308 lines
11 KiB
Python
308 lines
11 KiB
Python
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# Authors: Manoj Kumar mks542@nyu.edu
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# License: BSD 3 clause
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import numpy as np
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from scipy import optimize
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from ..base import BaseEstimator, RegressorMixin
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from ._base import LinearModel
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from ..utils import axis0_safe_slice
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from ..utils.validation import _check_sample_weight
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from ..utils.validation import _deprecate_positional_args
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from ..utils.extmath import safe_sparse_dot
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from ..utils.optimize import _check_optimize_result
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def _huber_loss_and_gradient(w, X, y, epsilon, alpha, sample_weight=None):
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"""Returns the Huber loss and the gradient.
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Parameters
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----------
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w : ndarray, shape (n_features + 1,) or (n_features + 2,)
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Feature vector.
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w[:n_features] gives the coefficients
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w[-1] gives the scale factor and if the intercept is fit w[-2]
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gives the intercept factor.
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X : ndarray of shape (n_samples, n_features)
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Input data.
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y : ndarray of shape (n_samples,)
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Target vector.
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epsilon : float
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Robustness of the Huber estimator.
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alpha : float
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Regularization parameter.
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sample_weight : ndarray of shape (n_samples,), default=None
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Weight assigned to each sample.
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Returns
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-------
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loss : float
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Huber loss.
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gradient : ndarray, shape (len(w))
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Returns the derivative of the Huber loss with respect to each
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coefficient, intercept and the scale as a vector.
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"""
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_, n_features = X.shape
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fit_intercept = (n_features + 2 == w.shape[0])
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if fit_intercept:
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intercept = w[-2]
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sigma = w[-1]
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w = w[:n_features]
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n_samples = np.sum(sample_weight)
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# Calculate the values where |y - X'w -c / sigma| > epsilon
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# The values above this threshold are outliers.
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linear_loss = y - safe_sparse_dot(X, w)
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if fit_intercept:
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linear_loss -= intercept
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abs_linear_loss = np.abs(linear_loss)
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outliers_mask = abs_linear_loss > epsilon * sigma
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# Calculate the linear loss due to the outliers.
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# This is equal to (2 * M * |y - X'w -c / sigma| - M**2) * sigma
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outliers = abs_linear_loss[outliers_mask]
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num_outliers = np.count_nonzero(outliers_mask)
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n_non_outliers = X.shape[0] - num_outliers
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# n_sq_outliers includes the weight give to the outliers while
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# num_outliers is just the number of outliers.
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outliers_sw = sample_weight[outliers_mask]
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n_sw_outliers = np.sum(outliers_sw)
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outlier_loss = (2. * epsilon * np.sum(outliers_sw * outliers) -
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sigma * n_sw_outliers * epsilon ** 2)
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# Calculate the quadratic loss due to the non-outliers.-
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# This is equal to |(y - X'w - c)**2 / sigma**2| * sigma
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non_outliers = linear_loss[~outliers_mask]
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weighted_non_outliers = sample_weight[~outliers_mask] * non_outliers
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weighted_loss = np.dot(weighted_non_outliers.T, non_outliers)
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squared_loss = weighted_loss / sigma
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if fit_intercept:
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grad = np.zeros(n_features + 2)
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else:
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grad = np.zeros(n_features + 1)
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# Gradient due to the squared loss.
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X_non_outliers = -axis0_safe_slice(X, ~outliers_mask, n_non_outliers)
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grad[:n_features] = (
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2. / sigma * safe_sparse_dot(weighted_non_outliers, X_non_outliers))
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# Gradient due to the linear loss.
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signed_outliers = np.ones_like(outliers)
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signed_outliers_mask = linear_loss[outliers_mask] < 0
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signed_outliers[signed_outliers_mask] = -1.0
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X_outliers = axis0_safe_slice(X, outliers_mask, num_outliers)
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sw_outliers = sample_weight[outliers_mask] * signed_outliers
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grad[:n_features] -= 2. * epsilon * (
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safe_sparse_dot(sw_outliers, X_outliers))
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# Gradient due to the penalty.
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grad[:n_features] += alpha * 2. * w
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# Gradient due to sigma.
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grad[-1] = n_samples
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grad[-1] -= n_sw_outliers * epsilon ** 2
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grad[-1] -= squared_loss / sigma
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# Gradient due to the intercept.
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if fit_intercept:
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grad[-2] = -2. * np.sum(weighted_non_outliers) / sigma
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grad[-2] -= 2. * epsilon * np.sum(sw_outliers)
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loss = n_samples * sigma + squared_loss + outlier_loss
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loss += alpha * np.dot(w, w)
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return loss, grad
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class HuberRegressor(LinearModel, RegressorMixin, BaseEstimator):
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"""Linear regression model that is robust to outliers.
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The Huber Regressor optimizes the squared loss for the samples where
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``|(y - X'w) / sigma| < epsilon`` and the absolute loss for the samples
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where ``|(y - X'w) / sigma| > epsilon``, where w and sigma are parameters
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to be optimized. The parameter sigma makes sure that if y is scaled up
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or down by a certain factor, one does not need to rescale epsilon to
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achieve the same robustness. Note that this does not take into account
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the fact that the different features of X may be of different scales.
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This makes sure that the loss function is not heavily influenced by the
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outliers while not completely ignoring their effect.
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Read more in the :ref:`User Guide <huber_regression>`
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.. versionadded:: 0.18
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Parameters
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----------
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epsilon : float, greater than 1.0, default=1.35
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The parameter epsilon controls the number of samples that should be
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classified as outliers. The smaller the epsilon, the more robust it is
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to outliers.
