Inzynierka/Lib/site-packages/sklearn/linear_model/tests/test_least_angle.py

import warnings

import numpy as np
import pytest
from scipy import linalg
from sklearn.base import clone
from sklearn.model_selection import train_test_split
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import StandardScaler
from sklearn.utils._testing import assert_allclose
from sklearn.utils._testing import assert_array_almost_equal
from sklearn.utils._testing import ignore_warnings
from sklearn.utils._testing import TempMemmap
from sklearn.utils import check_random_state
from sklearn.exceptions import ConvergenceWarning
from sklearn import linear_model, datasets
from sklearn.linear_model._least_angle import _lars_path_residues
from sklearn.linear_model import LassoLarsIC, lars_path
from sklearn.linear_model import Lars, LassoLars, LarsCV, LassoLarsCV

# TODO: use another dataset that has multiple drops
diabetes = datasets.load_diabetes()
X, y = diabetes.data, diabetes.target
G = np.dot(X.T, X)
Xy = np.dot(X.T, y)
n_samples = y.size

# TODO(1.4): 'normalize' to be removed
filterwarnings_normalize = pytest.mark.filterwarnings(
    "ignore:'normalize' was deprecated"
)


# TODO(1.4) 'normalize' to be removed
@pytest.mark.parametrize(
    "LeastAngleModel", [Lars, LassoLars, LarsCV, LassoLarsCV, LassoLarsIC]
)
@pytest.mark.parametrize(
    "normalize, n_warnings", [(True, 1), (False, 1), ("deprecated", 0)]
)
def test_assure_warning_when_normalize(LeastAngleModel, normalize, n_warnings):
    # check that we issue a FutureWarning when normalize was set
    rng = check_random_state(0)
    n_samples = 200
    n_features = 2
    X = rng.randn(n_samples, n_features)
    X[X < 0.1] = 0.0
    y = rng.rand(n_samples)

    model = LeastAngleModel(normalize=normalize)
    with warnings.catch_warnings(record=True) as rec:
        warnings.simplefilter("always", FutureWarning)
        model.fit(X, y)

    assert len([w.message for w in rec]) == n_warnings


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_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 _assert_same_lars_path_result(output1, output2):
    assert len(output1) == len(output2)
    for o1, o2 in zip(output1, output2):
        assert_allclose(o1, o2)


@pytest.mark.parametrize("method", ["lar", "lasso"])
@pytest.mark.parametrize("return_path", [True, False])
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_x_none_gram_none_raises_value_error():
    # Test that lars_path with no X and Gram raises exception
    Xy = np.dot(X.T, y)
    with pytest.raises(ValueError):
        linear_model.lars_path(None, y, Gram=None, Xy=Xy)


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)


# TODO(1.4): 'normalize' to be removed
@filterwarnings_normalize
@pytest.mark.filterwarnings("ignore: `rcond` parameter will change")
# numpy deprecation
def test_lars_lstsq():
    # Test that Lars gives least square solution at the end
    # of the path
    X1 = 3 * X  # use un-normalized dataset
    clf = linear_model.LassoLars(alpha=0.0)
    clf.fit(X1, y)
    coef_lstsq = np.linalg.lstsq(X1, y, rcond=None)[0]
    assert_array_almost_equal(clf.coef_, coef_lstsq)


@pytest.mark.filterwarnings("ignore:`rcond` parameter will change")
# numpy deprecation
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])


def test_collinearity():
    # Check that lars_path is robust to collinearity in input
    X = np.array([[3.0, 3.0, 1.0], [2.0, 2.0, 0.0], [1.0, 1.0, 0]])
    y = np.array([1.0, 0.0, 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() < 1.0  # 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=0.0,
        method="lasso",
        verbose=0,
        max_iter=500,
    )
    assert_array_almost_equal(coef_path_, np.zeros_like(coef_path_))


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_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]


