""" Least Angle Regression algorithm. See the documentation on the Generalized Linear Model for a complete discussion. """ # Authors: The scikit-learn developers # SPDX-License-Identifier: BSD-3-Clause import sys import warnings from math import log from numbers import Integral, Real import numpy as np from scipy import interpolate, linalg from scipy.linalg.lapack import get_lapack_funcs from ..base import MultiOutputMixin, RegressorMixin, _fit_context from ..exceptions import ConvergenceWarning from ..model_selection import check_cv # mypy error: Module 'sklearn.utils' has no attribute 'arrayfuncs' from ..utils import ( # type: ignore Bunch, arrayfuncs, as_float_array, check_random_state, ) from ..utils._metadata_requests import ( MetadataRouter, MethodMapping, _raise_for_params, _routing_enabled, process_routing, ) from ..utils._param_validation import Hidden, Interval, StrOptions, validate_params from ..utils.parallel import Parallel, delayed from ..utils.validation import validate_data from ._base import LinearModel, LinearRegression, _preprocess_data SOLVE_TRIANGULAR_ARGS = {"check_finite": False} @validate_params( { "X": [np.ndarray, None], "y": [np.ndarray, None], "Xy": [np.ndarray, None], "Gram": [StrOptions({"auto"}), "boolean", np.ndarray, None], "max_iter": [Interval(Integral, 0, None, closed="left")], "alpha_min": [Interval(Real, 0, None, closed="left")], "method": [StrOptions({"lar", "lasso"})], "copy_X": ["boolean"], "eps": [Interval(Real, 0, None, closed="neither"), None], "copy_Gram": ["boolean"], "verbose": ["verbose"], "return_path": ["boolean"], "return_n_iter": ["boolean"], "positive": ["boolean"], }, prefer_skip_nested_validation=True, ) def lars_path( X, y, Xy=None, *, Gram=None, max_iter=500, alpha_min=0, method="lar", copy_X=True, eps=np.finfo(float).eps, copy_Gram=True, verbose=0, return_path=True, return_n_iter=False, positive=False, ): """Compute Least Angle Regression or Lasso path using the LARS algorithm. 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='lar', 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 : None or ndarray of shape (n_samples, n_features) Input data. If X is `None`, Gram must also be `None`. If only the Gram matrix is available, use `lars_path_gram` instead. y : None or ndarray of shape (n_samples,) Input targets. Xy : array-like of shape (n_features,), default=None `Xy = X.T @ y` that can be precomputed. It is useful only when the Gram matrix is precomputed. Gram : None, 'auto', bool, ndarray of shape (n_features, n_features), \ default=None Precomputed Gram matrix `X.T @ X`, if `'auto'`, the Gram matrix is precomputed from the given X, if there are more samples than features. max_iter : int, default=500 Maximum number of iterations to perform, set to infinity for no limit. alpha_min : float, default=0 Minimum correlation along the path. It corresponds to the regularization parameter `alpha` in the Lasso. method : {'lar', 'lasso'}, default='lar' Specifies the returned model. Select `'lar'` for Least Angle Regression, `'lasso'` for the Lasso. copy_X : bool, default=True If `False`, `X` is overwritten. eps : float, default=np.finfo(float).eps 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_Gram : bool, default=True If `False`, `Gram` is overwritten. verbose : int, default=0 Controls output verbosity. return_path : bool, default=True If `True`, returns the entire path, else returns only the last point of the path. return_n_iter : bool, default=False Whether to return the number of iterations. positive : bool, 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 : ndarray of 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 : ndarray of shape (n_alphas,) Indices of active variables at the end of the path. coefs : ndarray of 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 -------- lars_path_gram : Compute LARS path in the sufficient stats mode. lasso_path : Compute Lasso path with coordinate descent. LassoLars : Lasso model fit with Least Angle Regression a.k.a. Lars. Lars : Least Angle Regression model a.k.a. LAR. LassoLarsCV : Cross-validated Lasso, using the LARS algorithm. LarsCV : Cross-validated Least Angle Regression model. sklearn.decomposition.sparse_encode : Sparse coding. References ---------- .. [1] "Least Angle Regression", Efron et al. http://statweb.stanford.edu/~tibs/ftp/lars.pdf .. [2] `Wikipedia entry on the Least-angle regression `_ .. [3] `Wikipedia entry on the Lasso `_ Examples -------- >>> from sklearn.linear_model import lars_path >>> from sklearn.datasets import make_regression >>> X, y, true_coef = make_regression( ... n_samples=100, n_features=5, n_informative=2, coef=True, random_state=0 ... ) >>> true_coef array([ 0. , 0. , 0. , 97.9..., 45.7...]) >>> alphas, _, estimated_coef = lars_path(X, y) >>> alphas.shape (3,) >>> estimated_coef array([[ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 46.96..., 97.99...], [ 0. , 0. , 45.70...]]) """ if X is None and Gram is not None: raise ValueError( "X