def test_lasso_lars_vs_lasso_cd_early_stopping():
# Test that LassoLars and Lasso using coordinate descent give the
# same results when early stopping is used.
# (test : before, in the middle, and in the last part of the path)
alphas_min = [10, 0.9, 1e-4]
for alpha_min in alphas_min:
alphas, _, lasso_path = linear_model.lars_path(X,
y,
method='lasso',
alpha_min=alpha_min)
lasso_cd = linear_model.Lasso(fit_intercept=False, tol=1e-8)
lasso_cd.alpha = alphas[-1]
lasso_cd.fit(X, y)
error = linalg.norm(lasso_path[:, -1] - lasso_cd.coef_)
assert error < 0.01
# same test, with normalization
for alpha_min in alphas_min:
alphas, _, lasso_path = linear_model.lars_path(X,
y,
method='lasso',
alpha_min=alpha_min)
lasso_cd = linear_model.Lasso(normalize=True, tol=1e-8)
lasso_cd.alpha = alphas[-1]
lasso_cd.fit(X, y)
error = linalg.norm(lasso_path[:, -1] - lasso_cd.coef_)
assert error < 0.01
def test_lars_path_positive_constraint():
# this is the main test for the positive parameter on the lars_path method
# the estimator classes just make use of this function
# we do the test on the diabetes dataset
# ensure that we get negative coefficients when positive=False
# and all positive when positive=True
# for method 'lar' (default) and lasso
err_msg = "Positive constraint not supported for 'lar' coding method."
with pytest.raises(ValueError, match=err_msg):
linear_model.lars_path(diabetes['data'],
diabetes['target'],
method='lar',
positive=True)
method = 'lasso'
_, _, coefs = \
linear_model.lars_path(X, y, return_path=True, method=method,
positive=False)
assert coefs.min() < 0
_, _, coefs = \
linear_model.lars_path(X, y, return_path=True, method=method,
positive=True)
assert coefs.min() >= 0
def test_all_precomputed():
# Test that lars_path with precomputed Gram and Xy gives the right answer
G = np.dot(X.T, X)
Xy = np.dot(X.T, y)
for method in 'lar', 'lasso':
output = linear_model.lars_path(X, y, method=method)
output_pre = linear_model.lars_path(X, y, Gram=G, Xy=Xy, method=method)
for expected, got in zip(output, output_pre):
assert_array_almost_equal(expected, got)
def test_no_path():
# Test that the ``return_path=False`` option returns the correct output
alphas_, _, coef_path_ = linear_model.lars_path(X, y, method='lar')
alpha_, _, coef = linear_model.lars_path(X,
y,
method='lar',
return_path=False)
assert_array_almost_equal(coef, coef_path_[:, -1])
assert alpha_ == alphas_[-1]
def test_no_path_precomputed():
# Test that the ``return_path=False`` option with Gram remains correct
alphas_, _, coef_path_ = linear_model.lars_path(X, y, method='lar', Gram=G)
alpha_, _, coef = linear_model.lars_path(X,
y,
method='lar',
Gram=G,
return_path=False)
assert_array_almost_equal(coef, coef_path_[:, -1])
assert alpha_ == alphas_[-1]
def test_collinearity():
# Check that lars_path is robust to collinearity in input
X = np.array([[3., 3., 1.], [2., 2., 0.], [1., 1., 0]])
y = np.array([1., 0., 0])
rng = np.random.RandomState(0)
f = ignore_warnings
_, _, coef_path_ = f(linear_model.lars_path)(X, y, alpha_min=0.01)
assert not np.isnan(coef_path_).any()
residual = np.dot(X, coef_path_[:, -1]) - y
assert (residual**2).sum() < 1. # just make sure it's bounded
n_samples = 10
X = rng.rand(n_samples, 5)
y = np.zeros(n_samples)
_, _, coef_path_ = linear_model.lars_path(X,
y,
Gram='auto',
copy_X=False,
copy_Gram=False,
alpha_min=0.,
method='lasso',
verbose=0,
max_iter=500)
assert_array_almost_equal(coef_path_, np.zeros_like(coef_path_))
def test_lasso_lars_vs_lasso_cd_ill_conditioned():
