""" Least Angle Regression algorithm. See the documentation on the Generalized Linear Model for a complete discussion. """ from __future__ import print_function # Author: Fabian Pedregosa # Alexandre Gramfort # Gael Varoquaux # # License: BSD 3 clause from math import log import sys import warnings import numpy as np from scipy import linalg, interpolate from scipy.linalg.lapack import get_lapack_funcs from .base import LinearModel from ..base import RegressorMixin from ..utils import arrayfuncs, as_float_array, check_X_y, deprecated from ..model_selection import check_cv from ..exceptions import ConvergenceWarning from ..utils._joblib import Parallel, delayed from ..externals.six.moves import xrange from ..externals.six import string_types solve_triangular_args = {'check_finite': False} def lars_path(X, y, Xy=None, Gram=None, max_iter=500, alpha_min=0, method='lar', copy_X=True, eps=np.finfo(np.float).eps, copy_Gram=True, verbose=0, return_path=True, return_n_iter=False, positive=False): """Compute Least Angle Regression or Lasso path using LARS algorithm [1] The optimization objective for the case method='lasso' is:: (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1 in the case of method='lars', the objective function is only known in the form of an implicit equation (see discussion in [1]) Read more in the :ref:`User Guide `. Parameters ----------- X : array, shape: (n_samples, n_features) Input data. y : array, shape: (n_samples) Input targets. Xy : array-like, shape (n_samples,) or (n_samples, n_targets), \ optional Xy = np.dot(X.T, y) that can be precomputed. It is useful only when the Gram matrix is precomputed. Gram : None, 'auto', array, shape: (n_features, n_features), optional Precomputed Gram matrix (X' * X), if ``'auto'``, the Gram matrix is precomputed from the given X, if there are more samples than features. max_iter : integer, optional (default=500) Maximum number of iterations to perform, set to infinity for no limit. alpha_min : float, optional (default=0) Minimum correlation along the path. It corresponds to the regularization parameter alpha parameter in the Lasso. method : {'lar', 'lasso'}, optional (default='lar') Specifies the returned model. Select ``'lar'`` for Least Angle Regression, ``'lasso'`` for the Lasso. copy_X : bool, optional (default=True) If ``False``, ``X`` is overwritten. eps : float, optional (default=``np.finfo(np.float).eps``) The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. copy_Gram : bool, optional (default=True) If ``False``, ``Gram`` is overwritten. verbose : int (default=0) Controls output verbosity. return_path : bool, optional (default=True) If ``return_path==True`` returns the entire path, else returns only the last point of the path. return_n_iter : bool, optional (default=False) Whether to return the number of iterations. positive : boolean (default=False) Restrict coefficients to be >= 0. This option is only allowed with method 'lasso'. Note that the model coefficients will not converge to the ordinary-least-squares solution for small values of alpha. Only coefficients up to the smallest alpha value (``alphas_[alphas_ > 0.].min()`` when fit_path=True) reached by the stepwise Lars-Lasso algorithm are typically in congruence with the solution of the coordinate descent lasso_path function. Returns -------- alphas : array, shape: [n_alphas + 1] Maximum of covariances (in absolute value) at each iteration. ``n_alphas`` is either ``max_iter``, ``n_features`` or the number of nodes in the path with ``alpha >= alpha_min``, whichever is smaller. active : array, shape [n_alphas] Indices of active variables at the end of the path. coefs : array, shape (n_features, n_alphas + 1) Coefficients along the path n_iter : int Number of iterations run. Returned only if return_n_iter is set to True. See also -------- lasso_path LassoLars Lars LassoLarsCV LarsCV sklearn.decomposition.sparse_encode References ---------- .. [1] "Least Angle Regression", Effron et al. http://statweb.stanford.edu/~tibs/ftp/lars.pdf .. [2] `Wikipedia entry on the Least-angle regression `_ .. [3] `Wikipedia entry on the Lasso `_ """ if method == 'lar' and positive: warnings.warn('positive option is broken for Least' ' Angle Regression (LAR). Use method="lasso".' ' This option will be removed in version 0.22.', DeprecationWarning) n_features = X.shape[1] n_samples = y.size max_features = min(max_iter, n_features) if return_path: coefs = np.zeros((max_features + 1, n_features)) alphas = np.zeros(max_features + 1) else: coef, prev_coef = np.zeros(n_features), np.zeros(n_features) alpha, prev_alpha = np.array([0.]), np.array([0.]) # better ideas? n_iter, n_active = 0, 0 active, indices = list(), np.arange(n_features) # holds the sign of covariance sign_active = np.empty(max_features, dtype=np.int8) drop = False # will hold the cholesky factorization. Only lower part is # referenced. L = np.empty((max_features, max_features), dtype=X.dtype) swap, nrm2 = linalg.get_blas_funcs(('swap', 'nrm2'), (X,)) solve_cholesky, = get_lapack_funcs(('potrs',), (X,)) if Gram is None or Gram is False: Gram = None if copy_X: # force copy. setting the array to be fortran-ordered # speeds up the calculation of the (partial) Gram matrix # and allows to easily swap columns X = X.copy('F') elif isinstance(Gram, string_types) and Gram == 'auto' or Gram is True: if Gram is True or X.shape[0] > X.shape[1]: Gram = np.dot(X.T, X) else: Gram = None elif copy_Gram: Gram = Gram.copy() if Xy is None: Cov = np.dot(X.T, y) else: Cov = Xy.copy() if verbose: if verbose > 1: