import numpy as np
import warnings
from scipy.linalg import cho_solve
from scipy.linalg import solve_triangular
import sklearn
from sklearn.gaussian_process import GaussianProcessRegressor as sk_GaussianProcessRegressor
from sklearn.utils import check_array
from .kernels import ConstantKernel
from .kernels import Sum
from .kernels import RBF
from .kernels import WhiteKernel
def _param_for_white_kernel_in_Sum(kernel, kernel_str=""):
"""
Check if a WhiteKernel exists in a Sum Kernel
and if it does return the corresponding key in
`kernel.get_params()`
"""
if kernel_str != "":
kernel_str = kernel_str + "__"
if isinstance(kernel, Sum):
for param, child in kernel.get_params(deep=False).items():
if isinstance(child, WhiteKernel):
return True, kernel_str + param
else:
present, child_str = _param_for_white_kernel_in_Sum(
child, kernel_str + param)
if present:
return True, child_str
return False, "_"
[docs]class GaussianProcessRegressor(sk_GaussianProcessRegressor):
"""
GaussianProcessRegressor that allows noise tunability.
The implementation is based on Algorithm 2.1 of Gaussian Processes
for Machine Learning (GPML) by Rasmussen and Williams.
In addition to standard scikit-learn estimator API,
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.
Parameters
----------
kernel : kernel object
The kernel specifying the covariance function of the GP. If None is
passed, the kernel "1.0 * RBF(1.0)" is used as default. Note that
the kernel's hyperparameters are optimized during fitting.
alpha : float or array-like, optional (default: 1e-10)
Value added to the diagonal of the kernel matrix during fitting.
Larger values correspond to increased noise level in the observations
and reduce potential numerical issue during fitting. 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. Note that this
is equivalent to adding a WhiteKernel with c=alpha. Allowing to specify
the noise level directly as a parameter is mainly for convenience and
for consistency with Ridge.
optimizer : string or callable, optional (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' is the objective function to be maximized, which
# takes the hyperparameters theta as 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 'fmin_l_bfgs_b' algorithm from scipy.optimize
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, optional (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 : boolean, optional (default: False)
Whether the target values y are normalized, i.e., the mean of the
observed target values become zero. This parameter should be set to
True if the target values' mean is expected to differ considerable from
zero. When enabled, the normalization effectively modifies the GP's
prior based on the data, which contradicts the likelihood principle;
normalization is thus disabled per default.
copy_X_train : bool, optional (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 : integer or numpy.RandomState, optional
The generator used to initialize the centers. If an integer is
given, it fixes the seed. Defaults to the global numpy random
number generator.
noise : string, "gaussian", optional
If set to "gaussian", then it is assumed that `y` is a noisy
estimate of `f(x)` where the noise is gaussian.
Attributes
----------
X_train_ : array-like, shape = (n_samples, n_features)
Feature values in training data (also required for prediction)
y_train_ : array-like, shape = (n_samples, [n_output_dims])
Target values in training data (also required for prediction)
kernel_ kernel object
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, shape = (n_samples, n_samples)
Lower-triangular Cholesky decomposition of the kernel in ``X_train_``
alpha_ : array-like, 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``
noise_ : float
Estimate of the gaussian noise. Useful only when noise is set to
"gaussian".
"""
[docs] 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,
noise=None):
self.noise = noise
super(GaussianProcessRegressor, self).__init__(
kernel=kernel, alpha=alpha, optimizer=optimizer,
n_restarts_optimizer=n_restarts_optimizer,
normalize_y=normalize_y, copy_X_train=copy_X_train,
random_state=random_state)
[docs] def fit(self, X, y):
"""Fit Gaussian process regression model.
Parameters
----------
X : array-like, shape = (n_samples, n_features)
Training data
y : array-like, shape = (n_samples, [n_output_dims])
Target values
Returns
-------
self
Returns an instance of self.
"""
if isinstance(self.noise, str) and self.noise != "gaussian":
raise ValueError("expected noise to be 'gaussian', got %s"
% self.noise)
if self.kernel is None:
self.kernel = ConstantKernel(1.0, constant_value_bounds="fixed") \
* RBF(1.0, length_scale_bounds="fixed")
if self.noise == "gaussian":
self.kernel = self.kernel + WhiteKernel()
elif self.noise:
self.kernel = self.kernel + WhiteKernel(
noise_level=self.noise, noise_level_bounds="fixed"
)
super(GaussianProcessRegressor, self).fit(X, y)
self.noise_ = None
if self.noise:
