1、高斯過程：

scikit-learn (sklearn) 官方文檔

scikit-learn (sklearn) 官方文檔中文版

scikit-learn (sklearn) 官方文檔中文版（1.7. 高斯過程）

其他介紹：

A Visual Exploration of Gaussian Processes

看得見的高斯過程：這是一份直觀的入門解讀（上面中文翻譯-機器之心）

Introduction to Gaussian Processes - Part I

從數學到實現，全面回顧高斯過程中的函數最優化（機器之心）

淺談高斯過程回歸

相關paper：

Gaussian Processes for Regression A Quick Introduction, M.Ebden, August 2008.

[RW2006] Carl Eduard Rasmussen and Christopher K.I. Williams, “Gaussian Processes for Machine Learning”, MIT Press 2006.

2、python實現：

（例1）

            
              import numpy as np

import matplotlib.pyplot as plt

from sklearn.gaussian_process import GaussianProcessRegressor

from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C

from mpl_toolkits.mplot3d import Axes3D

# 創建數據集

test = np.array([[2004, 98.31]])

data = np.array([

    [2001, 100.83, 410], [2005, 90.9, 500], [2007, 130.03, 550], [2004, 78.88, 410], [2006, 74.22, 460],

    [2005, 90.4, 497], [1983, 64.59, 370], [2000, 164.06, 610], [2003, 147.5, 560], [2003, 58.51, 408],

    [1999, 95.11, 565], [2000, 85.57, 430], [1995, 66.44, 378], [2003, 94.27, 498], [2007, 125.1, 760],

    [2006, 111.2, 730], [2008, 88.99, 430], [2005, 92.13, 506], [2008, 101.35, 405], [2000, 158.9, 615]])

# 核函數的取值

kernel = C(0.1, (0.001, 0.1)) * RBF(0.5, (1e-4, 10))

# 創建高斯過程回歸,并訓練

reg = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=10, alpha=0.1)

reg.fit(data[:, :-1], data[:, -1])

# 創建一個作圖用的網格的測試數據，數據位線性，x為【1982，2009】間隔位0.5；y為【57.5，165】間隔位0.5

x_min, x_max = data[:, 0].min() - 1, data[:, 0].max() + 1

y_min, y_max = data[:, 1].min() - 1, data[:, 1].max() + 1

xset, yset = np.meshgrid(np.arange(x_min, x_max, 0.5), np.arange(y_min, y_max, 0.5))

# 查看網格測試數據輸出結果，并返回標準差。

output, err = reg.predict(np.c_[xset.ravel(), yset.ravel()], return_std=True)

output, err = output.reshape(xset.shape), err.reshape(xset.shape)

sigma = np.sum(reg.predict(data[:, :-1], return_std=True)[1])

up, down = output * (1 + 1.96 * err), output * (1 - 1.96 * err)

# 作圖，并畫出

fig = plt.figure(figsize=(10.5, 5))

ax1 = fig.add_subplot(121, projection='3d')

surf = ax1.plot_wireframe(xset, yset, output, rstride=10, cstride=2, antialiased=True)

surf_u = ax1.plot_wireframe(xset, yset, up, colors='lightgreen', linewidths=1,

                            rstride=10, cstride=2, antialiased=True)

surf_d = ax1.plot_wireframe(xset, yset, down, colors='lightgreen', linewidths=1,

                            rstride=10, cstride=2, antialiased=True)

ax1.scatter(data[:, 0], data[:, 1], data[:, 2], c='red')

ax1.set_title('House Price at (2004, 98.31): {0:.2f}$*10^4$ RMB'.format(reg.predict(test)[0]))

ax1.set_xlabel('Year')

ax1.set_ylabel('Area, $m^2$')

plt.show()

            
          

結果：

（例2）

            
              import numpy as np
import matplotlib.pyplot as plt
from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C
from sklearn.gaussian_process import GaussianProcessRegressor

# Build a model
kernel = C(1.0, (1e-3, 1e3)) * RBF(10, (0.5, 2))
gp = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=9)

# Some data
xobs = np.array([[1], [1.5], [-3]])
yobs = np.array([3, 0, 1])

# Fit the model to the data (optimize hyper parameters)
gp.fit(xobs, yobs)

# Plot points and predictions
x_set = np.arange(-6, 6, 0.1)
x_set = np.array([[i] for i in x_set])
means, sigmas = gp.predict(x_set, return_std=True)

plt.figure(figsize=(8, 5))
plt.errorbar(x_set, means, yerr=sigmas, alpha=0.5)
plt.plot(x_set, means, 'g', linewidth=4)

colors = ['g', 'r', 'b', 'k']
for c in colors:
    y_set = gp.sample_y(x_set, random_state=np.random.randint(1000))
    plt.plot(x_set, y_set, c + '--', alpha=0.5)

plt.show()