# Test lasso lars on a very ill-conditioned design, and check that
# it does not blow up, and stays somewhat close to a solution given
# by the coordinate descent solver
# Also test that lasso_path (using lars_path output style) gives
# the same result as lars_path and previous lasso output style
# under these conditions.
rng = np.random.RandomState(42)
# Generate data
n, m = 70, 100
k = 5
X = rng.randn(n, m)
w = np.zeros((m, 1))
i = np.arange(0, m)
rng.shuffle(i)
supp = i[:k]
w[supp] = np.sign(rng.randn(k, 1)) * (rng.rand(k, 1) + 1)
y = np.dot(X, w)
sigma = 0.2
y += sigma * rng.rand(*y.shape)
y = y.squeeze()
lars_alphas, _, lars_coef = linear_model.lars_path(X, y, method='lasso')
_, lasso_coef2, _ = linear_model.lasso_path(X,
y,
alphas=lars_alphas,
tol=1e-6,
fit_intercept=False)
assert_array_almost_equal(lars_coef, lasso_coef2, decimal=1)
def test_simple():
# Principle of Lars is to keep covariances tied and decreasing
# also test verbose output
from io import StringIO
import sys
old_stdout = sys.stdout
try:
sys.stdout = StringIO()
_, _, coef_path_ = linear_model.lars_path(X,
y,
method='lar',
verbose=10)
sys.stdout = old_stdout
for i, coef_ in enumerate(coef_path_.T):
res = y - np.dot(X, coef_)
cov = np.dot(X.T, res)
C = np.max(abs(cov))
eps = 1e-3
ocur = len(cov[C - eps < abs(cov)])
if i < X.shape[1]:
assert ocur == i + 1
else:
# no more than max_pred variables can go into the active set
assert ocur == X.shape[1]
finally:
sys.stdout = old_stdout
def test_lars_path_gram_equivalent(method, return_path):
_assert_same_lars_path_result(
linear_model.lars_path_gram(Xy=Xy,
Gram=G,
n_samples=n_samples,
method=method,
return_path=return_path),
linear_model.lars_path(X,
y,
Gram=G,
method=method,
return_path=return_path))
def test_no_path_all_precomputed():
# Test that the ``return_path=False`` option with Gram and Xy remains
# correct
X, y = 3 * diabetes.data, diabetes.target
G = np.dot(X.T, X)
Xy = np.dot(X.T, y)
alphas_, _, coef_path_ = linear_model.lars_path(X,
y,
method='lasso',
Xy=Xy,
Gram=G,
alpha_min=0.9)
alpha_, _, coef = linear_model.lars_path(X,
y,
method='lasso',
Gram=G,
Xy=Xy,
alpha_min=0.9,
return_path=False)
assert_array_almost_equal(coef, coef_path_[:, -1])
assert alpha_ == alphas_[-1]
def test_lasso_lars_vs_lasso_cd():
# Test that LassoLars and Lasso using coordinate descent give the
# same results.
X = 3 * diabetes.data
alphas, _, lasso_path = linear_model.lars_path(X, y, method='lasso')
lasso_cd = linear_model.Lasso(fit_intercept=False, tol=1e-8)
for c, a in zip(lasso_path.T, alphas):
if a == 0:
continue
lasso_cd.alpha = a
lasso_cd.fit(X, y)
error = linalg.norm(c - lasso_cd.coef_)
assert error < 0.01
# similar test, with the classifiers
for alpha in np.linspace(1e-2, 1 - 1e-2, 20):
clf1 = linear_model.LassoLars(alpha=alpha, normalize=False).fit(X, y)
clf2 = linear_model.Lasso(alpha=alpha, tol=1e-8,
normalize=False).fit(X, y)
err = linalg.norm(clf1.coef_ - clf2.coef_)
assert err < 1e-3
# same test, with normalized data
X = diabetes.data
alphas, _, lasso_path = linear_model.lars_path(X, y, method='lasso')
lasso_cd = linear_model.Lasso(fit_intercept=False,
normalize=True,
tol=1e-8)
for c, a in zip(lasso_path.T, alphas):
if a == 0:
continue
lasso_cd.alpha = a
lasso_cd.fit(X, y)
error = linalg.norm(c - lasso_cd.coef_)
assert error < 0.01
def test_simple_precomputed():
# The same, with precomputed Gram matrix
_, _, coef_path_ = linear_model.lars_path(X, y, Gram=G, method='lar')
for i, coef_ in enumerate(coef_path_.T):
res = y - np.dot(X, coef_)
cov = np.dot(X.T, res)
C = np.max(abs(cov))
eps = 1e-3
ocur = len(cov[C - eps < abs(cov)])
if i < X.shape[1]:
assert ocur == i + 1
else:
# no more than max_pred variables can go into the active set
assert ocur == X.shape[1]
def test_lasso_path_return_models_vs_new_return_gives_same_coefficients():