cannot be None if Gram is not None" "Use lars_path_gram to avoid passing X and y." ) return _lars_path_solver( X=X, y=y, Xy=Xy, Gram=Gram, n_samples=None, max_iter=max_iter, alpha_min=alpha_min, method=method, copy_X=copy_X, eps=eps, copy_Gram=copy_Gram, verbose=verbose, return_path=return_path, return_n_iter=return_n_iter, positive=positive, ) @validate_params( { "Xy": [np.ndarray], "Gram": [np.ndarray], "n_samples": [Interval(Integral, 0, None, closed="left")], "max_iter": [Interval(Integral, 0, None, closed="left")], "alpha_min": [Interval(Real, 0, None, closed="left")], "method": [StrOptions({"lar", "lasso"})], "copy_X": ["boolean"], "eps": [Interval(Real, 0, None, closed="neither"), None], "copy_Gram": ["boolean"], "verbose": ["verbose"], "return_path": ["boolean"], "return_n_iter": ["boolean"], "positive": ["boolean"], }, prefer_skip_nested_validation=True, ) def lars_path_gram( Xy, Gram, *, n_samples, max_iter=500, alpha_min=0, method="lar", copy_X=True, eps=np.finfo(float).eps, copy_Gram=True, verbose=0, return_path=True, return_n_iter=False, positive=False, ): """The lars_path in the sufficient stats mode. 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='lar', 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 ---------- Xy : ndarray of shape (n_features,) `Xy = X.T @ y`. Gram : ndarray of shape (n_features, n_features) `Gram = X.T @ X`. n_samples : int Equivalent size of sample. max_iter : int, default=500 Maximum number of iterations to perform, set to infinity for no limit. alpha_min : float, default=0 Minimum correlation along the path. It corresponds to the regularization parameter alpha parameter in the Lasso. method : {'lar', 'lasso'}, default='lar' Specifies the returned model. Select `'lar'` for Least Angle Regression, ``'lasso'`` for the Lasso. copy_X : bool, default=True If `False`, `X` is overwritten. eps : float, default=np.finfo(float).eps 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_Gram : bool, default=True If `False`, `Gram` is overwritten. verbose : int, default=0 Controls output verbosity. return_path : bool, default=True If `return_path==True` returns the entire path, else returns only the last point of the path. return_n_iter : bool, default=False Whether to return the number of iterations. positive : bool, 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 : ndarray of 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 : ndarray of shape (n_alphas,) Indices of active variables at the end of the path. coefs : ndarray of 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 -------- lars_path_gram : Compute LARS path. lasso_path : Compute Lasso path with coordinate descent. LassoLars : Lasso model fit with Least Angle Regression a.k.a. Lars. Lars : Least Angle Regression model a.k.a. LAR. LassoLarsCV : Cross-validated Lasso, using the LARS algorithm. LarsCV : Cross-validated Least Angle Regression model. sklearn.decomposition.sparse_encode : Sparse coding. References ---------- .. [1] "Least Angle Regression", Efron et al. http://statweb.stanford.edu/~tibs/ftp/lars.pdf .. [2] `Wikipedia entry on the Least-angle regression `_ .. [3] `Wikipedia entry on the Lasso `_ Examples -------- >>> from sklearn.linear_model import lars_path_gram >>> from sklearn.datasets import make_regression >>> X, y, true_coef = make_regression( ... n_samples=100, n_features=5, n_informative=2, coef=True, random_state=0 ... ) >>> true_coef array([ 0. , 0. , 0. , 97.9..., 45.7...]) >>> alphas, _, estimated_coef = lars_path_gram(X.T @ y, X.T @ X, n_samples=100) >>> alphas.shape (3,) >>> estimated_coef array([[ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 0. , 0. ], [ 0. , 46.96..., 97.99...], [ 0. , 0. , 45.70...]]) """ return _lars_path_solver( X=None, y=None, Xy=Xy, Gram=Gram, n_samples=n_samples, max_iter=max_iter, alpha_min=alpha_min, method=method, copy_X=copy_X, eps=eps, copy_Gram=copy_Gram, verbose=verbose, return_path=return_path, return_n_iter=return_n_iter, positive=positive, ) def _lars_path_solver( X, y, Xy=None, Gram=None, n_samples=None, max_iter=500, alpha_min=0, method="lar", copy_X=True, eps=np.finfo(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='lar', 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 : None or ndarray of shape (n_samples, n_features) Input data. Note that if X is None then Gram must be specified, i.e., cannot be None or False. y : None or ndarray of shape (n_samples,) Input targets. Xy : array-like of shape (n_features,), default=None `Xy = np.dot(X.T, y)` that can be precomputed. It is useful only when the Gram matrix is precomputed. Gram : None, 'auto' or array-like of shape (n_features, n_features), \ default=None Precomputed Gram matrix `(X' * X)`, if ``'auto'``, the Gram matrix is precomputed from the given X, if there are more samples than features. n_samples : int or float, default=None Equivalent size of sample. If `None`, it will be `n_samples`. max_iter : int, default=500 Maximum number of iterations to perform, set to