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"""
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 <bayesian_regression>`.
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.
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.dot(Vh.T,
Vh / (eigen_vals_ +
lambda_ / alpha_)[:, np.newaxis])
coef_ = np.dot(coef_, XT_y)
else:
coef_ = np.dot(X.T, np.dot(
U / (eigen_vals_ + lambda_ / alpha_)[None, :], U.T))
coef_ = np.dot(coef_, 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 <bayesian_regression>`.
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``.
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
<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.
"""
@_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)
# 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.dot(
sigma_, np.dot(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