# Test that lasso_path with lars_path style output gives the
# same result
# Some toy data
X = np.array([[1, 2, 3.1], [2.3, 5.4, 4.3]]).T
y = np.array([1, 2, 3.1])
alphas = [5., 1., .5]
# Use lars_path and lasso_path(new output) with 1D linear interpolation
# to compute the same path
alphas_lars, _, coef_path_lars = lars_path(X, y, method='lasso')
coef_path_cont_lars = interpolate.interp1d(alphas_lars[::-1],
coef_path_lars[:, ::-1])
alphas_lasso2, coef_path_lasso2, _ = lasso_path(X,
y,
alphas=alphas,
return_models=False)
coef_path_cont_lasso = interpolate.interp1d(alphas_lasso2[::-1],
coef_path_lasso2[:, ::-1])
assert_array_almost_equal(coef_path_cont_lasso(alphas),
coef_path_cont_lars(alphas),
decimal=1)
def compute_bench(samples_range, features_range):
it = 0
results = defaultdict(lambda: [])
max_it = len(samples_range) * len(features_range)
for n_samples in samples_range:
for n_features in features_range:
it += 1
print('====================')
print('Iteration %03d of %03d' % (it, max_it))
print('====================')
dataset_kwargs = {
'n_samples': n_samples,
'n_features': n_features,
'n_informative': n_features // 10,
'effective_rank': min(n_samples, n_features) / 10,
#'effective_rank': None,
'bias': 0.0,
}
print("n_samples: %d" % n_samples)
print("n_features: %d" % n_features)
X, y = make_regression(**dataset_kwargs)
gc.collect()
print("benchmarking lars_path (with Gram):", end='')
sys.stdout.flush()
tstart = time()
G = np.dot(X.T, X) # precomputed Gram matrix
Xy = np.dot(X.T, y)
lars_path_gram(Xy=Xy, Gram=G, n_samples=y.size, method='lasso')
delta = time() - tstart
print("%0.3fs" % delta)
results['lars_path (with Gram)'].append(delta)
gc.collect()
print("benchmarking lars_path (without Gram):", end='')
sys.stdout.flush()
tstart = time()
lars_path(X, y, method='lasso')
delta = time() - tstart
print("%0.3fs" % delta)
results['lars_path (without Gram)'].append(delta)
gc.collect()
print("benchmarking lasso_path (with Gram):", end='')
sys.stdout.flush()
tstart = time()
lasso_path(X, y, precompute=True)
delta = time() - tstart
print("%0.3fs" % delta)
results['lasso_path (with Gram)'].append(delta)
gc.collect()
print("benchmarking lasso_path (without Gram):", end='')
sys.stdout.flush()
tstart = time()
lasso_path(X, y, precompute=False)
delta = time() - tstart
print("%0.3fs" % delta)
results['lasso_path (without Gram)'].append(delta)
return results
def test_lasso_lars_vs_lasso_cd_positive():
# Test that LassoLars and Lasso using coordinate descent give the
# same results when using the positive option
# This test is basically a copy of the above with additional positive
# option. However for the middle part, the comparison of coefficient values
# for a range of alphas, we had to make an adaptations. See below.
# not normalized data
X = 3 * diabetes.data
alphas, _, lasso_path = linear_model.lars_path(X,
y,
method='lasso',
positive=True)
lasso_cd = linear_model.Lasso(fit_intercept=False, tol=1e-8, positive=True)
for c, a in zip(lasso_path.T, alphas):
if a == 0:
continue
lasso_cd.alpha = a
lasso_cd.fit(X, y)
error = linalg.norm(c - lasso_cd.coef_)
assert error < 0.01
# The range of alphas chosen for coefficient comparison here is restricted
# as compared with the above test without the positive option. This is due
# to the circumstance that the Lars-Lasso algorithm does not converge to
# the least-squares-solution for small alphas, see 'Least Angle Regression'
# by Efron et al 2004. The coefficients are typically in congruence up to
# the smallest alpha reached by the Lars-Lasso algorithm and start to
# diverge thereafter. See
# https://gist.github.com/michigraber/7e7d7c75eca694c7a6ff
for alpha in np.linspace(6e-1, 1 - 1e-2, 20):
clf1 = linear_model.LassoLars(fit_intercept=False,
alpha=alpha,
normalize=False,
positive=True).fit(X, y)
clf2 = linear_model.Lasso(fit_intercept=False,
alpha=alpha,
tol=1e-8,
normalize=False,
