""" Generalized Linear models. """ # Author: Alexandre Gramfort # Fabian Pedregosa # Olivier Grisel # Vincent Michel # Peter Prettenhofer # Mathieu Blondel # Lars Buitinck # Maryan Morel # Giorgio Patrini # License: BSD 3 clause from __future__ import division from abc import ABCMeta, abstractmethod import numbers import warnings import numpy as np import scipy.sparse as sp from scipy import linalg from scipy import sparse from ..externals import six from ..utils._joblib import Parallel, delayed from ..base import BaseEstimator, ClassifierMixin, RegressorMixin from ..utils import check_array, check_X_y from ..utils.validation import FLOAT_DTYPES from ..utils import check_random_state from ..utils.extmath import safe_sparse_dot from ..utils.sparsefuncs import mean_variance_axis, inplace_column_scale from ..utils.fixes import sparse_lsqr from ..utils.seq_dataset import ArrayDataset, CSRDataset from ..utils.validation import check_is_fitted from ..exceptions import NotFittedError from ..preprocessing.data import normalize as f_normalize # TODO: bayesian_ridge_regression and bayesian_regression_ard # should be squashed into its respective objects. SPARSE_INTERCEPT_DECAY = 0.01 # For sparse data intercept updates are scaled by this decay factor to avoid # intercept oscillation. def make_dataset(X, y, sample_weight, random_state=None): """Create ``Dataset`` abstraction for sparse and dense inputs. This also returns the ``intercept_decay`` which is different for sparse datasets. Parameters ---------- X : array_like, shape (n_samples, n_features) Training data y : array_like, shape (n_samples, ) Target values. sample_weight : numpy array of shape (n_samples,) The weight of each sample random_state : int, RandomState instance or None (default) Determines random number generation for dataset shuffling and noise. Pass an int for reproducible output across multiple function calls. See :term:`Glossary `. Returns ------- dataset The ``Dataset`` abstraction intercept_decay The intercept decay """ rng = check_random_state(random_state) # seed should never be 0 in SequentialDataset seed = rng.randint(1, np.iinfo(np.int32).max) if sp.issparse(X): dataset = CSRDataset(X.data, X.indptr, X.indices, y, sample_weight, seed=seed) intercept_decay = SPARSE_INTERCEPT_DECAY else: dataset = ArrayDataset(X, y, sample_weight, seed=seed) intercept_decay = 1.0 return dataset, intercept_decay def _preprocess_data(X, y, fit_intercept, normalize=False, copy=True, sample_weight=None, return_mean=False, check_input=True): """ Centers data to have mean zero along axis 0. If fit_intercept=False or if the X is a sparse matrix, no centering is done, but normalization can still be applied. The function returns the statistics necessary to reconstruct the input data, which are X_offset, y_offset, X_scale, such that the output X = (X - X_offset) / X_scale X_scale is the L2 norm of X - X_offset. If sample_weight is not None, then the weighted mean of X and y is zero, and not the mean itself. If return_mean=True, the mean, eventually weighted, is returned, independently of whether X was centered (option used for optimization with sparse data in coordinate_descend). This is here because nearly all linear models will want their data to be centered. This function also systematically makes y consistent with X.dtype """ if isinstance(sample_weight, numbers.Number): sample_weight = None if check_input: X = check_array(X, copy=copy, accept_sparse=['csr', 'csc'], dtype=FLOAT_DTYPES) elif copy: if sp.issparse(X): X = X.copy() else: X = X.copy(order='K') y = np.asarray(y, dtype=X.dtype) if fit_intercept: if sp.issparse(X): X_offset, X_var = mean_variance_axis(X, axis=0) if not return_mean: X_offset[:] = X.dtype.type(0) if normalize: # TODO: f_normalize