616 lines
23 KiB
Python
616 lines
23 KiB
Python
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"""
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Generalized Linear Models with Exponential Dispersion Family
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"""
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# Author: Christian Lorentzen <lorentzen.ch@googlemail.com>
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# some parts and tricks stolen from other sklearn files.
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# License: BSD 3 clause
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import numbers
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import numpy as np
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import scipy.optimize
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from ...base import BaseEstimator, RegressorMixin
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from ...utils import check_array, check_X_y
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from ...utils.optimize import _check_optimize_result
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from ...utils.validation import check_is_fitted, _check_sample_weight
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from ..._loss.glm_distribution import (
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ExponentialDispersionModel,
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TweedieDistribution,
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EDM_DISTRIBUTIONS
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)
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from .link import (
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BaseLink,
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IdentityLink,
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LogLink,
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)
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def _safe_lin_pred(X, coef):
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"""Compute the linear predictor taking care if intercept is present."""
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if coef.size == X.shape[1] + 1:
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return X @ coef[1:] + coef[0]
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else:
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return X @ coef
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def _y_pred_deviance_derivative(coef, X, y, weights, family, link):
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"""Compute y_pred and the derivative of the deviance w.r.t coef."""
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lin_pred = _safe_lin_pred(X, coef)
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y_pred = link.inverse(lin_pred)
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d1 = link.inverse_derivative(lin_pred)
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temp = d1 * family.deviance_derivative(y, y_pred, weights)
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if coef.size == X.shape[1] + 1:
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devp = np.concatenate(([temp.sum()], temp @ X))
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else:
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devp = temp @ X # same as X.T @ temp
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return y_pred, devp
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class GeneralizedLinearRegressor(BaseEstimator, RegressorMixin):
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"""Regression via a penalized Generalized Linear Model (GLM).
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GLMs based on a reproductive Exponential Dispersion Model (EDM) aim at
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fitting and predicting the mean of the target y as y_pred=h(X*w).
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Therefore, the fit minimizes the following objective function with L2
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priors as regularizer::
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1/(2*sum(s)) * deviance(y, h(X*w); s)
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+ 1/2 * alpha * |w|_2
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with inverse link function h and s=sample_weight.
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The parameter ``alpha`` corresponds to the lambda parameter in glmnet.
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Read more in the :ref:`User Guide <Generalized_linear_regression>`.
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Parameters
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----------
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alpha : float, default=1
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Constant that multiplies the penalty term and thus determines the
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regularization strength. ``alpha = 0`` is equivalent to unpenalized
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GLMs. In this case, the design matrix `X` must have full column rank
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(no collinearities).
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fit_intercept : bool, default=True
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Specifies if a constant (a.k.a. bias or intercept) should be
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added to the linear predictor (X @ coef + intercept).
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family : {'normal', 'poisson', 'gamma', 'inverse-gaussian'} \
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or an ExponentialDispersionModel instance, default='normal'
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The distributional assumption of the GLM, i.e. which distribution from
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the EDM, specifies the loss function to be minimized.
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link : {'auto', 'identity', 'log'} or an instance of class BaseLink, \
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default='auto'
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The link function of the GLM, i.e. mapping from linear predictor
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`X @ coeff + intercept` to prediction `y_pred`. Option 'auto' sets
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the link depending on the chosen family as follows:
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- 'identity' for Normal distribution
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- 'log' for Poisson, Gamma and Inverse Gaussian distributions
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solver : 'lbfgs', default='lbfgs'
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Algorithm to use in the optimization problem:
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'lbfgs'
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Calls scipy's L-BFGS-B optimizer.
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max_iter : int, default=100
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The maximal number of iterations for the solver.
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tol : float, default=1e-4
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Stopping criterion. For the lbfgs solver,
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the iteration will stop when ``max{|g_j|, j = 1, ..., d} <= tol``
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where ``g_j`` is the j-th component of the gradient (derivative) of
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the objective function.
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warm_start : bool, default=False
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If set to ``True``, reuse the solution of the previous call to ``fit``
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as initialization for ``coef_`` and ``intercept_``.
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verbose : int, default=0
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For the lbfgs solver set verbose to any positive number for verbosity.
