build_K(xtrain, xtrain, hyp, K) # writes inside K
Ky = K + hyp[-1]*np.eye(ntrain)
Kyinv = invert(Ky, 4, 1e-6) # using gp_functions.invert
ntest = 300
xtest = np.linspace(0, 1, ntest)
ftest = f(xtest)
Ks = np.empty((ntrain, ntest)) # train-test
Kss = np.empty((ntest, ntest)) # test-test
build_K(xtrain, xtest, hyp, Ks)
build_K(xtest, xtest, hyp, Kss)
fmean = Ks.T.dot(Kyinv.dot(ytrain)) # predictive mean
Ef, varf = predict_f(hyp, xtrain.reshape(-1, 1),
ytrain.reshape(-1, 1), xtest.reshape(-1, 1), neig=8)# posterior
# Estimation and variance
varf = np.diag(varf)
# we keep only the diag because the variance is on it, the other terms are covariance
plt.figure()
plt.plot(xtrain, ytrain, 'kx')
plt.plot(xtest, ftest, 'm-')
plt.plot(xtest, fmean, 'r--')
axes = plt.gca()
axes.set_ylim([-1.5, 1])
plt.title('Random Gaussian Process with '+ str(ntrain) + ' observation(s) hyp = [0.1, 1e-4]')
plt.colorbar()
# Compute C
C = np.block([[K + (sigma_n / sigma_f) ** 2 * np.eye(n), np.transpose(K_star)], [K_star, K_star2]])
# Plot C
plt.imshow(C)
plt.colorbar()
# Plot function f(x)
plt.figure()
plt.plot(x, f_x, 'r')
# plot of n_prior samples from prior distribution (100 samples). The mean is zero and the covariance is given by K_star2
n_prior = 100
for i in range(0, n_prior):
f_star = np.random.multivariate_normal(np.zeros(n_star), K_star2)
plt.plot(x_star, f_star, 'b', linewidth=0.5)
# Compute posterior mean and covariance.
t = time.time()
f_bar_star, cov_f_star = predict_f(a, x, y, x_star, RBF, return_full_cov=True, neig=8)
elapsed = time.time() - t
print(elapsed)
# plot the covariance matrix
plt.imshow(cov_f_star)
# Plot of n_prior samples from the posterior distribution and of the true function
for i in range(n_prior):
f_posterior = np.random.multivariate_normal(f_bar_star[:, 0], cov_f_star)
plt.figure()
plt.plot(x_star, f_posterior)
plt.fill_between(x_star.flatten(),
(f_bar_star.flatten() + 2 * np.sqrt(np.diag(cov_f_star))),
(f_bar_star.flatten() - 2 * np.sqrt(np.diag(cov_f_star))))