_likelihood_value_ = -np.min(lml_values)
else:
self.log_marginal_likelihood_value_ = self.log_marginal_likelihood(
self.kernel_.theta, clone_kernel=False
)
# Precompute quantities required for predictions which are independent
# of actual query points
# Alg. 2.1, page 19, line 2 -> L = cholesky(K + sigma^2 I)
K = self.kernel_(self.X_train_)
K[np.diag_indices_from(K)] += self.alpha
try:
self.L_ = cholesky(K, lower=GPR_CHOLESKY_LOWER, check_finite=False)
except np.linalg.LinAlgError as exc:
exc.args = (
f"The kernel, {self.kernel_}, is not returning a positive "
"definite matrix. Try gradually increasing the 'alpha' "
"parameter of your GaussianProcessRegressor estimator.",
) + exc.args
raise
# Alg 2.1, page 19, line 3 -> alpha = L^T \ (L \ y)
self.alpha_ = cho_solve(
(self.L_, GPR_CHOLESKY_LOWER),
self.y_train_,
check_finite=False,
)
return self
def predict(self, X, return_std=False, return_cov=False):
"""Predict using the Gaussian process regression model.
We can also predict based on an unfitted model by using the GP prior.
In addition to the mean of the predictive distribution, optionally also
returns its standard deviation (`return_std=True`) or covariance
(`return_cov=True`). Note that at most one of the two can be requested.
Parameters
----------
X : array-like of shape (n_samples, n_features) or list of object
Query points where the GP is evaluated.
return_std : bool, default=False
If True, the standard-deviation of the predictive distribution at
the query points is returned along with the mean.
return_cov : bool, default=False
If True, the covariance of the joint predictive distribution at
the query points is returned along with the mean.
Returns
-------
y_mean : ndarray of shape (n_samples,) or (n_samples, n_targets)
Mean of predictive distribution a query points.
y_std : ndarray of shape (n_samples,), optional
Standard deviation of predictive distribution at query points.
Only returned when `return_std` is True.
y_cov : ndarray of shape (n_samples, n_samples), optional
Covariance of joint predictive distribution a query points.
Only returned when `return_cov` is True.
"""
if return_std and return_cov:
raise RuntimeError(
"At most one of return_std or return_cov can be requested."
)
if self.kernel is None or self.kernel.requires_vector_input:
dtype, ensure_2d = "numeric", True
else:
dtype, ensure_2d = None, False
X = self._validate_data(X, ensure_2d=ensure_2d, dtype=dtype, reset=False)
if not hasattr(self, "X_train_"): # Unfitted;predict based on GP prior
if self.kernel is None:
kernel = C(1.0, constant_value_bounds="fixed") * RBF(
1.0, length_scale_bounds="fixed"
)
else:
kernel = self.kernel
y_mean = np.zeros(X.shape[0])
if return_cov:
y_cov = kernel(X)
return y_mean, y_cov
elif return_std:
y_var = kernel.diag(X)
return y_mean, np.sqrt(y_var)
else:
return y_mean
else: # Predict based on GP posterior
# Alg 2.1, page 19, line 4 -> f*_bar = K(X_test, X_train) . alpha
K_trans = self.kernel_(X, self.X_train_)
y_mean = K_trans @ self.alpha_
# undo normalisation
y_mean = self._y_train_std * y_mean + self._y_train_mean
# Alg 2.1, page 19, line 5 -> v = L \ K(X_test, X_train)^T
V = solve_triangular(
self.L_, K_trans.T, lower=GPR_CHOLESKY_LOWER, check_finite=False
)
if return_cov:
# Alg 2.1, page 19, line 6 -> K(X_test, X_test) - v^T. v
y_cov = self.kernel_(X) - V.T @ V
# undo normalisation
y_cov = y_cov * self._y_train_std ** 2
return y_mean, y_cov
elif return_std:
# Compute variance of predictive distribution
# Use einsum to avoid explicitly forming the large matrix
# V^T @ V just to extract its diagonal afterward.
y_var = self.kernel_.diag(X)
y_var -= np.einsum("ij,ji->i", V.T, V)
# Check if any of the variances is negative because of
# numerical issues. If yes: set the variance to 0.
y_var_negative = y_var < 0
if np.any(y_var_negative):
warnings.warn(
"Predicted variances smaller than 0. "
"Setting those variances to 0."
)
y_var[y_var_negative] = 0.0
# undo normalisation
y_var = y_var * self._y_train_std ** 2
return y_mean, np.sqrt(y_var)
else:
return y_mean
[docs] def sample_y(self, X, n_samples=1, random_state=0):
"""Draw samples from Gaussian process and evaluate at X.
Parameters
----------
X : array-like of shape (n_samples_X, n_features) or list of object
Query points where the GP is evaluated.
n_samples : int, default=1
Number of samples drawn from the Gaussian process per query point.
random_state : int, RandomState instance or None, default=0
Determines random number generation to randomly draw samples.
Pass an int for reproducible results across multiple function
calls.
See :term:`Glossary <random_state>`.
Returns
-------
y_samples : ndarray of shape (n_samples_X, n_samples), or \
(n_samples_X, n_targets, n_samples)
Values of n_samples samples drawn from Gaussian process and
evaluated at query points.
"""
rng = check_random_state(random_state)
y_mean, y_cov = self.predict(X, return_cov=True)
if y_mean.ndim == 1:
y_samples = rng.multivariate_normal(y_mean, y_cov, n_samples).T
else:
y_samples = [
rng.multivariate_normal(y_mean[:, i], y_cov, n_samples).T[:, np.newaxis]
for i in range(y_mean.shape[1])
]
y_samples = np.hstack(y_samples)
return y_samples
[docs] def log_marginal_likelihood(
self, theta=None, eval_gradient=False, clone_kernel=True
):
"""Return log-marginal likelihood of theta for training data.