@filterwarnings_normalize
@pytest.mark.parametrize(
    "classifier", [linear_model.Lars, linear_model.LarsCV, linear_model.LassoLarsIC]
)
def test_lars_precompute(classifier):
    # Check for different values of precompute
    G = np.dot(X.T, X)

    clf = classifier(precompute=G)
    output_1 = ignore_warnings(clf.fit)(X, y).coef_
    for precompute in [True, False, "auto", None]:
        clf = classifier(precompute=precompute)
        output_2 = clf.fit(X, y).coef_
        assert_array_almost_equal(output_1, output_2, decimal=8)


def test_singular_matrix():
    # Test when input is a singular matrix
    X1 = np.array([[1, 1.0], [1.0, 1.0]])
    y1 = np.array([1, 1])
    _, _, coef_path = linear_model.lars_path(X1, y1)
    assert_array_almost_equal(coef_path.T, [[0, 0], [1, 0]])


def test_rank_deficient_design():
    # consistency test that checks that LARS Lasso is handling rank
    # deficient input data (with n_features < rank) in the same way
    # as coordinate descent Lasso
    y = [5, 0, 5]
    for X in ([[5, 0], [0, 5], [10, 10]], [[10, 10, 0], [1e-32, 0, 0], [0, 0, 1]]):
        # To be able to use the coefs to compute the objective function,
        # we need to turn off normalization
        lars = linear_model.LassoLars(0.1)
        coef_lars_ = lars.fit(X, y).coef_
        obj_lars = 1.0 / (2.0 * 3.0) * linalg.norm(
            y - np.dot(X, coef_lars_)
        ) ** 2 + 0.1 * linalg.norm(coef_lars_, 1)
        coord_descent = linear_model.Lasso(0.1, tol=1e-6)
        coef_cd_ = coord_descent.fit(X, y).coef_
        obj_cd = (1.0 / (2.0 * 3.0)) * linalg.norm(
            y - np.dot(X, coef_cd_)
        ) ** 2 + 0.1 * linalg.norm(coef_cd_, 1)
        assert obj_lars < obj_cd * (1.0 + 1e-8)


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).fit(X, y)
        clf2 = linear_model.Lasso(alpha=alpha, tol=1e-8).fit(X, y)
        err = linalg.norm(clf1.coef_ - clf2.coef_)
        assert err < 1e-3

    # same test, with normalized data
    X = diabetes.data
    X = X - X.sum(axis=0)
    X /= np.linalg.norm(X, axis=0)
    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


@filterwarnings_normalize
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]

    X = diabetes.data

    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
    X = diabetes.data - diabetes.data.sum(axis=0)
    X /= np.linalg.norm(X, axis=0)

    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(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


@filterwarnings_normalize
def test_lasso_lars_path_length():
    # Test that the path length of the LassoLars is right
    lasso = linear_model.LassoLars()
    lasso.fit(X, y)
    lasso2 = linear_model.LassoLars(alpha=lasso.alphas_[2])
    lasso2.fit(X, y)
    assert_array_almost_equal(lasso.alphas_[:3], lasso2.alphas_)
    # Also check that the sequence of alphas is always decreasing
    assert np.all(np.diff(lasso.alphas_) < 0)


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)

    assert_array_almost_equal(lars_coef, lasso_coef2, decimal=1)


def test_lasso_lars_vs_lasso_cd_ill_conditioned2():
    # Create an ill-conditioned situation in which the LARS has to go
    # far in the path to converge, and check that LARS and coordinate
    # descent give the same answers
    # Note it used to be the case that Lars had to use the drop for good
    # strategy for this but this is no longer the case with the
    # equality_tolerance checks
    X = [[1e20, 1e20, 0], [-1e-32, 0, 0], [1, 1, 1]]
    y = [10, 10, 1]
    alpha = 0.0001

    def objective_function(coef):
        return 1.0 / (2.0 * len(X)) * linalg.norm(
            y - np.dot(X, coef)
        ) ** 2 + alpha * linalg.norm(coef, 1)