            
          

結果：

（例3）

            
              import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import spline
from scipy.linalg import cho_solve
from numpy.linalg import cholesky
from itertools import cycle


class SimpleGP():
    """ One dimensional Gaussian Process class.  Uses
    squared exponential covariance form.

    parameters
    ----------
    width_scale : float, positive
        Same as sigma in (4) of post

    length_scale : float, positive
        Same as l in (4) of post

    noise : float
        Added to diagonal of covariance, useful for
        improving convergence
    """

    def __init__(self, width_scale, length_scale, noise=10 ** (-6)):
        self.width_scale = width_scale
        self.length_scale = length_scale
        self.noise = noise

    def _exponential_cov(self, x1, x2):
        """
        Return covariance matrix for two arrays,
        with i-j element = cov(x_1i, x_2j).

        parameters
        ----------
        x1, x2: np.array
            arrays containing x locations
        """
        return (self.width_scale ** 2) * np.exp(
            - np.subtract.outer(x1, x2) ** 2 / (2 * self.length_scale ** 2))

    def fit(self, sample_x, sample_y, sample_s):
        """
        Save for later use the Cholesky matrix
        associated with the inverse that appears
        in (5) of post. Also evaluate the weighted
        y vector that appears in that equation.

        parameters
        ----------
        sample_x : np.array
            locations where we have sampled

        sample_y : np.array
            y values observed at each sample location

        sample_s : np.array
            array of stds for each sample
        """

        self.sample_x = np.array(sample_x)

        S = self._exponential_cov(sample_x, sample_x)
        d = np.diag(np.array(sample_s) ** 2 + self.noise)

        self.lower_cholesky = cholesky(S + d)
        self.weighted_sample_y = cho_solve(
            (self.lower_cholesky, True), sample_y)

    def interval(self, test_x):
        """
        Obtain the one-sigam confidence interval
        for a set of test points

        parameters
        ----------
        test_x : np.array
            locations where we want to test
        """
        test_x = np.array([test_x]).flatten()
        means, stds = [], []
        for row in test_x:
            S0 = self._exponential_cov(row, self.sample_x)
            v = cho_solve((self.lower_cholesky, True), S0)
            means.append(np.dot(S0, self.weighted_sample_y))
            stds.append(np.sqrt(self.width_scale ** 2 - np.dot(S0, v)))
        return means, stds

    def sample(self, test_x, samples=1):
        """
        Obtain function samples from the posterior

        parameters
        ----------
        test_x : np.array
            locations where we want to test

        samples : int
            Number of samples to take
        """
        S0 = self._exponential_cov(test_x, self.sample_x)
        # construct covariance matrix of sampled points.
        m = []
        for row in S0:
            m.append(cho_solve((self.lower_cholesky, True), row))
        cov = self._exponential_cov(test_x, test_x) - np.dot(S0, np.array(m).T)
        mean = np.dot(S0, self.weighted_sample_y)
        return np.random.multivariate_normal(mean, cov, samples)


# Insert data here.
sample_x = [0.5, 2, -2]
sample_y = [2, 1.5, -0.5]
sample_s = [0.01, 0.05, 0.125]

WIDTH_SCALE = 1
LENGTH_SCALE = 1
SAMPLES = 8
model = SimpleGP(WIDTH_SCALE, LENGTH_SCALE)
model.fit(sample_x, sample_y, sample_s)

test_x = np.arange(-5, 5, .1)
means, stds = model.interval(test_x)
samples = model.sample(test_x, SAMPLES)

# plots here.
fig, ax = plt.subplots(figsize=(12, 5))
ax.set_ylim([-3, 3])
ax.axis('off')

colors = cycle(['g', 'b', 'k', 'y', 'c', 'r', 'm'])
plt.errorbar(test_x, means, yerr=stds,
             ecolor='g', linewidth=1.5,
             elinewidth=0.5, alpha=0.75)

for sample, c in zip(samples, colors):
    plt.plot(test_x, sample, c, linewidth=2. * np.random.rand(), alpha=0.5)

plt.show()

            
          

結果：

C++庫：

github地址：https://github.com/resibots/limbo

Documentation：http://www.resibots.eu/limbo

參考：

python.sklearn.gaussian_process高斯過程回歸的調用?

https://github.com/EFavDB/gaussian_processes/blob/master/GP_example.ipynb