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)
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)
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]))
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_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 lml > last_lml - np.finfo(np.float32).eps
last_lml = lml
# medium term irregularity
k3 = 0.66**2 \
* RationalQuadratic(length_scale=1.2, alpha=0.78)
k4 = 0.18**2 * RBF(length_scale=0.134) \
+ WhiteKernel(noise_level=0.19**2) # noise terms
kernel_gpml = k1 + k2 + k3 + k4
gp = GaussianProcessRegressor(kernel=kernel_gpml,
alpha=0,
optimizer=None,
normalize_y=True)
gp.fit(X, y)
print("GPML kernel: %s" % gp.kernel_)
print("Log-marginal-likelihood: %.3f" %
gp.log_marginal_likelihood(gp.kernel_.theta))
# Kernel with optimized parameters
k1 = 50.0**2 * RBF(length_scale=50.0) # long term smooth rising trend
k2 = 2.0**2 * RBF(length_scale=100.0) \
* ExpSineSquared(length_scale=1.0, periodicity=1.0,
periodicity_bounds="fixed") # seasonal component
# medium term irregularities
k3 = 0.5**2 * RationalQuadratic(length_scale=1.0, alpha=1.0)
k4 = 0.1**2 * RBF(length_scale=0.1) \
+ WhiteKernel(noise_level=0.1**2,
noise_level_bounds=(1e-3, np.inf)) # noise terms
kernel = k1 + k2 + k3 + k4
gp = GaussianProcessRegressor(kernel=kernel, alpha=0, normalize_y=True)
gp.fit(X, y)
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) == gpr.log_marginal_likelihood())
def test_lml_improving(kernel):
# 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))
plt.figure()
kernel = 1.0 * RBF(length_scale=100.0, length_scale_bounds=(1e-2, 1e3)) \
+ WhiteKernel(noise_level=1, noise_level_bounds=(1e-10, 1e+1))
gp = GaussianProcessRegressor(kernel=kernel, alpha=0.0).fit(X, y)
X_ = np.linspace(0, 5, 100)
y_mean, y_cov = gp.predict(X_[:, np.newaxis], return_cov=True)
plt.plot(X_, y_mean, 'k', lw=3, zorder=9)
plt.fill_between(X_,
y_mean - np.sqrt(np.diag(y_cov)),
y_mean + np.sqrt(np.diag(y_cov)),
alpha=0.5,
color='k')
plt.plot(X_, 0.5 * np.sin(3 * X_), 'r', lw=3, zorder=9)
plt.scatter(X[:, 0], y, c='r', s=50, zorder=10, edgecolors=(0, 0, 0))
plt.title("Initial: %s\nOptimum: %s\nLog-Marginal-Likelihood: %s" %
(kernel, gp.kernel_, gp.log_marginal_likelihood(gp.kernel_.theta)))
plt.tight_layout()
# Second run
plt.figure()
kernel = 1.0 * RBF(length_scale=1.0, length_scale_bounds=(1e-2, 1e3)) \
+ WhiteKernel(noise_level=1e-5, noise_level_bounds=(1e-10, 1e+1))
gp = GaussianProcessRegressor(kernel=kernel, alpha=0.0).fit(X, y)
X_ = np.linspace(0, 5, 100)
y_mean, y_cov = gp.predict(X_[:, np.newaxis], return_cov=True)
plt.plot(X_, y_mean, 'k', lw=3, zorder=9)
plt.fill_between(X_,
y_mean - np.sqrt(np.diag(y_cov)),
y_mean + np.sqrt(np.diag(y_cov)),
alpha=0.5,
color='k')