""" Generalized Linear Models. """ # Author: Alexandre Gramfort # Fabian Pedregosa # Olivier Grisel # Vincent Michel # Peter Prettenhofer # Mathieu Blondel # Lars Buitinck # Maryan Morel # Giorgio Patrini # Maria Telenczuk # License: BSD 3 clause import numbers import warnings from abc import ABCMeta, abstractmethod from numbers import Integral import numpy as np import scipy.sparse as sp from scipy import linalg, optimize, sparse from scipy.sparse.linalg import lsqr from scipy.special import expit from ..base import ( BaseEstimator, ClassifierMixin, MultiOutputMixin, RegressorMixin, _fit_context, ) from ..preprocessing._data import _is_constant_feature from ..utils import check_array, check_random_state from ..utils._array_api import get_namespace from ..utils._seq_dataset import ( ArrayDataset32, ArrayDataset64, CSRDataset32, CSRDataset64, ) from ..utils.extmath import _incremental_mean_and_var, safe_sparse_dot from ..utils.parallel import Parallel, delayed from ..utils.sparsefuncs import inplace_column_scale, mean_variance_axis from ..utils.validation import FLOAT_DTYPES, _check_sample_weight, check_is_fitted # 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. # TODO(1.4): remove # parameter 'normalize' should be removed from linear models def _deprecate_normalize(normalize, estimator_name): """Normalize is to be deprecated from linear models and a use of a pipeline with a StandardScaler is to be recommended instead. Here the appropriate message is selected to be displayed to the user depending on the default normalize value (as it varies between the linear models and normalize value selected by the user). Parameters ---------- normalize : bool, normalize value passed by the user estimator_name : str name of the linear estimator which calls this function. The name will be used for writing the deprecation warnings Returns ------- normalize : bool, normalize value which should further be used by the estimator at this stage of the depreciation process Notes ----- This function should be completely removed in 1.4. """ if normalize not in [True, False, "deprecated"]: raise ValueError( "Leave 'normalize' to its default value or set it to True or False" ) if normalize == "deprecated": _normalize = False else: _normalize = normalize pipeline_msg = ( "If you wish to scale the data, use Pipeline with a StandardScaler " "in a preprocessing stage. To reproduce the previous behavior:\n\n" "from sklearn.pipeline import make_pipeline\n\n" "model = make_pipeline(StandardScaler(with_mean=False), " f"{estimator_name}())\n\n" "If you wish to pass a sample_weight parameter, you need to pass it " "as a fit parameter to each step of the pipeline as follows:\n\n" "kwargs = {s[0] + '__sample_weight': sample_weight for s " "in model.steps}\n" "model.fit(X, y, **kwargs)\n\n" ) alpha_msg = "" if "LassoLars" in estimator_name: alpha_msg = "Set parameter alpha to: original_alpha * np.sqrt(n_samples). " if normalize != "deprecated" and normalize: warnings.warn( "'normalize' was deprecated in version 1.2 and will be removed in 1.4.\n" + pipeline_msg + alpha_msg, FutureWarning, ) elif not normalize: warnings.warn( ( "'normalize' was deprecated in version 1.2 and will be " "removed in 1.4. " "Please leave the normalize parameter to its default value to " "silence this warning. The default behavior of this estimator " "is to not do any normalization. If normalization is needed " "please use sklearn.preprocessing.StandardScaler instead." ), FutureWarning, ) return _normalize 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 random sampling. It is not used for dataset shuffling. 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 SequentialDataset64 seed = rng.randint(1, np.iinfo(np.int32).max) if X.dtype == np.float32: CSRData = CSRDataset32 ArrayData = ArrayDataset32 else: CSRData = CSRDataset64 ArrayData = ArrayDataset64 if sp.issparse(X): dataset = CSRData(X.data, X.indptr, X.indices, y, sample_weight, seed=seed) intercept_decay = SPARSE_INTERCEPT_DECAY else: X = np.ascontiguousarray(X) dataset = ArrayData(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, copy_y=True, sample_weight=None, check_input=True, ): """Center and scale data. 