""" Various bayesian regression """ from __future__ import print_function # Authors: V. Michel, F. Pedregosa, A. Gramfort # License: BSD 3 clause from math import log import numpy as np from scipy import linalg from scipy.linalg import pinvh from .base import LinearModel, _rescale_data from ..base import RegressorMixin from ..utils.extmath import fast_logdet from ..utils import check_X_y ############################################################################### # BayesianRidge regression class BayesianRidge(LinearModel, RegressorMixin): """Bayesian ridge regression Fit a Bayesian ridge model and optimize the regularization parameters lambda (precision of the weights) and alpha (precision of the noise). Read more in the :ref:`User Guide `. Parameters ---------- n_iter : int, optional Maximum number of iterations. Default is 300. tol : float, optional Stop the algorithm if w has converged. Default is 1.e-3. alpha_1 : float, optional Hyper-parameter : shape parameter for the Gamma distribution prior over the alpha parameter. Default is 1.e-6 alpha_2 : float, optional Hyper-parameter : inverse scale parameter (rate parameter) for the Gamma distribution prior over the alpha parameter. Default is 1.e-6. lambda_1 : float, optional Hyper-parameter : shape parameter for the Gamma distribution prior over the lambda parameter. Default is 1.e-6. lambda_2 : float, optional Hyper-parameter : inverse scale parameter (rate parameter) for the Gamma distribution prior over the lambda parameter. Default is 1.e-6 compute_score : boolean, optional If True, compute the objective function at each step of the model. Default is False fit_intercept : boolean, optional whether to calculate the intercept for this model. If set to false, no intercept will be used in calculations (e.g. data is expected to be already centered). Default is True. normalize : boolean, optional, default False This parameter is ignored when ``fit_intercept`` is set to False. If True, the regressors X will be normalized before regression by subtracting the mean and dividing by the l2-norm. If you wish to standardize, please use :class:`sklearn.preprocessing.StandardScaler` before calling ``fit`` on an estimator with ``normalize=False``. copy_X : boolean, optional, default True If True, X will be copied; else, it may be overwritten. verbose : boolean, optional, default False Verbose mode when fitting the model. Attributes ---------- coef_ : array, shape = (n_features) Coefficients of the regression model (mean of distribution) alpha_ : float estimated precision of the noise. lambda_ : float estimated precision of the weights. sigma_ : array, shape = (n_features, n_features) estimated variance-covariance matrix of the weights scores_ : float if computed, value of the objective function (to be maximized) Examples -------- >>> from sklearn import linear_model >>> clf = linear_model.BayesianRidge() >>> clf.fit([[0,0], [1, 1], [2, 2]], [0, 1, 2]) ... # doctest: +NORMALIZE_WHITESPACE BayesianRidge(alpha_1=1e-06, alpha_2=1e-06, compute_score=False, copy_X=True, fit_intercept=True, lambda_1=1e-06, lambda_2=1e-06, n_iter=300, normalize=False, tol=0.001, verbose=False) >>> clf.predict([[1, 1]]) array([1.]) Notes ----- For an example, see :ref:`examples/linear_model/plot_bayesian_ridge.py `. References ---------- D. J. C. MacKay, Bayesian Interpolation, Computation and Neural Systems, Vol. 4, No. 3, 1992. R. Salakhutdinov, Lecture notes on Statistical Machine Learning, http://www.utstat.toronto.edu/~rsalakhu/sta4273/notes/Lecture2.pdf#page=15 Their beta is our ``self.alpha_`` Their alpha is our ``self.lambda_`` """ def __init__(self, n_iter=300, tol=1.e-3, alpha_1=1.e-6, alpha_2=1.e-6, lambda_1=1.e-6, lambda_2=1.e-6, compute_score=False, fit_intercept=True, normalize=False, copy_X=True, verbose=False): self.n_iter = n_iter self.tol = tol self.alpha_1 = alpha_1 self.alpha_2 = alpha_2 self.lambda_1 = lambda_1 self.lambda_2 = lambda_2 self.compute_score = compute_score self.fit_intercept = fit_intercept self.normalize = normalize self.copy_X = copy_X self.verbose = verbose def fit(self, X, y, sample_weight=None): """Fit the model Parameters ---------- X : numpy array