print("Step\t\tAdded\t\tDropped\t\tActive set size\t\tC") else: sys.stdout.write('.') sys.stdout.flush() tiny32 = np.finfo(np.float32).tiny # to avoid division by 0 warning equality_tolerance = np.finfo(np.float32).eps while True: if Cov.size: if positive: C_idx = np.argmax(Cov) else: C_idx = np.argmax(np.abs(Cov)) C_ = Cov[C_idx] if positive: C = C_ else: C = np.fabs(C_) else: C = 0. if return_path: alpha = alphas[n_iter, np.newaxis] coef = coefs[n_iter] prev_alpha = alphas[n_iter - 1, np.newaxis] prev_coef = coefs[n_iter - 1] alpha[0] = C / n_samples if alpha[0] <= alpha_min + equality_tolerance: # early stopping if abs(alpha[0] - alpha_min) > equality_tolerance: # interpolation factor 0 <= ss < 1 if n_iter > 0: # In the first iteration, all alphas are zero, the formula # below would make ss a NaN ss = ((prev_alpha[0] - alpha_min) / (prev_alpha[0] - alpha[0])) coef[:] = prev_coef + ss * (coef - prev_coef) alpha[0] = alpha_min if return_path: coefs[n_iter] = coef break if n_iter >= max_iter or n_active >= n_features: break if not drop: ########################################################## # Append x_j to the Cholesky factorization of (Xa * Xa') # # # # ( L 0 ) # # L -> ( ) , where L * w = Xa' x_j # # ( w z ) and z = ||x_j|| # # # ########################################################## if positive: sign_active[n_active] = np.ones_like(C_) else: sign_active[n_active] = np.sign(C_) m, n = n_active, C_idx + n_active Cov[C_idx], Cov[0] = swap(Cov[C_idx], Cov[0]) indices[n], indices[m] = indices[m], indices[n] Cov_not_shortened = Cov Cov = Cov[1:] # remove Cov[0] if Gram is None: X.T[n], X.T[m] = swap(X.T[n], X.T[m]) c = nrm2(X.T[n_active]) ** 2 L[n_active, :n_active] = \ np.dot(X.T[n_active], X.T[:n_active].T) else: # swap does only work inplace if matrix is fortran # contiguous ... Gram[m], Gram[n] = swap(Gram[m], Gram[n]) Gram[:, m], Gram[:, n] = swap(Gram[:, m], Gram[:, n]) c = Gram[n_active, n_active] L[n_active, :n_active] = Gram[n_active, :n_active] # Update the cholesky decomposition for the Gram matrix if n_active: linalg.solve_triangular(L[:n_active, :n_active], L[n_active, :n_active], trans=0, lower=1, overwrite_b=True, **solve_triangular_args) v = np.dot(L[n_active, :n_active], L[n_active, :n_active]) diag = max(np.sqrt(np.abs(c - v)), eps) L[n_active, n_active] = diag if diag < 1e-7: # The system is becoming too ill-conditioned. # We have degenerate vectors in our active set. # We'll 'drop for good' the last regressor added. # Note: this case is very rare. It is no longer triggered by # the test suite. The `equality_tolerance` margin added in 0.16 # to get early stopping to work consistently on all versions of # Python including 32 bit Python under Windows seems to make it # very difficult to trigger the 'drop for good' strategy. warnings.warn('Regressors in active set degenerate. ' 'Dropping a regressor, after %i iterations, ' 'i.e. alpha=%.3e, ' 'with an active set of %i regressors, and ' 'the smallest cholesky pivot element being %.3e.' ' Reduce max_iter or increase eps parameters.' % (n_iter, alpha, n_active, diag), ConvergenceWarning) # XXX: need to figure a 'drop for good' way Cov = Cov_not_shortened Cov[0] = 0 Cov[C_idx], Cov[0] = swap(Cov[C_idx], Cov[0]) continue active.append(indices[n_active]) n_active += 1 if verbose > 1: print("%s\t\t%s\t\t%s\t\t%s\t\t%s" % (n_iter, active[-1], '', n_active, C)) if method == 'lasso' and n_iter > 0 and prev_alpha[0] < alpha[0]: # alpha is increasing. This is because the updates of Cov are # bringing in too much numerical error that is greater than # than the remaining correlation with the # regressors. Time to bail out warnings.warn('Early stopping the lars path, as the residues ' 'are small and the current value of alpha is no ' 'longer well controlled. %i iterations, alpha=%.3e, ' 'previous alpha=%.3e, with an active set of %i ' 'regressors.' % (n_iter, alpha, prev_alpha, n_active), ConvergenceWarning) break # least squares solution least_squares, info = solve_cholesky(L[:n_active, :n_active], sign_active[:n_active], lower=True) if least_squares.size == 1 and least_squares == 0: # This happens because sign_active[:n_active] = 0 least_squares[...] = 1 AA = 1. else: # is this really needed ? AA = 1. / np.sqrt(np.sum(least_squares * sign_active[:n_active])) if not np.isfinite(AA): # L is too ill-conditioned i = 0 L_ = L[:n_active, :n_active].copy() while not np.isfinite(AA): L_.flat[::n_active + 1] += (2 ** i) * eps least_squares, info = solve_cholesky( L_, sign_active[:n_active], lower=True) tmp = max(np.sum(least_squares * sign_active[:n_active]), eps) AA = 1. / np.sqrt(tmp) i += 1 least_squares *= AA if Gram is None: # equiangular direction of variables in the active set eq_dir = np.dot(X.T[:n_active].T, least_squares) # correlation between each unactive variables and # eqiangular vector corr_eq_dir = np.dot(X.T[n_active:], eq_dir) else: # if huge number of features, this takes 50% of time, I # think could be avoided if we just update it using an # orthogonal (QR) decomposition of X corr_eq_dir = np.dot(Gram[:n_active, n_active:].T, least_squares) g1 = arrayfuncs.min_pos((C - Cov) / (AA - corr_eq_dir + tiny32)) if positive: gamma_ = min(g1, C / AA) else: g2 = arrayfuncs.min_pos((C + Cov) / (AA + corr_eq_dir + tiny32)) gamma_ = min(g1, g2, C / AA) # TODO: better names for these variables: z drop = False z = -coef[active] / (least_squares + tiny32) z_pos = arrayfuncs.min_pos(z) if z_pos < gamma_: # some coefficients have changed sign idx = np.where(z == z_pos)[0][::-1] # update the sign, important