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max_iter : int, default=100
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Maximum number of iterations that
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``scipy.optimize.minimize(method="L-BFGS-B")`` should run for.
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alpha : float, default=0.0001
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Regularization parameter.
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warm_start : bool, default=False
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This is useful if the stored attributes of a previously used model
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has to be reused. If set to False, then the coefficients will
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be rewritten for every call to fit.
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See :term:`the Glossary <warm_start>`.
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fit_intercept : bool, default=True
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Whether or not to fit the intercept. This can be set to False
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if the data is already centered around the origin.
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tol : float, default=1e-05
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The iteration will stop when
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``max{|proj g_i | i = 1, ..., n}`` <= ``tol``
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where pg_i is the i-th component of the projected gradient.
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Attributes
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----------
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coef_ : array, shape (n_features,)
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Features got by optimizing the Huber loss.
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intercept_ : float
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Bias.
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scale_ : float
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The value by which ``|y - X'w - c|`` is scaled down.
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n_iter_ : int
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Number of iterations that
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``scipy.optimize.minimize(method="L-BFGS-B")`` has run for.
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.. versionchanged:: 0.20
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In SciPy <= 1.0.0 the number of lbfgs iterations may exceed
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``max_iter``. ``n_iter_`` will now report at most ``max_iter``.
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outliers_ : array, shape (n_samples,)
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A boolean mask which is set to True where the samples are identified
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as outliers.
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Examples
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--------
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>>> import numpy as np
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>>> from sklearn.linear_model import HuberRegressor, LinearRegression
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>>> from sklearn.datasets import make_regression
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>>> rng = np.random.RandomState(0)
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>>> X, y, coef = make_regression(
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... n_samples=200, n_features=2, noise=4.0, coef=True, random_state=0)
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>>> X[:4] = rng.uniform(10, 20, (4, 2))
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>>> y[:4] = rng.uniform(10, 20, 4)
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>>> huber = HuberRegressor().fit(X, y)
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>>> huber.score(X, y)
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-7.284...
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>>> huber.predict(X[:1,])
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array([806.7200...])
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>>> linear = LinearRegression().fit(X, y)
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>>> print("True coefficients:", coef)
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True coefficients: [20.4923... 34.1698...]
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>>> print("Huber coefficients:", huber.coef_)
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Huber coefficients: [17.7906... 31.0106...]
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>>> print("Linear Regression coefficients:", linear.coef_)
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Linear Regression coefficients: [-1.9221... 7.0226...]
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References
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----------
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.. [1] Peter J. Huber, Elvezio M. Ronchetti, Robust Statistics
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Concomitant scale estimates, pg 172
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.. [2] Art B. Owen (2006), A robust hybrid of lasso and ridge regression.
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https://statweb.stanford.edu/~owen/reports/hhu.pdf
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"""
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@_deprecate_positional_args
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def __init__(self, *, epsilon=1.35, max_iter=100, alpha=0.0001,
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warm_start=False, fit_intercept=True, tol=1e-05):
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self.epsilon = epsilon
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self.max_iter = max_iter
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self.alpha = alpha
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self.warm_start = warm_start
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self.fit_intercept = fit_intercept
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self.tol = tol
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def fit(self, X, y, sample_weight=None):
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"""Fit the model according to the given training data.
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Parameters
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----------
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X : array-like, shape (n_samples, n_features)
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Training vector, where n_samples in the number of samples and
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n_features is the number of features.
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y : array-like, shape (n_samples,)
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Target vector relative to X.
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sample_weight : array-like, shape (n_samples,)
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Weight given to each sample.
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Returns
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-------
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self : object
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"""
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X, y = self._validate_data(
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X, y, copy=False, accept_sparse=['csr'], y_numeric=True,
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dtype=[np.float64, np.float32])
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sample_weight = _check_sample_weight(sample_weight, X)
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if self.epsilon < 1.0:
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raise ValueError(
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"epsilon should be greater than or equal to 1.0, got %f"
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% self.epsilon)
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if self.warm_start and hasattr(self, 'coef_'):
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parameters = np.concatenate(
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(self.coef_, [self.intercept_, self.scale_]))
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else:
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if self.fit_intercept:
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parameters = np.zeros(X.shape[1] + 2)
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else:
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parameters = np.zeros(X.shape[1] + 1)
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# Make sure to initialize the scale parameter to a strictly
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# positive value:
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parameters[-1] = 1
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# Sigma or the scale factor should be non-negative.
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# Setting it to be zero might cause undefined bounds hence we set it
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# to a value close to zero.
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bounds = np.tile([-np.inf, np.inf], (parameters.shape[0], 1))
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bounds[-1][0] = np.finfo(np.float64).eps * 10
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opt_res = optimize.minimize(
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_huber_loss_and_gradient, parameters, method="L-BFGS-B", jac=True,
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args=(X, y, self.epsilon, self.alpha, sample_weight),
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options={"maxiter": self.max_iter, "gtol": self.tol, "iprint": -1},
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bounds=bounds)
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parameters = opt_res.x
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if opt_res.status == 2:
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raise ValueError("HuberRegressor convergence failed:"
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" l-BFGS-b solver terminated with %s"
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% opt_res.message)
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self.n_iter_ = _check_optimize_result("lbfgs", opt_res, self.max_iter)
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self.scale_ = parameters[-1]
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if self.fit_intercept:
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self.intercept_ = parameters[-2]
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else:
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self.intercept_ = 0.0
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self.coef_ = parameters[:X.shape[1]]
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residual = np.abs(
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y - safe_sparse_dot(X, self.coef_) - self.intercept_)
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self.outliers_ = residual > self.scale_ * self.epsilon
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return self
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