469 lines
17 KiB
Python
469 lines
17 KiB
Python
"""Testing for Gaussian process regression """
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# Author: Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
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# Modified by: Pete Green <p.l.green@liverpool.ac.uk>
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# License: BSD 3 clause
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import sys
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import numpy as np
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from scipy.optimize import approx_fprime
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import pytest
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from sklearn.gaussian_process import GaussianProcessRegressor
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from sklearn.gaussian_process.kernels \
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import RBF, ConstantKernel as C, WhiteKernel
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from sklearn.gaussian_process.kernels import DotProduct
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from sklearn.gaussian_process.tests._mini_sequence_kernel import MiniSeqKernel
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from sklearn.utils._testing \
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import (assert_array_less,
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assert_almost_equal, assert_raise_message,
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assert_array_almost_equal, assert_array_equal,
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assert_allclose)
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def f(x):
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return x * np.sin(x)
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X = np.atleast_2d([1., 3., 5., 6., 7., 8.]).T
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X2 = np.atleast_2d([2., 4., 5.5, 6.5, 7.5]).T
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y = f(X).ravel()
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fixed_kernel = RBF(length_scale=1.0, length_scale_bounds="fixed")
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kernels = [RBF(length_scale=1.0), fixed_kernel,
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RBF(length_scale=1.0, length_scale_bounds=(1e-3, 1e3)),
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C(1.0, (1e-2, 1e2)) *
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RBF(length_scale=1.0, length_scale_bounds=(1e-3, 1e3)),
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C(1.0, (1e-2, 1e2)) *
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RBF(length_scale=1.0, length_scale_bounds=(1e-3, 1e3)) +
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C(1e-5, (1e-5, 1e2)),
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C(0.1, (1e-2, 1e2)) *
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RBF(length_scale=1.0, length_scale_bounds=(1e-3, 1e3)) +
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C(1e-5, (1e-5, 1e2))]
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non_fixed_kernels = [kernel for kernel in kernels
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if kernel != fixed_kernel]
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@pytest.mark.parametrize('kernel', kernels)
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def test_gpr_interpolation(kernel):
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if sys.maxsize <= 2 ** 32 and sys.version_info[:2] == (3, 6):
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pytest.xfail("This test may fail on 32bit Py3.6")
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# Test the interpolating property for different kernels.
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gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
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y_pred, y_cov = gpr.predict(X, return_cov=True)
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assert_almost_equal(y_pred, y)
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assert_almost_equal(np.diag(y_cov), 0.)
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def test_gpr_interpolation_structured():
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# Test the interpolating property for different kernels.
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kernel = MiniSeqKernel(baseline_similarity_bounds='fixed')
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X = ['A', 'B', 'C']
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y = np.array([1, 2, 3])
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gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
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y_pred, y_cov = gpr.predict(X, return_cov=True)
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assert_almost_equal(kernel(X, eval_gradient=True)[1].ravel(),
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(1 - np.eye(len(X))).ravel())
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assert_almost_equal(y_pred, y)
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assert_almost_equal(np.diag(y_cov), 0.)
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@pytest.mark.parametrize('kernel', non_fixed_kernels)
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def test_lml_improving(kernel):
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if sys.maxsize <= 2 ** 32 and sys.version_info[:2] == (3, 6):
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pytest.xfail("This test may fail on 32bit Py3.6")
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# Test that hyperparameter-tuning improves log-marginal likelihood.
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gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
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assert (gpr.log_marginal_likelihood(gpr.kernel_.theta) >
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gpr.log_marginal_likelihood(kernel.theta))
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@pytest.mark.parametrize('kernel', kernels)
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def test_lml_precomputed(kernel):
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# Test that lml of optimized kernel is stored correctly.
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gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
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assert (gpr.log_marginal_likelihood(gpr.kernel_.theta) ==
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gpr.log_marginal_likelihood())
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@pytest.mark.parametrize('kernel', kernels)
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def test_lml_without_cloning_kernel(kernel):
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# Test that lml of optimized kernel is stored correctly.