infinity for no limit. alpha_min : float, default=0 Minimum correlation along the path. It corresponds to the regularization parameter alpha parameter in the Lasso. method : {'lar', 'lasso'}, default='lar' Specifies the returned model. Select ``'lar'`` for Least Angle Regression, ``'lasso'`` for the Lasso. copy_X : bool, default=True If ``False``, ``X`` is overwritten. eps : float, default=np.finfo(float).eps 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_Gram : bool, default=True If ``False``, ``Gram`` is overwritten. verbose : int, default=0 Controls output verbosity. return_path : bool, default=True If ``return_path==True`` returns the entire path, else returns only the last point of the path. return_n_iter : bool, default=False Whether to return the number of iterations. positive : bool, 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-like of 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-like of shape (n_alphas,) Indices of active variables at the end of the path. coefs : array-like of 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", Efron 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: raise ValueError("Positive constraint not supported for 'lar' coding method.") n_samples = n_samples if n_samples is not None else y.size if Xy is None: Cov = np.dot(X.T, y) else: Cov = Xy.copy() if Gram is None or Gram is False: Gram = None if X is None: raise ValueError("X and Gram cannot both be unspecified.") elif isinstance(Gram, str) 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 Gram is None: n_features = X.shape[1] else: n_features = Cov.shape[0] if Gram.shape != (n_features, n_features): raise ValueError("The shapes of the inputs Gram and Xy do not match.") if copy_X and X is not None and Gram is None: # 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") max_features = min(max_iter, n_features) dtypes = set(a.dtype for a in (X, y, Xy, Gram) if a is not None) if len(dtypes) == 1: # use the precision level of input data if it is consistent return_dtype = next(iter(dtypes)) else: # fallback to double precision otherwise return_dtype = np.float64 if return_path: coefs = np.zeros((max_features + 1, n_features), dtype=return_dtype) alphas = np.zeros(max_features + 1, dtype=return_dtype) else: coef, prev_coef = ( np.zeros(n_features, dtype=return_dtype), np.zeros(n_features, dtype=return_dtype), ) alpha, prev_alpha = ( np.array([0.0], dtype=return_dtype), np.array([0.0], dtype=return_dtype), ) # above 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. if Gram is None: L = np.empty((max_features, max_features), dtype=X.dtype) swap, nrm2 = linalg.get_blas_funcs(("swap", "nrm2"), (X,)) else: L = np.empty((max_features, max_features), dtype=Gram.dtype) swap, nrm2 = linalg.get_blas_funcs(("swap", "nrm2"), (Cov,)) (solve_cholesky,) = get_lapack_funcs(("potrs",), (L,)) 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 cov_precision = np.finfo(Cov.dtype).precision equality_tolerance = np.finfo(np.float32).eps if Gram is not None: Gram_copy = Gram.copy() Cov_copy = Cov.copy() 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 = 0.0 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. 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.item(), 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.item(), prev_alpha.item(), n_active), ConvergenceWarning, ) break # least squares solution least_squares, _ = 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 = 1.0 else: # is this really needed ? AA = 1.0 / 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, _ = solve_cholesky( L_, sign_active[:n_active], lower=True ) tmp = max(np.sum(least_squares * sign_active[:n_active]), eps) AA = 1.0 / 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) # Explicit rounding can be necessary to avoid `np.argmax(Cov)` yielding # unstable results because of rounding errors. np.around(corr_eq_dir, decimals=cov_precision, out=corr_eq_dir) 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 for ii in idx: arrayfuncs.cholesky_delete(L[:n_active, :n_active], ii) n_active -= 1 # 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 temp = Cov_copy[drop_idx] - np.dot(Gram_copy[drop_idx], coef) Cov = np.r_[temp, Cov] sign_active = np.delete(sign_active, idx) sign_active = np.append(sign_active, 0.0) # 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(MultiOutputMixin, RegressorMixin, LinearModel): """Least Angle Regression model a.k.a. LAR. Read more in the :ref:`User Guide `. 