Videre

linedance-app/venv/lib/python3.12/site-packages/sklearn/gaussian_process/tests/_mini_sequence_kernel.py

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+import numpy as np
+
+from sklearn.base import clone
+from sklearn.gaussian_process.kernels import (
+    GenericKernelMixin,
+    Hyperparameter,
+    Kernel,
+    StationaryKernelMixin,
+)
+
+
+class MiniSeqKernel(GenericKernelMixin, StationaryKernelMixin, Kernel):
+    """
+    A minimal (but valid) convolutional kernel for sequences of variable
+    length.
+    """
+
+    def __init__(self, baseline_similarity=0.5, baseline_similarity_bounds=(1e-5, 1)):
+        self.baseline_similarity = baseline_similarity
+        self.baseline_similarity_bounds = baseline_similarity_bounds
+
+    @property
+    def hyperparameter_baseline_similarity(self):
+        return Hyperparameter(
+            "baseline_similarity", "numeric", self.baseline_similarity_bounds
+        )
+
+    def _f(self, s1, s2):
+        return sum(
+            [1.0 if c1 == c2 else self.baseline_similarity for c1 in s1 for c2 in s2]
+        )
+
+    def _g(self, s1, s2):
+        return sum([0.0 if c1 == c2 else 1.0 for c1 in s1 for c2 in s2])
+
+    def __call__(self, X, Y=None, eval_gradient=False):
+        if Y is None:
+            Y = X
+
+        if eval_gradient:
+            return (
+                np.array([[self._f(x, y) for y in Y] for x in X]),
+                np.array([[[self._g(x, y)] for y in Y] for x in X]),
+            )
+        else:
+            return np.array([[self._f(x, y) for y in Y] for x in X])
+
+    def diag(self, X):
+        return np.array([self._f(x, x) for x in X])
+
+    def clone_with_theta(self, theta):
+        cloned = clone(self)
+        cloned.theta = theta
+        return cloned

linedance-app/venv/lib/python3.12/site-packages/sklearn/gaussian_process/tests/test_gpc.py

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+"""Testing for Gaussian process classification"""
+
+# Authors: The scikit-learn developers
+# SPDX-License-Identifier: BSD-3-Clause
+
+import warnings
+
+import numpy as np
+import pytest
+from scipy.optimize import approx_fprime
+
+from sklearn.exceptions import ConvergenceWarning
+from sklearn.gaussian_process import GaussianProcessClassifier
+from sklearn.gaussian_process.kernels import (
+    RBF,
+    CompoundKernel,
+    WhiteKernel,
+)
+from sklearn.gaussian_process.kernels import (
+    ConstantKernel as C,
+)
+from sklearn.gaussian_process.tests._mini_sequence_kernel import MiniSeqKernel
+from sklearn.utils._testing import assert_almost_equal, assert_array_equal
+
+
+def f(x):
+    return np.sin(x)
+
+
+X = np.atleast_2d(np.linspace(0, 10, 30)).T
+X2 = np.atleast_2d([2.0, 4.0, 5.5, 6.5, 7.5]).T
+y = np.array(f(X).ravel() > 0, dtype=int)
+fX = f(X).ravel()
+y_mc = np.empty(y.shape, dtype=int)  # multi-class
+y_mc[fX < -0.35] = 0
+y_mc[(fX >= -0.35) & (fX < 0.35)] = 1
+y_mc[fX > 0.35] = 2
+
+
+fixed_kernel = RBF(length_scale=1.0, length_scale_bounds="fixed")
+kernels = [
+    RBF(length_scale=0.1),
+    fixed_kernel,
+    RBF(length_scale=1.0, length_scale_bounds=(1e-3, 1e3)),
+    C(1.0, (1e-2, 1e2)) * RBF(length_scale=1.0, length_scale_bounds=(1e-3, 1e3)),
+]
+non_fixed_kernels = [kernel for kernel in kernels if kernel != fixed_kernel]
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_predict_consistent(kernel):
+    # Check binary predict decision has also predicted probability above 0.5.
+    gpc = GaussianProcessClassifier(kernel=kernel).fit(X, y)
+    assert_array_equal(gpc.predict(X), gpc.predict_proba(X)[:, 1] >= 0.5)
+
+
+def test_predict_consistent_structured():
+    # Check binary predict decision has also predicted probability above 0.5.
+    X = ["A", "AB", "B"]
+    y = np.array([True, False, True])
+    kernel = MiniSeqKernel(baseline_similarity_bounds="fixed")
+    gpc = GaussianProcessClassifier(kernel=kernel).fit(X, y)
+    assert_array_equal(gpc.predict(X), gpc.predict_proba(X)[:, 1] >= 0.5)
+
+
+@pytest.mark.parametrize("kernel", non_fixed_kernels)
+def test_lml_improving(kernel):
+    # Test that hyperparameter-tuning improves log-marginal likelihood.
+    gpc = GaussianProcessClassifier(kernel=kernel).fit(X, y)
+    assert gpc.log_marginal_likelihood(gpc.kernel_.theta) > gpc.log_marginal_likelihood(
+        kernel.theta
+    )
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_lml_precomputed(kernel):
+    # Test that lml of optimized kernel is stored correctly.
+    gpc = GaussianProcessClassifier(kernel=kernel).fit(X, y)
+    assert_almost_equal(
+        gpc.log_marginal_likelihood(gpc.kernel_.theta), gpc.log_marginal_likelihood(), 7
+    )
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_lml_without_cloning_kernel(kernel):
+    # Test that clone_kernel=False has side-effects of kernel.theta.
+    gpc = GaussianProcessClassifier(kernel=kernel).fit(X, y)
+    input_theta = np.ones(gpc.kernel_.theta.shape, dtype=np.float64)
+
+    gpc.log_marginal_likelihood(input_theta, clone_kernel=False)
+    assert_almost_equal(gpc.kernel_.theta, input_theta, 7)
+
+
+@pytest.mark.parametrize("kernel", non_fixed_kernels)
+def test_converged_to_local_maximum(kernel):
+    # Test that we are in local maximum after hyperparameter-optimization.
+    gpc = GaussianProcessClassifier(kernel=kernel).fit(X, y)
+
+    lml, lml_gradient = gpc.log_marginal_likelihood(gpc.kernel_.theta, True)
+
+    assert np.all(
+        (np.abs(lml_gradient) < 1e-4)
+        | (gpc.kernel_.theta == gpc.kernel_.bounds[:, 0])
+        | (gpc.kernel_.theta == gpc.kernel_.bounds[:, 1])
+    )
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_lml_gradient(kernel):
+    # Compare analytic and numeric gradient of log marginal likelihood.
+    gpc = GaussianProcessClassifier(kernel=kernel).fit(X, y)
+
+    lml, lml_gradient = gpc.log_marginal_likelihood(kernel.theta, True)
+    lml_gradient_approx = approx_fprime(
+        kernel.theta, lambda theta: gpc.log_marginal_likelihood(theta, False), 1e-10
+    )
+
+    assert_almost_equal(lml_gradient, lml_gradient_approx, 3)
+
+
+def test_random_starts(global_random_seed):
+    # Test that an increasing number of random-starts of GP fitting only
+    # increases the log marginal likelihood of the chosen theta.
+    n_samples, n_features = 25, 2
+    rng = np.random.RandomState(global_random_seed)
+    X = rng.randn(n_samples, n_features) * 2 - 1
+    y = (np.sin(X).sum(axis=1) + np.sin(3 * X).sum(axis=1)) > 0
+
+    kernel = C(1.0, (1e-2, 1e2)) * RBF(
+        length_scale=[1e-3] * n_features, length_scale_bounds=[(1e-4, 1e2)] * n_features
+    )
+    last_lml = -np.inf
+    for n_restarts_optimizer in range(5):
+        gp = GaussianProcessClassifier(
+            kernel=kernel,
+            n_restarts_optimizer=n_restarts_optimizer,
+            random_state=global_random_seed,
+        ).fit(X, y)
+        lml = gp.log_marginal_likelihood(gp.kernel_.theta)
+        assert lml > last_lml - np.finfo(np.float32).eps
+        last_lml = lml
+
+
+@pytest.mark.parametrize("kernel", non_fixed_kernels)
+def test_custom_optimizer(kernel, global_random_seed):
+    # Test that GPC can use externally defined optimizers.