could be used here as well but the function # inplace_csr_row_normalize_l2 must be changed such that it # can return also the norms computed internally # transform variance to norm in-place X_var *= X.shape[0] X_scale = np.sqrt(X_var, X_var) del X_var X_scale[X_scale == 0] = 1 inplace_column_scale(X, 1. / X_scale) else: X_scale = np.ones(X.shape[1], dtype=X.dtype) else: X_offset = np.average(X, axis=0, weights=sample_weight) X -= X_offset if normalize: X, X_scale = f_normalize(X, axis=0, copy=False, return_norm=True) else: X_scale = np.ones(X.shape[1], dtype=X.dtype) y_offset = np.average(y, axis=0, weights=sample_weight) y = y - y_offset else: X_offset = np.zeros(X.shape[1], dtype=X.dtype) X_scale = np.ones(X.shape[1], dtype=X.dtype) if y.ndim == 1: y_offset = X.dtype.type(0) else: y_offset = np.zeros(y.shape[1], dtype=X.dtype) return X, y, X_offset, y_offset, X_scale # TODO: _rescale_data should be factored into _preprocess_data. # Currently, the fact that sag implements its own way to deal with # sample_weight makes the refactoring tricky. def _rescale_data(X, y, sample_weight): """Rescale data so as to support sample_weight""" n_samples = X.shape[0] sample_weight = np.full(n_samples, sample_weight, dtype=np.array(sample_weight).dtype) sample_weight = np.sqrt(sample_weight) sw_matrix = sparse.dia_matrix((sample_weight, 0), shape=(n_samples, n_samples)) X = safe_sparse_dot(sw_matrix, X) y = safe_sparse_dot(sw_matrix, y) return X, y class LinearModel(six.with_metaclass(ABCMeta, BaseEstimator)): """Base class for Linear Models""" @abstractmethod def fit(self, X, y): """Fit model.""" def _decision_function(self, X): check_is_fitted(self, "coef_") X = check_array(X, accept_sparse=['csr', 'csc', 'coo']) return safe_sparse_dot(X, self.coef_.T, dense_output=True) + self.intercept_ def predict(self, X): """Predict using the linear model Parameters ---------- X : array_like or sparse matrix, shape (n_samples, n_features) Samples. Returns ------- C : array, shape (n_samples,) Returns predicted values. """ return self._decision_function(X) _preprocess_data = staticmethod(_preprocess_data) def _set_intercept(self, X_offset, y_offset, X_scale): """Set the intercept_ """ if self.fit_intercept: self.coef_ = self.coef_ / X_scale self.intercept_ = y_offset - np.dot(X_offset, self.coef_.T) else: self.intercept_ = 0. # XXX Should this derive from LinearModel? It should be a mixin, not an ABC. # Maybe the n_features checking can be moved to LinearModel. class LinearClassifierMixin(ClassifierMixin): """Mixin for linear classifiers. Handles prediction for sparse and dense X. """ def decision_function(self, X): """Predict confidence scores for samples. The confidence score for a sample is the signed distance of that sample to the hyperplane. Parameters ---------- X : array_like or sparse matrix, shape (n_samples, n_features) Samples. Returns ------- array, shape=(n_samples,) if n_classes == 2 else (n_samples, n_classes) Confidence scores per (sample, class) combination. In the binary case, confidence score for self.classes_[1] where >0 means this class would be predicted. """ if not hasattr(self, 'coef_') or self.coef_ is None: raise NotFittedError("This %(name)s instance is not fitted " "yet" % {'name': type(self).