sklearn.linear_model.ARDRegression

class sklearn.linear_model.ARDRegression(n_iter=300, tol=0.001, alpha_1=1e-06, alpha_2=1e-06, lambda_1=1e-06, lambda_2=1e-06, compute_score=False, threshold_lambda=10000.0, fit_intercept=True, normalize=False, copy_X=True, verbose=False)

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 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 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)

Notes

For an example, see 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.

Examples

>>> from sklearn import linear_model
>>> clf = linear_model.ARDRegression()
>>> clf.fit([[0,0], [1, 1], [2, 2]], [0, 1, 2])
... 
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.])

Methods

fit(X, y)	Fit the ARDRegression model according to the given training data and parameters.
get_params([deep])	Get parameters for this estimator.
predict(X[, return_std])	Predict using the linear model.
score(X, y[, sample_weight])	Returns the coefficient of determination R^2 of the prediction.
set_params(**params)	Set the parameters of this estimator.

__init__(n_iter=300, tol=0.001, alpha_1=1e-06, alpha_2=1e-06, lambda_1=1e-06, lambda_2=1e-06, compute_score=False, threshold_lambda=10000.0, fit_intercept=True, normalize=False, copy_X=True, verbose=False)

fit(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.

get_params(deep=True)

Get parameters for this estimator.

Parameters:

deep : boolean, optional

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

params : mapping of string to any

Parameter names mapped to their values.

predict(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.

score(X, y, sample_weight=None)

Returns the coefficient of determination R^2 of the prediction.

The coefficient R^2 is defined as (1 - u/v), where u is the residual sum of squares ((y_true - y_pred) ** 2).sum() and v is the total sum of squares ((y_true - y_true.mean()) ** 2).sum(). The best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse). A constant model that always predicts the expected value of y, disregarding the input features, would get a R^2 score of 0.0.

Parameters:

X : array-like, shape = (n_samples, n_features)

Test samples.

y : array-like, shape = (n_samples) or (n_samples, n_outputs)

True values for X.

sample_weight : array-like, shape = [n_samples], optional

Sample weights.

Returns:

score : float

R^2 of self.predict(X) wrt. y.

set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Returns:self :

Examples using sklearn.linear_model.ARDRegression

Automatic Relevance Determination Regression (ARD)