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Attributes
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----------
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coef_ : array of shape (n_features,)
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Estimated coefficients for the linear predictor (`X @ coef_ +
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intercept_`) in the GLM.
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intercept_ : float
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Intercept (a.k.a. bias) added to linear predictor.
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n_iter_ : int
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Actual number of iterations used in the solver.
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"""
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def __init__(self, *, alpha=1.0,
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fit_intercept=True, family='normal', link='auto',
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solver='lbfgs', max_iter=100, tol=1e-4, warm_start=False,
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verbose=0):
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self.alpha = alpha
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self.fit_intercept = fit_intercept
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self.family = family
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self.link = link
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self.solver = solver
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self.max_iter = max_iter
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self.tol = tol
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self.warm_start = warm_start
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self.verbose = verbose
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def fit(self, X, y, sample_weight=None):
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"""Fit a Generalized Linear Model.
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Parameters
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----------
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X : {array-like, sparse matrix} of shape (n_samples, n_features)
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Training data.
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y : array-like of shape (n_samples,)
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Target values.
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sample_weight : array-like of shape (n_samples,), default=None
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Sample weights.
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Returns
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-------
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self : returns an instance of self.
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"""
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if isinstance(self.family, ExponentialDispersionModel):
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self._family_instance = self.family
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elif self.family in EDM_DISTRIBUTIONS:
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self._family_instance = EDM_DISTRIBUTIONS[self.family]()
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else:
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raise ValueError(
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"The family must be an instance of class"
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" ExponentialDispersionModel or an element of"
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" ['normal', 'poisson', 'gamma', 'inverse-gaussian']"
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"; got (family={0})".format(self.family))
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# Guarantee that self._link_instance is set to an instance of
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# class BaseLink
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if isinstance(self.link, BaseLink):
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self._link_instance = self.link
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else:
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if self.link == 'auto':
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if isinstance(self._family_instance, TweedieDistribution):
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if self._family_instance.power <= 0:
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self._link_instance = IdentityLink()
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if self._family_instance.power >= 1:
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self._link_instance = LogLink()
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else:
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raise ValueError("No default link known for the "
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"specified distribution family. Please "
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"set link manually, i.e. not to 'auto'; "
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"got (link='auto', family={})"
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.format(self.family))
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elif self.link == 'identity':
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self._link_instance = IdentityLink()
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elif self.link == 'log':
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self._link_instance = LogLink()
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else:
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raise ValueError(
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"The link must be an instance of class Link or "
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"an element of ['auto', 'identity', 'log']; "
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"got (link={0})".format(self.link))
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if not isinstance(self.alpha, numbers.Number) or self.alpha < 0:
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raise ValueError("Penalty term must be a non-negative number;"
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" got (alpha={0})".format(self.alpha))
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if not isinstance(self.fit_intercept, bool):
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raise ValueError("The argument fit_intercept must be bool;"
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" got {0}".format(self.fit_intercept))
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if self.solver not in ['lbfgs']:
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raise ValueError("GeneralizedLinearRegressor supports only solvers"
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"'lbfgs'; got {0}".format(self.solver))
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solver = self.solver
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if (not isinstance(self.max_iter, numbers.Integral)
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or self.max_iter <= 0):
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raise ValueError("Maximum number of iteration must be a positive "
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"integer;"
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" got (max_iter={0!r})".format(self.max_iter))
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if not isinstance(self.tol, numbers.Number) or self.tol <= 0:
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raise ValueError("Tolerance for stopping criteria must be "
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"positive; got (tol={0!r})".format(self.tol))
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if not isinstance(self.warm_start, bool):
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raise ValueError("The argument warm_start must be bool;"
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" got {0}".format(self.warm_start))
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family = self._family_instance
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link = self._link_instance
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X, y = check_X_y(X, y, accept_sparse=['csc', 'csr'],
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dtype=[np.float64, np.float32],
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y_numeric=True, multi_output=False)
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weights = _check_sample_weight(sample_weight, X)
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_, n_features = X.shape
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if not np.all(family.in_y_range(y)):
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raise ValueError("Some value(s) of y are out of the valid "
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"range for family {0}"
