"""Gaussian processes classification.""" # 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 from scipy.optimize import fmin_l_bfgs_b from scipy.special import erf, expit from sklearn.base import BaseEstimator, ClassifierMixin, clone from sklearn.gaussian_process.kernels \ import RBF, CompoundKernel, ConstantKernel as C from sklearn.utils.validation import check_X_y, check_is_fitted, check_array from sklearn.utils import check_random_state from sklearn.preprocessing import LabelEncoder from sklearn.multiclass import OneVsRestClassifier, OneVsOneClassifier # Values required for approximating the logistic sigmoid by # error functions. coefs are obtained via: # x = np.array([0, 0.6, 2, 3.5, 4.5, np.inf]) # b = logistic(x) # A = (erf(np.dot(x, self.lambdas)) + 1) / 2 # coefs = lstsq(A, b)[0] LAMBDAS = np.array([0.41, 0.4, 0.37, 0.44, 0.39])[:, np.newaxis] COEFS = np.array([-1854.8214151, 3516.89893646, 221.29346712, 128.12323805, -2010.49422654])[:, np.newaxis] class _BinaryGaussianProcessClassifierLaplace(BaseEstimator): """Binary Gaussian process classification based on Laplace approximation. The implementation is based on Algorithm 3.1, 3.2, and 5.1 of ``Gaussian Processes for Machine Learning'' (GPML) by Rasmussen and Williams. Internally, the Laplace approximation is used for approximating the non-Gaussian posterior by a Gaussian. Currently, the implementation is restricted to using the logistic link function. .. versionadded:: 0.18 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. 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. max_iter_predict: int, optional (default: 100) The maximum number of iterations in Newton's method for approximating the posterior during predict. Smaller values will reduce computation time at the cost of worse results. warm_start : bool, optional (default: False) If warm-starts are enabled, the solution of the last Newton iteration on the Laplace approximation of the posterior mode is used as initialization for the next call of _posterior_mode(). This can speed up convergence when _posterior_mode is called several times on similar problems as in hyperparameter optimization. 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 : 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`. 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,) Target values in training data (also required for prediction) classes_ : array-like, shape = (n_classes,) Unique class labels. 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_ pi_ : array-like, shape = (n_samples,) The probabilities of the positive class for the training points X_train_ W_sr_ : array-like, shape = (n_samples,) Square root of W, the Hessian of log-likelihood of the latent function values for the observed labels. Since W is diagonal, only the diagonal of sqrt(W) is stored. log_marginal_likelihood_value_ : float The log-marginal-likelihood of ``self.kernel_.theta`` """ def __init__(self, kernel=None, optimizer="fmin_l_bfgs_b", n_restarts_optimizer=0, max_iter_predict=100, warm_start=False, copy_X_train=True, random_state=None): self.kernel = kernel self.optimizer = optimizer self.n_restarts_optimizer = n_restarts_optimizer self.max_iter_predict = max_iter_predict self.warm_start = warm_start self.copy_X_train = copy_X_train self.random_state = random_state def fit(self, X, y): """Fit Gaussian process classification model Parameters ---------- X : array-like, shape = (n_samples, n_features) Training data y : array-like, shape = (n_samples,) Target values, must be binary 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) self.X_train_ = np.copy(X) if self.copy_X_train else X # Encode class labels and check that it is a binary classification # problem label_encoder = LabelEncoder() self.y_train_ = label_encoder.fit_transform(y) self.classes_ = label_encoder.classes_ if self.classes_.size > 2: raise ValueError("%s supports only binary classification. " "y contains classes %s" % (self.__class__.__name__, self.classes_)) elif self.classes_.size == 1: raise ValueError("{0:s} requires 2 classes.".format( self.__class__.