    lars = linear_model.LassoLars(alpha=alpha)
    warning_message = "Regressors in active set degenerate."
    with pytest.warns(ConvergenceWarning, match=warning_message):
        lars.fit(X, y)
    lars_coef_ = lars.coef_
    lars_obj = objective_function(lars_coef_)

    coord_descent = linear_model.Lasso(alpha=alpha, tol=1e-4)
    cd_coef_ = coord_descent.fit(X, y).coef_
    cd_obj = objective_function(cd_coef_)

    assert lars_obj < cd_obj * (1.0 + 1e-8)


@filterwarnings_normalize
def test_lars_add_features():
    # assure that at least some features get added if necessary
    # test for 6d2b4c
    # Hilbert matrix
    n = 5
    H = 1.0 / (np.arange(1, n + 1) + np.arange(n)[:, np.newaxis])
    clf = linear_model.Lars(fit_intercept=False).fit(H, np.arange(n))
    assert np.all(np.isfinite(clf.coef_))


@filterwarnings_normalize
def test_lars_n_nonzero_coefs(verbose=False):
    lars = linear_model.Lars(n_nonzero_coefs=6, verbose=verbose)
    lars.fit(X, y)
    assert len(lars.coef_.nonzero()[0]) == 6
    # The path should be of length 6 + 1 in a Lars going down to 6
    # non-zero coefs
    assert len(lars.alphas_) == 7


@filterwarnings_normalize
@ignore_warnings
def test_multitarget():
    # Assure that estimators receiving multidimensional y do the right thing
    Y = np.vstack([y, y**2]).T
    n_targets = Y.shape[1]
    estimators = [
        linear_model.LassoLars(),
        linear_model.Lars(),
        # regression test for gh-1615
        linear_model.LassoLars(fit_intercept=False),
        linear_model.Lars(fit_intercept=False),
    ]

    for estimator in estimators:
        estimator.fit(X, Y)
        Y_pred = estimator.predict(X)
        alphas, active, coef, path = (
            estimator.alphas_,
            estimator.active_,
            estimator.coef_,
            estimator.coef_path_,
        )
        for k in range(n_targets):
            estimator.fit(X, Y[:, k])
            y_pred = estimator.predict(X)
            assert_array_almost_equal(alphas[k], estimator.alphas_)
            assert_array_almost_equal(active[k], estimator.active_)
            assert_array_almost_equal(coef[k], estimator.coef_)
            assert_array_almost_equal(path[k], estimator.coef_path_)
            assert_array_almost_equal(Y_pred[:, k], y_pred)


@filterwarnings_normalize
def test_lars_cv():
    # Test the LassoLarsCV object by checking that the optimal alpha
    # increases as the number of samples increases.
    # This property is not actually guaranteed in general and is just a
    # property of the given dataset, with the given steps chosen.
    old_alpha = 0
    lars_cv = linear_model.LassoLarsCV()
    for length in (400, 200, 100):
        X = diabetes.data[:length]
        y = diabetes.target[:length]
        lars_cv.fit(X, y)
        np.testing.assert_array_less(old_alpha, lars_cv.alpha_)
        old_alpha = lars_cv.alpha_
    assert not hasattr(lars_cv, "n_nonzero_coefs")


def test_lars_cv_max_iter(recwarn):
    warnings.simplefilter("always")
    with np.errstate(divide="raise", invalid="raise"):
        X = diabetes.data
        y = diabetes.target
        rng = np.random.RandomState(42)
        x = rng.randn(len(y))
        X = diabetes.data
        X = np.c_[X, x, x]  # add correlated features
        X = StandardScaler().fit_transform(X)
        lars_cv = linear_model.LassoLarsCV(max_iter=5, cv=5)
        lars_cv.fit(X, y)