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.format(family.__class__.__name__))
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# TODO: if alpha=0 check that X is not rank deficient
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# rescaling of sample_weight
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#
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# IMPORTANT NOTE: Since we want to minimize
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# 1/(2*sum(sample_weight)) * deviance + L2,
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# deviance = sum(sample_weight * unit_deviance),
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# we rescale weights such that sum(weights) = 1 and this becomes
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# 1/2*deviance + L2 with deviance=sum(weights * unit_deviance)
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weights = weights / weights.sum()
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if self.warm_start and hasattr(self, 'coef_'):
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if self.fit_intercept:
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coef = np.concatenate((np.array([self.intercept_]),
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self.coef_))
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else:
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coef = self.coef_
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else:
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if self.fit_intercept:
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coef = np.zeros(n_features+1)
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coef[0] = link(np.average(y, weights=weights))
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else:
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coef = np.zeros(n_features)
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# algorithms for optimization
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if solver == 'lbfgs':
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def func(coef, X, y, weights, alpha, family, link):
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y_pred, devp = _y_pred_deviance_derivative(
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coef, X, y, weights, family, link
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)
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dev = family.deviance(y, y_pred, weights)
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# offset if coef[0] is intercept
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offset = 1 if self.fit_intercept else 0
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coef_scaled = alpha * coef[offset:]
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obj = 0.5 * dev + 0.5 * (coef[offset:] @ coef_scaled)
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objp = 0.5 * devp
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objp[offset:] += coef_scaled
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return obj, objp
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args = (X, y, weights, self.alpha, family, link)
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opt_res = scipy.optimize.minimize(
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func, coef, method="L-BFGS-B", jac=True,
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options={
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"maxiter": self.max_iter,
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"iprint": (self.verbose > 0) - 1,
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"gtol": self.tol,
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"ftol": 1e3*np.finfo(float).eps,
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},
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args=args)
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self.n_iter_ = _check_optimize_result("lbfgs", opt_res)
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coef = opt_res.x
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if self.fit_intercept:
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self.intercept_ = coef[0]
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self.coef_ = coef[1:]
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else:
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# set intercept to zero as the other linear models do
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self.intercept_ = 0.
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self.coef_ = coef
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return self
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def _linear_predictor(self, X):
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"""Compute the linear_predictor = `X @ coef_ + intercept_`.
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Parameters
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----------
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X : {array-like, sparse matrix} of shape (n_samples, n_features)
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Samples.
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Returns
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-------
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y_pred : array of shape (n_samples,)
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Returns predicted values of linear predictor.
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"""
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check_is_fitted(self)
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X = check_array(X, accept_sparse=['csr', 'csc', 'coo'],
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dtype=[np.float64, np.float32], ensure_2d=True,
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allow_nd=False)
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return X @ self.coef_ + self.intercept_
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def predict(self, X):
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"""Predict using GLM with feature matrix X.
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Parameters
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----------
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X : {array-like, sparse matrix} of shape (n_samples, n_features)
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Samples.
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Returns
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-------
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y_pred : array of shape (n_samples,)
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Returns predicted values.
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"""
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# check_array is done in _linear_predictor
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eta = self._linear_predictor(X)
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y_pred = self._link_instance.inverse(eta)
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return y_pred
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def score(self, X, y, sample_weight=None):
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"""Compute D^2, the percentage of deviance explained.
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D^2 is a generalization of the coefficient of determination R^2.
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R^2 uses squared error and D^2 deviance. Note that those two are equal
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for ``family='normal'``.
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D^2 is defined as
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:math:`D^2 = 1-\\frac{D(y_{true},y_{pred})}{D_{null}}`,
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:math:`D_{null}` is the null deviance, i.e. the deviance of a model
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with intercept alone, which corresponds to :math:`y_{pred} = \\bar{y}`.
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The mean :math:`\\bar{y}` is averaged by sample_weight.
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Best possible score is 1.0 and it can be negative (because the model
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can be arbitrarily worse).
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Parameters
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----------
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X : {array-like, sparse matrix} of shape (n_samples, n_features)
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Test samples.
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y : array-like of shape (n_samples,)
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True values of target.
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sample_weight : array-like of shape (n_samples,), default=None
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Sample weights.
