""" Various bayesian regression """ # Authors: V. Michel, F. Pedregosa, A. Gramfort # License: BSD 3 clause from math import log import numpy as np from scipy import linalg from ._base import LinearModel, _rescale_data from ..base import RegressorMixin from ..utils.extmath import fast_logdet from scipy.linalg import pinvh from ..utils.validation import _check_sample_weight from ..utils.validation import _deprecate_positional_args ############################################################################### # BayesianRidge regression class BayesianRidge(RegressorMixin, LinearModel): """Bayesian ridge regression. Fit a Bayesian ridge model. See the Notes section for details on this implementation and the optimization of 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, default=300 Maximum number of iterations. Should be greater than or equal to 1. tol : float, default=1e-3 Stop the algorithm if w has converged. alpha_1 : float, default=1e-6 Hyper-parameter : shape parameter for the Gamma distribution prior over the alpha parameter. alpha_2 : float, default=1e-6 Hyper-parameter : inverse scale parameter (rate parameter) for the Gamma distribution prior over the alpha parameter. lambda_1 : float, default=1e-6 Hyper-parameter : shape parameter for the Gamma distribution prior over the lambda parameter. lambda_2 : float, default=1e-6 Hyper-parameter : inverse scale parameter (rate parameter) for the Gamma distribution prior over the lambda parameter. alpha_init : float, default=None Initial value for alpha (precision of the noise). If not set, alpha_init is 1/Var(y). .. versionadded:: 0.22 lambda_init : float, default=None Initial value for lambda (precision of the weights). If not set, lambda_init is 1. .. versionadded:: 0.22 compute_score : bool, default=False If True, compute the log marginal likelihood at each iteration of the optimization. fit_intercept : bool, default=True Whether to calculate the intercept for this model. The intercept is not treated as a probabilistic parameter and thus has no associated variance. If set to False, no intercept will be used in calculations (i.e. data is expected to be centered). normalize : bool, 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 : bool, default=True If True, X will be copied; else, it may be overwritten. verbose : bool, default=False Verbose mode when fitting the model. Attributes ---------- coef_ : array-like of shape (n_features,) Coefficients of the regression model (mean of distribution) intercept_ : float Independent term in decision function. Set to 0.0 if ``fit_intercept = False``. alpha_ : float Estimated precision of the noise. lambda_ : float Estimated precision of the weights. sigma_ : array-like of shape (n_features, n_features) Estimated variance-covariance matrix of the weights scores_ : array-like of shape (n_iter_+1,) If computed_score is True, value of the log marginal likelihood (to be maximized) at each iteration of the optimization. The array starts with the value of the log marginal likelihood obtained for the initial values of alpha and lambda and ends with the value obtained for the estimated alpha and lambda. n_iter_ : int The actual number of iterations to reach the stopping criterion. X_offset_ : float If `normalize=True`, offset subtracted for centering data to a zero mean. X_scale_ : float If `normalize=True`, parameter used to scale data to a unit standard deviation. Examples -------- >>> from sklearn import linear_model >>> clf = linear_model.BayesianRidge() >>> clf.fit([[0,0], [1, 1], [2, 2]], [0, 1, 2]) BayesianRidge() >>> clf.predict([[1, 1]]) array([1.]) Notes ----- There exist several strategies to perform Bayesian ridge regression. This implementation is based on the algorithm described in Appendix A of (Tipping, 2001) where updates of the regularization parameters are done as suggested in (MacKay, 1992). Note that according to A New View of Automatic Relevance Determination (Wipf and Nagarajan, 2008) these update rules do not guarantee that the marginal likelihood is increasing between two consecutive iterations of the optimization. References ---------- D. J. C. MacKay, Bayesian Interpolation, Computation and Neural Systems, Vol. 4, No. 3, 1992. M. E. Tipping, Sparse Bayesian Learning and the Relevance Vector Machine, Journal of Machine Learning Research, Vol. 1, 2001. """ @_deprecate_positional_args 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, alpha_init=None, lambda_init=None, 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.alpha_init = alpha_init self.lambda_init = lambda_init 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 : ndarray of shape (n_samples, n_features) Training data y : ndarray of shape (n_samples,) Target values. Will be cast to X's dtype if necessary sample_weight : ndarray of shape (n_samples,), default=None Individual weights for each sample .. versionadded:: 0.20 parameter *sample_weight* support to BayesianRidge. Returns ------- self : returns an instance of self. """ if self.n_iter < 1: raise ValueError('n_iter should be greater than or equal to 1.' ' Got {!r}.'