import warnings import numpy as np import packaging.version import sklearn from scipy.linalg import cho_solve, solve_triangular from sklearn.gaussian_process import ( GaussianProcessRegressor as sk_GaussianProcessRegressor, ) from sklearn.utils import check_array from .kernels import RBF, ConstantKernel, Sum, 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, "_" 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". """ 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().__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, ) 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 and not _param_for_white_kernel_in_Sum(self.kernel)[0]: 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().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 sklearn_version = packaging.version.Version(sklearn.__version__) if sklearn_version.major == 1 or ( sklearn_version.major == 0 and sklearn_version.minor >= 23 ): self.y_train_std_ = self._y_train_std self.y_train_mean_ = self._y_train_mean elif sklearn_version.major == 0 and sklearn_version.minor >= 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 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