"""
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
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 <bayesian_regression>`.
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
<sphx_glr_auto_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
[docs] def fit(self, X, y):
"""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
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)
self.X_offset_ = X_offset_
self.X_scale_ = X_scale_
n_samples, n_features = X.shape
# Initialization of the values of the parameters
alpha_ = 1. / np.var(y)
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_ = lambda_ * np.ones(n_features)
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
[docs] 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 <bayesian_regression>`.
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
<sphx_glr_auto_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
[docs] 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)
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
alpha_ = 1. / np.var(y)
lambda_ = np.ones(n_features)
self.scores_ = list()
coef_old_ = None
# Iterative procedure of ARDRegression
for iter_ in range(self.n_iter):
# Compute mu and sigma (using Woodbury matrix identity)
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]
coef_[keep_lambda] = alpha_ * np.dot(
sigma_, np.dot(X[:, keep_lambda].T, y))
# 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_)
self.coef_ = coef_
self.alpha_ = alpha_
self.sigma_ = sigma_
self.lambda_ = lambda_
self._set_intercept(X_offset_, y_offset_, X_scale_)
return self
[docs] 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