.format(self.n_iter)) X, y = self._validate_data(X, y, dtype=np.float64, y_numeric=True) if sample_weight is not None: sample_weight = _check_sample_weight(sample_weight, X, dtype=X.dtype) 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_ = self.alpha_init lambda_ = self.lambda_init if alpha_ is None: alpha_ = 1. / (np.var(y) + eps) if lambda_ is None: 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): # update posterior mean coef_ based on alpha_ and lambda_ and # compute corresponding rmse coef_, rmse_ = self._update_coef_(X, y, n_samples, n_features, XT_y, U, Vh, eigen_vals_, alpha_, lambda_) if self.compute_score: # compute the log marginal likelihood s = self._log_marginal_likelihood(n_samples, n_features, eigen_vals_, alpha_, lambda_, coef_, rmse_) self.scores_.append(s) # Update alpha and lambda according to (MacKay, 1992) 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)) # 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.n_iter_ = iter_ + 1 # return regularization parameters and corresponding posterior mean, # log marginal likelihood and posterior covariance self.alpha_ = alpha_ self.lambda_ = lambda_ self.coef_, rmse_ = self._update_coef_(X, y, n_samples, n_features, XT_y, U, Vh, eigen_vals_, alpha_, lambda_) if self.compute_score: # compute the log marginal likelihood s = self._log_marginal_likelihood(n_samples, n_features, eigen_vals_, alpha_, lambda_, coef_, rmse_) self.scores_.append(s) self.scores_ = np.array(self.scores_) # posterior covariance is given by 1/alpha_ * scaled_sigma_ scaled_sigma_ = np.dot(Vh.T, Vh / (eigen_vals_ + lambda_ / alpha_)[:, np.newaxis]) self.sigma_ = (1. / alpha_) * scaled_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} of shape (n_samples, n_features) Samples. return_std : bool, default=False Whether to return the standard deviation of posterior prediction. Returns ------- y_mean : array-like of shape (n_samples,) Mean of predictive distribution of query points. y_std : array-like of 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 def _update_coef_(self, X, y, n_samples, n_features, XT_y, U, Vh, eigen_vals_, alpha_, lambda_): """Update posterior mean and compute corresponding rmse. Posterior mean is given by coef_ = scaled_sigma_ * X.T * y where scaled_sigma_ = (lambda_/alpha_ * np.eye(n_features) + np.dot(X.T, X))^-1 """ if n_samples > n_features: coef_ = np.linalg.multi_dot([Vh.T, Vh / (eigen_vals_ + lambda_ / alpha_)[:, np.newaxis], XT_y]) else: coef_ = np.linalg.multi_dot([X.T, U / (eigen_vals_ + lambda_ / alpha_)[None, :], U.T, y]) rmse_ = np.sum((y - np.dot(X, coef_)) ** 2) return coef_, rmse_ def _log_marginal_likelihood(self, n_samples, n_features, eigen_vals, alpha_, lambda_, coef, rmse): """Log marginal likelihood.""" alpha_1 = self.alpha_1 alpha_2 = self.alpha_2 lambda_1 = self.lambda_1 lambda_2 = self.lambda_2 # compute the log of the determinant of the posterior covariance. # posterior covariance is given by # sigma = (lambda_ * np.eye(n_features) + alpha_ * np.dot(X.T, X))^-1 if n_samples > n_features: logdet_sigma = - np.sum(np.log(lambda_ + alpha_ * eigen_vals)) else: 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)) score = lambda_1 * log(lambda_) - lambda_2 * lambda_ score += alpha_1 * log(alpha_) - alpha_2 * alpha_ score += 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)) return score ############################################################################### # ARD (Automatic Relevance Determination) regression class ARDRegression(RegressorMixin, LinearModel): """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, default=300 Maximum number of iterations. tol : float, default=1e-3 Stop the algorithm if w has converged. alpha_1 : float, default=1e-6 Hyper-parameter : shape parameter for the Gamma distribution prior over the alpha parameter. alpha_2 : float, default=1e-6 Hyper-parameter : inverse scale parameter (rate parameter) for the Gamma distribution prior over the alpha parameter. lambda_1 : float, default=1e-6 Hyper-parameter : shape parameter for the Gamma distribution