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gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
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input_theta = np.ones(gpr.kernel_.theta.shape, dtype=np.float64)
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gpr.log_marginal_likelihood(input_theta, clone_kernel=False)
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assert_almost_equal(gpr.kernel_.theta, input_theta, 7)
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@pytest.mark.parametrize('kernel', non_fixed_kernels)
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def test_converged_to_local_maximum(kernel):
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# Test that we are in local maximum after hyperparameter-optimization.
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gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
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lml, lml_gradient = \
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gpr.log_marginal_likelihood(gpr.kernel_.theta, True)
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assert np.all((np.abs(lml_gradient) < 1e-4) |
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(gpr.kernel_.theta == gpr.kernel_.bounds[:, 0]) |
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(gpr.kernel_.theta == gpr.kernel_.bounds[:, 1]))
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@pytest.mark.parametrize('kernel', non_fixed_kernels)
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def test_solution_inside_bounds(kernel):
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# Test that hyperparameter-optimization remains in bounds#
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gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
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bounds = gpr.kernel_.bounds
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max_ = np.finfo(gpr.kernel_.theta.dtype).max
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tiny = 1e-10
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bounds[~np.isfinite(bounds[:, 1]), 1] = max_
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assert_array_less(bounds[:, 0], gpr.kernel_.theta + tiny)
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assert_array_less(gpr.kernel_.theta, bounds[:, 1] + tiny)
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@pytest.mark.parametrize('kernel', kernels)
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def test_lml_gradient(kernel):
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# Compare analytic and numeric gradient of log marginal likelihood.
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gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
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lml, lml_gradient = gpr.log_marginal_likelihood(kernel.theta, True)
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lml_gradient_approx = \
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approx_fprime(kernel.theta,
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lambda theta: gpr.log_marginal_likelihood(theta,
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False),
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1e-10)
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assert_almost_equal(lml_gradient, lml_gradient_approx, 3)
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@pytest.mark.parametrize('kernel', kernels)
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def test_prior(kernel):
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# Test that GP prior has mean 0 and identical variances.
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gpr = GaussianProcessRegressor(kernel=kernel)
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y_mean, y_cov = gpr.predict(X, return_cov=True)
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assert_almost_equal(y_mean, 0, 5)
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if len(gpr.kernel.theta) > 1:
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# XXX: quite hacky, works only for current kernels
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assert_almost_equal(np.diag(y_cov), np.exp(kernel.theta[0]), 5)
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else:
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assert_almost_equal(np.diag(y_cov), 1, 5)
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@pytest.mark.parametrize('kernel', kernels)
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def test_sample_statistics(kernel):
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# Test that statistics of samples drawn from GP are correct.
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gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
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y_mean, y_cov = gpr.predict(X2, return_cov=True)
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samples = gpr.sample_y(X2, 300000)
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# More digits accuracy would require many more samples
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assert_almost_equal(y_mean, np.mean(samples, 1), 1)
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assert_almost_equal(np.diag(y_cov) / np.diag(y_cov).max(),
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np.var(samples, 1) / np.diag(y_cov).max(), 1)
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def test_no_optimizer():
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# Test that kernel parameters are unmodified when optimizer is None.
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kernel = RBF(1.0)
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gpr = GaussianProcessRegressor(kernel=kernel, optimizer=None).fit(X, y)
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assert np.exp(gpr.kernel_.theta) == 1.0
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@pytest.mark.parametrize('kernel', kernels)
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def test_predict_cov_vs_std(kernel):
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if sys.maxsize <= 2 ** 32 and sys.version_info[:2] == (3, 6):
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pytest.xfail("This test may fail on 32bit Py3.6")
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# Test that predicted std.-dev. is consistent with cov's diagonal.
