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lars.py
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lars.py
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"""
Least Angle Regression algorithm. See the documentation on the
Generalized Linear Model for a complete discussion.
"""
# Author: Fabian Pedregosa <fabian.pedregosa@inria.fr>
# Alexandre Gramfort <alexandre.gramfort@inria.fr>
#
# License: BSD Style.
import numpy as np
from scipy import linalg
import scipy.sparse as sp # needed by LeastAngleRegression
from .base import LinearModel
from ..utils.fixes import copysign
from ..utils import arrayfuncs
def lars_path(X, y, Gram=None, max_iter=None, alpha_min=0,
method="lar", precompute=True):
""" Compute Least Angle Regression and LASSO path
Parameters
-----------
X: array, shape: (n, p)
Input data
y: array, shape: (n)
Input targets
max_iter: integer, optional
The number of 'kink' in the path
Gram: array, shape: (p, p), optional
Precomputed Gram matrix (X' * X)
alpha_min: float, optional
The minimum correlation along the path. It corresponds to
the regularization parameter alpha parameter in the Lasso.
method: 'lar' or 'lasso'
Specifies the problem solved: the LAR or its variant the
LASSO-LARS that gives the solution of the LASSO problem
for any regularization parameter.
Returns
--------
alphas: array, shape: (k)
The alphas along the path
active: array, shape (?)
Indices of active variables at the end of the path.
coefs: array, shape (p,k)
Coefficients along the path
Notes
------
http://en.wikipedia.org/wiki/Least-angle_regression
http://en.wikipedia.org/wiki/Lasso_(statistics)#LASSO_method
XXX : add reference papers
"""
# TODO: detect stationary points.
# Lasso variant
# store full path
X = np.atleast_2d(X)
y = np.atleast_1d(y)
n_samples, n_features = X.shape
if max_iter is None:
max_iter = min(n_samples, n_features)
max_pred = max_iter # OK for now
# because of some restrictions in Cython, boolean values are
# simulated using np.int8
beta = np.zeros ((max_iter + 1, X.shape[1]))
alphas = np.zeros (max_iter + 1)
n_iter, n_pred = 0, 0
active = list()
unactive = range (X.shape[1])
active_mask = np.zeros (X.shape[1], dtype=np.uint8)
# holds the sign of covariance
sign_active = np.empty (max_pred, dtype=np.int8)
Cov = np.empty (X.shape[1])
a = np.empty (X.shape[1])
drop = False
# will hold the cholesky factorization
# only lower part is referenced. We do not create it as
# empty array because chol_solve calls chkfinite on the
# whole array, which can cause problems.
L = np.zeros ((max_pred, max_pred), dtype=np.float64)
Xt = X.T
if Gram is not None:
res_init = np.dot (X.T, y)
while 1:
n_unactive = X.shape[1] - n_pred # number of unactive elements
if n_unactive:
# Calculate covariance matrix and get maximum
if Gram is None:
res = y - np.dot (X, beta[n_iter]) # there are better ways
arrayfuncs.dot_over (X.T, res, active_mask, np.False_, Cov)
else:
# could use dot_over
arrayfuncs.dot_over (Gram, beta[n_iter], active_mask, np.False_, a)
Cov = res_init[unactive] - a[:n_unactive]
imax = np.argmax (np.abs(Cov[:n_unactive])) #rename
C_ = Cov [imax]
# np.delete (Cov, imax) # very ugly, has to be fixed
else:
# special case when all elements are in the active set
if Gram is None:
res = y - np.dot (X, beta[n_iter])
C_ = np.dot (X.T[0], res)
else:
C_ = np.dot(Gram[0], beta[n_iter]) - res_init[0]
alpha = np.abs(C_) # ugly alpha vs alphas
alphas [n_iter] = alpha
if (n_iter >= max_iter or n_pred >= max_pred ):
break
if (alpha < alpha_min): break
if not drop:
imax = unactive.pop (imax)
# Update the Cholesky factorization of (Xa * Xa') #
# #
# ( L 0 ) #
# L -> ( ) , where L * w = b #
# ( w z ) z = 1 - ||w|| #
# #
# where u is the last added to the active set #
sign_active [n_pred] = np.sign (C_)
if Gram is None:
X_max = Xt[imax]
c = np.dot (X_max, X_max)
b = np.dot (X_max, X[:, active])
else:
c = Gram[imax, imax]
b = Gram[imax, active]
n_pred += 1
active.append(imax)
L [n_pred-1, n_pred-1] = c
if n_pred > 1:
# please refactor me, using linalg.solve is overkill
#L [n_pred-1, :n_pred-1] = linalg.solve (L[:n_pred-1, :n_pred-1], b)
arrayfuncs.solve_triangular (L[:n_pred-1, :n_pred-1],
b)
L [n_pred-1, :n_pred-1] = b[:]
v = np.dot(L [n_pred-1, :n_pred-1], L [n_pred - 1, :n_pred -1])
L [n_pred-1, n_pred-1] = np.sqrt (c - v)
# Now we go into the normal equations dance.
# (Golub & Van Loan, 1996)
b = copysign (C_.repeat(n_pred), sign_active[:n_pred])
b = linalg.cho_solve ((L[:n_pred, :n_pred], True), b)
C = A = np.abs(C_)
if Gram is None:
u = np.dot (Xt[active].T, b)
arrayfuncs.dot_over (X.T, u, active_mask, np.False_, a)
else:
# Not sure that this is not not buggy ...
arrayfuncs.dot_over (Gram[active].T, b, active_mask, np.False_, a)
# equation 2.13, there's probably a simpler way
g1 = (C - Cov[:n_unactive]) / (A - a[:n_unactive])
g2 = (C + Cov[:n_unactive]) / (A + a[:n_unactive])
if not drop:
# Quickfix
active_mask [imax] = np.True_
else:
drop = False
# one for the border cases
g = np.concatenate((g1, g2, [1.]))