for LAR sign_active[idx] = -sign_active[idx] if method == 'lasso': gamma_ = z_pos drop = True n_iter += 1 if return_path: if n_iter >= coefs.shape[0]: del coef, alpha, prev_alpha, prev_coef # resize the coefs and alphas array add_features = 2 * max(1, (max_features - n_active)) coefs = np.resize(coefs, (n_iter + add_features, n_features)) coefs[-add_features:] = 0 alphas = np.resize(alphas, n_iter + add_features) alphas[-add_features:] = 0 coef = coefs[n_iter] prev_coef = coefs[n_iter - 1] else: # mimic the effect of incrementing n_iter on the array references prev_coef = coef prev_alpha[0] = alpha[0] coef = np.zeros_like(coef) coef[active] = prev_coef[active] + gamma_ * least_squares # update correlations Cov -= gamma_ * corr_eq_dir # See if any coefficient has changed sign if drop and method == 'lasso': # handle the case when idx is not length of 1 [arrayfuncs.cholesky_delete(L[:n_active, :n_active], ii) for ii in idx] n_active -= 1 m, n = idx, n_active # handle the case when idx is not length of 1 drop_idx = [active.pop(ii) for ii in idx] if Gram is None: # propagate dropped variable for ii in idx: for i in range(ii, n_active): X.T[i], X.T[i + 1] = swap(X.T[i], X.T[i + 1]) # yeah this is stupid indices[i], indices[i + 1] = indices[i + 1], indices[i] # TODO: this could be updated residual = y - np.dot(X[:, :n_active], coef[active]) temp = np.dot(X.T[n_active], residual) Cov = np.r_[temp, Cov] else: for ii in idx: for i in range(ii, n_active): indices[i], indices[i + 1] = indices[i + 1], indices[i] Gram[i], Gram[i + 1] = swap(Gram[i], Gram[i + 1]) Gram[:, i], Gram[:, i + 1] = swap(Gram[:, i], Gram[:, i + 1]) # Cov_n = Cov_j + x_j * X + increment(betas) TODO: # will this still work with multiple drops ? # recompute covariance. Probably could be done better # wrong as Xy is not swapped with the rest of variables # TODO: this could be updated residual = y - np.dot(X, coef) temp = np.dot(X.T[drop_idx], residual) Cov = np.r_[temp, Cov] sign_active = np.delete(sign_active, idx) sign_active = np.append(sign_active, 0.) # just to maintain size if verbose > 1: print("%s\t\t%s\t\t%s\t\t%s\t\t%s" % (n_iter, '', drop_idx, n_active, abs(temp))) if return_path: # resize coefs in case of early stop alphas = alphas[:n_iter + 1] coefs = coefs[:n_iter + 1] if return_n_iter: return alphas, active, coefs.T, n_iter else: return alphas, active, coefs.T else: if return_n_iter: return alpha, active, coef, n_iter else: return alpha, active, coef ############################################################################### # Estimator classes class Lars(LinearModel, RegressorMixin): """Least Angle Regression model a.k.a. LAR Read more in the :ref:`User Guide `. Parameters ---------- fit_intercept : boolean 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). verbose : boolean or integer, optional Sets the verbosity amount normalize : boolean, optional, default True 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``. precompute : True | False | 'auto' | array-like Whether to use a precomputed Gram matrix to speed up calculations. If set to ``'auto'`` let us decide. The Gram matrix can also be passed as argument. n_nonzero_coefs : int, optional Target number of non-zero coefficients. Use ``np.inf`` for no limit. eps : float, optional The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Unlike the ``tol`` parameter in some iterative optimization-based algorithms, this parameter does not control the tolerance of the optimization. copy_X : boolean, optional, default True If ``True``, X will be copied; else, it may be overwritten. fit_path : boolean If True the full path is stored in the ``coef_path_`` attribute. If you compute the solution for a large problem or many targets, setting ``fit_path`` to ``False`` will lead to a speedup, especially with a small alpha. positive : boolean (default=False) Restrict coefficients to be >= 0. Be aware that you might want to remove fit_intercept which is set True by default. .. deprecated:: 0.20 The option is broken and deprecated. It will be removed in v0.22. Attributes ---------- alphas_ : array, shape (n_alphas + 1,) | list of n_targets such arrays Maximum of covariances (in absolute value) at each iteration. \ ``n_alphas`` is either ``n_nonzero_coefs`` or ``n_features``, \ whichever is smaller. active_ : list, length = n_alphas | list of n_targets such lists Indices of active variables at the end of the path. coef_path_ : array, shape (n_features, n_alphas + 1) \ | list of n_targets such arrays The varying values of the coefficients along the path. It is not present if the ``fit_path`` parameter is ``False``. coef_ : array, shape (n_features,) or (n_targets, n_features) Parameter vector (w in the formulation formula). intercept_ : float | array, shape (n_targets,) Independent term in decision function. n_iter_ : array-like or int The number of iterations taken by lars_path to find the grid of alphas for each target. Examples -------- >>> from sklearn import linear_model >>> reg = linear_model.Lars(n_nonzero_coefs=1) >>> reg.fit([[-1, 1], [0, 0], [1, 1]], [-1.1111, 0, -1.1111]) ... # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE Lars(copy_X=True, eps=..., fit_intercept=True, fit_path=True, n_nonzero_coefs=1, normalize=True, positive=False, precompute='auto', verbose=False) >>> print(reg.coef_) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE [ 0. -1.11...] See also -------- lars_path, LarsCV sklearn.decomposition.sparse_encode """ method = 'lar' def __init__(self, fit_intercept=True, verbose=False, normalize=True, precompute='auto', n_nonzero_coefs=500, eps=np.finfo(np.float).eps, copy_X=True, fit_path=True, positive=False): self.fit_intercept = fit_intercept self.verbose = verbose self.normalize = normalize self.precompute = precompute self.n_nonzero_coefs = n_nonzero_coefs self.positive = positive self.eps = eps self.copy_X = copy_X self.fit_path = fit_path @staticmethod def _get_gram(precompute, X, y): if (not hasattr(precompute, '__array__')) and ( (precompute is True) or (precompute == 'auto' and X.shape[0] > X.shape[1]) or (precompute == 'auto' and y.shape[1] > 1)): precompute = np.dot(X.T, X) return precompute def _fit(self, X, y, max_iter, alpha, fit_path, Xy=None): """Auxiliary method to fit the model using X, y as training data""" n_features = X.shape[1] X, y, X_offset, y_offset, X_scale = self._preprocess_data( X, y, self.fit_intercept, self.normalize, self.copy_X) if y.ndim == 1: y = y[:, np.newaxis] n_targets = y.shape[1] Gram = self._get_gram(self.precompute, X, y) self.alphas_ = [] self.n_iter_ = [] self.coef_ = np.empty((n_targets, n_features)) if fit_path: self.active_ = [] self.coef_path_ = [] for k in xrange(n_targets): this_Xy = None if Xy is None else Xy[:, k] alphas, active, coef_path, n_iter_ = lars_path( X, y[:, k], Gram=Gram, Xy=this_Xy, copy_X=self.copy_X, copy_Gram=True, alpha_min=alpha, method=self.method, verbose=max(0, self.verbose - 1), max_iter=max_iter, eps=self.eps, return_path=True, return_n_iter=True, positive=self.positive) self.alphas_.append(alphas) self.active_.append(active) self.n_iter_.append(n_iter_) self.coef_path_.append(coef_path) self.coef_[k] = coef_path[:, -1] if n_targets == 1: self.alphas_, self.active_, self.coef_path_, self.coef_ = [ a[0] for a in (self.alphas_, self.active_, self.coef_path_, self.coef_)] self.n_iter_ = self.n_iter_[0] else: for k in xrange(n_targets): this_Xy = None if Xy is None else Xy[:, k] alphas, _, self.coef_[k], n_iter_ = lars_path( X, y[:, k], Gram=Gram, Xy=this_Xy, copy_X=self.copy_X, copy_Gram=True, alpha_min=alpha, method=self.method, verbose=max(0, self.verbose - 1), max_iter=max_iter, eps=self.eps, return_path=False, return_n_iter=True, positive=self.positive) self.alphas_.append(alphas) self.n_iter_.append(n_iter_) if n_targets == 1: self.alphas_ = self.alphas_[0] self.n_iter_ = self.n_iter_[0] self._set_intercept(X_offset, y_offset, X_scale) return self def fit(self, X, y, Xy=None): """Fit the model using X, y as training data. Parameters ---------- X : array-like, shape (n_samples, n_features) Training data. y : array-like, shape (n_samples,) or (n_samples, n_targets) Target values. Xy : array-like, shape (n_samples,) or (n_samples, n_targets), \ optional Xy = np.dot(X.T, y) that can be precomputed. It is useful only when the Gram matrix is precomputed. Returns ------- self : object returns an instance of self. """ X, y = check_X_y(X, y, y_numeric=True, multi_output=True) alpha = getattr(self, 'alpha', 0.) if hasattr(self, 'n_nonzero_coefs'): alpha = 0. # n_nonzero_coefs parametrization takes priority max_iter = self.n_nonzero_coefs else: max_iter = self.max_iter self._fit(X, y, max_iter=max_iter, alpha=alpha, fit_path=self.fit_path, Xy=Xy) return self class LassoLars(Lars): """Lasso model fit with Least Angle Regression a.k.a. Lars It is a Linear Model trained with an L1 prior as regularizer. The optimization objective for Lasso is:: (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1 Read more in the :ref:`User Guide `. Parameters ---------- alpha : float Constant that multiplies the penalty term. Defaults to 1.0. ``alpha = 0`` is equivalent to an ordinary least square, solved by :class:`LinearRegression`. For numerical reasons, using ``alpha = 0`` with the LassoLars object is not advised and you should prefer the LinearRegression object. fit_intercept : boolean 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). verbose : boolean or integer, optional Sets the verbosity amount normalize : boolean, optional, default True 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``. precompute : True | False | 'auto' | array-like Whether to use a precomputed Gram matrix to speed up calculations. If set to ``'auto'`` let us decide. The Gram matrix can also be passed as argument. max_iter : integer, optional Maximum number of iterations to perform. eps : float, optional The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Unlike the ``tol`` parameter in some iterative optimization-based algorithms, this parameter does not control the tolerance of the optimization. copy_X : boolean, optional, default True If True, X will be copied; else, it may be overwritten. fit_path : boolean If ``True`` the full path is stored in the ``coef_path_`` attribute. If you compute the solution for a large problem or many targets, setting ``fit_path`` to ``False`` will lead to a speedup, especially with a small alpha. positive : boolean (default=False) Restrict coefficients to be >= 0. Be aware that you might want to remove fit_intercept which is set True by default. Under the positive restriction the model coefficients will not converge to the ordinary-least-squares solution for small values of alpha. Only coefficients up to the smallest alpha value (``alphas_[alphas_ > 0.].min()`` when fit_path=True) reached by the stepwise Lars-Lasso algorithm are typically in congruence with the solution of the coordinate descent Lasso estimator. Attributes ---------- alphas_ : array, shape (n_alphas + 1,) | list of n_targets such arrays Maximum of covariances (in absolute value) at each iteration. \ ``n_alphas`` is