Source code for sklearn.gaussian_process.gpr

"""Gaussian processes regression. """

# Authors: Jan Hendrik Metzen <>
# License: BSD 3 clause

import warnings
from operator import itemgetter

import numpy as np
from scipy.linalg import cholesky, cho_solve, solve_triangular
from scipy.optimize import fmin_l_bfgs_b

from sklearn.base import BaseEstimator, RegressorMixin, clone
from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C
from sklearn.utils import check_random_state
from sklearn.utils.validation import check_X_y, check_array
from sklearn.utils.deprecation import deprecated

class GaussianProcessRegressor(BaseEstimator, RegressorMixin):
    """Gaussian process regression (GPR).

    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,

       * 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

    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.
        This can also prevent a potential numerical issue during fitting, by
        ensuring that the calculated values form a positive definite matrix.
        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::


    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

    random_state : int, RandomState instance or None, optional (default: None)
        The generator used to initialize the centers. If int, random_state is
        the seed used by the random number generator; If RandomState instance,
        random_state is the random number generator; If None, the random number
        generator is the RandomState instance used by `np.random`.

    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``

    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

    @deprecated("Attribute rng was deprecated in version 0.19 and "
                "will be removed in 0.21.")
    def rng(self):
        return self._rng

    @deprecated("Attribute y_train_mean was deprecated in version 0.19 and "
                "will be removed in 0.21.")
    def y_train_mean(self):
        return self._y_train_mean

[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 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) X, y = check_X_y(X, y, multi_output=True, y_numeric=True) # Normalize target value if self.normalize_y: self._y_train_mean = np.mean(y, axis=0) # demean y y = y - self._y_train_mean else: self._y_train_mean = np.zeros(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 entries as y.(%d != %d)" % (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) return -lml, -grad else: return -self.log_marginal_likelihood(theta) # 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.log_marginal_likelihood_value_ = -np.min(lml_values) else: self.log_marginal_likelihood_value_ = \ self.log_marginal_likelihood(self.kernel_.theta) # Precompute quantities required for predictions which are independent # of actual query points K = self.kernel_(self.X_train_) K[np.diag_indices_from(K)] += self.alpha try: self.L_ = cholesky(K, lower=True) # Line 2 except np.linalg.LinAlgError as exc: exc.args = ("The kernel, %s, is not returning a " "positive definite matrix. Try gradually " "increasing the 'alpha' parameter of your " "GaussianProcessRegressor estimator." % self.kernel_,) + exc.args raise self.alpha_ = cho_solve((self.L_, True), self.y_train_) # Line 3 return self
[docs] 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, also 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, 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 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. """ if return_std and return_cov: raise RuntimeError( "Not returning standard deviation of predictions when " "returning full covariance.") X = check_array(X) 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 K_trans = self.kernel_(X, self.X_train_) y_mean = # Line 4 (y_mean = f_star) y_mean = self._y_train_mean + y_mean # undo normal. if return_cov: v = cho_solve((self.L_, True), K_trans.T) # Line 5 y_cov = self.kernel_(X) - # Line 6 return y_mean, y_cov elif return_std: # compute inverse K_inv of K based on its Cholesky # decomposition L and its inverse L_inv L_inv = solve_triangular(self.L_.T, np.eye(self.L_.shape[0])) K_inv = # Compute variance of predictive distribution y_var = self.kernel_.diag(X) y_var -= np.einsum("ij,ij->i",, K_inv), K_trans) # 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 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, shape = (n_samples_X, n_features) Query points where the GP samples are evaluated n_samples : int, default: 1 The number of samples drawn from the Gaussian process random_state : int, RandomState instance or None, optional (default=0) If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by `np.random`. Returns ------- y_samples : array, shape = (n_samples_X, [n_output_dims], 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
def log_marginal_likelihood(self, theta=None, eval_gradient=False): """Returns log-marginal likelihood of theta for training data. Parameters ---------- theta : array-like, shape = (n_kernel_params,) or 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. Returns ------- log_likelihood : float Log-marginal likelihood of theta for training data. log_likelihood_gradient : array, 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_ kernel = self.kernel_.clone_with_theta(theta) if eval_gradient: K, K_gradient = kernel(self.X_train_, eval_gradient=True) else: K = kernel(self.X_train_) K[np.diag_indices_from(K)] += self.alpha try: L = cholesky(K, lower=True) # Line 2 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] alpha = cho_solve((L, True), y_train) # Line 3 # Compute log-likelihood (compare line 7) 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) log_likelihood = log_likelihood_dims.sum(-1) # sum over dimensions if eval_gradient: # compare Equation 5.9 from GPML tmp = np.einsum("ik,jk->ijk", alpha, alpha) # k: output-dimension tmp -= cho_solve((L, True), np.eye(K.shape[0]))[:, :, np.newaxis] # Compute "0.5 * trace(" without # constructing the full matrix since only # its diagonal is required log_likelihood_gradient_dims = \ 0.5 * np.einsum("ijl,ijk->kl", tmp, K_gradient) log_likelihood_gradient = log_likelihood_gradient_dims.sum(-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": theta_opt, func_min, convergence_dict = \ fmin_l_bfgs_b(obj_func, initial_theta, bounds=bounds) if convergence_dict["warnflag"] != 0: warnings.warn("fmin_l_bfgs_b terminated abnormally with the " " state: %s" % convergence_dict) elif callable(self.optimizer): theta_opt, func_min = \ self.optimizer(obj_func, initial_theta, bounds=bounds) else: raise ValueError("Unknown optimizer %s." % self.optimizer) return theta_opt, func_min