__name__)) 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 = np.exp(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_) _, (self.pi_, self.W_sr_, self.L_, _, _) = \ self._posterior_mode(K, return_temporaries=True) return self def predict(self, X): """Perform classification on an array of test vectors X. Parameters ---------- X : array-like, shape = (n_samples, n_features) Returns ------- C : array, shape = (n_samples,) Predicted target values for X, values are from ``classes_`` """ check_is_fitted(self, ["X_train_", "y_train_", "pi_", "W_sr_", "L_"]) # As discussed on Section 3.4.2 of GPML, for making hard binary # decisions, it is enough to compute the MAP of the posterior and # pass it through the link function K_star = self.kernel_(self.X_train_, X) # K_star =k(x_star) f_star = K_star.T.dot(self.y_train_ - self.pi_) # Algorithm 3.2,Line 4 return np.where(f_star > 0, self.classes_[1], self.classes_[0]) def predict_proba(self, X): """Return probability estimates for the test vector X. Parameters ---------- X : array-like, shape = (n_samples, n_features) Returns ------- C : array-like, shape = (n_samples, n_classes) Returns the probability of the samples for each class in the model. The columns correspond to the classes in sorted order, as they appear in the attribute ``classes_``. """ check_is_fitted(self, ["X_train_", "y_train_", "pi_", "W_sr_", "L_"]) # Based on Algorithm 3.2 of GPML K_star = self.kernel_(self.X_train_, X) # K_star =k(x_star) f_star = K_star.T.dot(self.y_train_ - self.pi_) # Line 4 v = solve(self.L_, self.W_sr_[:, np.newaxis] * K_star) # Line 5 # Line 6 (compute np.diag(v.T.dot(v)) via einsum) var_f_star = self.kernel_.diag(X) - np.einsum("ij,ij->j", v, v) # Line 7: # Approximate \int log(z) * N(z | f_star, var_f_star) # Approximation is due to Williams & Barber, "Bayesian Classification # with Gaussian Processes", Appendix A: Approximate the logistic # sigmoid by a linear combination of 5 error functions. # For information on how this integral can be computed see # blitiri.blogspot.de/2012/11/gaussian-integral-of-error-function.html alpha = 1 / (2 * var_f_star) gamma = LAMBDAS * f_star integrals = np.sqrt(np.pi / alpha) \ * erf(gamma * np.sqrt(alpha / (alpha + LAMBDAS**2))) \ / (2 * np.sqrt(var_f_star * 2 * np.pi)) pi_star = (COEFS * integrals).sum(axis=0) + .5 * COEFS.sum() return np.vstack((1 - pi_star, pi_star)).T 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_) # Compute log-marginal-likelihood Z and also store some temporaries # which can be reused for computing Z's gradient Z, (pi, W_sr, L, b, a) = \ self._posterior_mode(K, return_temporaries=True) if not eval_gradient: return Z # Compute gradient based on Algorithm 5.1 of GPML d_Z = np.empty(theta.shape[0]) # XXX: Get rid of the np.diag() in the next line R = W_sr[:, np.newaxis] * cho_solve((L, True), np.diag(W_sr)) # Line 7 C = solve(L, W_sr[:, np.newaxis] * K) # Line 8 # Line 9: (use einsum to compute np.diag(C.T.dot(C)))) s_2 = -0.5 * (np.diag(K) - np.einsum('ij, ij -> j', C, C)) \ * (pi * (1 - pi) * (1 - 2 * pi)) # third derivative for j in range(d_Z.shape[0]): C = K_gradient[:, :, j] # Line 11 # Line 12: (R.T.ravel().dot(C.ravel()) = np.trace(R.dot(C))) s_1 = .5 * a.T.dot(C).dot(a) - .5 * R.T.ravel().dot(C.ravel()) b = C.dot(self.y_train_ - pi) # Line 13 s_3 = b - K.dot(R.dot(b)) # Line 14 d_Z[j] = s_1 + s_2.T.dot(s_3) # Line 15 return Z, d_Z def _posterior_mode(self, K, return_temporaries=False): """Mode-finding for binary Laplace GPC and fixed kernel. This approximates the posterior of the latent function values for given inputs and target observations with a Gaussian approximation and uses Newton's iteration to find the mode of this approximation. """ # Based on Algorithm 3.1 of GPML # If warm_start are enabled, we reuse the last solution for the # posterior mode as initialization; otherwise, we initialize with 0 if self.warm_start and hasattr(self, "f_cached") \ and self.f_cached.shape == self.y_train_.shape: f = self.f_cached else: f = np.zeros_like(self.y_train_, dtype=np.float64) # Use Newton's iteration method to find mode of Laplace approximation log_marginal_likelihood = -np.inf for _ in range(self.max_iter_predict): # Line 4 pi = expit(f) W = pi * (1 - pi) # Line 5 W_sr = np.sqrt(W) W_sr_K = W_sr[:, np.newaxis] * K B = np.eye(W.shape[0]) + W_sr_K * W_sr L = cholesky(B, lower=True) # Line 6 b = W * f + (self.y_train_ - pi) # Line 7 a = b - W_sr * cho_solve((L, True), W_sr_K.dot(b)) # Line 