"""The function to predict (classification will then consist in predicting
whether g(x) <= 0 or not)"""
return 5. - x[:, 1] - .5 * x[:, 0]**2.
# Design of experiments
X = np.array([[-4.61611719, -6.00099547], [4.10469096, 5.32782448],
[0.00000000, -0.50000000], [-6.17289014, -4.6984743],
[1.3109306, -6.93271427], [-5.03823144, 3.10584743],
[-2.87600388, 6.74310541], [5.21301203, 4.26386883]])
# Observations
y = g(X)
# Instanciate and fit Gaussian Process Model
gp = GaussianProcess(theta0=5e-1)
# Don't perform MLE or you'll get a perfect prediction for this simple example!
gp.fit(X, y)
# Evaluate real function, the prediction and its MSE on a grid
res = 50
x1, x2 = np.meshgrid(np.linspace(- lim, lim, res), \
np.linspace(- lim, lim, res))
xx = np.vstack([x1.reshape(x1.size), x2.reshape(x2.size)]).T
y_true = g(xx)
y_pred, MSE = gp.predict(xx, eval_MSE=True)
sigma = np.sqrt(MSE)
y_true = y_true.reshape((res, res))
y_pred = y_pred.reshape((res, res))
import numpy as np
from scikits.learn import datasets
from scikits.learn.gaussian_process import GaussianProcess
from scikits.learn.cross_val import cross_val_score, KFold
from scikits.learn.metrics import r2_score
# Load the dataset from scikits' data sets
diabetes = datasets.load_diabetes()
X, y = diabetes.data, diabetes.target
# Instanciate a GP model
gp = GaussianProcess(regr='constant',
corr='absolute_exponential',
theta0=[1e-4] * 10,
thetaL=[1e-12] * 10,
thetaU=[1e-2] * 10,
nugget=1e-2,
optimizer='Welch')
# Fit the GP model to the data performing maximum likelihood estimation
gp.fit(X, y)
# Deactivate maximum likelihood estimation for the cross-validation loop
gp.theta0 = gp.theta # Given correlation parameter = MLE
gp.thetaL, gp.thetaU = None, None # None bounds deactivate MLE
# Perform a cross-validation estimate of the coefficient of determination using
# the cross_val module using all CPUs available on the machine
K = 20 # folds
R2 = cross_val_score(gp, X, y=y, cv=KFold(y.size, K), n_jobs=-1).mean()
"""The function to predict."""
return x * np.sin(x)
# The design of experiments
X = np.atleast_2d([1., 3., 5., 6., 7., 8.]).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(0, 10, 1000)).T
# Instanciate a Gaussian Process model
gp = GaussianProcess(corr='cubic', theta0=1e-2, thetaL=1e-4, thetaU=1e-1, \
random_start=100)
# 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, MSE = gp.predict(x, eval_MSE=True)
sigma = np.sqrt(MSE)
# Plot the function, the prediction and the 95% confidence interval based on
# the MSE
fig = pl.figure()
pl.plot(x, f(x), 'r:', label=u'$f(x) = x\,\sin(x)$')
pl.plot(X, y, 'r.', markersize=10, label=u'Observations')
pl.plot(x, y_pred, 'b-', label=u'Prediction')
pl.fill(np.concatenate([x, x[::-1]]), \