# The noise component of this kernel should be set to zero
# while estimating K(X_test, X_test)
# Note that the term K(X, X) should include the noise but
# this (K(X, X))^-1y is precomputed as the attribute `alpha_`.
# (Notice the underscore).
# This has been described in Eq 2.24 of
# http://www.gaussianprocess.org/gpml/chapters/RW2.pdf
# Hence this hack
if isinstance(self.kernel_, WhiteKernel):
self.kernel_.set_params(noise_level=0.0)
else:
white_present, white_param = _param_for_white_kernel_in_Sum(
self.kernel_)
# This should always be true. Just in case.
if white_present:
noise_kernel = self.kernel_.get_params()[white_param]
self.noise_ = noise_kernel.noise_level
self.kernel_.set_params(
**{white_param: WhiteKernel(noise_level=0.0)})
# Precompute arrays needed at prediction
L_inv = solve_triangular(self.L_.T, np.eye(self.L_.shape[0]))
self.K_inv_ = L_inv.dot(L_inv.T)
# Fix deprecation warning #462
if int(sklearn.__version__[2:4]) >= 23:
self.y_train_std_ = self._y_train_std
self.y_train_mean_ = self._y_train_mean
elif int(sklearn.__version__[2:4]) >= 19:
self.y_train_mean_ = self._y_train_mean
self.y_train_std_ = 1
else:
self.y_train_mean_ = self.y_train_mean
self.y_train_std_ = 1
return self
[docs] def predict(self, X, return_std=False, return_cov=False,
return_mean_grad=False, return_std_grad=False):
"""
Predict output for X.
In addition to the mean of the predictive distribution, also its
standard deviation (return_std=True) or covariance (return_cov=True),
the gradient of the mean and the standard-deviation with respect to X
can be optionally provided.
Parameters
----------
X : array-like, shape = (n_samples, n_features)
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.
return_mean_grad : bool, default: False
Whether or not to return the gradient of the mean.
Only valid when X is a single point.
return_std_grad : bool, default: False
Whether or not to return the gradient of the std.
Only valid when X is a single point.
Returns
-------
y_mean : array, shape = (n_samples, [n_output_dims])
Mean of predictive distribution a query points
y_std : array, shape = (n_samples,), optional
Standard deviation of predictive distribution at query points.
Only returned when return_std is True.
y_cov : array, shape = (n_samples, n_samples), optional
Covariance of joint predictive distribution a query points.
Only returned when return_cov is True.
y_mean_grad : shape = (n_samples, n_features)
The gradient of the predicted mean
y_std_grad : shape = (n_samples, n_features)
The gradient of the predicted std.
"""
if return_std and return_cov:
raise RuntimeError(
"Not returning standard deviation of predictions when "
"returning full covariance.")
if return_std_grad and not return_std:
raise ValueError(
"Not returning std_gradient without returning "
"the std.")
X = check_array(X)
if X.shape[0] != 1 and (return_mean_grad or return_std_grad):
raise ValueError("Not implemented for n_samples > 1")
if not hasattr(self, "X_train_"): # Not fit; predict based on GP prior
y_mean = np.zeros(X.shape[0])
if return_cov:
y_cov = self.kernel(X)
return y_mean, y_cov
elif return_std:
y_var = self.kernel.diag(X)
return y_mean, np.sqrt(y_var)
else:
return y_mean
else: # Predict based on GP posterior
K_trans = self.kernel_(X, self.X_train_)
y_mean = K_trans.dot(self.alpha_) # Line 4 (y_mean = f_star)
# undo normalisation
y_mean = self.y_train_std_ * y_mean + self.y_train_mean_
if return_cov:
v = cho_solve((self.L_, True), K_trans.T) # Line 5
y_cov = self.kernel_(X) - K_trans.dot(v) # Line 6
# undo normalisation
y_cov = y_cov * self.y_train_std_**2
return y_mean, y_cov
elif return_std:
K_inv = self.K_inv_
# Compute variance of predictive distribution
y_var = self.kernel_.diag(X)
y_var -= np.einsum("ki,kj,ij->k", K_trans, K_trans, K_inv)
# 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
y_std = np.sqrt(y_var)
if return_mean_grad:
grad = self.kernel_.gradient_x(X[0], self.X_train_)
grad_mean = np.dot(grad.T, self.alpha_)
# undo normalisation
grad_mean = grad_mean * self.y_train_std_
if return_std_grad:
grad_std = np.zeros(X.shape[1])
if not np.allclose(y_std, grad_std):
grad_std = -np.dot(K_trans,
np.dot(K_inv, grad))[0] / y_std
# undo normalisation
grad_std = grad_std * self.y_train_std_**2
return y_mean, y_std, grad_mean, grad_std
if return_std:
return y_mean, y_std, grad_mean
else:
return y_mean, grad_mean
else:
if return_std:
return y_mean, y_std
else:
return y_mean