positive=True).fit(X, y)
err = linalg.norm(clf1.coef_ - clf2.coef_)
assert err < 1e-3
# normalized data
X = diabetes.data
alphas, _, lasso_path = linear_model.lars_path(X,
y,
method='lasso',
positive=True)
lasso_cd = linear_model.Lasso(fit_intercept=False,
normalize=True,
tol=1e-8,
positive=True)
for c, a in zip(lasso_path.T[:-1], alphas[:-1]): # don't include alpha=0
lasso_cd.alpha = a
lasso_cd.fit(X, y)
error = linalg.norm(c - lasso_cd.coef_)
assert error < 0.01
def compute_bench(samples_range, features_range):
it = 0
results = dict()
lars = np.empty((len(features_range), len(samples_range)))
lars_gram = lars.copy()
omp = lars.copy()
omp_gram = lars.copy()
max_it = len(samples_range) * len(features_range)
for i_s, n_samples in enumerate(samples_range):
for i_f, n_features in enumerate(features_range):
it += 1
n_informative = n_features / 10
print('====================')
print('Iteration %03d of %03d' % (it, max_it))
print('====================')
# dataset_kwargs = {
# 'n_train_samples': n_samples,
# 'n_test_samples': 2,
# 'n_features': n_features,
# 'n_informative': n_informative,
# 'effective_rank': min(n_samples, n_features) / 10,
# #'effective_rank': None,
# 'bias': 0.0,
# }
dataset_kwargs = {
'n_samples': 1,
'n_components': n_features,
'n_features': n_samples,
'n_nonzero_coefs': n_informative,
'random_state': 0
}
print("n_samples: %d" % n_samples)
print("n_features: %d" % n_features)
y, X, _ = make_sparse_coded_signal(**dataset_kwargs)
X = np.asfortranarray(X)
gc.collect()
print("benchmarking lars_path (with Gram):", end='')
sys.stdout.flush()
tstart = time()
G = np.dot(X.T, X) # precomputed Gram matrix
Xy = np.dot(X.T, y)
lars_path_gram(Xy=Xy, Gram=G, n_samples=y.size,
max_iter=n_informative)
delta = time() - tstart
print("%0.3fs" % delta)
lars_gram[i_f, i_s] = delta
gc.collect()
print("benchmarking lars_path (without Gram):", end='')
sys.stdout.flush()
tstart = time()
lars_path(X, y, Gram=None, max_iter=n_informative)
delta = time() - tstart
print("%0.3fs" % delta)
lars[i_f, i_s] = delta
gc.collect()
print("benchmarking orthogonal_mp (with Gram):", end='')
sys.stdout.flush()
tstart = time()
orthogonal_mp(X, y, precompute=True,
n_nonzero_coefs=n_informative)
delta = time() - tstart
print("%0.3fs" % delta)
omp_gram[i_f, i_s] = delta
gc.collect()
print("benchmarking orthogonal_mp (without Gram):", end='')
sys.stdout.flush()
tstart = time()
orthogonal_mp(X, y, precompute=False,
n_nonzero_coefs=n_informative)
delta = time() - tstart
print("%0.3fs" % delta)
omp[i_f, i_s] = delta
results['time(LARS) / time(OMP)\n (w/ Gram)'] = (lars_gram / omp_gram)
results['time(LARS) / time(OMP)\n (w/o Gram)'] = (lars / omp)
return results
def test_lasso_gives_lstsq_solution():
# Test that Lars Lasso gives least square solution at the end
# of the path
_, _, coef_path_ = linear_model.lars_path(X, y, method='lasso')
coef_lstsq = np.linalg.lstsq(X, y)[0]
assert_array_almost_equal(coef_lstsq, coef_path_[:, -1])
"""
print(__doc__)
# Author: Fabian Pedregosa <*****@*****.**>
# Alexandre Gramfort <*****@*****.**>
# License: BSD 3 clause
import numpy as np
import matplotlib.pyplot as plt
from mrex import linear_model
from mrex import datasets
X, y = datasets.load_diabetes(return_X_y=True)
print("Computing regularization path using the LARS ...")
_, _, coefs = linear_model.lars_path(X, y, method='lasso', verbose=True)
xx = np.sum(np.abs(coefs.T), axis=1)
xx /= xx[-1]
plt.plot(xx, coefs.T)
ymin, ymax = plt.ylim()
plt.vlines(xx, ymin, ymax, linestyle='dashed')
plt.xlabel('|coef| / max|coef|')
plt.ylabel('Coefficients')
plt.title('LASSO Path')
plt.axis('tight')
plt.show()
def test_singular_matrix():
# Test when input is a singular matrix
X1 = np.array([[1, 1.], [1., 1.]])
y1 = np.array([1, 1])
_, _, coef_path = linear_model.lars_path(X1, y1)
assert_array_almost_equal(coef_path.T, [[0, 0], [1, 0]])