    # Check that there is no warning in general and no ConvergenceWarning
    # in particular.
    # Materialize the string representation of the warning to get a more
    # informative error message in case of AssertionError.
    recorded_warnings = [str(w) for w in recwarn]
    assert len(recorded_warnings) == 0


def test_lasso_lars_ic():
    # Test the LassoLarsIC object by checking that
    # - some good features are selected.
    # - alpha_bic > alpha_aic
    # - n_nonzero_bic < n_nonzero_aic
    lars_bic = linear_model.LassoLarsIC("bic")
    lars_aic = linear_model.LassoLarsIC("aic")
    rng = np.random.RandomState(42)
    X = diabetes.data
    X = np.c_[X, rng.randn(X.shape[0], 5)]  # add 5 bad features
    X = StandardScaler().fit_transform(X)
    lars_bic.fit(X, y)
    lars_aic.fit(X, y)
    nonzero_bic = np.where(lars_bic.coef_)[0]
    nonzero_aic = np.where(lars_aic.coef_)[0]
    assert lars_bic.alpha_ > lars_aic.alpha_
    assert len(nonzero_bic) < len(nonzero_aic)
    assert np.max(nonzero_bic) < diabetes.data.shape[1]


def test_lars_path_readonly_data():
    # When using automated memory mapping on large input, the
    # fold data is in read-only mode
    # This is a non-regression test for:
    # https://github.com/scikit-learn/scikit-learn/issues/4597
    splitted_data = train_test_split(X, y, random_state=42)
    with TempMemmap(splitted_data) as (X_train, X_test, y_train, y_test):
        # The following should not fail despite copy=False
        _lars_path_residues(X_train, y_train, X_test, y_test, copy=False)


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


# now we gonna test the positive option for all estimator classes

default_parameter = {"fit_intercept": False}

estimator_parameter_map = {
    "LassoLars": {"alpha": 0.1},
    "LassoLarsCV": {},
    "LassoLarsIC": {},
}


@filterwarnings_normalize
def test_estimatorclasses_positive_constraint():
    # testing the transmissibility for the positive option of all estimator
    # classes in this same function here
    default_parameter = {"fit_intercept": False}

    estimator_parameter_map = {
        "LassoLars": {"alpha": 0.1},
        "LassoLarsCV": {},
        "LassoLarsIC": {},
    }
    for estname in estimator_parameter_map:
        params = default_parameter.copy()
        params.update(estimator_parameter_map[estname])
        estimator = getattr(linear_model, estname)(positive=False, **params)
        estimator.fit(X, y)
        assert estimator.coef_.min() < 0
        estimator = getattr(linear_model, estname)(positive=True, **params)
        estimator.fit(X, y)
        assert min(estimator.coef_) >= 0


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, positive=True
        ).fit(X, y)
        clf2 = linear_model.Lasso(
            fit_intercept=False, alpha=alpha, tol=1e-8, positive=True
        ).fit(X, y)
        err = linalg.norm(clf1.coef_ - clf2.coef_)
        assert err < 1e-3

    # normalized data
    X = diabetes.data - diabetes.data.sum(axis=0)
    X /= np.linalg.norm(X, axis=0)
    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[:-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 test_lasso_lars_vs_R_implementation():
    # Test that sklearn LassoLars implementation agrees with the LassoLars
    # implementation available in R (lars library) when fit_intercept=False.

    # Let's generate the data used in the bug report 7778
    y = np.array([-6.45006793, -3.51251449, -8.52445396, 6.12277822, -19.42109366])
    x = np.array(
        [
            [0.47299829, 0, 0, 0, 0],
            [0.08239882, 0.85784863, 0, 0, 0],
            [0.30114139, -0.07501577, 0.80895216, 0, 0],
            [-0.01460346, -0.1015233, 0.0407278, 0.80338378, 0],
            [-0.69363927, 0.06754067, 0.18064514, -0.0803561, 0.40427291],
        ]
    )