Parameters
----------
theta : array-like of shape (n_kernel_params,) default=None
Kernel hyperparameters for which the log-marginal likelihood is
evaluated. If None, the precomputed log_marginal_likelihood
of ``self.kernel_.theta`` is returned.
eval_gradient : bool, default=False
If True, the gradient of the log-marginal likelihood with respect
to the kernel hyperparameters at position theta is returned
additionally. If True, theta must not be None.
clone_kernel : bool, default=True
If True, the kernel attribute is copied. If False, the kernel
attribute is modified, but may result in a performance improvement.
Returns
-------
log_likelihood : float
Log-marginal likelihood of theta for training data.
log_likelihood_gradient : ndarray of shape (n_kernel_params,), optional
Gradient of the log-marginal likelihood with respect to the kernel
hyperparameters at position theta.
Only returned when eval_gradient is True.
"""
if theta is None:
if eval_gradient:
raise ValueError("Gradient can only be evaluated for theta!=None")
return self.log_marginal_likelihood_value_
if clone_kernel:
kernel = self.kernel_.clone_with_theta(theta)
else:
kernel = self.kernel_
kernel.theta = theta
if eval_gradient:
K, K_gradient = kernel(self.X_train_, eval_gradient=True)
else:
K = kernel(self.X_train_)
# Alg. 2.1, page 19, line 2 -> L = cholesky(K + sigma^2 I)
K[np.diag_indices_from(K)] += self.alpha
try:
L = cholesky(K, lower=GPR_CHOLESKY_LOWER, check_finite=False)
except np.linalg.LinAlgError:
return (-np.inf, np.zeros_like(theta)) if eval_gradient else -np.inf
# Support multi-dimensional output of self.y_train_
y_train = self.y_train_
if y_train.ndim == 1:
y_train = y_train[:, np.newaxis]
# Alg 2.1, page 19, line 3 -> alpha = L^T \ (L \ y)
alpha = cho_solve((L, GPR_CHOLESKY_LOWER), y_train, check_finite=False)
# Alg 2.1, page 19, line 7
# -0.5 . y^T . alpha - sum(log(diag(L))) - n_samples / 2 log(2*pi)
# y is originally thought to be a (1, n_samples) row vector. However,
# in multioutputs, y is of shape (n_samples, 2) and we need to compute
# y^T . alpha for each output, independently using einsum. Thus, it
# is equivalent to:
# for output_idx in range(n_outputs):
# log_likelihood_dims[output_idx] = (
# y_train[:, [output_idx]] @ alpha[:, [output_idx]]
# )
log_likelihood_dims = -0.5 * np.einsum("ik,ik->k", y_train, alpha)
log_likelihood_dims -= np.log(np.diag(L)).sum()
log_likelihood_dims -= K.shape[0] / 2 * np.log(2 * np.pi)
# the log likehood is sum-up across the outputs
log_likelihood = log_likelihood_dims.sum(axis=-1)
if eval_gradient:
# Eq. 5.9, p. 114, and footnote 5 in p. 114
# 0.5 * trace((alpha . alpha^T - K^-1) . K_gradient)
# alpha is supposed to be a vector of (n_samples,) elements. With
# multioutputs, alpha is a matrix of size (n_samples, n_outputs).
# Therefore, we want to construct a matrix of
# (n_samples, n_samples, n_outputs) equivalent to
# for output_idx in range(n_outputs):
# output_alpha = alpha[:, [output_idx]]
# inner_term[..., output_idx] = output_alpha @ output_alpha.T
inner_term = np.einsum("ik,jk->ijk", alpha, alpha)
# compute K^-1 of shape (n_samples, n_samples)
K_inv = cho_solve(
(L, GPR_CHOLESKY_LOWER), np.eye(K.shape[0]), check_finite=False
)
# create a new axis to use broadcasting between inner_term and
# K_inv
inner_term -= K_inv[..., np.newaxis]
# Since we are interested about the trace of
# inner_term @ K_gradient, we don't explicitly compute the
# matrix-by-matrix operation and instead use an einsum. Therefore
# it is equivalent to:
# for param_idx in range(n_kernel_params):
# for output_idx in range(n_output):
# log_likehood_gradient_dims[param_idx, output_idx] = (
# inner_term[..., output_idx] @
# K_gradient[..., param_idx]
# )
log_likelihood_gradient_dims = 0.5 * np.einsum(
"ijl,jik->kl", inner_term, K_gradient
)
# the log likehood gradient is the sum-up across the outputs
log_likelihood_gradient = log_likelihood_gradient_dims.sum(axis=-1)
if eval_gradient:
return log_likelihood, log_likelihood_gradient
else:
return log_likelihood
def _constrained_optimization(self, obj_func, initial_theta, bounds):
if self.optimizer == "fmin_l_bfgs_b":
opt_res = scipy.optimize.minimize(
obj_func,
initial_theta,
method="L-BFGS-B",
jac=True,
bounds=bounds,
)
_check_optimize_result("lbfgs", opt_res)
theta_opt, func_min = opt_res.x, opt_res.fun
elif callable(self.optimizer):
theta_opt, func_min = self.optimizer(obj_func, initial_theta, bounds=bounds)
else:
raise ValueError(f"Unknown optimizer {self.optimizer}.")
return theta_opt, func_min
def _more_tags(self):
return {"requires_fit": False}