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 fit_intercept=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 Returns ------- X_out : {ndarray, sparse matrix} of shape (n_samples, n_features) If copy=True a copy of the input X is triggered, otherwise operations are inplace. If input X is dense, then X_out is centered. If normalize is True, then X_out is rescaled (dense and sparse case) y_out : {ndarray, sparse matrix} of shape (n_samples,) or (n_samples, n_targets) Centered version of y. Likely performed inplace on input y. X_offset : ndarray of shape (n_features,) The mean per column of input X. y_offset : float or ndarray of shape (n_features,) X_scale : ndarray of shape (n_features,) The standard deviation per column of input X. """ if isinstance(sample_weight, numbers.Number): sample_weight = None if sample_weight is not None: sample_weight = np.asarray(sample_weight) if check_input: X = check_array(X, copy=copy, accept_sparse=["csr", "csc"], dtype=FLOAT_DTYPES) y = check_array(y, dtype=X.dtype, copy=copy_y, ensure_2d=False) else: y = y.astype(X.dtype, copy=copy_y) if copy: if sp.issparse(X): X = X.copy() else: X = X.copy(order="K") if fit_intercept: if sp.issparse(X): X_offset, X_var = mean_variance_axis(X, axis=0, weights=sample_weight) else: if normalize: X_offset, X_var, _ = _incremental_mean_and_var( X, last_mean=0.0, last_variance=0.0, last_sample_count=0.0, sample_weight=sample_weight, ) else: X_offset = np.average(X, axis=0, weights=sample_weight) X_offset = X_offset.astype(X.dtype, copy=False) X -= X_offset if normalize: X_var = X_var.astype(X.dtype, copy=False) # Detect constant features on the computed variance, before taking # the np.sqrt. Otherwise constant features cannot be detected with # sample weights. constant_mask = _is_constant_feature(X_var, X_offset, X.shape[0]) if sample_weight is None: X_var *= X.shape[0] else: X_var *= sample_weight.sum() X_scale = np.sqrt(X_var, out=X_var) X_scale[constant_mask] = 1.0 if sp.issparse(X): inplace_column_scale(X, 1.0 / X_scale) else: X /= X_scale else: X_scale = np.ones(X.shape[1], dtype=X.dtype) y_offset = np.average(y, axis=0, weights=sample_weight) 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, inplace=False): """Rescale data sample-wise by square root of sample_weight. For many linear models, this enables easy support for sample_weight because (y - X w)' S (y - X w) with S = diag(sample_weight) becomes ||y_rescaled - X_rescaled w||_2^2 when setting y_rescaled = sqrt(S) y X_rescaled = sqrt(S) X Returns ------- X_rescaled : {array-like, sparse matrix} y_rescaled : {array-like, sparse matrix} """ # Assume that _validate_data and _check_sample_weight have been called by # the caller. n_samples = X.shape[0] sample_weight_sqrt = np.sqrt(sample_weight) if sp.issparse(X) or sp.issparse(y): sw_matrix = sparse.dia_matrix( (sample_weight_sqrt, 0), shape=(n_samples, n_samples) ) if sp.issparse(X): X = safe_sparse_dot(sw_matrix, X) else: if inplace: X *= sample_weight_sqrt[:, np.newaxis] else: X = X * sample_weight_sqrt[:, np.newaxis] if sp.issparse(y): y = safe_sparse_dot(sw_matrix, y) else: if inplace: if y.ndim == 1: y *= sample_weight_sqrt else: y *= sample_weight_sqrt[:, np.newaxis] else: if y.ndim == 1: y = y * sample_weight_sqrt else: y = y * sample_weight_sqrt[:, np.newaxis] return X, y, sample_weight_sqrt class LinearModel(BaseEstimator, metaclass=ABCMeta): """Base class for Linear Models""" @abstractmethod def fit(self, X, y): """Fit model.""" def _decision_function(self, X): check_is_fitted(self) X = self._validate_data(X, accept_sparse=["csr", "csc", "coo"], reset=False) 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) def _set_intercept(self, X_offset, y_offset, X_scale): """Set the intercept_""" if self.fit_intercept: # We always want coef_.dtype=X.dtype. For instance, X.dtype can differ from # coef_.dtype if warm_start=True. self.coef_ = np.divide(self.coef_, X_scale, dtype=X_scale.dtype) self.intercept_ = y_offset - np.dot(X_offset, self.coef_.T) else: self.intercept_ = 0.0 def _more_tags(self): return {"requires_y": True} # 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 proportional to the signed distance of that sample to the hyperplane. Parameters ---------- X : {array-like, sparse matrix} of shape (n_samples, n_features) The data matrix for which we want to get the confidence scores. Returns ------- scores : ndarray of shape (n_samples,) or (n_samples, n_classes) Confidence scores per `(n_samples, n_classes)` combination. In the binary case, confidence score for `self.classes_[1]` where >0 means this class would be predicted. """ check_is_fitted(self) xp, _ = get_namespace(X) X = self._validate_data(X, accept_sparse="csr", reset=False) scores = safe_sparse_dot(X, self.coef_.T, dense_output=True) + self.intercept_ return xp.reshape(scores, (-1,)) if scores.shape[1] == 1 else scores def predict(self, X): """ Predict class labels for samples in X. Parameters ---------- X : {array-like, sparse matrix} of shape (n_samples, n_features) The data matrix for which