print(__doc__)

# Author: Vincent Dubourg <vincent.dubourg@gmail.com>
#         Jake Vanderplas <vanderplas@astro.washington.edu>
#         Jan Hendrik Metzen <jhm@informatik.uni-bremen.de>s
# License: BSD 3 clause

import numpy as np
from matplotlib import pyplot as plt

from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import Matern, DotProduct, RationalQuadratic, ExpSineSquared, RBF, ConstantKernel as C
np.random.seed(1)


def f(x):
    """The function to predict."""
    return x * x*np.sin(x)

# ----------------------------------------------------------------------
#  First the noiseless case
X = np.atleast_2d([-12., 0., 6., 7., 8., 9.5, 20., -20.]).T

# Observations
y = f(X).ravel()

# Mesh the input space for evaluations of the real function, the prediction and
# its MSE
x = np.atleast_2d(np.linspace(-100., 100, 10000)).T

# Instanciate a Gaussian Process model
kernel = Matern(nu=2.5)
print np.std(y)
#kernel = (DotProduct(sigma_0=0.)**2)*(Matern(nu=2.5)) + C(constant_value=np.mean(y))
kernel = DotProduct(sigma_0=10.)*(C(constant_value=np.std(y))**3)*(Matern(nu=2.5)) + C(constant_value=np.mean(y))
gp = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=9)

# Fit to data using Maximum Likelihood Estimation of the parameters
gp.fit(X, y)

# Make the prediction on the meshed x-axis (ask for MSE as well)
y_pred, sigma = gp.predict(x, return_std=True)

# Plot the function, the prediction and the 95% confidence interval based on
# the MSE
fig = plt.figure()
plt.plot(x, f(x), 'r:', label=u'$f(x) = x\,\sin(x)$')
plt.plot(X, y, 'r.', markersize=10, label=u'Observations')
plt.plot(x, y_pred, 'b-', label=u'Prediction')
plt.fill(np.concatenate([x, x[::-1]]),
         np.concatenate([y_pred - 1.9600 * sigma,
                        (y_pred + 1.9600 * sigma)[::-1]]),
         alpha=.5, fc='b', ec='None', label='95% confidence interval')
plt.xlabel('$x$')
plt.ylabel('$f(x)$')
plt.ylim(-1000, 1000)
plt.legend(loc='upper left')
'''
# ----------------------------------------------------------------------
# now the noisy case
X = np.linspace(0.1, 9.9, 20)
X = np.atleast_2d(X).T

# Observations and noise
y = f(X).ravel()
dy = 0.5 + 1.0 * np.random.random(y.shape)
noise = np.random.normal(0, dy)
y += noise

# Instanciate a Gaussian Process model
gp = GaussianProcessRegressor(kernel=kernel, alpha=(dy / y) ** 2,
                              n_restarts_optimizer=10)

# Fit to data using Maximum Likelihood Estimation of the parameters
gp.fit(X, y)

# Make the prediction on the meshed x-axis (ask for MSE as well)
y_pred, sigma = gp.predict(x, return_std=True)

# Plot the function, the prediction and the 95% confidence interval based on
# the MSE
fig = plt.figure()
plt.plot(x, f(x), 'r:', label=u'$f(x) = x\,\sin(x)$')
plt.errorbar(X.ravel(), y, dy, fmt='r.', markersize=10, label=u'Observations')
plt.plot(x, y_pred, 'b-', label=u'Prediction')
plt.fill(np.concatenate([x, x[::-1]]),
         np.concatenate([y_pred - 1.9600 * sigma,
                        (y_pred + 1.9600 * sigma)[::-1]]),
         alpha=.5, fc='b', ec='None', label='95% confidence interval')
plt.xlabel('$x$')
plt.ylabel('$f(x)$')
plt.ylim(-1000, 1000)
plt.legend(loc='upper left')
'''
plt.show()