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Returns
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-------
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score : float
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D^2 of self.predict(X) w.r.t. y.
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||
|
|
"""
|
||
|
|
# Note, default score defined in RegressorMixin is R^2 score.
|
||
|
|
# TODO: make D^2 a score function in module metrics (and thereby get
|
||
|
|
# input validation and so on)
|
||
|
|
weights = _check_sample_weight(sample_weight, X)
|
||
|
|
y_pred = self.predict(X)
|
||
|
|
dev = self._family_instance.deviance(y, y_pred, weights=weights)
|
||
|
|
y_mean = np.average(y, weights=weights)
|
||
|
|
dev_null = self._family_instance.deviance(y, y_mean, weights=weights)
|
||
|
|
return 1 - dev / dev_null
|
||
|
|
|
||
|
|
def _more_tags(self):
|
||
|
|
# create the _family_instance if fit wasn't called yet.
|
||
|
|
if hasattr(self, '_family_instance'):
|
||
|
|
_family_instance = self._family_instance
|
||
|
|
elif isinstance(self.family, ExponentialDispersionModel):
|
||
|
|
_family_instance = self.family
|
||
|
|
elif self.family in EDM_DISTRIBUTIONS:
|
||
|
|
_family_instance = EDM_DISTRIBUTIONS[self.family]()
|
||
|
|
else:
|
||
|
|
raise ValueError
|
||
|
|
return {"requires_positive_y": not _family_instance.in_y_range(-1.0)}
|
||
|
|
|
||
|
|
|
||
|
|
class PoissonRegressor(GeneralizedLinearRegressor):
|
||
|
|
"""Generalized Linear Model with a Poisson distribution.
|
||
|
|
|
||
|
|
Read more in the :ref:`User Guide <Generalized_linear_regression>`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
alpha : float, default=1
|
||
|
|
Constant that multiplies the penalty term and thus determines the
|
||
|
|
regularization strength. ``alpha = 0`` is equivalent to unpenalized
|
||
|
|
GLMs. In this case, the design matrix `X` must have full column rank
|
||
|
|
(no collinearities).
|
||
|
|
|
||
|
|
fit_intercept : bool, default=True
|
||
|
|
Specifies if a constant (a.k.a. bias or intercept) should be
|
||
|
|
added to the linear predictor (X @ coef + intercept).
|
||
|
|
|
||
|
|
max_iter : int, default=100
|
||
|
|
The maximal number of iterations for the solver.
|
||
|
|
|
||
|
|
tol : float, default=1e-4
|
||
|
|
Stopping criterion. For the lbfgs solver,
|
||
|
|
the iteration will stop when ``max{|g_j|, j = 1, ..., d} <= tol``
|
||
|
|
where ``g_j`` is the j-th component of the gradient (derivative) of
|
||
|
|
the objective function.
|
||
|
|
|
||
|
|
warm_start : bool, default=False
|
||
|
|
If set to ``True``, reuse the solution of the previous call to ``fit``
|
||
|
|
as initialization for ``coef_`` and ``intercept_`` .
|
||
|
|
|
||
|
|
verbose : int, default=0
|
||
|
|
For the lbfgs solver set verbose to any positive number for verbosity.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
coef_ : array of shape (n_features,)
|
||
|
|
Estimated coefficients for the linear predictor (`X @ coef_ +
|
||
|
|
intercept_`) in the GLM.
|
||
|
|
|
||
|
|
intercept_ : float
|
||
|
|
Intercept (a.k.a. bias) added to linear predictor.
|
||
|
|
|
||
|
|
n_iter_ : int
|
||
|
|
Actual number of iterations used in the solver.
|
||
|
|
"""
|
||
|
|
def __init__(self, *, alpha=1.0, fit_intercept=True, max_iter=100,
|
||
|
|
tol=1e-4, warm_start=False, verbose=0):
|
||
|
|
|
||
|
|
super().__init__(alpha=alpha, fit_intercept=fit_intercept,
|
||
|
|
family="poisson", link='log', max_iter=max_iter,
|
||
|
|
tol=tol, warm_start=warm_start, verbose=verbose)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def family(self):
|
||
|
|
# Make this attribute read-only to avoid mis-uses e.g. in GridSearch.
|
||
|
|
return "poisson"
|
||
|
|
|
||
|
|
@family.setter
|
||
|
|
def family(self, value):
|
||
|
|
if value != "poisson":
|
||
|
|
raise ValueError("PoissonRegressor.family must be 'poisson'!")