+    # Define a dummy optimizer that simply tests 10 random hyperparameters
+    def optimizer(obj_func, initial_theta, bounds):
+        rng = np.random.RandomState(global_random_seed)
+        theta_opt, func_min = (
+            initial_theta,
+            obj_func(initial_theta, eval_gradient=False),
+        )
+        for _ in range(10):
+            theta = np.atleast_1d(
+                rng.uniform(np.maximum(-2, bounds[:, 0]), np.minimum(1, bounds[:, 1]))
+            )
+            f = obj_func(theta, eval_gradient=False)
+            if f < func_min:
+                theta_opt, func_min = theta, f
+        return theta_opt, func_min
+
+    gpc = GaussianProcessClassifier(kernel=kernel, optimizer=optimizer)
+    gpc.fit(X, y_mc)
+    # Checks that optimizer improved marginal likelihood
+    assert gpc.log_marginal_likelihood(
+        gpc.kernel_.theta
+    ) >= gpc.log_marginal_likelihood(kernel.theta)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_multi_class(kernel):
+    # Test GPC for multi-class classification problems.
+    gpc = GaussianProcessClassifier(kernel=kernel)
+    gpc.fit(X, y_mc)
+
+    y_prob = gpc.predict_proba(X2)
+    assert_almost_equal(y_prob.sum(1), 1)
+
+    y_pred = gpc.predict(X2)
+    assert_array_equal(np.argmax(y_prob, 1), y_pred)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_multi_class_n_jobs(kernel):
+    # Test that multi-class GPC produces identical results with n_jobs>1.
+    gpc = GaussianProcessClassifier(kernel=kernel)
+    gpc.fit(X, y_mc)
+
+    gpc_2 = GaussianProcessClassifier(kernel=kernel, n_jobs=2)
+    gpc_2.fit(X, y_mc)
+
+    y_prob = gpc.predict_proba(X2)
+    y_prob_2 = gpc_2.predict_proba(X2)
+    assert_almost_equal(y_prob, y_prob_2)
+
+
+def test_warning_bounds():
+    kernel = RBF(length_scale_bounds=[1e-5, 1e-3])
+    gpc = GaussianProcessClassifier(kernel=kernel)
+    warning_message = (
+        "The optimal value found for dimension 0 of parameter "
+        "length_scale is close to the specified upper bound "
+        "0.001. Increasing the bound and calling fit again may "
+        "find a better value."
+    )
+    with pytest.warns(ConvergenceWarning, match=warning_message):
+        gpc.fit(X, y)
+
+    kernel_sum = WhiteKernel(noise_level_bounds=[1e-5, 1e-3]) + RBF(
+        length_scale_bounds=[1e3, 1e5]
+    )
+    gpc_sum = GaussianProcessClassifier(kernel=kernel_sum)
+    with warnings.catch_warnings(record=True) as record:
+        warnings.simplefilter("always")
+        gpc_sum.fit(X, y)
+
+        assert len(record) == 2
+
+        assert issubclass(record[0].category, ConvergenceWarning)
+        assert (
+            record[0].message.args[0] == "The optimal value found for "
+            "dimension 0 of parameter "
+            "k1__noise_level is close to the "
+            "specified upper bound 0.001. "
+            "Increasing the bound and calling "
+            "fit again may find a better value."
+        )
+
+        assert issubclass(record[1].category, ConvergenceWarning)
+        assert (
+            record[1].message.args[0] == "The optimal value found for "
+            "dimension 0 of parameter "
+            "k2__length_scale is close to the "
+            "specified lower bound 1000.0. "
+            "Decreasing the bound and calling "
+            "fit again may find a better value."
+        )
+
+    X_tile = np.tile(X, 2)
+    kernel_dims = RBF(length_scale=[1.0, 2.0], length_scale_bounds=[1e1, 1e2])
+    gpc_dims = GaussianProcessClassifier(kernel=kernel_dims)
+
+    with warnings.catch_warnings(record=True) as record:
+        warnings.simplefilter("always")
+        gpc_dims.fit(X_tile, y)
+
+        assert len(record) == 2
+
+        assert issubclass(record[0].category, ConvergenceWarning)
+        assert (
+            record[0].message.args[0] == "The optimal value found for "
+            "dimension 0 of parameter "
+            "length_scale is close to the "
+            "specified upper bound 100.0. "
+            "Increasing the bound and calling "
+            "fit again may find a better value."
+        )
+
+        assert issubclass(record[1].category, ConvergenceWarning)
+        assert (
+            record[1].message.args[0] == "The optimal value found for "
+            "dimension 1 of parameter "
+            "length_scale is close to the "
+            "specified upper bound 100.0. "
+            "Increasing the bound and calling "
+            "fit again may find a better value."
+        )
+
+
+@pytest.mark.parametrize(
+    "params, error_type, err_msg",
+    [
+        (
+            {"kernel": CompoundKernel(0)},
+            ValueError,
+            "kernel cannot be a CompoundKernel",
+        )
+    ],
+)
+def test_gpc_fit_error(params, error_type, err_msg):
+    """Check that expected error are raised during fit."""
+    gpc = GaussianProcessClassifier(**params)
+    with pytest.raises(error_type, match=err_msg):
+        gpc.fit(X, y)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_gpc_latent_mean_and_variance_shape(kernel):
+    """Checks that the latent mean and variance have the right shape."""
+    gpc = GaussianProcessClassifier(kernel=kernel)
+    gpc.fit(X, y)
+
+    # Check that the latent mean and variance have the right shape
+    latent_mean, latent_variance = gpc.latent_mean_and_variance(X)
+    assert latent_mean.shape == (X.shape[0],)
+    assert latent_variance.shape == (X.shape[0],)
+
+
+def test_gpc_latent_mean_and_variance_complain_on_more_than_2_classes():
+    """Checks that the latent mean and variance have the right shape."""
+    gpc = GaussianProcessClassifier(kernel=RBF())
+    gpc.fit(X, y_mc)
+
+    # Check that the latent mean and variance have the right shape
+    with pytest.raises(
+        ValueError,
+        match="Returning the mean and variance of the latent function f "
+        "is only supported for binary classification",
+    ):
+        gpc.latent_mean_and_variance(X)
+
+
+def test_latent_mean_and_variance_works_on_structured_kernels():
+    X = ["A", "AB", "B"]
+    y = np.array([True, False, True])
+    kernel = MiniSeqKernel(baseline_similarity_bounds="fixed")
+    gpc = GaussianProcessClassifier(kernel=kernel).fit(X, y)
+
+    gpc.latent_mean_and_variance(X)

linedance-app/venv/lib/python3.12/site-packages/sklearn/gaussian_process/tests/test_gpr.py

@@ -0,0 +1,861 @@
+"""Testing for Gaussian process regression"""
+
+# Authors: The scikit-learn developers
+# SPDX-License-Identifier: BSD-3-Clause
+
+import re
+import sys
+import warnings
+
+import numpy as np
+import pytest
+from scipy.optimize import approx_fprime
+
+from sklearn.exceptions import ConvergenceWarning
+from sklearn.gaussian_process import GaussianProcessRegressor
+from sklearn.gaussian_process.kernels import (
+    RBF,
+    DotProduct,
+    ExpSineSquared,
+    WhiteKernel,
+)
+from sklearn.gaussian_process.kernels import (
+    ConstantKernel as C,
+)
+from sklearn.gaussian_process.tests._mini_sequence_kernel import MiniSeqKernel
+from sklearn.utils._testing import (
+    assert_allclose,
+    assert_almost_equal,
+    assert_array_almost_equal,
+    assert_array_less,
+)
+
+
+def f(x):
+    return x * np.sin(x)
+
+
+X = np.atleast_2d([1.0, 3.0, 5.0, 6.0, 7.0, 8.0]).T
+X2 = np.atleast_2d([2.0, 4.0, 5.5, 6.5, 7.5]).T
+y = f(X).ravel()
+
+fixed_kernel = RBF(length_scale=1.0, length_scale_bounds="fixed")
+kernels = [
+    RBF(length_scale=1.0),
+    fixed_kernel,
+    RBF(length_scale=1.0, length_scale_bounds=(1e-3, 1e3)),
+    C(1.0, (1e-2, 1e2)) * RBF(length_scale=1.0, length_scale_bounds=(1e-3, 1e3)),
+    C(1.0, (1e-2, 1e2)) * RBF(length_scale=1.0, length_scale_bounds=(1e-3, 1e3))
+    + C(1e-5, (1e-5, 1e2)),
+    C(0.1, (1e-2, 1e2)) * RBF(length_scale=1.0, length_scale_bounds=(1e-3, 1e3))
+    + C(1e-5, (1e-5, 1e2)),
+]
+non_fixed_kernels = [kernel for kernel in kernels if kernel != fixed_kernel]
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_gpr_interpolation(kernel):
+    if sys.maxsize <= 2**32:
+        pytest.xfail("This test may fail on 32 bit Python")
+
+    # Test the interpolating property for different kernels.