g = g[g > 0.]
gamma_ = np.min (g)
if n_pred >= X.shape[1]:
gamma_ = 1.
if method == 'lasso':
z = - beta[n_iter, active] / b
z[z <= 0.] = np.inf
idx = np.argmin(z)
if z[idx] < gamma_:
gamma_ = z[idx]
drop = True
n_iter += 1
beta[n_iter, active] = beta[n_iter - 1, active] + gamma_ * b
if drop:
arrayfuncs.cholesky_delete (L[:n_pred, :n_pred], idx)
n_pred -= 1
drop_idx = active.pop (idx)
unactive.append(drop_idx)
active_mask[drop_idx] = False
sign_active = np.delete (sign_active, idx) # do an append to maintain size
sign_active = np.append (sign_active, 0.)
# should be done using cholesky deletes
if alpha < alpha_min: # interpolate
# interpolation factor 0 <= ss < 1
ss = (alphas[n_iter-1] - alpha_min) / (alphas[n_iter-1] - alphas[n_iter])
beta[n_iter] = beta[n_iter-1] + ss*(beta[n_iter] - beta[n_iter-1]);
alphas[n_iter] = alpha_min
alphas = alphas[:n_iter+1]
beta = beta[:n_iter+1]
return alphas, active, beta.T
class LARS (LinearModel):
""" Least Angle Regression model a.k.a. LAR
Parameters
----------
n_features : int, optional
Number of selected active features
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).
Attributes
----------
`coef_` : array, shape = [n_features]
parameter vector (w in the fomulation formula)
`intercept_` : float
independent term in decision function.
Examples
--------
>>> from scikits.learn import glm
>>> clf = glm.LARS(n_features=1)
>>> clf.fit([[-1,1], [0, 0], [1, 1]], [-1, 0, -1])
LARS(normalize=True, n_features=1)
>>> print clf.coef_
[ 0. -0.81649658]
Notes
-----
See also scikits.learn.glm.LassoLARS that fits a LASSO model
using a variant of Least Angle Regression
http://en.wikipedia.org/wiki/Least_angle_regression
See examples. XXX : add examples names
"""
def __init__(self, n_features, normalize=True):
self.n_features = n_features
self.normalize = normalize
self.coef_ = None
self.fit_intercept = True
def fit (self, X, y, Gram=None, **params):
self._set_params(**params)
# will only normalize non-zero columns
X = np.atleast_2d(X)
y = np.atleast_1d(y)
X, y, Xmean, Ymean = self._center_data(X, y)
if self.normalize:
norms = np.sqrt(np.sum(X**2, axis=0))
nonzeros = np.flatnonzero(norms)
X[:, nonzeros] /= norms[nonzeros]
method = 'lar'
alphas_, active, coef_path_ = lars_path(X, y, Gram=Gram,
max_iter=self.n_features, method=method)
self.coef_ = coef_path_[:,-1]
return self
class LassoLARS (LinearModel):
""" Lasso model fit with Least Angle Regression a.k.a. LARS
It is a Linear Model trained with an L1 prior as regularizer.
lasso).
Parameters
----------
alpha : float, optional
Constant that multiplies the L1 term. Defaults to 1.0
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).
Attributes
----------
`coef_` : array, shape = [n_features]
parameter vector (w in the fomulation formula)
`intercept_` : float
independent term in decision function.
Examples
--------
>>> from scikits.learn import glm
>>> clf = glm.LassoLARS(alpha=0.1)
>>> clf.fit([[-1,1], [0, 0], [1, 1]], [-1, 0, -1])
LassoLARS(normalize=True, alpha=0.1, max_iter=None, fit_intercept=True)
>>> print clf.coef_
[ 0. -0.51649658]
Notes
-----
See also scikits.learn.glm.Lasso that fits the same model using
an alternative optimization strategy called 'coordinate descent.'
"""
def __init__(self, alpha=1.0, max_iter=None, normalize=True,
fit_intercept=True):
""" XXX : add doc
# will only normalize non-zero columns
"""
self.alpha = alpha
self.normalize = normalize
self.coef_ = None
self.max_iter = max_iter
self.fit_intercept = fit_intercept
def fit (self, X, y, Gram=None, **params):
""" XXX : add doc
"""
self._set_params(**params)
X = np.atleast_2d(X)
y = np.atleast_1d(y)
X, y, Xmean, Ymean = self._center_data(X, y)
n_samples = X.shape[0]
alpha = self.alpha * n_samples # scale alpha with number of samples
# XXX : should handle also unnormalized datasets
if self.normalize:
norms = np.sqrt(np.sum(X**2, axis=0))
nonzeros = np.flatnonzero(norms)
X[:, nonzeros] /= norms[nonzeros]
method = 'lasso'
alphas_, active, coef_path_ = lars_path(X, y, Gram=Gram,
alpha_min=alpha, method=method,
max_iter=self.max_iter)
self.coef_ = coef_path_[:,-1]
self._set_intercept(Xmean, Ymean)
return self