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gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
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y_mean, y_cov = gpr.predict(X2, return_cov=True)
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y_mean, y_std = gpr.predict(X2, return_std=True)
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assert_almost_equal(np.sqrt(np.diag(y_cov)), y_std)
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def test_anisotropic_kernel():
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# Test that GPR can identify meaningful anisotropic length-scales.
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# We learn a function which varies in one dimension ten-times slower
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# than in the other. The corresponding length-scales should differ by at
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# least a factor 5
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rng = np.random.RandomState(0)
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X = rng.uniform(-1, 1, (50, 2))
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y = X[:, 0] + 0.1 * X[:, 1]
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kernel = RBF([1.0, 1.0])
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gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
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assert (np.exp(gpr.kernel_.theta[1]) >
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np.exp(gpr.kernel_.theta[0]) * 5)
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def test_random_starts():
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# Test that an increasing number of random-starts of GP fitting only
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# increases the log marginal likelihood of the chosen theta.
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n_samples, n_features = 25, 2
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rng = np.random.RandomState(0)
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X = rng.randn(n_samples, n_features) * 2 - 1
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y = np.sin(X).sum(axis=1) + np.sin(3 * X).sum(axis=1) \
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+ rng.normal(scale=0.1, size=n_samples)
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kernel = C(1.0, (1e-2, 1e2)) \
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* RBF(length_scale=[1.0] * n_features,
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length_scale_bounds=[(1e-4, 1e+2)] * n_features) \
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+ WhiteKernel(noise_level=1e-5, noise_level_bounds=(1e-5, 1e1))
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last_lml = -np.inf
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for n_restarts_optimizer in range(5):
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gp = GaussianProcessRegressor(
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kernel=kernel, n_restarts_optimizer=n_restarts_optimizer,
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random_state=0,).fit(X, y)
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lml = gp.log_marginal_likelihood(gp.kernel_.theta)
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assert lml > last_lml - np.finfo(np.float32).eps
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last_lml = lml
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@pytest.mark.parametrize('kernel', kernels)
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def test_y_normalization(kernel):
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"""
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Test normalization of the target values in GP
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Fitting non-normalizing GP on normalized y and fitting normalizing GP
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on unnormalized y should yield identical results. Note that, here,
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'normalized y' refers to y that has been made zero mean and unit
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variance.
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"""
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y_mean = np.mean(y)
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y_std = np.std(y)
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y_norm = (y - y_mean) / y_std
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# Fit non-normalizing GP on normalized y
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gpr = GaussianProcessRegressor(kernel=kernel)
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gpr.fit(X, y_norm)
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# Fit normalizing GP on unnormalized y
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gpr_norm = GaussianProcessRegressor(kernel=kernel, normalize_y=True)
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gpr_norm.fit(X, y)
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# Compare predicted mean, std-devs and covariances
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y_pred, y_pred_std = gpr.predict(X2, return_std=True)
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y_pred = y_pred * y_std + y_mean
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y_pred_std = y_pred_std * y_std
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y_pred_norm, y_pred_std_norm = gpr_norm.predict(X2, return_std=True)
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assert_almost_equal(y_pred, y_pred_norm)
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assert_almost_equal(y_pred_std, y_pred_std_norm)
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_, y_cov = gpr.predict(X2, return_cov=True)
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y_cov = y_cov * y_std**2
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_, y_cov_norm = gpr_norm.predict(X2, return_cov=True)
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assert_almost_equal(y_cov, y_cov_norm)
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def test_large_variance_y():
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"""
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Here we test that, when noramlize_y=True, our GP can produce a
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sensible fit to training data whose variance is significantly
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larger than unity. This test was made in response to issue #15612.
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GP predictions are verified against predictions that were made
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using GPy which, here, is treated as the 'gold standard'. Note that we
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only investigate the RBF kernel here, as that is what was used in the
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GPy implementation.
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The following code can be used to recreate the GPy data:
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--------------------------------------------------------------------------
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import GPy
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kernel_gpy = GPy.kern.RBF(input_dim=1, lengthscale=1.)