we want to get the predictions. Returns ------- y_pred : ndarray of shape (n_samples,) Vector containing the class labels for each sample. """ xp, _ = get_namespace(X) scores = self.decision_function(X) if len(scores.shape) == 1: indices = xp.astype(scores > 0, int) else: indices = xp.argmax(scores, axis=1) return xp.take(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) expit(prob, out=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: """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 Fitted estimator. """ msg = "Estimator, %(name)s, must be fitted before densifying." check_is_fitted(self, 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. Returns ------- self Fitted estimator. 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. """ msg = "Estimator, %(name)s, must be fitted before sparsifying." check_is_fitted(self, msg=msg) self.coef_ = sp.csr_matrix(self.coef_) return self class LinearRegression(MultiOutputMixin, RegressorMixin, LinearModel): """ Ordinary least squares Linear Regression. LinearRegression fits a linear model with coefficients w = (w1, ..., wp) to minimize the residual sum of squares between the observed targets in the dataset, and the targets predicted by the linear approximation. Parameters ---------- 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). copy_X : bool, default=True If True, X will be copied; else, it may be overwritten. n_jobs : int, default=None The number of jobs to use for the computation. This will only provide speedup in case of sufficiently large problems, that is if firstly `n_targets > 1` and secondly `X` is sparse or if `positive` is set to `True`. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. positive : bool, default=False When set to ``True``, forces the coefficients to be positive. This option is only supported for dense arrays. .. versionadded:: 0.24 Attributes ---------- coef_ : array of 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. rank_ : int Rank of matrix `X`. Only available when `X` is dense. singular_ : array of shape (min(X, y),) Singular values of `X`. Only available when `X` is dense. intercept_ : float or array of shape (n_targets,) Independent term in the linear model. Set to 0.0 if `fit_intercept = False`. n_features_in_ : int Number of features seen during :term:`fit`. .. versionadded:: 0.24 feature_names_in_ : ndarray of shape (`n_features_in_`,) Names of features seen during :term:`fit`. Defined only when `X` has feature names that are all strings. .. versionadded:: 1.0 See Also -------- Ridge : Ridge regression addresses some of the problems of Ordinary Least Squares by imposing a penalty on the size of the coefficients with l2 regularization. Lasso : The Lasso is a linear model that estimates sparse coefficients with l1 regularization. ElasticNet : Elastic-Net is a linear regression model trained with both l1 and l2 -norm regularization of the coefficients. Notes ----- From the implementation point of view, this is just plain Ordinary Least Squares (scipy.linalg.lstsq) or Non Negative Least Squares (scipy.optimize.nnls) wrapped as a predictor object. 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_ 3.0... >>> reg.predict(np.array([[3, 5]])) array([16.]) """ _parameter_constraints: dict = { "fit_intercept": ["boolean"], "copy_X": ["boolean"], "n_jobs": [None, Integral], "positive": ["boolean"], } def __init__( self, *, fit_intercept=True, copy_X=True, n_jobs=None, positive=False, ): self.fit_intercept = fit_intercept self.copy_X = copy_X self.n_jobs = n_jobs self.positive = positive @_fit_context(prefer_skip_nested_validation=True) def fit(self, X, y, sample_weight=None): """ Fit linear model. Parameters ---------- X : {array-like, sparse matrix} of shape (n_samples, n_features) Training data. y : array-like of shape (n_samples,) or (n_samples, n_targets) Target values. Will be cast to X's dtype if necessary. sample_weight : array-like of shape (n_samples,), default=None Individual weights for each sample. .. versionadded:: 0.17 parameter *sample_weight* support to LinearRegression. Returns ------- self : object Fitted Estimator. """ n_jobs_ = self.n_jobs accept_sparse = False if self.positive else ["csr", "csc", "coo"] X, y = self._validate_data( X, y, accept_sparse=accept_sparse, y_numeric=True, multi_output=True ) has_sw = sample_weight is not None if has_sw: sample_weight = _check_sample_weight( sample_weight, X, dtype=X.dtype, only_non_negative=True ) # Note that neither _rescale_data nor the rest of the fit method of # LinearRegression can benefit from in-place operations when X is a # sparse matrix. Therefore, let's not