prior over the lambda parameter. lambda_2 : float, default=1e-6 Hyper-parameter : inverse scale parameter (rate parameter) for the Gamma distribution prior over the lambda parameter. compute_score : bool, default=False If True, compute the objective function at each step of the model. threshold_lambda : float, default=10 000 threshold for removing (pruning) weights with high precision from the computation. fit_intercept : bool, default=True whether to calculate the intercept for this model. If set to false, no intercept will be used in calculations (i.e. data is expected to be centered). normalize : bool, 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 : bool, default=True If True, X will be copied; else, it may be overwritten. verbose : bool, default=False Verbose mode when fitting the model. Attributes ---------- coef_ : array-like of shape (n_features,) Coefficients of the regression model (mean of distribution) alpha_ : float estimated precision of the noise. lambda_ : array-like of shape (n_features,) estimated precisions of the weights. sigma_ : array-like of shape (n_features, n_features) estimated variance-covariance matrix of the weights scores_ : float if computed, value of the objective function (to be maximized) intercept_ : float Independent term in decision function. Set to 0.0 if ``fit_intercept = False``. X_offset_ : float If `normalize=True`, offset subtracted for centering data to a zero mean. X_scale_ : float If `normalize=True`, parameter used to scale data to a unit standard deviation. Examples -------- >>> from sklearn import linear_model >>> clf = linear_model.ARDRegression() >>> clf.fit([[0,0], [1, 1], [2, 2]], [0, 1, 2]) ARDRegression() >>> 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. """ @_deprecate_positional_args 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 of 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-like of shape (n_samples,) Target values (integers). Will be cast to X's dtype if necessary Returns ------- self : returns an instance of self. """ X, y = self._validate_data(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) self.X_offset_ = X_offset_ self.X_scale_ = X_scale_ # 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 def update_coeff(X, y, coef_, alpha_, keep_lambda, sigma_): coef_[keep_lambda] = alpha_ * np.linalg.multi_dot([ sigma_, X[:, keep_lambda].T, y]) return coef_ update_sigma = (self._update_sigma if n_samples >= n_features else self._update_sigma_woodbury) # Iterative procedure of ARDRegression for iter_ in range(self.n_iter): sigma_ = update_sigma(X, alpha_, lambda_, keep_lambda) 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_) if not keep_lambda.any(): break if keep_lambda.any(): # update sigma and mu using updated params from the last iteration sigma_ = update_sigma(X, alpha_, lambda_, keep_lambda) coef_ = update_coeff(X, y, coef_, alpha_, keep_lambda, sigma_) else: sigma_ = np.array([]).reshape(0, 0) self.coef_ = coef_ self.alpha_ = alpha_ self.sigma_ = sigma_ self.lambda_ = lambda_ self._set_intercept(X_offset_, y_offset_, X_scale_) return self def _update_sigma_woodbury(self, X, alpha_, lambda_, keep_lambda): # See slides as referenced in the docstring note # this function is used when n_samples < n_features and will invert # a matrix of shape (n_samples, n_samples) making use of the # woodbury formula: # https://en.wikipedia.org/wiki/Woodbury_matrix_identity n_samples = X.shape[0] X_keep = X[:, keep_lambda] inv_lambda = 1 / lambda_[keep_lambda].reshape(1, -1) sigma_ = pinvh( np.eye(n_samples) / alpha_ + np.dot(X_keep * inv_lambda, X_keep.T) ) sigma_ = np.dot(sigma_, X_keep * inv_lambda) sigma_ = - np.dot(inv_lambda.reshape(-1, 1) * X_keep.T, sigma_) sigma_[np.diag_indices(sigma_.shape[1])] += 1. / lambda_[keep_lambda] return sigma_ def _update_sigma(self, X, alpha_, lambda_, keep_lambda): # See slides as referenced in the docstring note # this function is used when n_samples >= n_features and will # invert a matrix of shape (n_features, n_features) X_keep = X[:, keep_lambda] gram = np.dot(X_keep.T, X_keep) eye = np.eye(gram.shape[0]) sigma_inv = lambda_[keep_lambda] * eye + alpha_ * gram sigma_ = pinvh(sigma_inv) return sigma_ 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} of shape (n_samples, n_features) Samples. return_std : bool, default=False Whether to return the standard deviation of posterior prediction. Returns ------- y_mean : array-like of shape (n_samples,) Mean of predictive distribution of query points. y_std : array-like of 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