8 f = K.dot(a) # Line 10: Compute log marginal likelihood in loop and use as # convergence criterion lml = -0.5 * a.T.dot(f) \ - np.log(1 + np.exp(-(self.y_train_ * 2 - 1) * f)).sum() \ - np.log(np.diag(L)).sum() # Check if we have converged (log marginal likelihood does # not decrease) # XXX: more complex convergence criterion if lml - log_marginal_likelihood < 1e-10: break log_marginal_likelihood = lml self.f_cached = f # Remember solution for later warm-starts if return_temporaries: return log_marginal_likelihood, (pi, W_sr, L, b, a) else: return log_marginal_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 class GaussianProcessClassifier(BaseEstimator, ClassifierMixin): """Gaussian process classification (GPC) based on Laplace approximation. The implementation is based on Algorithm 3.1, 3.2, and 5.1 of Gaussian Processes for Machine Learning (GPML) by Rasmussen and Williams. Internally, the Laplace approximation is used for approximating the non-Gaussian posterior by a Gaussian. Currently, the implementation is restricted to using the logistic link function. For multi-class classification, several binary one-versus rest classifiers are fitted. Note that this class thus does not implement a true multi-class Laplace approximation. 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. 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. max_iter_predict : int, optional (default: 100) The maximum number of iterations in Newton's method for approximating the posterior during predict. Smaller values will reduce computation time at the cost of worse results. warm_start : bool, optional (default: False) If warm-starts are enabled, the solution of the last Newton iteration on the Laplace approximation of the posterior mode is used as initialization for the next call of _posterior_mode(). This can speed up convergence when _posterior_mode is called several times on similar problems as in hyperparameter optimization. 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 : 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`. multi_class : string, default : "one_vs_rest" Specifies how multi-class classification problems are handled. Supported are "one_vs_rest" and "one_vs_one". In "one_vs_rest", one binary Gaussian process classifier is fitted for each class, which is trained to separate this class from the rest. In "one_vs_one", one binary Gaussian process classifier is fitted for each pair of classes, which is trained to separate these two classes. The predictions of these binary predictors are combined into multi-class predictions. Note that "one_vs_one" does not support predicting probability estimates. n_jobs : int, optional, default: 1 The number of jobs to use for the computation. If -1 all CPUs are used. If 1 is given, no parallel computing code is used at all, which is useful for debugging. For n_jobs below -1, (n_cpus + 1 + n_jobs) are used. Thus for n_jobs = -2, all CPUs but one are used. Attributes ---------- kernel_ : kernel object The kernel used for prediction. In case of binary classification, the structure of the kernel is the same as the one passed as parameter but with optimized hyperparameters. In case of multi-class classification, a CompoundKernel is returned which consists of the different kernels used in the one-versus-rest classifiers. log_marginal_likelihood_value_ : float The log-marginal-likelihood of ``self.kernel_.theta`` classes_ : array-like, shape = (n_classes,) Unique class labels. n_classes_ : int The number of classes in the training data .. versionadded:: 0.18 """ def __init__(self, kernel=None, optimizer="fmin_l_bfgs_b", n_restarts_optimizer=0, max_iter_predict=100, warm_start=False, copy_X_train=True, random_state=None, multi_class="one_vs_rest", n_jobs=1): self.kernel = kernel self.optimizer = optimizer self.n_restarts_optimizer = n_restarts_optimizer self.max_iter_predict = max_iter_predict self.warm_start = warm_start self.copy_X_train = copy_X_train self.random_state = random_state self.multi_class = multi_class self.n_jobs = n_jobs def fit(self, X, y): """Fit Gaussian process classification model Parameters ---------- X : array-like, shape = (n_samples, n_features) Training data y : array-like, shape = (n_samples,) Target values, must be binary Returns ------- self : returns an instance of self. """ X, y = check_X_y(X, y, multi_output=False) self.base_estimator_ = _BinaryGaussianProcessClassifierLaplace( self.kernel, self.optimizer, self.n_restarts_optimizer, self.max_iter_predict, self.warm_start, self.copy_X_train, self.random_state) self.classes_ = np.unique(y) self.n_classes_ = self.classes_.size if self.n_classes_ == 1: raise ValueError("GaussianProcessClassifier requires 2 or more " "distinct classes. Only class %s present." % self.classes_[0]) if self.n_classes_ > 2: if self.multi_class == "one_vs_rest": self.base_estimator_ = \ OneVsRestClassifier(self.base_estimator_, n_jobs=self.n_jobs) elif self.multi_class == "one_vs_one": self.base_estimator_ = \ OneVsOneClassifier(self.base_estimator_, n_jobs=self.n_jobs) else: raise ValueError("Unknown multi-class mode %s" % self.multi_class) self.base_estimator_.fit(X, y) if self.n_classes_ > 2: self.log_marginal_likelihood_value_ = np.mean( [estimator.log_marginal_likelihood() for estimator in self.base_estimator_.estimators_]) else: self.log_marginal_likelihood_value_ = \ self.base_estimator_.log_marginal_likelihood() return self def predict(self, X): """Perform classification on an array of test vectors X. Parameters ---------- X : array-like, shape = (n_samples, n_features) Returns ------- C : array, shape = (n_samples,) Predicted target values for X, values are from ``classes_`` """ check_is_fitted(self, ["classes_", "n_classes_"]) X = check_array(X) return self.base_estimator_.predict(X) def predict_proba(self, X): """Return probability estimates for the test vector X. Parameters ---------- X : array-like, shape = (n_samples, n_features) Returns ------- C : array-like, shape = (n_samples, n_classes) Returns the probability of the samples for each class in the model. The columns correspond to the classes in sorted order, as they appear in the attribute `classes_`. """ check_is_fitted(self, ["classes_", "n_classes_"]) if self.n_classes_ > 2 and self.multi_class == "one_vs_one": raise ValueError("one_vs_one multi-class mode does not support " "predicting probability estimates. Use " "one_vs_rest mode instead.") X = check_array(X) return self.base_estimator_.predict_proba(X) @property def kernel_(self): if self.n_classes_ == 2: return self.base_estimator_.kernel_ else: return CompoundKernel( [estimator.kernel_ for estimator in self.base_estimator_.estimators_]) def log_marginal_likelihood(self, theta=None, eval_gradient=False): """Returns log-marginal likelihood of theta for training data. In the case of multi-class classification, the mean log-marginal likelihood of the one-versus-rest classifiers are returned. Parameters ---------- theta : array-like, shape = (n_kernel_params,) or none Kernel hyperparameters for which the log-marginal likelihood is evaluated. In the case of multi-class classification, theta may be the hyperparameters of the compound kernel or of an individual kernel. In the latter case, all individual kernel get assigned the same theta values. 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. Note that gradient computation is not supported for non-binary classification. 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. """ check_is_fitted(self, ["classes_", "n_classes_"]) if theta is None: if eval_gradient: raise ValueError( "Gradient can only be evaluated for theta!=None") return self.log_marginal_likelihood_value_ theta = np.asarray(theta) if self.n_classes_ == 2: return self.base_estimator_.log_marginal_likelihood( theta, eval_gradient) else: if eval_gradient: raise NotImplementedError( "Gradient of log-marginal-likelihood not implemented for " "multi-class GPC.") estimators = self.base_estimator_.estimators_ n_dims = estimators[0].kernel_.n_dims if theta.shape[0] == n_dims: # use same theta for all sub-kernels return np.mean( [estimator.log_marginal_likelihood(theta) for i, estimator in enumerate(estimators)]) elif theta.shape[0] == n_dims * self.classes_.shape[0]: # theta for compound kernel return np.mean( [estimator.log_marginal_likelihood( theta[n_dims * i:n_dims * (i + 1)]) for i, estimator in enumerate(estimators)]) else: raise ValueError("Shape of theta must be either %d or %d. " "Obtained theta with shape %d." % (n_dims, n_dims * self.classes_.shape[0], theta.shape[0]))