    X = x.T

    # The R result was obtained using the following code:
    #
    # library(lars)
    # model_lasso_lars = lars(X, t(y), type="lasso", intercept=FALSE,
    #                         trace=TRUE, normalize=FALSE)
    # r = t(model_lasso_lars$beta)
    #

    r = np.array(
        [
            [
                0,
                0,
                0,
                0,
                0,
                -79.810362809499026,
                -83.528788732782829,
                -83.777653739190711,
                -83.784156932888934,
                -84.033390591756657,
            ],
            [0, 0, 0, 0, -0.476624256777266, 0, 0, 0, 0, 0.025219751009936],
            [
                0,
                -3.577397088285891,
                -4.702795355871871,
                -7.016748621359461,
                -7.614898471899412,
                -0.336938391359179,
                0,
                0,
                0.001213370600853,
                0.048162321585148,
            ],
            [
                0,
                0,
                0,
                2.231558436628169,
                2.723267514525966,
                2.811549786389614,
                2.813766976061531,
                2.817462468949557,
                2.817368178703816,
                2.816221090636795,
            ],
            [
                0,
                0,
                -1.218422599914637,
                -3.457726183014808,
                -4.021304522060710,
                -45.827461592423745,
                -47.776608869312305,
                -47.911561610746404,
                -47.914845922736234,
                -48.039562334265717,
            ],
        ]
    )

    model_lasso_lars = linear_model.LassoLars(alpha=0, fit_intercept=False)
    model_lasso_lars.fit(X, y)
    skl_betas = model_lasso_lars.coef_path_

    assert_array_almost_equal(r, skl_betas, decimal=12)


@filterwarnings_normalize
@pytest.mark.parametrize("copy_X", [True, False])
def test_lasso_lars_copyX_behaviour(copy_X):
    """
    Test that user input regarding copy_X is not being overridden (it was until
    at least version 0.21)

    """
    lasso_lars = LassoLarsIC(copy_X=copy_X, precompute=False)
    rng = np.random.RandomState(0)
    X = rng.normal(0, 1, (100, 5))
    X_copy = X.copy()
    y = X[:, 2]
    lasso_lars.fit(X, y)
    assert copy_X == np.array_equal(X, X_copy)


@filterwarnings_normalize
@pytest.mark.parametrize("copy_X", [True, False])
def test_lasso_lars_fit_copyX_behaviour(copy_X):
    """
    Test that user input to .fit for copy_X overrides default __init__ value

    """
    lasso_lars = LassoLarsIC(precompute=False)
    rng = np.random.RandomState(0)
    X = rng.normal(0, 1, (100, 5))
    X_copy = X.copy()
    y = X[:, 2]
    lasso_lars.fit(X, y, copy_X=copy_X)
    assert copy_X == np.array_equal(X, X_copy)


@filterwarnings_normalize
@pytest.mark.parametrize("est", (LassoLars(alpha=1e-3), Lars()))
def test_lars_with_jitter(est):
    # Test that a small amount of jitter helps stability,
    # using example provided in issue #2746

    X = np.array([[0.0, 0.0, 0.0, -1.0, 0.0], [0.0, -1.0, 0.0, 0.0, 0.0]])
    y = [-2.5, -2.5]
    expected_coef = [0, 2.5, 0, 2.5, 0]

    # set to fit_intercept to False since target is constant and we want check
    # the value of coef. coef would be all zeros otherwise.
    est.set_params(fit_intercept=False)
    est_jitter = clone(est).set_params(jitter=10e-8, random_state=0)

    est.fit(X, y)
    est_jitter.fit(X, y)

    assert np.mean((est.coef_ - est_jitter.coef_) ** 2) > 0.1
    np.testing.assert_allclose(est_jitter.coef_, expected_coef, rtol=1e-3)


def test_X_none_gram_not_none():
    with pytest.raises(ValueError, match="X cannot be None if Gram is not None"):
        lars_path(X=None, y=[1], Gram="not None")


def test_copy_X_with_auto_gram():
    # Non-regression test for #17789, `copy_X=True` and Gram='auto' does not
    # overwrite X
    rng = np.random.RandomState(42)
    X = rng.rand(6, 6)
    y = rng.rand(6)