__name__}) X = check_array(X, accept_sparse='csr') n_features = self.coef_.shape[1] if X.shape[1] != n_features: raise ValueError("X has %d features per sample; expecting %d" % (X.shape[1], n_features)) scores = safe_sparse_dot(X, self.coef_.T, dense_output=True) + self.intercept_ return scores.ravel() if scores.shape[1] == 1 else scores def predict(self, X): """Predict class labels for samples in X. Parameters ---------- X : array_like or sparse matrix, shape (n_samples, n_features) Samples. Returns ------- C : array, shape [n_samples] Predicted class label per sample. """ scores = self.decision_function(X) if len(scores.shape) == 1: indices = (scores > 0).astype(np.int) else: indices = scores.argmax(axis=1) return self.classes_[indices] def _predict_proba_lr(self, X): """Probability estimation for OvR logistic regression. Positive class probabilities are computed as 1. / (1. + np.exp(-self.decision_function(X))); multiclass is handled by normalizing that over all classes. """ prob = self.decision_function(X) prob *= -1 np.exp(prob, prob) prob += 1 np.reciprocal(prob, prob) if prob.ndim == 1: return np.vstack([1 - prob, prob]).T else: # OvR normalization, like LibLinear's predict_probability prob /= prob.sum(axis=1).reshape((prob.shape[0], -1)) return prob class SparseCoefMixin(object): """Mixin for converting coef_ to and from CSR format. L1-regularizing estimators should inherit this. """ def densify(self): """Convert coefficient matrix to dense array format. Converts the ``coef_`` member (back) to a numpy.ndarray. This is the default format of ``coef_`` and is required for fitting, so calling this method is only required on models that have previously been sparsified; otherwise, it is a no-op. Returns ------- self : estimator """ msg = "Estimator, %(name)s, must be fitted before densifying." check_is_fitted(self, "coef_", msg=msg) if sp.issparse(self.coef_): self.coef_ = self.coef_.toarray() return self def sparsify(self): """Convert coefficient matrix to sparse format. Converts the ``coef_`` member to a scipy.sparse matrix, which for L1-regularized models can be much more memory- and storage-efficient than the usual numpy.ndarray representation. The ``intercept_`` member is not converted. Notes ----- For non-sparse models, i.e. when there are not many zeros in ``coef_``, this may actually *increase* memory usage, so use this method with care. A rule of thumb is that the number of zero elements, which can be computed with ``(coef_ == 0).sum()``, must be more than 50% for this to provide significant benefits. After calling this method, further fitting with the partial_fit method (if any) will not work until you call densify. Returns ------- self : estimator """ msg = "Estimator, %(name)s, must be fitted before sparsifying." check_is_fitted(self, "coef_", msg=msg) self.coef_ = sp.csr_matrix(self.coef_) return self class LinearRegression(LinearModel, RegressorMixin): """ Ordinary least squares Linear Regression. Parameters ---------- fit_intercept : boolean, optional, default True 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). 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. n_jobs : int or None, optional (default=None) The number of jobs to use for the computation. This will only provide speedup for n_targets > 1 and sufficient large problems. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. Attributes ---------- coef_ : array, shape (n_features, ) or (n_targets, n_features) Estimated coefficients for the linear regression problem. If multiple targets are passed during the fit (y 2D), this is a 2D array of shape (n_targets, n_features), while if only one target is passed, this is a 1D array of length n_features. intercept_ : array Independent term in the linear model. Examples -------- >>> import numpy as np >>> from sklearn.linear_model import LinearRegression >>> X = np.array([[1, 1], [1, 2], [2, 2], [2, 3]]) >>> # y = 1 * x_0 + 2 * x_1 + 3 >>> y = np.dot(X, np.array([1, 2])) + 3 >>> reg = LinearRegression().fit(X, y) >>> reg.score(X, y) 1.0 >>> reg.coef_ array([1., 2.]) >>> reg.intercept_ # doctest: +ELLIPSIS 3.0000... >>> reg.predict(np.array([[3, 5]])) array([16.]) Notes ----- From the implementation point of view, this is just plain Ordinary Least Squares (scipy.linalg.lstsq) wrapped as a predictor object. """ def __init__(self, fit_intercept=True, normalize=False, copy_X=True, n_jobs=None): self.fit_intercept = fit_intercept self.normalize = normalize self.copy_X = copy_X self.n_jobs = n_jobs def fit(self, X, y, sample_weight=None): """ Fit linear model. Parameters ---------- X : array-like or sparse matrix, shape (n_samples, n_features) Training data y : array_like, shape (n_samples, n_targets) Target values. Will be cast to X's dtype if necessary sample_weight : numpy array of shape [n_samples] Individual weights for each sample .. versionadded:: 0.17 parameter *sample_weight* support to LinearRegression. Returns ------- self : returns an instance of self. """ n_jobs_ = self.n_jobs X, y = check_X_y(X, y, accept_sparse=['csr', 'csc', 'coo'], y_numeric=True, multi_output=True) if sample_weight is not None and np.atleast_1d(sample_weight).ndim > 1: raise ValueError("Sample weights must be 1D array or scalar") X, y, X_offset, y_offset, X_scale = self._preprocess_data( X, y, fit_intercept=self.fit_intercept, normalize=self.normalize, copy=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) if sp.issparse(X): if y.ndim < 2: out = sparse_lsqr(X, y) self.coef_ = out[0] self._residues = out[3] else: # sparse_lstsq cannot handle y with shape (M, K) outs = Parallel(n_jobs=n_jobs_)( delayed(sparse_lsqr)(X, y[:, j].ravel()) for j in range(y.shape[1])) self.coef_ = np.vstack([out[0] for out in outs]) self._residues = np.vstack([out[3] for out in outs]) else: self.coef_, self._residues, self.rank_, self.singular_ = \ linalg.lstsq(X, y) self.coef_ = self.coef_.T if y.ndim == 1: self.coef_ = np.ravel(self.coef_) self._set_intercept(X_offset, y_offset, X_scale) return self def _pre_fit(X, y, Xy, precompute, normalize, fit_intercept, copy, check_input=True): """Aux function used at beginning of fit in linear models""" n_samples, n_features = X.shape if sparse.isspmatrix(X): # copy is not needed here as X is not modified inplace when X is sparse precompute = False X, y, X_offset, y_offset, X_scale = _preprocess_data( X, y, fit_intercept=fit_intercept, normalize=normalize, copy=False, return_mean=True, check_input=check_input) else: # copy was done in fit if necessary X, y, X_offset, y_offset, X_scale = _preprocess_data( X, y, fit_intercept=fit_intercept, normalize=normalize, copy=copy, check_input=check_input) if hasattr(precompute, '__array__') and ( fit_intercept and not np.allclose(X_offset, np.zeros(n_features)) or normalize and not np.allclose(X_scale, np.ones(n_features))): warnings.warn("Gram matrix was provided but X was centered" " to fit intercept, " "or X was normalized : recomputing Gram matrix.", UserWarning) # recompute Gram precompute = 'auto' Xy = None # precompute if n_samples > n_features if isinstance(precompute, six.string_types) and precompute == 'auto': precompute = (n_samples > n_features) if precompute is True: # make sure that the 'precompute' array is contiguous. precompute = np.empty(shape=(n_features, n_features), dtype=X.dtype, order='C') np.dot(X.T, X, out=precompute) if not hasattr(precompute, '__array__'): Xy = None # cannot use Xy if precompute is not Gram if hasattr(precompute, '__array__') and Xy is None: common_dtype = np.find_common_type([X.dtype, y.dtype], []) if y.ndim == 1: # Xy is 1d, make sure it is contiguous. Xy = np.empty(shape=n_features, dtype=common_dtype, order='C') np.dot(X.T, y, out=Xy) else: # Make sure that Xy is always F contiguous even if X or y are not # contiguous: the goal is to make it fast to extract the data for a # specific target. n_targets = y.shape[1] Xy = np.empty(shape=(n_features, n_targets), dtype=common_dtype, order='F') np.dot(y.T, X, out=Xy.T) return X, y, X_offset, y_offset, X_scale, precompute, Xy