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). verbose : bool or int, default=False Sets the verbosity amount. precompute : bool, 'auto' or array-like , default='auto' 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, default=500 Target number of non-zero coefficients. Use ``np.inf`` for no limit. eps : float, default=np.finfo(float).eps 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 : bool, default=True If ``True``, X will be copied; else, it may be overwritten. fit_path : bool, default=True 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. jitter : float, default=None Upper bound on a uniform noise parameter to be added to the `y` values, to satisfy the model's assumption of one-at-a-time computations. Might help with stability. .. versionadded:: 0.23 random_state : int, RandomState instance or None, default=None Determines random number generation for jittering. Pass an int for reproducible output across multiple function calls. See :term:`Glossary `. Ignored if `jitter` is None. .. versionadded:: 0.23 Attributes ---------- alphas_ : array-like of shape (n_alphas + 1,) or list of 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 ``alpha >= alpha_min``, whichever is smaller. If this is a list of array-like, the length of the outer list is `n_targets`. active_ : list of shape (n_alphas,) or list of such lists Indices of active variables at the end of the path. If this is a list of list, the length of the outer list is `n_targets`. coef_path_ : array-like of shape (n_features, n_alphas + 1) or list \ of such arrays The varying values of the coefficients along the path. It is not present if the ``fit_path`` parameter is ``False``. If this is a list of array-like, the length of the outer list is `n_targets`. coef_ : array-like of shape (n_features,) or (n_targets, n_features) Parameter vector (w in the formulation formula). intercept_ : float or array-like of 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. 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 -------- lars_path: Compute Least Angle Regression or Lasso path using LARS algorithm. LarsCV : Cross-validated Least Angle Regression model. sklearn.decomposition.sparse_encode : Sparse coding. 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]) Lars(n_nonzero_coefs=1) >>> print(reg.coef_) [ 0. -1.11...] """ _parameter_constraints: dict = { "fit_intercept": ["boolean"], "verbose": ["verbose"], "precompute": ["boolean", StrOptions({"auto"}), np.ndarray, Hidden(None)], "n_nonzero_coefs": [Interval(Integral, 1, None, closed="left")], "eps": [Interval(Real, 0, None, closed="left")], "copy_X": ["boolean"], "fit_path": ["boolean"], "jitter": [Interval(Real, 0, None, closed="left"), None], "random_state": ["random_state"], } method = "lar" positive = False def __init__( self, *, fit_intercept=True, verbose=False, precompute="auto", n_nonzero_coefs=500, eps=np.finfo(float).eps, copy_X=True, fit_path=True, jitter=None, random_state=None, ): self.fit_intercept = fit_intercept self.verbose = verbose self.precompute = precompute self.n_nonzero_coefs = n_nonzero_coefs self.eps = eps self.copy_X = copy_X self.fit_path = fit_path self.jitter = jitter self.random_state = random_state @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 = _preprocess_data( X, y, fit_intercept=self.fit_intercept, copy=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), dtype=X.dtype) if fit_path: self.active_ = [] self.coef_path_ = [] for k in range(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 range(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 @_fit_context(prefer_skip_nested_validation=True) def fit(self, X, y, Xy=None): """Fit the model using X, y as training data. Parameters ---------- X : array-like of shape (n_samples, n_features) Training data. y : array-like of shape (n_samples,) or (n_samples, n_targets) Target values. Xy : array-like of shape (n_features,) or (n_features, n_targets), \ default=None 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 = validate_data( self, X, y, force_writeable=True, y_numeric=True, multi_output=True ) alpha = getattr(self, "alpha", 0.0) if hasattr(self, "n_nonzero_coefs"): alpha = 0.0 # n_nonzero_coefs parametrization takes priority max_iter = self.n_nonzero_coefs else: max_iter = self.max_iter if self.jitter is not None: rng = check_random_state(self.random_state) noise = rng.uniform(high=self.jitter, size=len(y)) y = y + noise 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, default=1.0 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 : 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). verbose : bool or int, default=False Sets the verbosity amount. precompute : bool, 'auto' or array-like, default='auto' 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 : int, default=500 Maximum number of iterations to perform. eps : float, default=np.finfo(float).eps 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 : bool, default=True If True, X will be copied; else, it may be overwritten. fit_path : bool, default=True 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 : bool, 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. jitter : float, default=None Upper bound on a uniform noise parameter to be added to the `y` values, to satisfy the model's assumption of one-at-a-time computations. Might help with stability. .. versionadded:: 0.23 random_state : int, RandomState instance or None, default=None Determines random number generation for jittering. Pass