copy X when it is sparse. copy_X_in_preprocess_data = self.copy_X and not sp.issparse(X) X, y, X_offset, y_offset, X_scale = _preprocess_data( X, y, fit_intercept=self.fit_intercept, copy=copy_X_in_preprocess_data, sample_weight=sample_weight, ) if has_sw: # Sample weight can be implemented via a simple rescaling. Note # that we safely do inplace rescaling when _preprocess_data has # already made a copy if requested. X, y, sample_weight_sqrt = _rescale_data( X, y, sample_weight, inplace=copy_X_in_preprocess_data ) if self.positive: if y.ndim < 2: self.coef_ = optimize.nnls(X, y)[0] else: # scipy.optimize.nnls cannot handle y with shape (M, K) outs = Parallel(n_jobs=n_jobs_)( delayed(optimize.nnls)(X, y[:, j]) for j in range(y.shape[1]) ) self.coef_ = np.vstack([out[0] for out in outs]) elif sp.issparse(X): X_offset_scale = X_offset / X_scale if has_sw: def matvec(b): return X.dot(b) - sample_weight_sqrt * b.dot(X_offset_scale) def rmatvec(b): return X.T.dot(b) - X_offset_scale * b.dot(sample_weight_sqrt) else: def matvec(b): return X.dot(b) - b.dot(X_offset_scale) def rmatvec(b): return X.T.dot(b) - X_offset_scale * b.sum() X_centered = sparse.linalg.LinearOperator( shape=X.shape, matvec=matvec, rmatvec=rmatvec ) if y.ndim < 2: self.coef_ = lsqr(X_centered, y)[0] else: # sparse_lstsq cannot handle y with shape (M, K) outs = Parallel(n_jobs=n_jobs_)( delayed(lsqr)(X_centered, y[:, j].ravel()) for j in range(y.shape[1]) ) self.coef_ = np.vstack([out[0] for out in outs]) else: self.coef_, _, 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 _check_precomputed_gram_matrix( X, precompute, X_offset, X_scale, rtol=None, atol=1e-5 ): """Computes a single element of the gram matrix and compares it to the corresponding element of the user supplied gram matrix. If the values do not match a ValueError will be thrown. Parameters ---------- X : ndarray of shape (n_samples, n_features) Data array. precompute : array-like of shape (n_features, n_features) User-supplied gram matrix. X_offset : ndarray of shape (n_features,) Array of feature means used to center design matrix. X_scale : ndarray of shape (n_features,) Array of feature scale factors used to normalize design matrix. rtol : float, default=None Relative tolerance; see numpy.allclose If None, it is set to 1e-4 for arrays of dtype numpy.float32 and 1e-7 otherwise. atol : float, default=1e-5 absolute tolerance; see :func`numpy.allclose`. Note that the default here is more tolerant than the default for :func:`numpy.testing.assert_allclose`, where `atol=0`. Raises ------ ValueError Raised when the provided Gram matrix is not consistent. """ n_features = X.shape[1] f1 = n_features // 2 f2 = min(f1 + 1, n_features - 1) v1 = (X[:, f1] - X_offset[f1]) * X_scale[f1] v2 = (X[:, f2] - X_offset[f2]) * X_scale[f2] expected = np.dot(v1, v2) actual = precompute[f1, f2] dtypes = [precompute.dtype, expected.dtype] if rtol is None: rtols = [1e-4 if dtype == np.float32 else 1e-7 for dtype in dtypes] rtol = max(rtols) if not np.isclose(expected, actual, rtol=rtol, atol=atol): raise ValueError( "Gram matrix passed in via 'precompute' parameter " "did not pass validation when a single element was " "checked - please check that it was computed " f"properly. For element ({f1},{f2}) we computed " f"{expected} but the user-supplied value was " f"{actual}." ) def _pre_fit( X, y, Xy, precompute, normalize, fit_intercept, copy, check_input=True, sample_weight=None, ): """Function used at beginning of fit in linear models with L1 or L0 penalty. This function applies _preprocess_data and additionally computes the gram matrix `precompute` as needed as well as `Xy`. """ n_samples, n_features = X.shape if sparse.issparse(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, check_input=check_input, sample_weight=sample_weight, ) 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, sample_weight=sample_weight, ) # Rescale only in dense case. Sparse cd solver directly deals with # sample_weight. if sample_weight is not None: # This triggers copies anyway. X, y, _ = _rescale_data(X, y, sample_weight=sample_weight) # FIXME: 'normalize' to be removed in 1.4 if hasattr(precompute, "__array__"): if ( 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 elif check_input: # If we're going to use the user's precomputed gram matrix, we # do a quick check to make sure its not totally bogus. _check_precomputed_gram_matrix(X, precompute, X_offset, X_scale) # precompute if n_samples > n_features if isinstance(precompute, str) 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.result_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