either ``max_iter``, ``n_features``, or the number of \ nodes in the path with correlation greater than ``alpha``, whichever \ is smaller. active_ : list, length = n_alphas | list of n_targets such lists Indices of active variables at the end of the path. coef_path_ : array, shape (n_features, n_alphas + 1) or list If a list is passed it's expected to be one of n_targets such arrays. The varying values of the coefficients along the path. It is not present if the ``fit_path`` parameter is ``False``. coef_ : array, shape (n_features,) or (n_targets, n_features) Parameter vector (w in the formulation formula). intercept_ : float | array, shape (n_targets,) Independent term in decision function. n_iter_ : array-like or int. The number of iterations taken by lars_path to find the grid of alphas for each target. Examples -------- >>> from sklearn import linear_model >>> reg = linear_model.LassoLars(alpha=0.01) >>> reg.fit([[-1, 1], [0, 0], [1, 1]], [-1, 0, -1]) ... # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE LassoLars(alpha=0.01, copy_X=True, eps=..., fit_intercept=True, fit_path=True, max_iter=500, normalize=True, positive=False, precompute='auto', verbose=False) >>> print(reg.coef_) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE [ 0. -0.963257...] See also -------- lars_path lasso_path Lasso LassoCV LassoLarsCV LassoLarsIC sklearn.decomposition.sparse_encode """ method = 'lasso' def __init__(self, alpha=1.0, fit_intercept=True, verbose=False, normalize=True, precompute='auto', max_iter=500, eps=np.finfo(np.float).eps, copy_X=True, fit_path=True, positive=False): self.alpha = alpha self.fit_intercept = fit_intercept self.max_iter = max_iter self.verbose = verbose self.normalize = normalize self.positive = positive self.precompute = precompute self.copy_X = copy_X self.eps = eps self.fit_path = fit_path ############################################################################### # Cross-validated estimator classes def _check_copy_and_writeable(array, copy=False): if copy or not array.flags.writeable: return array.copy() return array def _lars_path_residues(X_train, y_train, X_test, y_test, Gram=None, copy=True, method='lars', verbose=False, fit_intercept=True, normalize=True, max_iter=500, eps=np.finfo(np.float).eps, positive=False): """Compute the residues on left-out data for a full LARS path Parameters ----------- X_train : array, shape (n_samples, n_features) The data to fit the LARS on y_train : array, shape (n_samples) The target variable to fit LARS on X_test : array, shape (n_samples, n_features) The data to compute the residues on y_test : array, shape (n_samples) The target variable to compute the residues on Gram : None, 'auto', array, shape: (n_features, n_features), optional Precomputed Gram matrix (X' * X), if ``'auto'``, the Gram matrix is precomputed from the given X, if there are more samples than features copy : boolean, optional Whether X_train, X_test, y_train and y_test should be copied; if False, they may be overwritten. method : 'lar' | 'lasso' Specifies the returned model. Select ``'lar'`` for Least Angle Regression, ``'lasso'`` for the Lasso. verbose : integer, optional Sets the amount of verbosity fit_intercept : boolean 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). positive : boolean (default=False) Restrict coefficients to be >= 0. Be aware that you might want to remove fit_intercept which is set True by default. See reservations for using this option in combination with method 'lasso' for expected small values of alpha in the doc of LassoLarsCV and LassoLarsIC. normalize : boolean, optional, default True 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``. max_iter : integer, optional Maximum number of iterations to perform. eps : float, optional The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Unlike the ``tol`` parameter in some iterative optimization-based algorithms, this parameter does not control the tolerance of the optimization. Returns -------- alphas : array, shape (n_alphas,) Maximum of covariances (in absolute value) at each iteration. ``n_alphas`` is either ``max_iter`` or ``n_features``, whichever is smaller. active : list Indices of active variables at the end of the path. coefs : array, shape (n_features, n_alphas) Coefficients along the path residues : array, shape (n_alphas, n_samples) Residues of the prediction on the test data """ X_train = _check_copy_and_writeable(X_train, copy) y_train = _check_copy_and_writeable(y_train, copy) X_test = _check_copy_and_writeable(X_test, copy) y_test = _check_copy_and_writeable(y_test, copy) if fit_intercept: X_mean = X_train.mean(axis=0) X_train -= X_mean X_test -= X_mean y_mean = y_train.mean(axis=0) y_train = as_float_array(y_train, copy=False) y_train -= y_mean y_test = as_float_array(y_test, copy=False) y_test -= y_mean if normalize: norms = np.sqrt(np.sum(X_train ** 2, axis=0)) nonzeros = np.flatnonzero(norms) X_train[:, nonzeros] /= norms[nonzeros] alphas, active, coefs = lars_path( X_train, y_train, Gram=Gram, copy_X=False, copy_Gram=False, method=method, verbose=max(0, verbose - 1), max_iter=max_iter, eps=eps, positive=positive) if normalize: coefs[nonzeros] /= norms[nonzeros][:, np.newaxis] residues = np.dot(X_test, coefs) - y_test[:, np.newaxis] return alphas, active, coefs, residues.T class LarsCV(Lars): """Cross-validated Least Angle Regression model. See glossary entry for :term:`cross-validation estimator`. Read more in the :ref:`User Guide `. Parameters ---------- fit_intercept : boolean 