+    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
+    y_pred, y_cov = gpr.predict(X, return_cov=True)
+
+    assert_almost_equal(y_pred, y)
+    assert_almost_equal(np.diag(y_cov), 0.0)
+
+
+def test_gpr_interpolation_structured():
+    # Test the interpolating property for different kernels.
+    kernel = MiniSeqKernel(baseline_similarity_bounds="fixed")
+    X = ["A", "B", "C"]
+    y = np.array([1, 2, 3])
+    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
+    y_pred, y_cov = gpr.predict(X, return_cov=True)
+
+    assert_almost_equal(
+        kernel(X, eval_gradient=True)[1].ravel(), (1 - np.eye(len(X))).ravel()
+    )
+    assert_almost_equal(y_pred, y)
+    assert_almost_equal(np.diag(y_cov), 0.0)
+
+
+@pytest.mark.parametrize("kernel", non_fixed_kernels)
+def test_lml_improving(kernel):
+    if sys.maxsize <= 2**32:
+        pytest.xfail("This test may fail on 32 bit Python")
+
+    # Test that hyperparameter-tuning improves log-marginal likelihood.
+    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
+    assert gpr.log_marginal_likelihood(gpr.kernel_.theta) > gpr.log_marginal_likelihood(
+        kernel.theta
+    )
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_lml_precomputed(kernel):
+    # Test that lml of optimized kernel is stored correctly.
+    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
+    assert gpr.log_marginal_likelihood(gpr.kernel_.theta) == pytest.approx(
+        gpr.log_marginal_likelihood()
+    )
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_lml_without_cloning_kernel(kernel):
+    # Test that lml of optimized kernel is stored correctly.
+    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
+    input_theta = np.ones(gpr.kernel_.theta.shape, dtype=np.float64)
+
+    gpr.log_marginal_likelihood(input_theta, clone_kernel=False)
+    assert_almost_equal(gpr.kernel_.theta, input_theta, 7)
+
+
+@pytest.mark.parametrize("kernel", non_fixed_kernels)
+def test_converged_to_local_maximum(kernel):
+    # Test that we are in local maximum after hyperparameter-optimization.
+    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
+
+    lml, lml_gradient = gpr.log_marginal_likelihood(gpr.kernel_.theta, True)
+
+    assert np.all(
+        (np.abs(lml_gradient) < 1e-4)
+        | (gpr.kernel_.theta == gpr.kernel_.bounds[:, 0])
+        | (gpr.kernel_.theta == gpr.kernel_.bounds[:, 1])
+    )
+
+
+@pytest.mark.parametrize("kernel", non_fixed_kernels)
+def test_solution_inside_bounds(kernel):
+    # Test that hyperparameter-optimization remains in bounds#
+    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
+
+    bounds = gpr.kernel_.bounds
+    max_ = np.finfo(gpr.kernel_.theta.dtype).max
+    tiny = 1e-10
+    bounds[~np.isfinite(bounds[:, 1]), 1] = max_
+
+    assert_array_less(bounds[:, 0], gpr.kernel_.theta + tiny)
+    assert_array_less(gpr.kernel_.theta, bounds[:, 1] + tiny)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_lml_gradient(kernel):
+    # Compare analytic and numeric gradient of log marginal likelihood.
+    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
+
+    lml, lml_gradient = gpr.log_marginal_likelihood(kernel.theta, True)
+    lml_gradient_approx = approx_fprime(
+        kernel.theta, lambda theta: gpr.log_marginal_likelihood(theta, False), 1e-10
+    )
+
+    assert_almost_equal(lml_gradient, lml_gradient_approx, 3)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_prior(kernel):
+    # Test that GP prior has mean 0 and identical variances.
+    gpr = GaussianProcessRegressor(kernel=kernel)
+
+    y_mean, y_cov = gpr.predict(X, return_cov=True)
+
+    assert_almost_equal(y_mean, 0, 5)
+    if len(gpr.kernel.theta) > 1:
+        # XXX: quite hacky, works only for current kernels
+        assert_almost_equal(np.diag(y_cov), np.exp(kernel.theta[0]), 5)
+    else:
+        assert_almost_equal(np.diag(y_cov), 1, 5)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_sample_statistics(kernel):
+    # Test that statistics of samples drawn from GP are correct.
+    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
+
+    y_mean, y_cov = gpr.predict(X2, return_cov=True)
+
+    samples = gpr.sample_y(X2, 300000)
+
+    # More digits accuracy would require many more samples
+    assert_almost_equal(y_mean, np.mean(samples, 1), 1)
+    assert_almost_equal(
+        np.diag(y_cov) / np.diag(y_cov).max(),
+        np.var(samples, 1) / np.diag(y_cov).max(),
+        1,
+    )
+
+
+def test_no_optimizer():
+    # Test that kernel parameters are unmodified when optimizer is None.
+    kernel = RBF(1.0)
+    gpr = GaussianProcessRegressor(kernel=kernel, optimizer=None).fit(X, y)
+    assert np.exp(gpr.kernel_.theta) == 1.0
+
+
+@pytest.mark.parametrize("kernel", kernels)
+@pytest.mark.parametrize("target", [y, np.ones(X.shape[0], dtype=np.float64)])
+def test_predict_cov_vs_std(kernel, target):
+    if sys.maxsize <= 2**32:
+        pytest.xfail("This test may fail on 32 bit Python")
+
+    # Test that predicted std.-dev. is consistent with cov's diagonal.
+    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
+    y_mean, y_cov = gpr.predict(X2, return_cov=True)
+    y_mean, y_std = gpr.predict(X2, return_std=True)
+    assert_almost_equal(np.sqrt(np.diag(y_cov)), y_std)
+
+
+def test_anisotropic_kernel():
+    # Test that GPR can identify meaningful anisotropic length-scales.
+    # We learn a function which varies in one dimension ten-times slower
+    # than in the other. The corresponding length-scales should differ by at
+    # least a factor 5
+    rng = np.random.RandomState(0)
+    X = rng.uniform(-1, 1, (50, 2))
+    y = X[:, 0] + 0.1 * X[:, 1]
+
+    kernel = RBF([1.0, 1.0])
+    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
+    assert np.exp(gpr.kernel_.theta[1]) > np.exp(gpr.kernel_.theta[0]) * 5
+
+
+def test_random_starts():
+    # Test that an increasing number of random-starts of GP fitting only
+    # increases the log marginal likelihood of the chosen theta.
+    n_samples, n_features = 25, 2
+    rng = np.random.RandomState(0)
+    X = rng.randn(n_samples, n_features) * 2 - 1
+    y = (
+        np.sin(X).sum(axis=1)
+        + np.sin(3 * X).sum(axis=1)
+        + rng.normal(scale=0.1, size=n_samples)
+    )
+
+    kernel = C(1.0, (1e-2, 1e2)) * RBF(
+        length_scale=[1.0] * n_features, length_scale_bounds=[(1e-4, 1e2)] * n_features
+    ) + WhiteKernel(noise_level=1e-5, noise_level_bounds=(1e-5, 1e1))
+    last_lml = -np.inf
+    for n_restarts_optimizer in range(5):
+        gp = GaussianProcessRegressor(
+            kernel=kernel,
+            n_restarts_optimizer=n_restarts_optimizer,
+            random_state=0,
+        ).fit(X, y)
+        lml = gp.log_marginal_likelihood(gp.kernel_.theta)
+        assert lml > last_lml - np.finfo(np.float32).eps
+        last_lml = lml
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_y_normalization(kernel):
+    """
+    Test normalization of the target values in GP
+
+    Fitting non-normalizing GP on normalized y and fitting normalizing GP
+    on unnormalized y should yield identical results. Note that, here,
+    'normalized y' refers to y that has been made zero mean and unit
+    variance.
