ガウス過程回帰

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.plot(X, y, label=r"$f(x) = x \sin(x)$", linestyle="dotted")
plt.legend()
plt.xlabel("$x$")
plt.ylabel("$f(x)$")
_ = plt.title("True generative process")

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))
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="Observations",
)
plt.plot(X, mean_prediction, label="Mean prediction")
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% confidence interval",
)
plt.legend()
plt.xlabel("$x$")
plt.ylabel("$f(x)$")
_ = plt.title("Gaussian process regression on a noisy dataset")