an int for reproducible output across multiple function calls. See :term:`Glossary `. Ignored if `jitter` is None. .. versionadded:: 0.23 Attributes ---------- alphas_ : array-like of shape (n_alphas + 1,) or list of 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 ``alpha >= alpha_min``, whichever is smaller. If this is a list of array-like, the length of the outer list is `n_targets`. active_ : list of length n_alphas or list of such lists Indices of active variables at the end of the path. If this is a list of list, the length of the outer list is `n_targets`. coef_path_ : array-like of shape (n_features, n_alphas + 1) or list \ of such arrays 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``. If this is a list of array-like, the length of the outer list is `n_targets`. coef_ : array-like of shape (n_features,) or (n_targets, n_features) Parameter vector (w in the formulation formula). intercept_ : float or array-like of 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. 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 -------- lars_path : Compute Least Angle Regression or Lasso path using LARS algorithm. lasso_path : Compute Lasso path with coordinate descent. Lasso : Linear Model trained with L1 prior as regularizer (aka the Lasso). LassoCV : Lasso linear model with iterative fitting along a regularization path. LassoLarsCV: Cross-validated Lasso, using the LARS algorithm. LassoLarsIC : Lasso model fit with Lars using BIC or AIC for model selection. sklearn.decomposition.sparse_encode : Sparse coding. Examples -------- >>> from sklearn import linear_model >>> reg = linear_model.LassoLars(alpha=0.01) >>> reg.fit([[-1, 1], [0, 0], [1, 1]], [-1, 0, -1]) LassoLars(alpha=0.01) >>> print(reg.coef_) [ 0. -0.955...] """ _parameter_constraints: dict = { **Lars._parameter_constraints, "alpha": [Interval(Real, 0, None, closed="left")], "max_iter": [Interval(Integral, 0, None, closed="left")], "positive": ["boolean"], } _parameter_constraints.pop("n_nonzero_coefs") method = "lasso" def __init__( self, alpha=1.0, *, fit_intercept=True, verbose=False, precompute="auto", max_iter=500, eps=np.finfo(float).eps, copy_X=True, fit_path=True, positive=False, jitter=None, random_state=None, ): self.alpha = alpha self.fit_intercept = fit_intercept self.max_iter = max_iter self.verbose = verbose self.positive = positive self.precompute = precompute self.copy_X = copy_X self.eps = eps self.fit_path = fit_path self.jitter = jitter self.random_state = random_state ############################################################################### # 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="lar", verbose=False, fit_intercept=True, max_iter=500, eps=np.finfo(float).eps, positive=False, ): """Compute the residues on left-out data for a full LARS path Parameters ----------- X_train : array-like of shape (n_samples, n_features) The data to fit the LARS on y_train : array-like of shape (n_samples,) The target variable to fit LARS on X_test : array-like of shape (n_samples, n_features) The data to compute the residues on y_test : array-like of shape (n_samples,) The target variable to compute the residues on Gram : None, 'auto' or array-like of shape (n_features, n_features), \ default=None Precomputed Gram matrix (X' * X), if ``'auto'``, the Gram matrix is precomputed from the given X, if there are more samples than features copy : bool, default=True Whether X_train, X_test, y_train and y_test should be copied; if False, they may be overwritten. method : {'lar' , 'lasso'}, default='lar' Specifies the returned model. Select ``'lar'`` for Least Angle Regression, ``'lasso'`` for the Lasso. verbose : bool or int, default=False Sets the amount of verbosity 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). positive : bool, 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. max_iter : int, default=500 Maximum number of iterations to perform. eps : float, default=np.finfo(float).eps 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-like of 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-like of shape (n_features, n_alphas) Coefficients along the path residues : array-like of 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 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, ) 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 : 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). verbose : bool or int, default=False Sets the verbosity amount. max_iter : int, default=500 Maximum number of iterations to perform. precompute : bool, 'auto' or array-like , default='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, default=None Determines the cross-validation splitting strategy. Possible inputs for cv are: - None, to use the default 5-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:`~sklearn.model_selection.KFold` is used. Refer :ref:`User Guide ` for the various cross-validation strategies that can be used here. .. versionchanged:: 0.22 ``cv`` default value if None