+
+    """
+
+    y_mean = np.mean(y)
+    y_std = np.std(y)
+    y_norm = (y - y_mean) / y_std
+
+    # Fit non-normalizing GP on normalized y
+    gpr = GaussianProcessRegressor(kernel=kernel)
+    gpr.fit(X, y_norm)
+
+    # Fit normalizing GP on unnormalized y
+    gpr_norm = GaussianProcessRegressor(kernel=kernel, normalize_y=True)
+    gpr_norm.fit(X, y)
+
+    # Compare predicted mean, std-devs and covariances
+    y_pred, y_pred_std = gpr.predict(X2, return_std=True)
+    y_pred = y_pred * y_std + y_mean
+    y_pred_std = y_pred_std * y_std
+    y_pred_norm, y_pred_std_norm = gpr_norm.predict(X2, return_std=True)
+
+    assert_almost_equal(y_pred, y_pred_norm)
+    assert_almost_equal(y_pred_std, y_pred_std_norm)
+
+    _, y_cov = gpr.predict(X2, return_cov=True)
+    y_cov = y_cov * y_std**2
+    _, y_cov_norm = gpr_norm.predict(X2, return_cov=True)
+
+    assert_almost_equal(y_cov, y_cov_norm)
+
+
+def test_large_variance_y():
+    """
+    Here we test that, when noramlize_y=True, our GP can produce a
+    sensible fit to training data whose variance is significantly
+    larger than unity. This test was made in response to issue #15612.
+
+    GP predictions are verified against predictions that were made
+    using GPy which, here, is treated as the 'gold standard'. Note that we
+    only investigate the RBF kernel here, as that is what was used in the
+    GPy implementation.
+
+    The following code can be used to recreate the GPy data:
+
+    --------------------------------------------------------------------------
+    import GPy
+
+    kernel_gpy = GPy.kern.RBF(input_dim=1, lengthscale=1.)
+    gpy = GPy.models.GPRegression(X, np.vstack(y_large), kernel_gpy)
+    gpy.optimize()
+    y_pred_gpy, y_var_gpy = gpy.predict(X2)
+    y_pred_std_gpy = np.sqrt(y_var_gpy)
+    --------------------------------------------------------------------------
+    """
+
+    # Here we utilise a larger variance version of the training data
+    y_large = 10 * y
+
+    # Standard GP with normalize_y=True
+    RBF_params = {"length_scale": 1.0}
+    kernel = RBF(**RBF_params)
+    gpr = GaussianProcessRegressor(kernel=kernel, normalize_y=True)
+    gpr.fit(X, y_large)
+    y_pred, y_pred_std = gpr.predict(X2, return_std=True)
+
+    # 'Gold standard' mean predictions from GPy
+    y_pred_gpy = np.array(
+        [15.16918303, -27.98707845, -39.31636019, 14.52605515, 69.18503589]
+    )
+
+    # 'Gold standard' std predictions from GPy
+    y_pred_std_gpy = np.array(
+        [7.78860962, 3.83179178, 0.63149951, 0.52745188, 0.86170042]
+    )
+
+    # Based on numerical experiments, it's reasonable to expect our
+    # GP's mean predictions to get within 7% of predictions of those
+    # made by GPy.
+    assert_allclose(y_pred, y_pred_gpy, rtol=0.07, atol=0)
+
+    # Based on numerical experiments, it's reasonable to expect our
+    # GP's std predictions to get within 15% of predictions of those
+    # made by GPy.
+    assert_allclose(y_pred_std, y_pred_std_gpy, rtol=0.15, atol=0)
+
+
+def test_y_multioutput():
+    # Test that GPR can deal with multi-dimensional target values
+    y_2d = np.vstack((y, y * 2)).T
+
+    # Test for fixed kernel that first dimension of 2d GP equals the output
+    # of 1d GP and that second dimension is twice as large
+    kernel = RBF(length_scale=1.0)
+
+    gpr = GaussianProcessRegressor(kernel=kernel, optimizer=None, normalize_y=False)
+    gpr.fit(X, y)
+
+    gpr_2d = GaussianProcessRegressor(kernel=kernel, optimizer=None, normalize_y=False)
+    gpr_2d.fit(X, y_2d)
+
+    y_pred_1d, y_std_1d = gpr.predict(X2, return_std=True)
+    y_pred_2d, y_std_2d = gpr_2d.predict(X2, return_std=True)
+    _, y_cov_1d = gpr.predict(X2, return_cov=True)
+    _, y_cov_2d = gpr_2d.predict(X2, return_cov=True)
+
+    assert_almost_equal(y_pred_1d, y_pred_2d[:, 0])
+    assert_almost_equal(y_pred_1d, y_pred_2d[:, 1] / 2)
+
+    # Standard deviation and covariance do not depend on output
+    for target in range(y_2d.shape[1]):
+        assert_almost_equal(y_std_1d, y_std_2d[..., target])
+        assert_almost_equal(y_cov_1d, y_cov_2d[..., target])
+
+    y_sample_1d = gpr.sample_y(X2, n_samples=10)
+    y_sample_2d = gpr_2d.sample_y(X2, n_samples=10)
+
+    assert y_sample_1d.shape == (5, 10)
+    assert y_sample_2d.shape == (5, 2, 10)
+    # Only the first target will be equal
+    assert_almost_equal(y_sample_1d, y_sample_2d[:, 0, :])
+
+    # Test hyperparameter optimization
+    for kernel in kernels:
+        gpr = GaussianProcessRegressor(kernel=kernel, normalize_y=True)
+        gpr.fit(X, y)
+
+        gpr_2d = GaussianProcessRegressor(kernel=kernel, normalize_y=True)
+        gpr_2d.fit(X, np.vstack((y, y)).T)
+
+        assert_almost_equal(gpr.kernel_.theta, gpr_2d.kernel_.theta, 4)
+
+
+@pytest.mark.parametrize("kernel", non_fixed_kernels)
+def test_custom_optimizer(kernel):
+    # Test that GPR can use externally defined optimizers.
+    # Define a dummy optimizer that simply tests 50 random hyperparameters
+    def optimizer(obj_func, initial_theta, bounds):
+        rng = np.random.RandomState(0)
+        theta_opt, func_min = (
+            initial_theta,
+            obj_func(initial_theta, eval_gradient=False),
+        )
+        for _ in range(50):
+            theta = np.atleast_1d(
+                rng.uniform(np.maximum(-2, bounds[:, 0]), np.minimum(1, bounds[:, 1]))
+            )
+            f = obj_func(theta, eval_gradient=False)
+            if f < func_min:
+                theta_opt, func_min = theta, f
+        return theta_opt, func_min
+
+    gpr = GaussianProcessRegressor(kernel=kernel, optimizer=optimizer)
+    gpr.fit(X, y)
+    # Checks that optimizer improved marginal likelihood
+    assert gpr.log_marginal_likelihood(gpr.kernel_.theta) > gpr.log_marginal_likelihood(
+        gpr.kernel.theta
+    )
+
+
+def test_gpr_correct_error_message():
+    X = np.arange(12).reshape(6, -1)
+    y = np.ones(6)
+    kernel = DotProduct()
+    gpr = GaussianProcessRegressor(kernel=kernel, alpha=0.0)
+    message = (
+        "The kernel, %s, is not returning a "
+        "positive definite matrix. Try gradually increasing "
+        "the 'alpha' parameter of your "
+        "GaussianProcessRegressor estimator." % kernel
+    )
+    with pytest.raises(np.linalg.LinAlgError, match=re.escape(message)):
+        gpr.fit(X, y)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_duplicate_input(kernel):
+    # Test GPR can handle two different output-values for the same input.