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gpy = GPy.models.GPRegression(X, np.vstack(y_large), kernel_gpy)
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gpy.optimize()
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y_pred_gpy, y_var_gpy = gpy.predict(X2)
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y_pred_std_gpy = np.sqrt(y_var_gpy)
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--------------------------------------------------------------------------
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"""
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# Here we utilise a larger variance version of the training data
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y_large = 10 * y
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# Standard GP with normalize_y=True
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RBF_params = {'length_scale': 1.0}
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kernel = RBF(**RBF_params)
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gpr = GaussianProcessRegressor(kernel=kernel, normalize_y=True)
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gpr.fit(X, y_large)
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y_pred, y_pred_std = gpr.predict(X2, return_std=True)
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# 'Gold standard' mean predictions from GPy
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y_pred_gpy = np.array([15.16918303,
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-27.98707845,
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-39.31636019,
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14.52605515,
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69.18503589])
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# 'Gold standard' std predictions from GPy
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y_pred_std_gpy = np.array([7.78860962,
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3.83179178,
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0.63149951,
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0.52745188,
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0.86170042])
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# Based on numerical experiments, it's reasonable to expect our
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# GP's mean predictions to get within 7% of predictions of those
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# made by GPy.
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assert_allclose(y_pred, y_pred_gpy, rtol=0.07, atol=0)
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# Based on numerical experiments, it's reasonable to expect our
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# GP's std predictions to get within 15% of predictions of those
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# made by GPy.
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assert_allclose(y_pred_std, y_pred_std_gpy, rtol=0.15, atol=0)
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def test_y_multioutput():
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# Test that GPR can deal with multi-dimensional target values
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y_2d = np.vstack((y, y * 2)).T
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# Test for fixed kernel that first dimension of 2d GP equals the output
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# of 1d GP and that second dimension is twice as large
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kernel = RBF(length_scale=1.0)
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gpr = GaussianProcessRegressor(kernel=kernel, optimizer=None,
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normalize_y=False)
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gpr.fit(X, y)
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gpr_2d = GaussianProcessRegressor(kernel=kernel, optimizer=None,
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normalize_y=False)
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gpr_2d.fit(X, y_2d)
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y_pred_1d, y_std_1d = gpr.predict(X2, return_std=True)
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y_pred_2d, y_std_2d = gpr_2d.predict(X2, return_std=True)
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_, y_cov_1d = gpr.predict(X2, return_cov=True)
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_, y_cov_2d = gpr_2d.predict(X2, return_cov=True)
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assert_almost_equal(y_pred_1d, y_pred_2d[:, 0])
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assert_almost_equal(y_pred_1d, y_pred_2d[:, 1] / 2)
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# Standard deviation and covariance do not depend on output
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assert_almost_equal(y_std_1d, y_std_2d)
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assert_almost_equal(y_cov_1d, y_cov_2d)
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y_sample_1d = gpr.sample_y(X2, n_samples=10)
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y_sample_2d = gpr_2d.sample_y(X2, n_samples=10)
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assert_almost_equal(y_sample_1d, y_sample_2d[:, 0])
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# Test hyperparameter optimization
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for kernel in kernels:
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gpr = GaussianProcessRegressor(kernel=kernel, normalize_y=True)
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gpr.fit(X, y)
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gpr_2d = GaussianProcessRegressor(kernel=kernel, normalize_y=True)
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gpr_2d.fit(X, np.vstack((y, y)).T)
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assert_almost_equal(gpr.kernel_.theta, gpr_2d.kernel_.theta, 4)
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@pytest.mark.parametrize('kernel', non_fixed_kernels)
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def test_custom_optimizer(kernel):
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# Test that GPR can use externally defined optimizers.