changed from 3-fold to 5-fold. max_n_alphas : int, default=1000 The maximum number of points on the path used to compute the residuals in the cross-validation. n_jobs : int or None, 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, default=np.finfo(float).eps 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 : bool, default=True If ``True``, X will be copied; else, it may be overwritten. Attributes ---------- active_ : list of length n_alphas or list of such lists Indices of active variables at the end of the path. If this is a list of lists, the outer list length is `n_targets`. coef_ : array-like of shape (n_features,) parameter vector (w in the formulation formula) intercept_ : float independent term in decision function coef_path_ : array-like of shape (n_features, n_alphas) the varying values of the coefficients along the path alpha_ : float the estimated regularization parameter alpha alphas_ : array-like of shape (n_alphas,) the different values of alpha along the path cv_alphas_ : array-like of shape (n_cv_alphas,) all the values of alpha along the path for the different folds mse_path_ : array-like of 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. 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 -------- lars_path : Compute Least Angle Regression or Lasso path using LARS algorithm. lasso_path : Compute Lasso path with coordinate descent. Lasso : Linear Model trained with L1 prior as regularizer (aka the Lasso). LassoCV : Lasso linear model with iterative fitting along a regularization path. LassoLars : Lasso model fit with Least Angle Regression a.k.a. Lars. LassoLarsIC : Lasso model fit with Lars using BIC or AIC for model selection. sklearn.decomposition.sparse_encode : Sparse coding. Notes ----- In `fit`, once the best parameter `alpha` is found through cross-validation, the model is fit again using the entire training set. 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) 0.9996... >>> reg.alpha_ np.float64(0.2961...) >>> reg.predict(X[:1,]) array([154.3996...]) """ _parameter_constraints: dict = { **Lars._parameter_constraints, "max_iter": [Interval(Integral, 0, None, closed="left")], "cv": ["cv_object"], "max_n_alphas": [Interval(Integral, 1, None, closed="left")], "n_jobs": [Integral, None], } for parameter in ["n_nonzero_coefs", "jitter", "fit_path", "random_state"]: _parameter_constraints.pop(parameter) method = "lar" def __init__( self, *, fit_intercept=True, verbose=False, max_iter=500, precompute="auto", cv=None, max_n_alphas=1000, n_jobs=None, eps=np.finfo(float).eps, copy_X=True, ): self.max_iter = max_iter self.cv = cv self.max_n_alphas = max_n_alphas self.n_jobs = n_jobs super().__init__( fit_intercept=fit_intercept, verbose=verbose, precompute=precompute, n_nonzero_coefs=500, eps=eps, copy_X=copy_X, fit_path=True, ) def __sklearn_tags__(self): tags = super().__sklearn_tags__() tags.target_tags.multi_output = False return tags @_fit_context(prefer_skip_nested_validation=True) def fit(self, X, y, **params): """Fit the model using X, y as training data. Parameters ---------- X : array-like of shape (n_samples, n_features) Training data. y : array-like of shape (n_samples,) Target values. **params : dict, default=None Parameters to be passed to the CV splitter. .. versionadded:: 1.4 Only available if `enable_metadata_routing=True`, which can be set by using ``sklearn.set_config(enable_metadata_routing=True)``. See :ref:`Metadata Routing User Guide ` for more details. Returns ------- self : object Returns an instance of self. """ _raise_for_params(params, self, "fit") X, y = validate_data(self, X, y, force_writeable=True, 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) if _routing_enabled(): routed_params = process_routing(self, "fit", **params) else: routed_params = Bunch(splitter=Bunch(split={})) # 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), fit_intercept=self.fit_intercept, max_iter=self.max_iter, eps=self.eps, positive=self.positive, ) for train, test in cv.split(X, y, **routed_params.splitter.split) ) 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, _, _, 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 using best_alpha # 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 def get_metadata_routing(self): """Get metadata routing of this object. Please check :ref:`User Guide ` on how the routing mechanism works. .. versionadded:: 1.4 Returns ------- routing : MetadataRouter A :class:`~sklearn.utils.metadata_routing.MetadataRouter` encapsulating routing information. """ router = MetadataRouter(owner=self.__class__.