+    gpr_equal_inputs = GaussianProcessRegressor(kernel=kernel, alpha=1e-2)
+    gpr_similar_inputs = GaussianProcessRegressor(kernel=kernel, alpha=1e-2)
+
+    X_ = np.vstack((X, X[0]))
+    y_ = np.hstack((y, y[0] + 1))
+    gpr_equal_inputs.fit(X_, y_)
+
+    X_ = np.vstack((X, X[0] + 1e-15))
+    y_ = np.hstack((y, y[0] + 1))
+    gpr_similar_inputs.fit(X_, y_)
+
+    X_test = np.linspace(0, 10, 100)[:, None]
+    y_pred_equal, y_std_equal = gpr_equal_inputs.predict(X_test, return_std=True)
+    y_pred_similar, y_std_similar = gpr_similar_inputs.predict(X_test, return_std=True)
+
+    assert_almost_equal(y_pred_equal, y_pred_similar)
+    assert_almost_equal(y_std_equal, y_std_similar)
+
+
+def test_no_fit_default_predict():
+    # Test that GPR predictions without fit does not break by default.
+    default_kernel = C(1.0, constant_value_bounds="fixed") * RBF(
+        1.0, length_scale_bounds="fixed"
+    )
+    gpr1 = GaussianProcessRegressor()
+    _, y_std1 = gpr1.predict(X, return_std=True)
+    _, y_cov1 = gpr1.predict(X, return_cov=True)
+
+    gpr2 = GaussianProcessRegressor(kernel=default_kernel)
+    _, y_std2 = gpr2.predict(X, return_std=True)
+    _, y_cov2 = gpr2.predict(X, return_cov=True)
+
+    assert_array_almost_equal(y_std1, y_std2)
+    assert_array_almost_equal(y_cov1, y_cov2)
+
+
+def test_warning_bounds():
+    kernel = RBF(length_scale_bounds=[1e-5, 1e-3])
+    gpr = GaussianProcessRegressor(kernel=kernel)
+    warning_message = (
+        "The optimal value found for dimension 0 of parameter "
+        "length_scale is close to the specified upper bound "
+        "0.001. Increasing the bound and calling fit again may "
+        "find a better value."
+    )
+    with pytest.warns(ConvergenceWarning, match=warning_message):
+        gpr.fit(X, y)
+
+    kernel_sum = WhiteKernel(noise_level_bounds=[1e-5, 1e-3]) + RBF(
+        length_scale_bounds=[1e3, 1e5]
+    )
+    gpr_sum = GaussianProcessRegressor(kernel=kernel_sum)
+    with warnings.catch_warnings(record=True) as record:
+        warnings.simplefilter("always")
+        gpr_sum.fit(X, y)
+
+        assert len(record) == 2
+
+        assert issubclass(record[0].category, ConvergenceWarning)
+        assert (
+            record[0].message.args[0] == "The optimal value found for "
+            "dimension 0 of parameter "
+            "k1__noise_level is close to the "
+            "specified upper bound 0.001. "
+            "Increasing the bound and calling "
+            "fit again may find a better value."
+        )
+
+        assert issubclass(record[1].category, ConvergenceWarning)
+        assert (
+            record[1].message.args[0] == "The optimal value found for "
+            "dimension 0 of parameter "
+            "k2__length_scale is close to the "
+            "specified lower bound 1000.0. "
+            "Decreasing the bound and calling "
+            "fit again may find a better value."
+        )
+
+    X_tile = np.tile(X, 2)
+    kernel_dims = RBF(length_scale=[1.0, 2.0], length_scale_bounds=[1e1, 1e2])
+    gpr_dims = GaussianProcessRegressor(kernel=kernel_dims)
+
+    with warnings.catch_warnings(record=True) as record:
+        warnings.simplefilter("always")
+        gpr_dims.fit(X_tile, y)
+
+        assert len(record) == 2
+
+        assert issubclass(record[0].category, ConvergenceWarning)
+        assert (
+            record[0].message.args[0] == "The optimal value found for "
+            "dimension 0 of parameter "
+            "length_scale is close to the "
+            "specified lower bound 10.0. "
+            "Decreasing the bound and calling "
+            "fit again may find a better value."
+        )
+
+        assert issubclass(record[1].category, ConvergenceWarning)
+        assert (
+            record[1].message.args[0] == "The optimal value found for "
+            "dimension 1 of parameter "
+            "length_scale is close to the "
+            "specified lower bound 10.0. "
+            "Decreasing the bound and calling "
+            "fit again may find a better value."
+        )
+
+
+def test_bound_check_fixed_hyperparameter():
+    # Regression test for issue #17943
+    # Check that having a hyperparameter with fixed bounds doesn't cause an
+    # error
+    k1 = 50.0**2 * RBF(length_scale=50.0)  # long term smooth rising trend
+    k2 = ExpSineSquared(
+        length_scale=1.0, periodicity=1.0, periodicity_bounds="fixed"
+    )  # seasonal component
+    kernel = k1 + k2
+    GaussianProcessRegressor(kernel=kernel).fit(X, y)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_constant_target(kernel):
+    """Check that the std. dev. is affected to 1 when normalizing a constant
+    feature.
+    Non-regression test for:
+    https://github.com/scikit-learn/scikit-learn/issues/18318
+    NaN where affected to the target when scaling due to null std. dev. with
+    constant target.
+    """
+    y_constant = np.ones(X.shape[0], dtype=np.float64)
+
+    gpr = GaussianProcessRegressor(kernel=kernel, normalize_y=True)
+    gpr.fit(X, y_constant)
+    assert gpr._y_train_std == pytest.approx(1.0)
+
+    y_pred, y_cov = gpr.predict(X, return_cov=True)
+    assert_allclose(y_pred, y_constant)
+    # set atol because we compare to zero
+    assert_allclose(np.diag(y_cov), 0.0, atol=1e-9)
+
+    # Test multi-target data
+    n_samples, n_targets = X.shape[0], 2
+    rng = np.random.RandomState(0)
+    y = np.concatenate(
+        [
+            rng.normal(size=(n_samples, 1)),  # non-constant target
+            np.full(shape=(n_samples, 1), fill_value=2),  # constant target
+        ],
+        axis=1,
+    )
+
+    gpr.fit(X, y)
+    Y_pred, Y_cov = gpr.predict(X, return_cov=True)
+
+    assert_allclose(Y_pred[:, 1], 2)
+    assert_allclose(np.diag(Y_cov[..., 1]), 0.0, atol=1e-9)
+
+    assert Y_pred.shape == (n_samples, n_targets)
+    assert Y_cov.shape == (n_samples, n_samples, n_targets)
+
+
+def test_gpr_consistency_std_cov_non_invertible_kernel():
+    """Check the consistency between the returned std. dev. and the covariance.
+    Non-regression test for:
+    https://github.com/scikit-learn/scikit-learn/issues/19936
+    Inconsistencies were observed when the kernel cannot be inverted (or
+    numerically stable).
+    """
+    kernel = C(8.98576054e05, (1e-12, 1e12)) * RBF(
+        [5.91326520e02, 1.32584051e03], (1e-12, 1e12)
+    ) + WhiteKernel(noise_level=1e-5)
+    gpr = GaussianProcessRegressor(kernel=kernel, alpha=0, optimizer=None)
+    X_train = np.array(
+        [
+            [0.0, 0.0],
+            [1.54919334, -0.77459667],
+            [-1.54919334, 0.0],
+            [0.0, -1.54919334],
+            [0.77459667, 0.77459667],
+            [-0.77459667, 1.54919334],
+        ]
+    )
+    y_train = np.array(
+        [
+            [-2.14882017e-10],
+            [-4.66975823e00],
+            [4.01823986e00],
+            [-1.30303674e00],
+            [-1.35760156e00],
+            [3.31215668e00],
+        ]
+    )
+    gpr.fit(X_train, y_train)
+    X_test = np.array(
+        [
+            [-1.93649167, -1.93649167],
+            [1.93649167, -1.93649167],
+            [-1.93649167, 1.93649167],
+            [1.93649167, 1.93649167],
+        ]
+    )
+    pred1, std = gpr.predict(X_test, return_std=True)
+    pred2, cov = gpr.predict(X_test, return_cov=True)
+    assert_allclose(std, np.sqrt(np.diagonal(cov)), rtol=1e-5)
+
+
+@pytest.mark.parametrize(
+    "params, TypeError, err_msg",
+    [
+        (
+            {"alpha": np.zeros(100)},
+            ValueError,
+            "alpha must be a scalar or an array with same number of entries as y",
+        ),
+        (
+            {
+                "kernel": WhiteKernel(noise_level_bounds=(-np.inf, np.inf)),
+                "n_restarts_optimizer": 2,
+            },
+            ValueError,
+            "requires that all bounds are finite",
+        ),
+    ],
+)
+def test_gpr_fit_error(params, TypeError, err_msg):
+    """Check that expected error are raised during fit."""