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). verbose : boolean or integer, optional Sets the verbosity amount max_iter : integer, optional Maximum number of iterations to perform. normalize : boolean, optional, default True 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``. precompute : True | False | 'auto' | array-like Whether to use a precomputed Gram matrix to speed up calculations. If set to ``'auto'`` let us decide. The Gram matrix cannot be passed as argument since we will use only subsets of X. cv : int, cross-validation generator or an iterable, optional Determines the cross-validation splitting strategy. Possible inputs for cv are: - None, to use the default 3-fold cross-validation, - integer, to specify the number of folds. - :term:`CV splitter`, - An iterable yielding (train, test) splits as arrays of indices. For integer/None inputs, :class:`KFold` is used. Refer :ref:`User Guide ` for the various cross-validation strategies that can be used here. .. versionchanged:: 0.20 ``cv`` default value if None will change from 3-fold to 5-fold in v0.22. max_n_alphas : integer, optional The maximum number of points on the path used to compute the residuals in the cross-validation n_jobs : int or None, optional (default=None) Number of CPUs to use during the cross validation. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. eps : float, optional The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. copy_X : boolean, optional, default True If ``True``, X will be copied; else, it may be overwritten. positive : boolean (default=False) Restrict coefficients to be >= 0. Be aware that you might want to remove fit_intercept which is set True by default. .. deprecated:: 0.20 The option is broken and deprecated. It will be removed in v0.22. Attributes ---------- coef_ : array, shape (n_features,) parameter vector (w in the formulation formula) intercept_ : float independent term in decision function coef_path_ : array, shape (n_features, n_alphas) the varying values of the coefficients along the path alpha_ : float the estimated regularization parameter alpha alphas_ : array, shape (n_alphas,) the different values of alpha along the path cv_alphas_ : array, shape (n_cv_alphas,) all the values of alpha along the path for the different folds mse_path_ : array, shape (n_folds, n_cv_alphas) the mean square error on left-out for each fold along the path (alpha values given by ``cv_alphas``) n_iter_ : array-like or int the number of iterations run by Lars with the optimal alpha. Examples -------- >>> from sklearn.linear_model import LarsCV >>> from sklearn.datasets import make_regression >>> X, y = make_regression(n_samples=200, noise=4.0, random_state=0) >>> reg = LarsCV(cv=5).fit(X, y) >>> reg.score(X, y) # doctest: +ELLIPSIS 0.9996... >>> reg.alpha_ 0.0254... >>> reg.predict(X[:1,]) array([154.0842...]) See also -------- lars_path, LassoLars, LassoLarsCV """ method = 'lar' def __init__(self, fit_intercept=True, verbose=False, max_iter=500, normalize=True, precompute='auto', cv='warn', max_n_alphas=1000, n_jobs=None, eps=np.finfo(np.float).eps, copy_X=True, positive=False): self.max_iter = max_iter self.cv = cv self.max_n_alphas = max_n_alphas self.n_jobs = n_jobs super(LarsCV, self).__init__(fit_intercept=fit_intercept, verbose=verbose, normalize=normalize, precompute=precompute, n_nonzero_coefs=500, eps=eps, copy_X=copy_X, fit_path=True, positive=positive) def fit(self, X, y): """Fit the model using X, y as training data. Parameters ---------- X : array-like, shape (n_samples, n_features) Training data. y : array-like, shape (n_samples,) Target values. Returns ------- self : object returns an instance of self. """ X, y = check_X_y(X, y, y_numeric=True) X = as_float_array(X, copy=self.copy_X) y = as_float_array(y, copy=self.copy_X) # init cross-validation generator cv = check_cv(self.cv, classifier=False) # As we use cross-validation, the Gram matrix is not precomputed here Gram = self.precompute if hasattr(Gram, '__array__'): warnings.warn("Parameter 'precompute' cannot be an array in " "%s. Automatically switch to 'auto' instead." % self.__class__.__name__) Gram = 'auto' cv_paths = Parallel(n_jobs=self.n_jobs, verbose=self.verbose)( delayed(_lars_path_residues)( X[train], y[train], X[test], y[test], Gram=Gram, copy=False, method=self.method, verbose=max(0, self.verbose - 1), normalize=self.normalize, fit_intercept=self.fit_intercept, max_iter=self.max_iter, eps=self.eps, positive=self.positive) for train, test in cv.split(X, y)) all_alphas = np.concatenate(list(zip(*cv_paths))[0]) # Unique also sorts all_alphas = np.unique(all_alphas) # Take at most max_n_alphas values stride = int(max(1, int(len(all_alphas) / float(self.max_n_alphas)))) all_alphas = all_alphas[::stride] mse_path = np.empty((len(all_alphas), len(cv_paths))) for index, (alphas, active, coefs, residues) in enumerate(cv_paths): alphas = alphas[::-1] residues = residues[::-1] if alphas[0] != 0: alphas = np.r_[0, alphas] residues = np.r_[residues[0, np.newaxis], residues] if alphas[-1] != all_alphas[-1]: alphas = np.r_[alphas, all_alphas[-1]] residues = np.r_[residues, residues[-1, np.newaxis]] this_residues = interpolate.interp1d(alphas, residues, axis=0)(all_alphas) this_residues **= 2 mse_path[:, index] = np.mean(this_residues, axis=-1) mask = np.all(np.isfinite(mse_path), axis=-1) all_alphas = all_alphas[mask] mse_path = mse_path[mask] # Select the alpha that minimizes left-out error i_best_alpha = np.argmin(mse_path.mean(axis=-1)) best_alpha = all_alphas[i_best_alpha] # Store our parameters self.alpha_ = best_alpha self.cv_alphas_ = all_alphas self.mse_path_ = mse_path # Now compute the full model # it will call a lasso internally when self if LassoLarsCV # as self.method == 'lasso' self._fit(X, y, max_iter=self.max_iter, alpha=best_alpha, Xy=None, fit_path=True) return self @property @deprecated("Attribute alpha is deprecated in 0.19 and " "will be removed in 0.21. See ``alpha_`` instead") def alpha(self): # impedance matching for the above Lars.fit (should not be documented) return self.alpha_ class LassoLarsCV(LarsCV): """Cross-validated Lasso, using the LARS algorithm. See glossary entry for :term:`cross-validation estimator`. The optimization objective for Lasso is:: (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1 Read more in the :ref:`User Guide `. Parameters ---------- fit_intercept : boolean 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). verbose : boolean or integer, optional Sets the verbosity amount max_iter : integer, optional Maximum number of iterations to perform. normalize : boolean, optional, default True 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``. precompute : True | False | 'auto' Whether to use a precomputed Gram matrix to speed up calculations. If set to ``'auto'`` let us decide. The Gram matrix cannot be passed as argument since we will use only subsets of X. cv : int, cross-validation generator or an iterable, optional Determines the cross-validation splitting strategy. Possible inputs for cv are: - None, to use the default 3-fold cross-validation, - integer, to specify the number of folds. - :term:`CV splitter`, - An iterable yielding (train, test) splits as arrays of indices. For integer/None inputs, :class:`KFold` is used. Refer :ref:`User Guide ` for the various cross-validation strategies that can be used here. .. versionchanged:: 0.20 ``cv`` default value if None will change from 3-fold to 5-fold in v0.22. max_n_alphas : integer, optional The maximum number of points on the path used to compute the residuals in the cross-validation n_jobs : int or None, optional (default=None) Number of CPUs to use during the cross validation. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. See :term:`Glossary ` for more details. eps : float, optional The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. copy_X : boolean, optional, default True If True, X will be copied; else, it may be overwritten. positive : boolean (default=False) Restrict coefficients to be >= 0. Be aware that you might want to remove fit_intercept which is set True by default. Under the positive restriction the model coefficients do not converge to the ordinary-least-squares solution for small values of alpha. Only coefficients up to the smallest alpha value (``alphas_[alphas_ > 0.].min()`` when fit_path=True) reached by the stepwise Lars-Lasso algorithm are typically in congruence with the solution of the coordinate descent Lasso estimator. As a consequence using LassoLarsCV only makes sense for problems where a sparse solution is expected and/or reached. Attributes ---------- coef_ : array, shape (n_features,) parameter vector (w in the formulation formula) intercept_ : float independent term in decision function. coef_path_ : array, shape (n_features, n_alphas) the varying values of the coefficients along the path alpha_ : float the estimated regularization parameter alpha alphas_ : array, shape (n_alphas,) the different values of alpha along the path cv_alphas_ : array, shape (n_cv_alphas,) all the values of alpha along the path for the different folds mse_path_ : array, shape (n_folds, n_cv_alphas) the mean square error on left-out for each fold along the path (alpha values given by ``cv_alphas``) n_iter_ : array-like or int the number of iterations run by Lars with the optimal alpha. Examples -------- >>> from sklearn.linear_model import LassoLarsCV >>> from sklearn.datasets import make_regression >>> X, y = make_regression(noise=4.0, random_state=0) >>> reg = LassoLarsCV(cv=5).fit(X, y) >>> reg.score(X, y) # doctest: +ELLIPSIS 0.9992... >>> reg.alpha_ 0.0484... >>> reg.predict(X[:1,]) array([-77.8723...]) Notes ----- The object solves the same problem as the LassoCV object. However, unlike the LassoCV, it find the relevant alphas values by itself. In general, because of this property, it will be more stable. However, it is more fragile to heavily multicollinear datasets. It is more efficient than the LassoCV if only a small number of features are selected compared to the total number, for instance if there are very few samples compared to the number of features. See also -------- lars_path, LassoLars, LarsCV, LassoCV """ method = 'lasso' def __init__(self, fit_intercept=True, verbose=False, max_iter=500, normalize=True, precompute='auto', cv='warn', max_n_alphas=1000, n_jobs=None, eps=np.finfo(np.float).eps, copy_X=True, positive=False): self.fit_intercept = fit_intercept self.verbose = verbose self.max_iter = max_iter self.normalize = normalize self.precompute = precompute self.cv = cv self.max_n_alphas = max_n_alphas self.n_jobs = n_jobs self.eps = eps self.copy_X = copy_X self.positive = positive # XXX : we don't use super(LarsCV, self).