    X_before = X.copy()
    linear_model.lars_path(X, y, Gram="auto", copy_X=True, method="lasso")
    # X did not change
    assert_allclose(X, X_before)


@pytest.mark.parametrize(
    "LARS, has_coef_path, args",
    (
        (Lars, True, {}),
        (LassoLars, True, {}),
        (LassoLarsIC, False, {}),
        (LarsCV, True, {}),
        # max_iter=5 is for avoiding ConvergenceWarning
        (LassoLarsCV, True, {"max_iter": 5}),
    ),
)
@pytest.mark.parametrize("dtype", (np.float32, np.float64))
@filterwarnings_normalize
def test_lars_dtype_match(LARS, has_coef_path, args, dtype):
    # The test ensures that the fit method preserves input dtype
    rng = np.random.RandomState(0)
    X = rng.rand(20, 6).astype(dtype)
    y = rng.rand(20).astype(dtype)

    model = LARS(**args)
    model.fit(X, y)
    assert model.coef_.dtype == dtype
    if has_coef_path:
        assert model.coef_path_.dtype == dtype
    assert model.intercept_.dtype == dtype


@pytest.mark.parametrize(
    "LARS, has_coef_path, args",
    (
        (Lars, True, {}),
        (LassoLars, True, {}),
        (LassoLarsIC, False, {}),
        (LarsCV, True, {}),
        # max_iter=5 is for avoiding ConvergenceWarning
        (LassoLarsCV, True, {"max_iter": 5}),
    ),
)
@filterwarnings_normalize
def test_lars_numeric_consistency(LARS, has_coef_path, args):
    # The test ensures numerical consistency between trained coefficients
    # of float32 and float64.
    rtol = 1e-5
    atol = 1e-5

    rng = np.random.RandomState(0)
    X_64 = rng.rand(10, 6)
    y_64 = rng.rand(10)

    model_64 = LARS(**args).fit(X_64, y_64)
    model_32 = LARS(**args).fit(X_64.astype(np.float32), y_64.astype(np.float32))

    assert_allclose(model_64.coef_, model_32.coef_, rtol=rtol, atol=atol)
    if has_coef_path:
        assert_allclose(model_64.coef_path_, model_32.coef_path_, rtol=rtol, atol=atol)
    assert_allclose(model_64.intercept_, model_32.intercept_, rtol=rtol, atol=atol)


@pytest.mark.parametrize("criterion", ["aic", "bic"])
def test_lassolarsic_alpha_selection(criterion):
    """Check that we properly compute the AIC and BIC score.

    In this test, we reproduce the example of the Fig. 2 of Zou et al.
    (reference [1] in LassoLarsIC) In this example, only 7 features should be
    selected.
    """
    model = make_pipeline(StandardScaler(), LassoLarsIC(criterion=criterion))
    model.fit(X, y)

    best_alpha_selected = np.argmin(model[-1].criterion_)
    assert best_alpha_selected == 7


@pytest.mark.parametrize("fit_intercept", [True, False])
def test_lassolarsic_noise_variance(fit_intercept):
    """Check the behaviour when `n_samples` < `n_features` and that one needs
    to provide the noise variance."""
    rng = np.random.RandomState(0)
    X, y = datasets.make_regression(
        n_samples=10, n_features=11 - fit_intercept, random_state=rng
    )

    model = make_pipeline(StandardScaler(), LassoLarsIC(fit_intercept=fit_intercept))

    err_msg = (
        "You are using LassoLarsIC in the case where the number of samples is smaller"
        " than the number of features"
    )
    with pytest.raises(ValueError, match=err_msg):
        model.fit(X, y)

    model.set_params(lassolarsic__noise_variance=1.0)
    model.fit(X, y).predict(X)