不同核的先验和后验高斯过程图示

此示例说明了具有不同核的 GaussianProcessRegressor 的先验和后验。均值、标准差和 5 个样本显示了先验和后验分布。

在这里，我们仅提供一些说明。要了解有关核公式的更多信息，请参阅用户指南。

# Authors: Jan Hendrik Metzen <[email protected]>
#          Guillaume Lemaitre <[email protected]>
# License: BSD 3 clause

辅助函数

在介绍可用于高斯过程的每个单独核之前，我们将定义一个辅助函数，允许我们绘制从高斯过程中提取的样本。

此函数将采用 GaussianProcessRegressor 模型，并将从高斯过程中绘制样本。如果模型未拟合，则从先验分布中提取样本，而在模型拟合后，从后验分布中提取样本。

import matplotlib.pyplot as plt
import numpy as np


def plot_gpr_samples(gpr_model, n_samples, ax):
    """Plot samples drawn from the Gaussian process model.

    If the Gaussian process model is not trained then the drawn samples are
    drawn from the prior distribution. Otherwise, the samples are drawn from
    the posterior distribution. Be aware that a sample here corresponds to a
    function.

    Parameters
    ----------
    gpr_model : `GaussianProcessRegressor`
        A :class:`~sklearn.gaussian_process.GaussianProcessRegressor` model.
    n_samples : int
        The number of samples to draw from the Gaussian process distribution.
    ax : matplotlib axis
        The matplotlib axis where to plot the samples.
    """
    x = np.linspace(0, 5, 100)
    X = x.reshape(-1, 1)

    y_mean, y_std = gpr_model.predict(X, return_std=True)
    y_samples = gpr_model.sample_y(X, n_samples)

    for idx, single_prior in enumerate(y_samples.T):
        ax.plot(
            x,
            single_prior,
            linestyle="--",
            alpha=0.7,
            label=f"Sampled function #{idx + 1}",
        )
    ax.plot(x, y_mean, color="black", label="Mean")
    ax.fill_between(
        x,
        y_mean - y_std,
        y_mean + y_std,
        alpha=0.1,
        color="black",
        label=r"$\pm$ 1 std. dev.",
    )
    ax.set_xlabel("x")
    ax.set_ylabel("y")
    ax.set_ylim([-3, 3])

数据集和高斯过程生成

我们将创建一个训练数据集，我们将在不同的部分中使用它。

rng = np.random.RandomState(4)
X_train = rng.uniform(0, 5, 10).reshape(-1, 1)
y_train = np.sin((X_train[:, 0] - 2.5) ** 2)
n_samples = 5

核手册

在本节中，我们将说明从具有不同核的高斯过程的先验和后验分布中提取的一些样本。

径向基函数核

from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RBF

kernel = 1.0 * RBF(length_scale=1.0, length_scale_bounds=(1e-1, 10.0))
gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

# plot prior
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
axs[0].set_title("Samples from prior distribution")

# plot posterior
gpr.fit(X_train, y_train)
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
axs[1].set_title("Samples from posterior distribution")

fig.suptitle("Radial Basis Function kernel", fontsize=18)
plt.tight_layout()

print(f"Kernel parameters before fit:\n{kernel})")
print(
    f"Kernel parameters after fit: \n{gpr.kernel_} \n"
    f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
)

Kernel parameters before fit:
1**2 * RBF(length_scale=1))
Kernel parameters after fit:
0.594**2 * RBF(length_scale=0.279)
Log-likelihood: -0.067

有理二次核

from sklearn.gaussian_process.kernels import RationalQuadratic

kernel = 1.0 * RationalQuadratic(length_scale=1.0, alpha=0.1, alpha_bounds=(1e-5, 1e15))
gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