of shape [n_samples,n_features] Training data y : numpy array of shape [n_samples] Target values. Will be cast to X's dtype if necessary sample_weight : numpy array of shape [n_samples] Individual weights for each sample .. versionadded:: 0.20 parameter *sample_weight* support to BayesianRidge. Returns ------- self : returns an instance of self. """ X, y = check_X_y(X, y, dtype=np.float64, y_numeric=True) X, y, X_offset_, y_offset_, X_scale_ = self._preprocess_data( X, y, self.fit_intercept, self.normalize, self.copy_X, sample_weight=sample_weight) if sample_weight is not None: # Sample weight can be implemented via a simple rescaling. X, y = _rescale_data(X, y, sample_weight) self.X_offset_ = X_offset_ self.X_scale_ = X_scale_ n_samples, n_features = X.shape # Initialization of the values of the parameters eps = np.finfo(np.float64).eps # Add `eps` in the denominator to omit division by zero if `np.var(y)` # is zero alpha_ = 1. / (np.var(y) + eps) lambda_ = 1. verbose = self.verbose lambda_1 = self.lambda_1 lambda_2 = self.lambda_2 alpha_1 = self.alpha_1 alpha_2 = self.alpha_2 self.scores_ = list() coef_old_ = None XT_y = np.dot(X.T, y) U, S, Vh = linalg.svd(X, full_matrices=False) eigen_vals_ = S ** 2 # Convergence loop of the bayesian ridge regression for iter_ in range(self.n_iter): # Compute mu and sigma # sigma_ = lambda_ / alpha_ * np.eye(n_features) + np.dot(X.T, X) # coef_ = sigma_^-1 * XT * y if n_samples > n_features: coef_ = np.dot(Vh.T, Vh / (eigen_vals_ + lambda_ / alpha_)[:, np.newaxis]) coef_ = np.dot(coef_, XT_y) if self.compute_score: logdet_sigma_ = - np.sum( np.log(lambda_ + alpha_ * eigen_vals_)) else: coef_ = np.dot(X.T, np.dot( U / (eigen_vals_ + lambda_ / alpha_)[None, :], U.T)) coef_ = np.dot(coef_, y) if self.compute_score: logdet_sigma_ = np.full(n_features, lambda_, dtype=np.array(lambda_).dtype) logdet_sigma_[:n_samples] += alpha_ * eigen_vals_ logdet_sigma_ = - np.sum(np.log(logdet_sigma_)) # Preserve the alpha and lambda values that were used to # calculate the final coefficients self.alpha_ = alpha_ self.lambda_ = lambda_ # Update alpha and lambda rmse_ = np.sum((y - np.dot(X, coef_)) ** 2) gamma_ = (np.sum((alpha_ * eigen_vals_) / (lambda_ + alpha_ * eigen_vals_))) lambda_ = ((gamma_ + 2 * lambda_1) / (np.sum(coef_ ** 2) + 2 * lambda_2)) alpha_ = ((n_samples - gamma_ + 2 * alpha_1) / (rmse_ + 2 * alpha_2)) # Compute the objective function if self.compute_score: s = lambda_1 * log(lambda_) - lambda_2 * lambda_ s += alpha_1 * log(alpha_) - alpha_2 * alpha_ s += 0.5 * (n_features * log(lambda_) + n_samples * log(alpha_) - alpha_ * rmse_ - (lambda_ * np.sum(coef_ ** 2)) - logdet_sigma_ - n_samples * log(2 * np.pi)) self.scores_.append(s) # Check for convergence if iter_ != 0 and np.sum(np.abs(coef_old_ - coef_)) < self.tol: if verbose: print("Convergence after ", str(iter_), " iterations") break coef_old_ = np.copy(coef_) self.coef_ = coef_ sigma_ = np.dot(Vh.T, Vh / (eigen_vals_ + lambda_ / alpha_)[:, np.newaxis]) self.sigma_ = (1. / alpha_) * sigma_ self._set_intercept(X_offset_, y_offset_, X_scale_) return self def predict(self, X, return_std=False): """Predict using the linear model. In addition to the mean of the predictive distribution, also its standard deviation can be returned. Parameters ---------- X : {array-like, sparse matrix}, shape = (n_samples, n_features) Samples. return_std : boolean, optional Whether to return the standard deviation of posterior prediction. Returns ------- y_mean : array, shape = (n_samples,) Mean of predictive distribution of query points. y_std : array, shape = (n_samples,) Standard deviation of predictive distribution of query points. """ y_mean = self._decision_function(X) if return_std is False: return y_mean else: if self.normalize: X = (X - self.X_offset_) / self.X_scale_ sigmas_squared_data = (np.dot(X, self.sigma_) * X).sum(axis=1) y_std = np.sqrt(sigmas_squared_data + (1. / self.alpha_)) return y_mean, y_std ############################################################################### # ARD (Automatic Relevance Determination) regression class ARDRegression(LinearModel, RegressorMixin): """Bayesian ARD regression. Fit the weights