__init__ # to avoid setting n_nonzero_coefs class LassoLarsIC(LassoLars): """Lasso model fit with Lars using BIC or AIC for model selection The optimization objective for Lasso is:: (1 / (2 * n_samples)) * ||y - Xw||^2_2 + alpha * ||w||_1 AIC is the Akaike information criterion and BIC is the Bayes Information criterion. Such criteria are useful to select the value of the regularization parameter by making a trade-off between the goodness of fit and the complexity of the model. A good model should explain well the data while being simple. Read more in the :ref:`User Guide `. Parameters ---------- criterion : 'bic' | 'aic' The type of criterion to use. fit_intercept : boolean 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). verbose : boolean or integer, optional Sets the verbosity amount normalize : boolean, optional, default True 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``. precompute : True | False | 'auto' | array-like Whether to use a precomputed Gram matrix to speed up calculations. If set to ``'auto'`` let us decide. The Gram matrix can also be passed as argument. max_iter : integer, optional Maximum number of iterations to perform. Can be used for early stopping. eps : float, optional The machine-precision regularization in the computation of the Cholesky diagonal factors. Increase this for very ill-conditioned systems. Unlike the ``tol`` parameter in some iterative optimization-based algorithms, this parameter does not control the tolerance of the optimization. copy_X : boolean, optional, default True If True, X will be copied; else, it may be overwritten. positive : boolean (default=False) Restrict coefficients to be >= 0. Be aware that you might want to remove fit_intercept which is set True by default. Under the positive restriction the model coefficients do not converge to the ordinary-least-squares solution for small values of alpha. Only coefficients up to the smallest alpha value (``alphas_[alphas_ > 0.].min()`` when fit_path=True) reached by the stepwise Lars-Lasso algorithm are typically in congruence with the solution of the coordinate descent Lasso estimator. As a consequence using LassoLarsIC only makes sense for problems where a sparse solution is expected and/or reached. Attributes ---------- coef_ : array, shape (n_features,) parameter vector (w in the formulation formula) intercept_ : float independent term in decision function. alpha_ : float the alpha parameter chosen by the information criterion n_iter_ : int number of iterations run by lars_path to find the grid of alphas. criterion_ : array, shape (n_alphas,) The value of the information criteria ('aic', 'bic') across all alphas. The alpha which has the smallest information criterion is chosen. This value is larger by a factor of ``n_samples`` compared to Eqns. 2.15 and 2.16 in (Zou et al, 2007). Examples -------- >>> from sklearn import linear_model >>> reg = linear_model.LassoLarsIC(criterion='bic') >>> reg.fit([[-1, 1], [0, 0], [1, 1]], [-1.1111, 0, -1.1111]) ... # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE LassoLarsIC(copy_X=True, criterion='bic', eps=..., fit_intercept=True, max_iter=500, normalize=True, positive=False, precompute='auto', verbose=False) >>> print(reg.coef_) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE [ 0. -1.11...] Notes ----- The estimation of the number of degrees of freedom is given by: "On the degrees of freedom of the lasso" Hui Zou, Trevor Hastie, and Robert Tibshirani Ann. Statist. Volume 35, Number 5 (2007), 2173-2192. https://en.wikipedia.org/wiki/Akaike_information_criterion https://en.wikipedia.org/wiki/Bayesian_information_criterion See also -------- lars_path, LassoLars, LassoLarsCV """ def __init__(self, criterion='aic', fit_intercept=True, verbose=False, normalize=True, precompute='auto', max_iter=500, eps=np.finfo(np.float).eps, copy_X=True, positive=False): self.criterion = criterion self.fit_intercept = fit_intercept self.positive = positive self.max_iter = max_iter self.verbose = verbose self.normalize = normalize self.copy_X = copy_X self.precompute = precompute self.eps = eps self.fit_path = True def fit(self, X, y, copy_X=True): """Fit the model using X, y as training data. Parameters ---------- X : array-like, shape (n_samples, n_features) training data. y : array-like, shape (n_samples,) target values. Will be cast to X's dtype if necessary copy_X : boolean, optional, default True If ``True``, X will be copied; else, it may be overwritten. Returns ------- self : object returns an instance of self. """ X, y = check_X_y(X, y, y_numeric=True) X, y, Xmean, ymean, Xstd = LinearModel._preprocess_data( X, y, self.fit_intercept, self.normalize, self.copy_X) max_iter = self.max_iter Gram = self.precompute alphas_, active_, coef_path_, self.n_iter_ = lars_path( X, y, Gram=Gram, copy_X=copy_X, copy_Gram=True, alpha_min=0.0, method='lasso', verbose=self.verbose, max_iter=max_iter, eps=self.eps, return_n_iter=True, positive=self.positive) n_samples = X.shape[0] if self.criterion == 'aic': K = 2 # AIC elif self.criterion == 'bic': K = log(n_samples) # BIC else: raise ValueError('criterion should be either bic or aic') R = y[:, np.newaxis] - np.dot(X, coef_path_) # residuals mean_squared_error = np.mean(R ** 2, axis=0) sigma2 = np.var(y) df = np.zeros(coef_path_.shape[1], dtype=np.int) # Degrees of freedom for k, coef in enumerate(coef_path_.T): mask = np.abs(coef) > np.finfo(coef.dtype).eps if not np.any(mask): continue # get the number of degrees of freedom equal to: # Xc = X[:, mask] # Trace(Xc * inv(Xc.T, Xc) * Xc.T) ie the number of non-zero coefs df[k] = np.sum(mask) self.alphas_ = alphas_ eps64 = np.finfo('float64').eps self.criterion_ = (n_samples * mean_squared_error / (sigma2 + eps64) + K * df) # Eqns. 2.15--16 in (Zou et al, 2007) n_best = np.argmin(self.criterion_) self.alpha_ = alphas_[n_best] self.coef_ = coef_path_[:, n_best] self._set_intercept(Xmean, ymean, Xstd) return self