"""Gaussian processes regression.""" # Authors: Jan Hendrik Metzen # Modified by: Pete Green # License: BSD 3 clause import warnings from operator import itemgetter import numpy as np from scipy.linalg import cholesky, cho_solve 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.validation import check_array from ..utils.optimize import _check_optimize_result from ..utils.validation import _deprecate_positional_args class GaussianProcessRegressor(MultiOutputMixin, RegressorMixin, BaseEstimator): """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, 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 `. .. 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 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' is the objective function to be minimized, 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 'L-BGFS-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 the target values y are normalized, the mean and variance of the target values are set equal to 0 and 1 respectively. 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 `. 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`` 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...])) """ @_deprecate_positional_args 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 : 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) if self.kernel_.requires_vector_input: X, y = self._validate_data(X, y, multi_output=True, y_numeric=True, ensure_2d=True, dtype="numeric") else: X, y = self._validate_data(X, y, multi_output=True, y_numeric=True, ensure_2d=False, dtype=None) # 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 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, 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 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 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 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, [n_output_dims]) 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( "Not returning standard deviation of predictions when " "returning full covariance.") if self.kernel is None or self.kernel.requires_vector_input: X = check_array(X, ensure_2d=True, dtype="numeric") else: X = check_array(X, ensure_2d=False, dtype=None) 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 = 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: # Solve K @ V = K_trans.T 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: # Solve K @ V = K_trans.T V = cho_solve((self.L_, True), K_trans.T) # Line 5 # Compute variance of predictive distribution # Use einsum to avoid explicitly forming the large matrix # K_trans @ V just to extract its diagonal afterward. y_var = self.kernel_.diag(X) y_var -= np.einsum("ij,ji->i", K_trans, 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 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, n_features) or list of object Query points where the GP is evaluated. n_samples : int, default=1 The number of samples drawn from the Gaussian process 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 `. Returns ------- y_samples : ndarray of 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, clone_kernel=True): """Returns 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_) 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(tmp.dot(K_gradient))" without # constructing the full matrix tmp.dot(K_gradient) since only # its diagonal is required log_likelihood_gradient_dims = \ 0.5 * np.einsum("ijl,jik->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": 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("Unknown optimizer %s." % self.optimizer) return theta_opt, func_min def _more_tags(self): return {'requires_fit': False}