|
||
|
|
|
||
|
|
|
||
|
|
class GammaRegressor(GeneralizedLinearRegressor):
|
||
|
|
"""Generalized Linear Model with a Gamma distribution.
|
||
|
|
|
||
|
|
Read more in the :ref:`User Guide <Generalized_linear_regression>`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
alpha : float, default=1
|
||
|
|
Constant that multiplies the penalty term and thus determines the
|
||
|
|
regularization strength. ``alpha = 0`` is equivalent to unpenalized
|
||
|
|
GLMs. In this case, the design matrix `X` must have full column rank
|
||
|
|
(no collinearities).
|
||
|
|
|
||
|
|
fit_intercept : bool, default=True
|
||
|
|
Specifies if a constant (a.k.a. bias or intercept) should be
|
||
|
|
added to the linear predictor (X @ coef + intercept).
|
||
|
|
|
||
|
|
max_iter : int, default=100
|
||
|
|
The maximal number of iterations for the solver.
|
||
|
|
|
||
|
|
tol : float, default=1e-4
|
||
|
|
Stopping criterion. For the lbfgs solver,
|
||
|
|
the iteration will stop when ``max{|g_j|, j = 1, ..., d} <= tol``
|
||
|
|
where ``g_j`` is the j-th component of the gradient (derivative) of
|
||
|
|
the objective function.
|
||
|
|
|
||
|
|
warm_start : bool, default=False
|
||
|
|
If set to ``True``, reuse the solution of the previous call to ``fit``
|
||
|
|
as initialization for ``coef_`` and ``intercept_`` .
|
||
|
|
|
||
|
|
verbose : int, default=0
|
||
|
|
For the lbfgs solver set verbose to any positive number for verbosity.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
coef_ : array of shape (n_features,)
|
||
|
|
Estimated coefficients for the linear predictor (`X * coef_ +
|
||
|
|
intercept_`) in the GLM.
|
||
|
|
|
||
|
|
intercept_ : float
|
||
|
|
Intercept (a.k.a. bias) added to linear predictor.
|
||
|
|
|
||
|
|
n_iter_ : int
|
||
|
|
Actual number of iterations used in the solver.
|
||
|
|
"""
|
||
|
|
def __init__(self, *, alpha=1.0, fit_intercept=True, max_iter=100,
|
||
|
|
tol=1e-4, warm_start=False, verbose=0):
|
||
|
|
|
||
|
|
super().__init__(alpha=alpha, fit_intercept=fit_intercept,
|
||
|
|
family="gamma", link='log', max_iter=max_iter,
|
||
|
|
tol=tol, warm_start=warm_start, verbose=verbose)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def family(self):
|
||
|
|
# Make this attribute read-only to avoid mis-uses e.g. in GridSearch.
|
||
|
|
return "gamma"
|
||
|
|
|
||
|
|
@family.setter
|
||
|
|
def family(self, value):
|
||
|
|
if value != "gamma":
|
||
|
|
raise ValueError("GammaRegressor.family must be 'gamma'!")
|
||
|
|
|
||
|
|
|
||
|
|
class TweedieRegressor(GeneralizedLinearRegressor):
|
||
|
|
"""Generalized Linear Model with a Tweedie distribution.
|
||
|
|
|
||
|
|
This estimator can be used to model different GLMs depending on the
|
||
|
|
``power`` parameter, which determines the underlying distribution.
|
||
|
|
|
||
|
|
Read more in the :ref:`User Guide <Generalized_linear_regression>`.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
power : float, default=0
|
||
|
|
The power determines the underlying target distribution according
|
||
|
|
to the following table:
|
||
|
|
|
||
|
|
+-------+------------------------+
|
||
|
|
| Power | Distribution |
|
||
|
|
+=======+========================+
|
||
|
|
| 0 | Normal |
|
||
|
|
+-------+------------------------+
|
||
|
|
| 1 | Poisson |
|
||
|
|
+-------+------------------------+
|
||
|
|
| (1,2) | Compound Poisson Gamma |
|
||
|
|
+-------+------------------------+
|
||
|
|
| 2 | Gamma |
|
||
|
|
+-------+------------------------+
|
||
|
|
| 3 | Inverse Gaussian |
|
||
|
|
+-------+------------------------+
|
||
|
|
|
||
|
|
For ``0 < power < 1``, no distribution exists.
