"""Testing for Gaussian process regression """

# Author: Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>
# License: BSD 3 clause

import numpy as np

from scipy.optimize import approx_fprime

import pytest

from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels \
    import RBF, ConstantKernel as C, WhiteKernel
from sklearn.gaussian_process.kernels import DotProduct

from sklearn.utils.testing \
    import (assert_true, assert_greater, assert_array_less,
            assert_almost_equal, assert_equal, assert_raise_message,
            assert_array_almost_equal, assert_array_equal)


def f(x):
    return x * np.sin(x)


X = np.atleast_2d([1., 3., 5., 6., 7., 8.]).T
X2 = np.atleast_2d([2., 4., 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):
    # 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.)


@pytest.mark.parametrize('kernel', non_fixed_kernels)
def test_lml_improving(kernel):
    # Test that hyperparameter-tuning improves log-marginal likelihood.
    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
    assert_greater(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_equal(gpr.log_marginal_likelihood(gpr.kernel_.theta),
                 gpr.log_marginal_likelihood())


@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_true(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_equal(np.exp(gpr.kernel_.theta), 1.0)


@pytest.mark.parametrize('kernel', kernels)
def test_predict_cov_vs_std(kernel):
    # 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_greater(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, 1e+2)] * 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_greater(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
    y_mean = y.mean(0)
    y_norm = y - y_mean

    # 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_mean + y_pred
    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_norm = gpr_norm.predict(X2, return_cov=True)
    assert_almost_equal(y_cov, y_cov_norm)


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
    assert_almost_equal(y_std_1d, y_std_2d)
    assert_almost_equal(y_cov_1d, y_cov_2d)

    y_sample_1d = gpr.sample_y(X2, n_samples=10)
    y_sample_2d = gpr_2d.sample_y(X2, n_samples=10)
    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_greater(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)
    assert_raise_message(np.linalg.LinAlgError,
                         "The kernel, %s, is not returning a "
                         "positive definite matrix. Try gradually increasing "
                         "the 'alpha' parameter of your "
                         "GaussianProcessRegressor estimator."
                         % kernel, 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)


@pytest.mark.parametrize('kernel', kernels)
def test_K_inv_reset(kernel):
    y2 = f(X2).ravel()

    # Test that self._K_inv is reset after a new fit
    gpr = GaussianProcessRegressor(kernel=kernel).fit(X, y)
    assert hasattr(gpr, '_K_inv')
    assert gpr._K_inv is None
    gpr.predict(X, return_std=True)
    assert gpr._K_inv is not None
    gpr.fit(X2, y2)
    assert gpr._K_inv is None
    gpr.predict(X2, return_std=True)
    gpr2 = GaussianProcessRegressor(kernel=kernel).fit(X2, y2)
    gpr2.predict(X2, return_std=True)
    # the value of K_inv should be independent of the first fit
    assert_array_equal(gpr._K_inv, gpr2._K_inv)