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# Define a dummy optimizer that simply tests 50 random hyperparameters
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def optimizer(obj_func, initial_theta, bounds):
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rng = np.random.RandomState(0)
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theta_opt, func_min = \
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initial_theta, obj_func(initial_theta, eval_gradient=False)
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for _ in range(50):
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theta = np.atleast_1d(rng.uniform(np.maximum(-2, bounds[:, 0]),
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np.minimum(1, bounds[:, 1])))
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f = obj_func(theta, eval_gradient=False)
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if f < func_min:
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theta_opt, func_min = theta, f
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return theta_opt, func_min
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gpr = GaussianProcessRegressor(kernel=kernel, optimizer=optimizer)
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gpr.fit(X, y)
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# Checks that optimizer improved marginal likelihood
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assert (gpr.log_marginal_likelihood(gpr.kernel_.theta) >
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gpr.log_marginal_likelihood(gpr.kernel.theta))
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def test_gpr_correct_error_message():
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X = np.arange(12).reshape(6, -1)
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y = np.ones(6)
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kernel = DotProduct()
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gpr = GaussianProcessRegressor(kernel=kernel, alpha=0.0)
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assert_raise_message(np.linalg.LinAlgError,
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"The kernel, %s, is not returning a "
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"positive definite matrix. Try gradually increasing "
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"the 'alpha' parameter of your "
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"GaussianProcessRegressor estimator."
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% kernel, gpr.fit, X, y)
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@pytest.mark.parametrize('kernel', kernels)
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def test_duplicate_input(kernel):
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# Test GPR can handle two different output-values for the same input.
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gpr_equal_inputs = GaussianProcessRegressor(kernel=kernel, alpha=1e-2)
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gpr_similar_inputs = GaussianProcessRegressor(kernel=kernel, alpha=1e-2)
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X_ = np.vstack((X, X[0]))
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y_ = np.hstack((y, y[0] + 1))
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gpr_equal_inputs.fit(X_, y_)
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X_ = np.vstack((X, X[0] + 1e-15))
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y_ = np.hstack((y, y[0] + 1))
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gpr_similar_inputs.fit(X_, y_)
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X_test = np.linspace(0, 10, 100)[:, None]
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y_pred_equal, y_std_equal = \
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gpr_equal_inputs.predict(X_test, return_std=True)
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y_pred_similar, y_std_similar = \
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gpr_similar_inputs.predict(X_test, return_std=True)
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assert_almost_equal(y_pred_equal, y_pred_similar)
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assert_almost_equal(y_std_equal, y_std_similar)
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def test_no_fit_default_predict():
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# Test that GPR predictions without fit does not break by default.
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default_kernel = (C(1.0, constant_value_bounds="fixed") *
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RBF(1.0, length_scale_bounds="fixed"))
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gpr1 = GaussianProcessRegressor()
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_, y_std1 = gpr1.predict(X, return_std=True)
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_, y_cov1 = gpr1.predict(X, return_cov=True)
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gpr2 = GaussianProcessRegressor(kernel=default_kernel)
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_, y_std2 = gpr2.predict(X, return_std=True)
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_, y_cov2 = gpr2.predict(X, return_cov=True)
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assert_array_almost_equal(y_std1, y_std2)
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assert_array_almost_equal(y_cov1, y_cov2)
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@pytest.mark.parametrize('kernel', kernels)
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def test_K_inv_reset(kernel):
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y2 = f(X2).ravel()
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# Test that self._K_inv is reset after a new fit
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gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
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assert hasattr(gpr, '_K_inv')
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assert gpr._K_inv is None
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gpr.predict(X, return_std=True)
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assert gpr._K_inv is not None
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gpr.fit(X2, y2)
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assert gpr._K_inv is None
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gpr.predict(X2, return_std=True)
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gpr2 = GaussianProcessRegressor(kernel=kernel).fit(X2, y2)
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gpr2.predict(X2, return_std=True)
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# the value of K_inv should be independent of the first fit
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assert_array_equal(gpr._K_inv, gpr2._K_inv)
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