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Iso-probability lines for Gaussian Processes classification (GPC)#

A two-dimensional classification example showing iso-probability lines for the predicted probabilities.

plot gpc isoprobability

Learned kernel: 0.0256**2 * DotProduct(sigma_0=5.72) ** 2

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
importnumpyasnp
frommatplotlibimport cm
frommatplotlibimport pyplot as plt
fromsklearn.gaussian_processimport GaussianProcessClassifier
fromsklearn.gaussian_process.kernelsimport ConstantKernel as C
fromsklearn.gaussian_process.kernelsimport DotProduct
# A few constants
lim = 8
defg(x):
"""The function to predict (classification will then consist in predicting
 whether g(x) <= 0 or not)"""
 return 5.0 - x[:, 1] - 0.5 * x[:, 0] ** 2.0
# 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 = np.array (g(X) > 0, dtype=int)
# Instantiate and fit Gaussian Process Model
kernel = C (0.1, (1e-5, np.inf )) * DotProduct (sigma_0=0.1) ** 2
gp = GaussianProcessClassifier (kernel=kernel)
gp.fit(X, y)
print("Learned kernel: %s " % gp.kernel_)
# Evaluate real function and the predicted probability
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_prob = gp.predict_proba(xx)[:, 1]
y_true = y_true.reshape((res, res))
y_prob = y_prob.reshape((res, res))
# Plot the probabilistic classification iso-values
fig = plt.figure (1)
ax = fig.gca()
ax.axes.set_aspect("equal")
plt.xticks ([])
plt.yticks ([])
ax.set_xticklabels([])
ax.set_yticklabels([])
plt.xlabel ("$x_1$")
plt.ylabel ("$x_2$")
cax = plt.imshow (y_prob, cmap=cm.gray_r, alpha=0.8, extent=(-lim, lim, -lim, lim))
norm = plt.matplotlib.colors.Normalize(vmin=0.0, vmax=0.9)
cb = plt.colorbar (cax, ticks=[0.0, 0.2, 0.4, 0.6, 0.8, 1.0], norm=norm)
cb.set_label(r"${\rm \mathbb{P}}\left[\widehat{G}(\mathbf{x}) \leq 0\right]$")
plt.clim (0, 1)
plt.plot (X[y <= 0, 0], X[y <= 0, 1], "r.", markersize=12)
plt.plot (X[y > 0, 0], X[y > 0, 1], "b.", markersize=12)
plt.contour (x1, x2, y_true, [0.0], colors="k", linestyles="dashdot")
cs = plt.contour (x1, x2, y_prob, [0.666], colors="b", linestyles="solid")
plt.clabel (cs, fontsize=11)
cs = plt.contour (x1, x2, y_prob, [0.5], colors="k", linestyles="dashed")
plt.clabel (cs, fontsize=11)
cs = plt.contour (x1, x2, y_prob, [0.334], colors="r", linestyles="solid")
plt.clabel (cs, fontsize=11)
plt.show ()