"""Gaussian processes regression.""" # Authors: Jan Hendrik Metzen # Modified by: Pete Green # License: BSD 3 clause import warnings from numbers import Integral, Real from operator import itemgetter import numpy as np import scipy.optimize from scipy.linalg import cho_solve, cholesky, solve_triangular from ..base import BaseEstimator, MultiOutputMixin, RegressorMixin, _fit_context, clone from ..preprocessing._data import _handle_zeros_in_scale from ..utils import check_random_state from ..utils._param_validation import Interval, StrOptions from ..utils.optimize import _check_optimize_result from .kernels import RBF, Kernel from .kernels import ConstantKernel as C GPR_CHOLESKY_LOWER = True class GaussianProcessRegressor(MultiOutputMixin, RegressorMixin, BaseEstimator): """Gaussian process regression (GPR). The implementation is based on Algorithm 2.1 of [RW2006]_. 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. To learn the difference between a point-estimate approach vs. a more Bayesian modelling approach, refer to the example entitled :ref:`sphx_glr_auto_examples_gaussian_process_plot_compare_gpr_krr.py`. 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 :class:`~sklearn.linear_model.Ridge`. optimizer : "fmin_l_bfgs_b", callable or None, 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 normalize 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. n_targets : int, default=None The number of dimensions of the target values. Used to decide the number of outputs when sampling from the prior distributions (i.e. calling :meth:`sample_y` before :meth:`fit`). This parameter is ignored once :meth:`fit` has been called. .. versionadded:: 1.3 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``. 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 ---------- .. [RW2006] `Carl E. Rasmussen and Christopher K.I. Williams, "Gaussian Processes for Machine Learning", MIT Press 2006 `_ 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...])) """ _parameter_constraints: dict = { "kernel": [None, Kernel], "alpha": [Interval(Real, 0, None, closed="left"), np.ndarray], "optimizer": [StrOptions({"fmin_l_bfgs_b"}), callable, None], "n_restarts_optimizer": [Interval(Integral, 0, None, closed="left")], "normalize_y": ["boolean"], "copy_X_train": ["boolean"], "n_targets": [Interval(Integral, 1, None, closed="left"), None], "random_state": ["random_state"], } def __init__( self, kernel=None, *, alpha=1e-10, optimizer="fmin_l_bfgs_b", n_restarts_optimizer=0, normalize_y=False, copy_X_train=True, n_targets=None, 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.n_targets = n_targets self.random_state = random_state @_fit_context(prefer_skip_nested_validation=True) 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, ) n_targets_seen = y.shape[1] if y.ndim > 1 else 1 if self.n_targets is not None and n_targets_seen != self.n_targets: raise ValueError( "The number of targets seen in `y` is different from the parameter " f"`n_targets`. Got {n_targets_seen} != {self.n_targets}." ) # 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: shape_y_stats = (y.shape[1],) if y.ndim == 2 else 1 self._y_train_mean = np.zeros(shape=shape_y_stats) self._y_train_std = np.ones(shape=shape_y_stats) 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,) or (n_samples, n_targets), 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) or \ (n_samples, n_samples, n_targets), 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 n_targets = self.n_targets if self.n_targets is not None else 1 y_mean = np.zeros(shape=(X.shape[0], n_targets)).squeeze() if return_cov: y_cov = kernel(X) if n_targets > 1: y_cov = np.repeat( np.expand_dims(y_cov, -1), repeats=n_targets, axis=-1 ) return y_mean, y_cov elif return_std: y_var = kernel.diag(X) if n_targets > 1: y_var = np.repeat( np.expand_dims(y_var, -1), repeats=n_targets, axis=-1 ) 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 # if y_mean has shape (n_samples, 1), reshape to (n_samples,) if y_mean.ndim > 1 and y_mean.shape[1] == 1: y_mean = np.squeeze(y_mean, axis=1) # 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 = np.outer(y_cov, self._y_train_std**2).reshape( *y_cov.shape, -1 ) # if y_cov has shape (n_samples, n_samples, 1), reshape to # (n_samples, n_samples) if y_cov.shape[2] == 1: y_cov = np.squeeze(y_cov, axis=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).copy() 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 = np.outer(y_var, self._y_train_std**2).reshape( *y_var.shape, -1 ) # if y_var has shape (n_samples, 1), reshape to (n_samples,) if y_var.shape[1] == 1: y_var = np.squeeze(y_var, axis=1) 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_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 `. 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[:, target], y_cov[..., target], n_samples ).T[:, np.newaxis] for target 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 ): """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}