of a regression model, using an ARD prior. The weights of the regression model are assumed to be in Gaussian distributions. Also estimate the parameters lambda (precisions of the distributions of the weights) and alpha (precision of the distribution of the noise). The estimation is done by an iterative procedures (Evidence Maximization) Read more in the :ref:`User Guide `. Parameters ---------- n_iter : int, optional Maximum number of iterations. Default is 300 tol : float, optional Stop the algorithm if w has converged. Default is 1.e-3. alpha_1 : float, optional Hyper-parameter : shape parameter for the Gamma distribution prior over the alpha parameter. Default is 1.e-6. alpha_2 : float, optional Hyper-parameter : inverse scale parameter (rate parameter) for the Gamma distribution prior over the alpha parameter. Default is 1.e-6. lambda_1 : float, optional Hyper-parameter : shape parameter for the Gamma distribution prior over the lambda parameter. Default is 1.e-6. lambda_2 : float, optional Hyper-parameter : inverse scale parameter (rate parameter) for the Gamma distribution prior over the lambda parameter. Default is 1.e-6. compute_score : boolean, optional If True, compute the objective function at each step of the model. Default is False. threshold_lambda : float, optional threshold for removing (pruning) weights with high precision from the computation. Default is 1.e+4. fit_intercept : boolean, optional whether to calculate the intercept for this model. If set to false, no intercept will be used in calculations (e.g. data is expected to be already centered). Default is True. normalize : boolean, optional, default False This parameter is ignored when ``fit_intercept`` is set to False. If True, the regressors X will be normalized before regression by subtracting the mean and dividing by the l2-norm. If you wish to standardize, please use :class:`sklearn.preprocessing.StandardScaler` before calling ``fit`` on an estimator with ``normalize=False``. copy_X : boolean, optional, default True. If True, X will be copied; else, it may be overwritten. verbose : boolean, optional, default False Verbose mode when fitting the model. Attributes ---------- coef_ : array, shape = (n_features) Coefficients of the regression model (mean of distribution) alpha_ : float estimated precision of the noise. lambda_ : array, shape = (n_features) estimated precisions of the weights. sigma_ : array, shape = (n_features, n_features) estimated variance-covariance matrix of the weights scores_ : float if computed, value of the objective function (to be maximized) Examples -------- >>> from sklearn import linear_model >>> clf = linear_model.ARDRegression() >>> clf.fit([[0,0], [1, 1], [2, 2]], [0, 1, 2]) ... # doctest: +NORMALIZE_WHITESPACE ARDRegression(alpha_1=1e-06, alpha_2=1e-06, compute_score=False, copy_X=True, fit_intercept=True, lambda_1=1e-06, lambda_2=1e-06, n_iter=300, normalize=False, threshold_lambda=10000.0, tol=0.001, verbose=False) >>> clf.predict([[1, 1]]) array([1.]) Notes ----- For an example, see :ref:`examples/linear_model/plot_ard.py `. References ---------- D. J. C. MacKay, Bayesian nonlinear modeling for the prediction competition, ASHRAE Transactions, 1994. R. Salakhutdinov, Lecture notes on Statistical Machine Learning, http://www.utstat.toronto.edu/~rsalakhu/sta4273/notes/Lecture2.pdf#page=15 Their beta is our ``self.alpha_`` Their alpha is our ``self.lambda_`` ARD is a little different than the slide: only dimensions/features for which ``self.lambda_ < self.threshold_lambda`` are kept and the rest are discarded. """ def __init__(self, n_iter=300, tol=1.e-3, alpha_1=1.e-6, alpha_2=1.e-6, lambda_1=1.e-6, lambda_2=1.e-6, compute_score=False, threshold_lambda=1.e+4, fit_intercept=True, normalize=False, copy_X=True, verbose=False): self.n_iter = n_iter self.tol = tol self.fit_intercept = fit_intercept self.normalize = normalize self.alpha_1 = alpha_1 self.alpha_2 = alpha_2 self.lambda_1 = lambda_1 self.lambda_2 = lambda_2 self.compute_score = compute_score self.threshold_lambda = threshold_lambda self.copy_X = copy_X self.verbose = verbose def fit(self, X, y): """Fit the ARDRegression model according