+    gpr = GaussianProcessRegressor(**params)
+    with pytest.raises(TypeError, match=err_msg):
+        gpr.fit(X, y)
+
+
+def test_gpr_lml_error():
+    """Check that we raise the proper error in the LML method."""
+    gpr = GaussianProcessRegressor(kernel=RBF()).fit(X, y)
+
+    err_msg = "Gradient can only be evaluated for theta!=None"
+    with pytest.raises(ValueError, match=err_msg):
+        gpr.log_marginal_likelihood(eval_gradient=True)
+
+
+def test_gpr_predict_error():
+    """Check that we raise the proper error during predict."""
+    gpr = GaussianProcessRegressor(kernel=RBF()).fit(X, y)
+
+    err_msg = "At most one of return_std or return_cov can be requested."
+    with pytest.raises(RuntimeError, match=err_msg):
+        gpr.predict(X, return_cov=True, return_std=True)
+
+
+@pytest.mark.parametrize("normalize_y", [True, False])
+@pytest.mark.parametrize("n_targets", [None, 1, 10])
+def test_predict_shapes(normalize_y, n_targets):
+    """Check the shapes of y_mean, y_std, and y_cov in single-output
+    (n_targets=None) and multi-output settings, including the edge case when
+    n_targets=1, where the sklearn convention is to squeeze the predictions.
+
+    Non-regression test for:
+    https://github.com/scikit-learn/scikit-learn/issues/17394
+    https://github.com/scikit-learn/scikit-learn/issues/18065
+    https://github.com/scikit-learn/scikit-learn/issues/22174
+    """
+    rng = np.random.RandomState(1234)
+
+    n_features, n_samples_train, n_samples_test = 6, 9, 7
+
+    y_train_shape = (n_samples_train,)
+    if n_targets is not None:
+        y_train_shape = y_train_shape + (n_targets,)
+
+    # By convention single-output data is squeezed upon prediction
+    y_test_shape = (n_samples_test,)
+    if n_targets is not None and n_targets > 1:
+        y_test_shape = y_test_shape + (n_targets,)
+
+    X_train = rng.randn(n_samples_train, n_features)
+    X_test = rng.randn(n_samples_test, n_features)
+    y_train = rng.randn(*y_train_shape)
+
+    model = GaussianProcessRegressor(normalize_y=normalize_y)
+    model.fit(X_train, y_train)
+
+    y_pred, y_std = model.predict(X_test, return_std=True)
+    _, y_cov = model.predict(X_test, return_cov=True)
+
+    assert y_pred.shape == y_test_shape
+    assert y_std.shape == y_test_shape
+    assert y_cov.shape == (n_samples_test,) + y_test_shape
+
+
+@pytest.mark.parametrize("normalize_y", [True, False])
+@pytest.mark.parametrize("n_targets", [None, 1, 10])
+def test_sample_y_shapes(normalize_y, n_targets):
+    """Check the shapes of y_samples in single-output (n_targets=0) and
+    multi-output settings, including the edge case when n_targets=1, where the
+    sklearn convention is to squeeze the predictions.
+
+    Non-regression test for:
+    https://github.com/scikit-learn/scikit-learn/issues/22175
+    """
+    rng = np.random.RandomState(1234)
+
+    n_features, n_samples_train = 6, 9
+    # Number of spatial locations to predict at
+    n_samples_X_test = 7
+    # Number of sample predictions per test point
+    n_samples_y_test = 5
+
+    y_train_shape = (n_samples_train,)
+    if n_targets is not None:
+        y_train_shape = y_train_shape + (n_targets,)
+
+    # By convention single-output data is squeezed upon prediction
+    if n_targets is not None and n_targets > 1:
+        y_test_shape = (n_samples_X_test, n_targets, n_samples_y_test)
+    else:
+        y_test_shape = (n_samples_X_test, n_samples_y_test)
+
+    X_train = rng.randn(n_samples_train, n_features)
+    X_test = rng.randn(n_samples_X_test, n_features)
+    y_train = rng.randn(*y_train_shape)
+
+    model = GaussianProcessRegressor(normalize_y=normalize_y)
+
+    # FIXME: before fitting, the estimator does not have information regarding
+    # the number of targets and default to 1. This is inconsistent with the shape
+    # provided after `fit`. This assert should be made once the following issue
+    # is fixed:
+    # https://github.com/scikit-learn/scikit-learn/issues/22430
+    # y_samples = model.sample_y(X_test, n_samples=n_samples_y_test)
+    # assert y_samples.shape == y_test_shape
+
+    model.fit(X_train, y_train)
+
+    y_samples = model.sample_y(X_test, n_samples=n_samples_y_test)
+    assert y_samples.shape == y_test_shape
+
+
+@pytest.mark.parametrize("n_targets", [None, 1, 2, 3])
+@pytest.mark.parametrize("n_samples", [1, 5])
+def test_sample_y_shape_with_prior(n_targets, n_samples):
+    """Check the output shape of `sample_y` is consistent before and after `fit`."""
+    rng = np.random.RandomState(1024)
+
+    X = rng.randn(10, 3)
+    y = rng.randn(10, n_targets if n_targets is not None else 1)
+
+    model = GaussianProcessRegressor(n_targets=n_targets)
+    shape_before_fit = model.sample_y(X, n_samples=n_samples).shape
+    model.fit(X, y)
+    shape_after_fit = model.sample_y(X, n_samples=n_samples).shape
+    assert shape_before_fit == shape_after_fit
+
+
+@pytest.mark.parametrize("n_targets", [None, 1, 2, 3])
+def test_predict_shape_with_prior(n_targets):
+    """Check the output shape of `predict` with prior distribution."""
+    rng = np.random.RandomState(1024)
+
+    n_sample = 10
+    X = rng.randn(n_sample, 3)
+    y = rng.randn(n_sample, n_targets if n_targets is not None else 1)
+
+    model = GaussianProcessRegressor(n_targets=n_targets)
+    mean_prior, cov_prior = model.predict(X, return_cov=True)
+    _, std_prior = model.predict(X, return_std=True)
+
+    model.fit(X, y)
+    mean_post, cov_post = model.predict(X, return_cov=True)
+    _, std_post = model.predict(X, return_std=True)
+
+    assert mean_prior.shape == mean_post.shape
+    assert cov_prior.shape == cov_post.shape
+    assert std_prior.shape == std_post.shape
+
+
+def test_n_targets_error():
+    """Check that an error is raised when the number of targets seen at fit is
+    inconsistent with n_targets.
+    """
+    rng = np.random.RandomState(0)
+    X = rng.randn(10, 3)
+    y = rng.randn(10, 2)
+
+    model = GaussianProcessRegressor(n_targets=1)
+    with pytest.raises(ValueError, match="The number of targets seen in `y`"):
+        model.fit(X, y)
+
+
+class CustomKernel(C):
+    """
+    A custom kernel that has a diag method that returns the first column of the
+    input matrix X. This is a helper for the test to check that the input
+    matrix X is not mutated.
+    """
+
+    def diag(self, X):
+        return X[:, 0]
+
+
+def test_gpr_predict_input_not_modified():
+    """
+    Check that the input X is not modified by the predict method of the
+    GaussianProcessRegressor when setting return_std=True.
+
+    Non-regression test for:
+    https://github.com/scikit-learn/scikit-learn/issues/24340
+    """
+    gpr = GaussianProcessRegressor(kernel=CustomKernel()).fit(X, y)
+
+    X2_copy = np.copy(X2)
+    _, _ = gpr.predict(X2, return_std=True)
+
+    assert_allclose(X2, X2_copy)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_gpr_predict_no_cov_no_std_return(kernel):
+    """
+    Check that only y_mean is returned when return_cov=False and
+    return_std=False.
+    """
+    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
+    y_pred = gpr.predict(X, return_cov=False, return_std=False)
+
+    assert_allclose(y_pred, y)

linedance-app/venv/lib/python3.12/site-packages/sklearn/gaussian_process/tests/test_kernels.py

@@ -0,0 +1,403 @@
+"""Testing for kernels for Gaussian processes."""
