ガウス過程回帰

ガウス過程というランダムな関数の確率分布を利用した回帰モデル

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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")

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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")

参考文献

• ガウス過程回帰モデルの基礎 — ごちきか

• 1.7. Gaussian Processes — scikit-learn 1.3.2 documentation