|
||
|
|
|
||
|
|
alpha : float, default=1
|
||
|
|
Constant that multiplies the penalty term and thus determines the
|
||
|
|
regularization strength. ``alpha = 0`` is equivalent to unpenalized
|
||
|
|
GLMs. In this case, the design matrix `X` must have full column rank
|
||
|
|
(no collinearities).
|
||
|
|
|
||
|
|
link : {'auto', 'identity', 'log'}, default='auto'
|
||
|
|
The link function of the GLM, i.e. mapping from linear predictor
|
||
|
|
`X @ coeff + intercept` to prediction `y_pred`. Option 'auto' sets
|
||
|
|
the link depending on the chosen family as follows:
|
||
|
|
|
||
|
|
- 'identity' for Normal distribution
|
||
|
|
- 'log' for Poisson, Gamma and Inverse Gaussian distributions
|
||
|
|
|
||
|
|
fit_intercept : bool, default=True
|
||
|
|
Specifies if a constant (a.k.a. bias or intercept) should be
|
||
|
|
added to the linear predictor (X @ coef + intercept).
|
||
|
|
|
||
|
|
max_iter : int, default=100
|
||
|
|
The maximal number of iterations for the solver.
|
||
|
|
|
||
|
|
tol : float, default=1e-4
|
||
|
|
Stopping criterion. For the lbfgs solver,
|
||
|
|
the iteration will stop when ``max{|g_j|, j = 1, ..., d} <= tol``
|
||
|
|
where ``g_j`` is the j-th component of the gradient (derivative) of
|
||
|
|
the objective function.
|
||
|
|
|
||
|
|
warm_start : bool, default=False
|
||
|
|
If set to ``True``, reuse the solution of the previous call to ``fit``
|
||
|
|
as initialization for ``coef_`` and ``intercept_`` .
|
||
|
|
|
||
|
|
verbose : int, default=0
|
||
|
|
For the lbfgs solver set verbose to any positive number for verbosity.
|
||
|
|
|
||
|
|
Attributes
|
||
|
|
----------
|
||
|
|
coef_ : array of shape (n_features,)
|
||
|
|
Estimated coefficients for the linear predictor (`X @ coef_ +
|
||
|
|
intercept_`) in the GLM.
|
||
|
|
|
||
|
|
intercept_ : float
|
||
|
|
Intercept (a.k.a. bias) added to linear predictor.
|
||
|
|
|
||
|
|
n_iter_ : int
|
||
|
|
Actual number of iterations used in the solver.
|
||
|
|
"""
|
||
|
|
def __init__(self, *, power=0.0, alpha=1.0, fit_intercept=True,
|
||
|
|
link='auto', max_iter=100, tol=1e-4,
|
||
|
|
warm_start=False, verbose=0):
|
||
|
|
|
||
|
|
super().__init__(alpha=alpha, fit_intercept=fit_intercept,
|
||
|
|
family=TweedieDistribution(power=power), link=link,
|
||
|
|
max_iter=max_iter, tol=tol,
|
||
|
|
warm_start=warm_start, verbose=verbose)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def family(self):
|
||
|
|
# We use a property with a setter to make sure that the family is
|
||
|
|
# always a Tweedie distribution, and that self.power and
|
||
|
|
# self.family.power are identical by construction.
|
||
|
|
dist = TweedieDistribution(power=self.power)
|
||
|
|
# TODO: make the returned object immutable
|
||
|
|
return dist
|
||
|
|
|
||
|
|
@family.setter
|
||
|
|
def family(self, value):
|
||
|
|
if isinstance(value, TweedieDistribution):
|
||
|
|
self.power = value.power
|
||
|
|
else:
|
||
|
|
raise TypeError("TweedieRegressor.family must be of type "
|
||
|
|
"TweedieDistribution!")
|