# plot prior
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
axs[0].set_title("Samples from prior distribution")

# plot posterior
gpr.fit(X_train, y_train)
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
axs[1].set_title("Samples from posterior distribution")

fig.suptitle("Rational Quadratic kernel", fontsize=18)
plt.tight_layout()

print(f"Kernel parameters before fit:\n{kernel})")
print(
    f"Kernel parameters after fit: \n{gpr.kernel_} \n"
    f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
)

Kernel parameters before fit:
1**2 * RationalQuadratic(alpha=0.1, length_scale=1))
Kernel parameters after fit:
0.594**2 * RationalQuadratic(alpha=1.05e+06, length_scale=0.279)
Log-likelihood: -0.067

指数正弦平方核

from sklearn.gaussian_process.kernels import ExpSineSquared

kernel = 1.0 * ExpSineSquared(
    length_scale=1.0,
    periodicity=3.0,
    length_scale_bounds=(0.1, 10.0),
    periodicity_bounds=(1.0, 10.0),
)
gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

# plot prior
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
axs[0].set_title("Samples from prior distribution")

# plot posterior
gpr.fit(X_train, y_train)
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
axs[1].set_title("Samples from posterior distribution")

fig.suptitle("Exp-Sine-Squared kernel", fontsize=18)
plt.tight_layout()

print(f"Kernel parameters before fit:\n{kernel})")
print(
    f"Kernel parameters after fit: \n{gpr.kernel_} \n"
    f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
)

Kernel parameters before fit:
1**2 * ExpSineSquared(length_scale=1, periodicity=3))
Kernel parameters after fit:
0.799**2 * ExpSineSquared(length_scale=0.791, periodicity=2.87)
Log-likelihood: 3.394

点积核

from sklearn.gaussian_process.kernels import ConstantKernel, DotProduct

kernel = ConstantKernel(0.1, (0.01, 10.0)) * (
    DotProduct(sigma_0=1.0, sigma_0_bounds=(0.1, 10.0)) ** 2
)
gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

# plot prior
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
axs[0].set_title("Samples from prior distribution")

# plot posterior
gpr.fit(X_train, y_train)
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
axs[1].set_title("Samples from posterior distribution")

fig.suptitle("Dot-product kernel", fontsize=18)
plt.tight_layout()

/home/circleci/project/sklearn/gaussian_process/_gpr.py:659: ConvergenceWarning:

lbfgs failed to converge (status=2):
ABNORMAL_TERMINATION_IN_LNSRCH.

Increase the number of iterations (max_iter) or scale the data as shown in:
    https://scikit-learn.cn/stable/modules/preprocessing.html

/home/circleci/project/sklearn/gaussian_process/_gpr.py:477: UserWarning:

Predicted variances smaller than 0. Setting those variances to 0.

print(f"Kernel parameters before fit:\n{kernel})")
print(
    f"Kernel parameters after fit: \n{gpr.kernel_} \n"
    f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
)

Kernel parameters before fit:
0.316**2 * DotProduct(sigma_0=1) ** 2)
Kernel parameters after fit:
2.68**2 * DotProduct(sigma_0=8.47) ** 2
Log-likelihood: -7337046907.481

Matérn 核

from sklearn.gaussian_process.kernels import Matern

kernel = 1.0 * Matern(length_scale=1.0, length_scale_bounds=(1e-1, 10.0), nu=1.5)
gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

# plot prior
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
axs[0].set_title("Samples from prior distribution")

# plot posterior
gpr.fit(X_train, y_train)
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
axs[1].set_title("Samples from posterior distribution")

fig.suptitle("Matérn kernel", fontsize=18)
plt.tight_layout()

print(f"Kernel parameters before fit:\n{kernel})")
print(
    f"Kernel parameters after fit: \n{gpr.kernel_} \n"
    f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
)

Kernel parameters before fit:
1**2 * Matern(length_scale=1, nu=1.5))
Kernel parameters after fit:
0.609**2 * Matern(length_scale=0.484, nu=1.5)
Log-likelihood: -1.185

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