to the given training data and parameters. Iterative procedure to maximize the evidence Parameters ---------- X : array-like, shape = [n_samples, n_features] Training vector, where n_samples in the number of samples and n_features is the number of features. y : array, shape = [n_samples] Target values (integers). Will be cast to X's dtype if necessary Returns ------- self : returns an instance of self. """ X, y = check_X_y(X, y, dtype=np.float64, y_numeric=True, ensure_min_samples=2) n_samples, n_features = X.shape coef_ = np.zeros(n_features) X, y, X_offset_, y_offset_, X_scale_ = self._preprocess_data( X, y, self.fit_intercept, self.normalize, self.copy_X) # Launch the convergence loop keep_lambda = np.ones(n_features, dtype=bool) lambda_1 = self.lambda_1 lambda_2 = self.lambda_2 alpha_1 = self.alpha_1 alpha_2 = self.alpha_2 verbose = self.verbose # Initialization of the values of the parameters eps = np.finfo(np.float64).eps # Add `eps` in the denominator to omit division by zero if `np.var(y)` # is zero alpha_ = 1. / (np.var(y) + eps) lambda_ = np.ones(n_features) self.scores_ = list() coef_old_ = None # Compute sigma and mu (using Woodbury matrix identity) def update_sigma(X, alpha_, lambda_, keep_lambda, n_samples): sigma_ = pinvh(np.eye(n_samples) / alpha_ + np.dot(X[:, keep_lambda] * np.reshape(1. / lambda_[keep_lambda], [1, -1]), X[:, keep_lambda].T)) sigma_ = np.dot(sigma_, X[:, keep_lambda] * np.reshape(1. / lambda_[keep_lambda], [1, -1])) sigma_ = - np.dot(np.reshape(1. / lambda_[keep_lambda], [-1, 1]) * X[:, keep_lambda].T, sigma_) sigma_.flat[::(sigma_.shape[1] + 1)] += 1. / lambda_[keep_lambda] return sigma_ def update_coeff(X, y, coef_, alpha_, keep_lambda, sigma_): coef_[keep_lambda] = alpha_ * np.dot( sigma_, np.dot(X[:, keep_lambda].T, y)) return coef_ # Iterative procedure of ARDRegression for iter_ in range(self.n_iter): sigma_ = update_sigma(X, alpha_, lambda_, keep_lambda, n_samples) coef_ = update_coeff(X, y, coef_, alpha_, keep_lambda, sigma_) # Update alpha and lambda rmse_ = np.sum((y - np.dot(X, coef_)) ** 2) gamma_ = 1. - lambda_[keep_lambda] * np.diag(sigma_) lambda_[keep_lambda] = ((gamma_ + 2. * lambda_1) / ((coef_[keep_lambda]) ** 2 + 2. * lambda_2)) alpha_ = ((n_samples - gamma_.sum() + 2. * alpha_1) / (rmse_ + 2. * alpha_2)) # Prune the weights with a precision over a threshold keep_lambda = lambda_ < self.threshold_lambda coef_[~keep_lambda] = 0 # Compute the objective function if self.compute_score: s = (lambda_1 * np.log(lambda_) - lambda_2 * lambda_).sum() s += alpha_1 * log(alpha_) - alpha_2 * alpha_ s += 0.5 * (fast_logdet(sigma_) + n_samples * log(alpha_) + np.sum(np.log(lambda_))) s -= 0.5 * (alpha_ * rmse_ + (lambda_ * coef_ ** 2).sum()) self.scores_.append(s) # Check for convergence if iter_ > 0 and np.sum(np.abs(coef_old_ - coef_)) < self.tol: if verbose: print("Converged after %s iterations" % iter_) break coef_old_ = np.copy(coef_) # update sigma and mu using updated parameters from the last iteration sigma_ = update_sigma(X, alpha_, lambda_, keep_lambda, n_samples) coef_ = update_coeff(X, y, coef_, alpha_, keep_lambda, sigma_) self.coef_ = coef_ self.alpha_ = alpha_ self.sigma_ = sigma_ self.lambda_ = lambda_ self._set_intercept(X_offset_, y_offset_, X_scale_) return self def predict(self, X, return_std=False): """Predict using the linear model. In addition to the mean of the predictive distribution, also its standard deviation can be returned. Parameters ---------- X : {array-like, sparse matrix}, shape = (n_samples, n_features) Samples. return_std : boolean, optional Whether to return the standard deviation of posterior prediction. Returns ------- y_mean : array, shape = (n_samples,) Mean of predictive distribution of query points. y_std : array, shape = (n_samples,) Standard deviation of predictive distribution of query points. """ y_mean = self._decision_function(X) if return_std is False: return y_mean else: if self.normalize: X = (X - self.X_offset_) / self.X_scale_ X = X[:, self.lambda_ < self.threshold_lambda] sigmas_squared_data = (np.dot(X, self.sigma_) * X).sum(axis=1) y_std = np.sqrt(sigmas_squared_data + (1. / self.alpha_)) return y_mean, y_std