+
+# Authors: The scikit-learn developers
+# SPDX-License-Identifier: BSD-3-Clause
+
+from inspect import signature
+
+import numpy as np
+import pytest
+
+from sklearn.base import clone
+from sklearn.gaussian_process.kernels import (
+    RBF,
+    CompoundKernel,
+    ConstantKernel,
+    DotProduct,
+    Exponentiation,
+    ExpSineSquared,
+    KernelOperator,
+    Matern,
+    PairwiseKernel,
+    RationalQuadratic,
+    WhiteKernel,
+    _approx_fprime,
+)
+from sklearn.metrics.pairwise import (
+    PAIRWISE_KERNEL_FUNCTIONS,
+    euclidean_distances,
+    pairwise_kernels,
+)
+from sklearn.utils._testing import (
+    assert_allclose,
+    assert_almost_equal,
+    assert_array_almost_equal,
+    assert_array_equal,
+)
+
+X = np.random.RandomState(0).normal(0, 1, (5, 2))
+Y = np.random.RandomState(0).normal(0, 1, (6, 2))
+# Set shared test data as read-only to avoid unintentional in-place
+# modifications that would introduce side-effects between tests.
+X.flags.writeable = False
+Y.flags.writeable = False
+
+kernel_rbf_plus_white = RBF(length_scale=2.0) + WhiteKernel(noise_level=3.0)
+kernels = [
+    RBF(length_scale=2.0),
+    RBF(length_scale_bounds=(0.5, 2.0)),
+    ConstantKernel(constant_value=10.0),
+    2.0 * RBF(length_scale=0.33, length_scale_bounds="fixed"),
+    2.0 * RBF(length_scale=0.5),
+    kernel_rbf_plus_white,
+    2.0 * RBF(length_scale=[0.5, 2.0]),
+    2.0 * Matern(length_scale=0.33, length_scale_bounds="fixed"),
+    2.0 * Matern(length_scale=0.5, nu=0.5),
+    2.0 * Matern(length_scale=1.5, nu=1.5),
+    2.0 * Matern(length_scale=2.5, nu=2.5),
+    2.0 * Matern(length_scale=[0.5, 2.0], nu=0.5),
+    3.0 * Matern(length_scale=[2.0, 0.5], nu=1.5),
+    4.0 * Matern(length_scale=[0.5, 0.5], nu=2.5),
+    RationalQuadratic(length_scale=0.5, alpha=1.5),
+    ExpSineSquared(length_scale=0.5, periodicity=1.5),
+    DotProduct(sigma_0=2.0),
+    DotProduct(sigma_0=2.0) ** 2,
+    RBF(length_scale=[2.0]),
+    Matern(length_scale=[2.0]),
+]
+for metric in PAIRWISE_KERNEL_FUNCTIONS:
+    if metric in ["additive_chi2", "chi2"]:
+        continue
+    kernels.append(PairwiseKernel(gamma=1.0, metric=metric))
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_kernel_gradient(kernel):
+    # Compare analytic and numeric gradient of kernels.
+    kernel = clone(kernel)  # make tests independent of one-another
+    K, K_gradient = kernel(X, eval_gradient=True)
+
+    assert K_gradient.shape[0] == X.shape[0]
+    assert K_gradient.shape[1] == X.shape[0]
+    assert K_gradient.shape[2] == kernel.theta.shape[0]
+
+    def eval_kernel_for_theta(theta):
+        kernel_clone = kernel.clone_with_theta(theta)
+        K = kernel_clone(X, eval_gradient=False)
+        return K
+
+    K_gradient_approx = _approx_fprime(kernel.theta, eval_kernel_for_theta, 1e-10)
+
+    assert_almost_equal(K_gradient, K_gradient_approx, 4)
+
+
+@pytest.mark.parametrize(
+    "kernel",
+    [
+        kernel
+        for kernel in kernels
+        # skip non-basic kernels
+        if not (isinstance(kernel, (KernelOperator, Exponentiation)))
+    ],
+)
+def test_kernel_theta(kernel):
+    # Check that parameter vector theta of kernel is set correctly.
+    kernel = clone(kernel)  # make tests independent of one-another
+    theta = kernel.theta
+    _, K_gradient = kernel(X, eval_gradient=True)
+
+    # Determine kernel parameters that contribute to theta
+    init_sign = signature(kernel.__class__.__init__).parameters.values()
+    args = [p.name for p in init_sign if p.name != "self"]
+    theta_vars = map(
+        lambda s: s[0 : -len("_bounds")], filter(lambda s: s.endswith("_bounds"), args)
+    )
+    assert set(hyperparameter.name for hyperparameter in kernel.hyperparameters) == set(
+        theta_vars
+    )
+
+    # Check that values returned in theta are consistent with
+    # hyperparameter values (being their logarithms)
+    for i, hyperparameter in enumerate(kernel.hyperparameters):
+        assert theta[i] == np.log(getattr(kernel, hyperparameter.name))
+
+    # Fixed kernel parameters must be excluded from theta and gradient.
+    for i, hyperparameter in enumerate(kernel.hyperparameters):
+        # create copy with certain hyperparameter fixed
+        params = kernel.get_params()
+        params[hyperparameter.name + "_bounds"] = "fixed"
+        kernel_class = kernel.__class__
+        new_kernel = kernel_class(**params)
+        # Check that theta and K_gradient are identical with the fixed
+        # dimension left out
+        _, K_gradient_new = new_kernel(X, eval_gradient=True)
+        assert theta.shape[0] == new_kernel.theta.shape[0] + 1
+        assert K_gradient.shape[2] == K_gradient_new.shape[2] + 1
+        if i > 0:
+            assert theta[:i] == new_kernel.theta[:i]
+            assert_array_equal(K_gradient[..., :i], K_gradient_new[..., :i])
+        if i + 1 < len(kernel.hyperparameters):
+            assert theta[i + 1 :] == new_kernel.theta[i:]
+            assert_array_equal(K_gradient[..., i + 1 :], K_gradient_new[..., i:])
+
+    # Check that values of theta are modified correctly
+    for i, hyperparameter in enumerate(kernel.hyperparameters):
+        theta[i] = np.log(42)
+        kernel.theta = theta
+        assert_almost_equal(getattr(kernel, hyperparameter.name), 42)
+
+        setattr(kernel, hyperparameter.name, 43)
+        assert_almost_equal(kernel.theta[i], np.log(43))
+
+
+@pytest.mark.parametrize(
+    "kernel",
+    [
+        kernel
+        for kernel in kernels
+        # Identity is not satisfied on diagonal
+        if kernel != kernel_rbf_plus_white
+    ],
+)
+def test_auto_vs_cross(kernel):
+    kernel = clone(kernel)  # make tests independent of one-another
+    # Auto-correlation and cross-correlation should be consistent.
+    K_auto = kernel(X)
+    K_cross = kernel(X, X)
+    assert_almost_equal(K_auto, K_cross, 5)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_kernel_diag(kernel):
+    kernel = clone(kernel)  # make tests independent of one-another
+    # Test that diag method of kernel returns consistent results.
+    K_call_diag = np.diag(kernel(X))
+    K_diag = kernel.diag(X)
+    assert_almost_equal(K_call_diag, K_diag, 5)
+
+
+def test_kernel_operator_commutative():
+    # Adding kernels and multiplying kernels should be commutative.
+    # Check addition
+    assert_almost_equal((RBF(2.0) + 1.0)(X), (1.0 + RBF(2.0))(X))
+
+    # Check multiplication
+    assert_almost_equal((3.0 * RBF(2.0))(X), (RBF(2.0) * 3.0)(X))
+
+
+def test_kernel_anisotropic():
+    # Anisotropic kernel should be consistent with isotropic kernels.