__name__).add( splitter=check_cv(self.cv), method_mapping=MethodMapping().add(caller="fit", callee="split"), ) return router 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 : 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). verbose : bool or int, default=False Sets the verbosity amount. max_iter : int, default=500 Maximum number of iterations to perform. precompute : bool or 'auto' , default='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, default=None Determines the cross-validation splitting strategy. Possible inputs for cv are: - None, to use the default 5-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:`~sklearn.model_selection.KFold` is used. Refer :ref:`User Guide ` for the various cross-validation strategies that can be used here. .. versionchanged:: 0.22 ``cv`` default value if None changed from 3-fold to 5-fold. max_n_alphas : int, default=1000 The maximum number of points on the path used to compute the residuals in the cross-validation. n_jobs : int or None, 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, default=np.finfo(float).eps 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 : bool, default=True If True, X will be copied; else, it may be overwritten. positive : bool, 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-like of shape (n_features,) parameter vector (w in the formulation formula) intercept_ : float independent term in decision function. coef_path_ : array-like of shape (n_features, n_alphas) the varying values of the coefficients along the path alpha_ : float the estimated regularization parameter alpha alphas_ : array-like of shape (n_alphas,) the different values of alpha along the path cv_alphas_ : array-like of shape (n_cv_alphas,) all the values of alpha along the path for the different folds mse_path_ : array-like of 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. active_ : list of int Indices of active variables at the end of the path. 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 -------- lars_path : Compute Least Angle Regression or Lasso path using LARS algorithm. lasso_path : Compute Lasso path with coordinate descent. Lasso : Linear Model trained with L1 prior as regularizer (aka the Lasso). LassoCV : Lasso linear model with iterative fitting along a regularization path. LassoLars : Lasso model fit with Least Angle Regression a.k.a. Lars. LassoLarsIC : Lasso model fit with Lars using BIC or AIC for model selection. sklearn.decomposition.sparse_encode : Sparse coding. Notes ----- The object solves the same problem as the :class:`~sklearn.linear_model.LassoCV` object. However, unlike the :class:`~sklearn.linear_model.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 :class:`~sklearn.linear_model.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. In `fit`, once the best parameter `alpha` is found through cross-validation, the model is fit again using the entire training set. 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) 0.9993... >>> reg.alpha_ np.float64(0.3972...) >>> reg.predict(X[:1,]) array([-78.4831...]) """ _parameter_constraints = { **LarsCV._parameter_constraints, "positive": ["boolean"], } method = "lasso" def __init__( self, *, fit_intercept=True, verbose=False, max_iter=500, precompute="auto", cv=None, max_n_alphas=1000, n_jobs=None, eps=np.finfo(float).eps, copy_X=True, positive=False, ): self.fit_intercept = fit_intercept self.verbose = verbose self.max_iter = max_iter 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().__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 [2]_ and BIC is the Bayes Information criterion [3]_. 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 : {'aic', 'bic'}, default='aic' The type of criterion to use. 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). verbose : bool or int, default=False Sets the verbosity amount. precompute : bool, 'auto' or array-like, default='auto' 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 : int, default=500 Maximum number of iterations to perform. Can be used for early stopping. eps : float, default=np.finfo(float).eps 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 : bool, default=True If True, X will be copied; else, it may be overwritten. positive : bool, 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. noise_variance : float, default=None The estimated noise variance of the data. If `None`, an unbiased estimate is computed by an OLS model. However, it is only possible in the case where `n_samples > n_features + fit_intercept`. .. versionadded:: 1.1 Attributes ---------- coef_ : array-like of 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 alphas_ : array-like of shape (n_alphas + 1,) or list of 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 ``alpha >= alpha_min``, whichever is smaller. If a list, it will be of length `n_targets`. n_iter_ : int number of iterations run by lars_path to find the grid of alphas. criterion_ : array-like of shape (n_alphas,) The value of the information criteria ('aic', 'bic') across all alphas. The alpha which has the smallest information criterion is chosen, as specified in [1]_. noise_variance_ : float The estimated noise variance from the data used to compute the criterion. .. versionadded:: 1.1 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 -------- lars_path : Compute Least Angle Regression or Lasso path using LARS algorithm. lasso_path : Compute Lasso path with coordinate descent. Lasso : Linear Model trained with L1 prior as regularizer (aka the Lasso). LassoCV : Lasso linear model with iterative fitting along a regularization path. LassoLars : Lasso model fit with Least Angle Regression a.k.a. Lars. LassoLarsCV: Cross-validated Lasso, using the LARS algorithm. sklearn.decomposition.sparse_encode : Sparse coding. Notes ----- The number of degrees of freedom is computed as in [1]_. To have more details regarding the mathematical formulation of the AIC and BIC criteria, please refer to :ref:`User Guide `. References ---------- .. [1] :arxiv:`Zou, Hui, Trevor Hastie, and Robert Tibshirani. "On the degrees of freedom of the lasso." The Annals of Statistics 35.5 (2007): 2173-2192. <0712.0881>` .. [2] `Wikipedia entry on the Akaike information criterion `_ .. [3] `Wikipedia entry on the Bayesian information criterion `_ Examples -------- >>> from sklearn import linear_model >>> reg = linear_model.LassoLarsIC(criterion='bic') >>> X = [[-2, 2], [-1, 1], [0, 0], [1, 1], [2, 2]] >>> y = [-2.2222, -1.1111, 0, -1.1111, -2.2222] >>> reg.fit(X, y) LassoLarsIC(criterion='bic') >>> print(reg.coef_) [ 0. -1.11...] """ _parameter_constraints: dict = { **LassoLars._parameter_constraints, "criterion": [StrOptions({"aic", "bic"})], "noise_variance": [Interval(Real, 0, None, closed="left"), None], } for parameter in ["jitter", "fit_path", "alpha", "random_state"]: _parameter_constraints.pop(parameter) def __init__( self, criterion="aic", *, fit_intercept=True, verbose=False, precompute="auto", max_iter=500, eps=np.finfo(float).eps, copy_X=True, positive=False, noise_variance=None, ): self.criterion = criterion self.fit_intercept = fit_intercept self.positive = positive self.max_iter = max_iter self.verbose = verbose self.copy_X = copy_X self.precompute = precompute self.eps = eps self.fit_path = True self.noise_variance = noise_variance def __sklearn_tags__(self): tags = super().__sklearn_tags__() tags.target_tags.multi_output = False return tags @_fit_context(prefer_skip_nested_validation=True) def fit(self, X, y, copy_X=None): """Fit the model using X, y as training data. Parameters ---------- X : array-like of shape (n_samples, n_features) Training data. y : array-like of shape (n_samples,) Target values. Will be cast to X's dtype if necessary. copy_X : bool, default=None If provided, this parameter will override the choice of copy_X made at instance creation. If ``True``, X will be copied; else, it may be overwritten. Returns ------- self : object Returns an instance of self. """ if copy_X is None: copy_X = self.copy_X X, y = validate_data(self, X, y, force_writeable=True, y_numeric=True) X, y, Xmean, ymean, Xstd = _preprocess_data( X, y, fit_intercept=self.fit_intercept, copy=copy_X ) Gram = self.precompute alphas_, _, 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=self.max_iter, eps=self.eps, return_n_iter=True, positive=self.positive, ) n_samples = X.shape[0] if self.criterion == "aic": criterion_factor = 2 elif self.criterion == "bic": criterion_factor = log(n_samples) else: raise ValueError( f"criterion should be either bic or aic, got {self.criterion!r}" ) residuals = y[:, np.newaxis] - np.dot(X, coef_path_) residuals_sum_squares = np.sum(residuals**2, axis=0) degrees_of_freedom = np.zeros(coef_path_.shape[1], dtype=int) 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 degrees_of_freedom[k] = np.sum(mask) self.alphas_ = alphas_ if self.noise_variance is None: self.noise_variance_ = self._estimate_noise_variance( X, y, positive=self.positive ) else: self.noise_variance_ = self.noise_variance self.criterion_ = ( n_samples * np.log(2 * np.pi * self.noise_variance_) + residuals_sum_squares / self.noise_variance_ + criterion_factor * degrees_of_freedom ) 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 def _estimate_noise_variance(self, X, y, positive): """Compute an estimate of the variance with an OLS model. Parameters ---------- X : ndarray of shape (n_samples, n_features) Data to be fitted by the OLS model. We expect the data to be centered. y : ndarray of shape (n_samples,) Associated target. positive : bool, default=False Restrict coefficients to be >= 0. This should be inline with the `positive` parameter from `LassoLarsIC`. Returns ------- noise_variance : float An estimator of the noise variance of an OLS model. """ if X.shape[0] <= X.shape[1] + self.fit_intercept: raise ValueError( f"You are using {self.__class__.__name__} in the case where the number " "of samples is smaller than the number of features. In this setting, " "getting a good estimate for the variance of the noise is not " "possible. Provide an estimate of the noise variance in the " "constructor." ) # X and y are already centered and we don't need to fit with an intercept ols_model = LinearRegression(positive=positive, fit_intercept=False) y_pred = ols_model.fit(X, y).predict(X) return np.sum((y - y_pred) ** 2) / ( X.shape[0] - X.shape[1] - self.fit_intercept )