20、噪声数据集上的高斯过程回归
import numpy as np
X = np.linspace(start=0, stop=10, num=1_000).reshape(-1, 1)
y = np.squeeze(X * np.sin(X))
import matplotlib.pyplot as plt
plt.rcParams['font.sans-serif'] = ['SimHei']
plt.rcParams['axes.unicode_minus'] = False
rng = np.random.RandomState(1)
training_indices = rng.choice(np.arange(y.size), size=6, replace=False)
X_train, y_train = X[training_indices], y[training_indices]
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RBF
kernel = 1 * RBF(length_scale=1.0, length_scale_bounds=(1e-2, 1e2))
gaussian_process = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=9)
gaussian_process.fit(X_train, y_train)
mean_prediction, std_prediction = gaussian_process.predict(X, return_std=True)
noise_std = 0.75
y_train_noisy = y_train + rng.normal(loc=0.0, scale=noise_std, size=y_train.shape)
gaussian_process = GaussianProcessRegressor(
kernel=kernel, alpha=noise_std**2, n_restarts_optimizer=9
)
gaussian_process.fit(X_train, y_train_noisy)
mean_prediction, std_prediction = gaussian_process.predict(X, return_std=True)
plt.plot(X, y, label=r"$f(x) = x \sin(x)$", linestyle="dotted")
plt.errorbar(
X_train,
y_train_noisy,
noise_std,
linestyle="None",
color="tab:blue",
marker=".",
markersize=10,
label="观察值",
)
plt.plot(X, mean_prediction, label="均值预测")
plt.fill_between(
X.ravel(),
mean_prediction - 1.96 * std_prediction,
mean_prediction + 1.96 * std_prediction,
color="tab:orange",
alpha=0.5,
label=r"95% 置信区间",
)
plt.legend()
plt.xlabel("$x$")
plt.ylabel("$f(x)$")
_ = plt.title("噪声数据集上的高斯过程回归")