+    kernel = 3.0 * RBF([0.5, 2.0])
+
+    K = kernel(X)
+    X1 = X.copy()
+    X1[:, 0] *= 4
+    K1 = 3.0 * RBF(2.0)(X1)
+    assert_almost_equal(K, K1)
+
+    X2 = X.copy()
+    X2[:, 1] /= 4
+    K2 = 3.0 * RBF(0.5)(X2)
+    assert_almost_equal(K, K2)
+
+    # Check getting and setting via theta
+    kernel.theta = kernel.theta + np.log(2)
+    assert_array_equal(kernel.theta, np.log([6.0, 1.0, 4.0]))
+    assert_array_equal(kernel.k2.length_scale, [1.0, 4.0])
+
+
+@pytest.mark.parametrize(
+    "kernel", [kernel for kernel in kernels if kernel.is_stationary()]
+)
+def test_kernel_stationary(kernel):
+    kernel = clone(kernel)  # make tests independent of one-another
+    # Test stationarity of kernels.
+    K = kernel(X, X + 1)
+    assert_almost_equal(K[0, 0], np.diag(K))
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_kernel_input_type(kernel):
+    kernel = clone(kernel)  # make tests independent of one-another
+    # Test whether kernels is for vectors or structured data
+    if isinstance(kernel, Exponentiation):
+        assert kernel.requires_vector_input == kernel.kernel.requires_vector_input
+    if isinstance(kernel, KernelOperator):
+        assert kernel.requires_vector_input == (
+            kernel.k1.requires_vector_input or kernel.k2.requires_vector_input
+        )
+
+
+def test_compound_kernel_input_type():
+    kernel = CompoundKernel([WhiteKernel(noise_level=3.0)])
+    assert not kernel.requires_vector_input
+
+    kernel = CompoundKernel([WhiteKernel(noise_level=3.0), RBF(length_scale=2.0)])
+    assert kernel.requires_vector_input
+
+
+def check_hyperparameters_equal(kernel1, kernel2):
+    # Check that hyperparameters of two kernels are equal
+    for attr in set(dir(kernel1) + dir(kernel2)):
+        if attr.startswith("hyperparameter_"):
+            attr_value1 = getattr(kernel1, attr)
+            attr_value2 = getattr(kernel2, attr)
+            assert attr_value1 == attr_value2
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_kernel_clone(kernel):
+    kernel = clone(kernel)  # make tests independent of one-another
+    # Test that sklearn's clone works correctly on kernels.
+    kernel_cloned = clone(kernel)
+
+    # XXX: Should this be fixed?
+    # This differs from the sklearn's estimators equality check.
+    assert kernel == kernel_cloned
+    assert id(kernel) != id(kernel_cloned)
+
+    # Check that all constructor parameters are equal.
+    assert kernel.get_params() == kernel_cloned.get_params()
+
+    # Check that all hyperparameters are equal.
+    check_hyperparameters_equal(kernel, kernel_cloned)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_kernel_clone_after_set_params(kernel):
+    kernel = clone(kernel)  # make tests independent of one-another
+    # This test is to verify that using set_params does not
+    # break clone on kernels.
+    # This used to break because in kernels such as the RBF, non-trivial
+    # logic that modified the length scale used to be in the constructor
+    # See https://github.com/scikit-learn/scikit-learn/issues/6961
+    # for more details.
+    bounds = (1e-5, 1e5)
+    kernel_cloned = clone(kernel)
+    params = kernel.get_params()
+    # RationalQuadratic kernel is isotropic.
+    isotropic_kernels = (ExpSineSquared, RationalQuadratic)
+    if "length_scale" in params and not isinstance(kernel, isotropic_kernels):
+        length_scale = params["length_scale"]
+        if np.iterable(length_scale):
+            # XXX unreached code as of v0.22
+            params["length_scale"] = length_scale[0]
+            params["length_scale_bounds"] = bounds
+        else:
+            params["length_scale"] = [length_scale] * 2
+            params["length_scale_bounds"] = bounds * 2
+        kernel_cloned.set_params(**params)
+        kernel_cloned_clone = clone(kernel_cloned)
+        assert kernel_cloned_clone.get_params() == kernel_cloned.get_params()
+        assert id(kernel_cloned_clone) != id(kernel_cloned)
+        check_hyperparameters_equal(kernel_cloned, kernel_cloned_clone)
+
+
+def test_matern_kernel():
+    # Test consistency of Matern kernel for special values of nu.
+    K = Matern(nu=1.5, length_scale=1.0)(X)
+    # the diagonal elements of a matern kernel are 1
+    assert_array_almost_equal(np.diag(K), np.ones(X.shape[0]))
+    # matern kernel for coef0==0.5 is equal to absolute exponential kernel
+    K_absexp = np.exp(-euclidean_distances(X, X, squared=False))
+    K = Matern(nu=0.5, length_scale=1.0)(X)
+    assert_array_almost_equal(K, K_absexp)
+    # matern kernel with coef0==inf is equal to RBF kernel
+    K_rbf = RBF(length_scale=1.0)(X)
+    K = Matern(nu=np.inf, length_scale=1.0)(X)
+    assert_array_almost_equal(K, K_rbf)
+    assert_allclose(K, K_rbf)
+    # test that special cases of matern kernel (coef0 in [0.5, 1.5, 2.5])
+    # result in nearly identical results as the general case for coef0 in
+    # [0.5 + tiny, 1.5 + tiny, 2.5 + tiny]
+    tiny = 1e-10
+    for nu in [0.5, 1.5, 2.5]:
+        K1 = Matern(nu=nu, length_scale=1.0)(X)
+        K2 = Matern(nu=nu + tiny, length_scale=1.0)(X)
+        assert_array_almost_equal(K1, K2)
+    # test that coef0==large is close to RBF
+    large = 100
+    K1 = Matern(nu=large, length_scale=1.0)(X)
+    K2 = RBF(length_scale=1.0)(X)
+    assert_array_almost_equal(K1, K2, decimal=2)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_kernel_versus_pairwise(kernel):
+    kernel = clone(kernel)  # make tests independent of one-another
+    # Check that GP kernels can also be used as pairwise kernels.
+
+    # Test auto-kernel
+    if kernel != kernel_rbf_plus_white:
+        # For WhiteKernel: k(X) != k(X,X). This is assumed by
+        # pairwise_kernels
+        K1 = kernel(X)
+        K2 = pairwise_kernels(X, metric=kernel)
+        assert_array_almost_equal(K1, K2)
+
+    # Test cross-kernel
+    K1 = kernel(X, Y)
+    K2 = pairwise_kernels(X, Y, metric=kernel)
+    assert_array_almost_equal(K1, K2)
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_set_get_params(kernel):
+    kernel = clone(kernel)  # make tests independent of one-another
+    # Check that set_params()/get_params() is consistent with kernel.theta.
+
+    # Test get_params()
+    index = 0
+    params = kernel.get_params()
+    for hyperparameter in kernel.hyperparameters:
+        if isinstance("string", type(hyperparameter.bounds)):
+            if hyperparameter.bounds == "fixed":
+                continue
+        size = hyperparameter.n_elements
+        if size > 1:  # anisotropic kernels
+            assert_almost_equal(
+                np.exp(kernel.theta[index : index + size]), params[hyperparameter.name]
+            )
+            index += size
+        else:
+            assert_almost_equal(
+                np.exp(kernel.theta[index]), params[hyperparameter.name]
+            )
+            index += 1
+    # Test set_params()
+    index = 0
+    value = 10  # arbitrary value
+    for hyperparameter in kernel.hyperparameters:
+        if isinstance("string", type(hyperparameter.bounds)):
+            if hyperparameter.bounds == "fixed":
+                continue
+        size = hyperparameter.n_elements
+        if size > 1:  # anisotropic kernels
+            kernel.set_params(**{hyperparameter.name: [value] * size})
+            assert_almost_equal(
+                np.exp(kernel.theta[index : index + size]), [value] * size
+            )
+            index += size
+        else:
+            kernel.set_params(**{hyperparameter.name: value})
+            assert_almost_equal(np.exp(kernel.theta[index]), value)
+            index += 1
+
+
+@pytest.mark.parametrize("kernel", kernels)
+def test_repr_kernels(kernel):
+    kernel = clone(kernel)  # make tests independent of one-another
+    # Smoke-test for repr in kernels.
+
+    repr(kernel)
+
+
+def test_rational_quadratic_kernel():
+    kernel = RationalQuadratic(length_scale=[1.0, 1.0])
+    message = (
+        "RationalQuadratic kernel only supports isotropic "
+        "version, please use a single "
+        "scalar for length_scale"
+    )
+    with pytest.raises(AttributeError, match=message):
+        kernel(X)