import numpy as np
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA, KernelPCA
from sklearn.datasets import make_circles
np.random.seed(0)
X,y=make_circles(n_samples=100,factor=.3,noise=.03)
kpca=KernelPCA(kernel="rbf",fit_inverse_transform=True,gamma=10)
X_kpca=kpca.fit_transform(X)
X_back=kpca.inverse_transform(X_kpca)
pca=PCA()
X_pca=pca.fit_transform(X)
#绘制结果
plt.figure()
plt.subplot(2,2,1,aspect='equal')
plt.title("Original space")
reds=y==0
blues=y==1
plt.scatter(X[reds,0],X[reds,1],c="red",s=20,edgecolor='k')
plt.scatter(X[blues,0],X[blues,1],c="blue",s=20,edgecolor='k')
plt.xlabel("$x_1$")
plt.ylabel("$x_2$")
X1, X2=np.meshgrid(np.linspace(-1.5,1.5,50),np.linspace(-1.5,1.5,50))
X_grid=np.array([np.ravel(X1),np.ravel(X2)]).T
#第一主分量上的投影
Z_grid=kpca.transform(X_grid)[:,0].reshape(X1.shape)
plt.contour(X1,X2,Z_grid,colors='grey',linewidths=1,origin='lower')
plt.subplot(2,2,2,aspect='equal')
plt.scatter(X_pca[reds,0],X_pca[reds,1],c="red",s=20,edgecolor='k')
plt.scatter(X_pca[blues,0],X_pca[blues,1],c="blue",s=20,edgecolor='k')
plt.title("Projection by PCA")
plt.xlabel("1st principal component")
plt.ylabel("2nd component")
plt.subplot(2,2,3,aspect='equal')
plt.scatter(X_kpca[reds,0],X_kpca[reds,1],c="red",s=20,edgecolor='k')
plt.scatter(X_kpca[blues,0],X_kpca[blues,1],c="blue",s=20,edgecolor='k')
plt.title("Projection by KPCA")
plt.xlabel("1st principal component in space induced by $phi$")
plt.ylabel("2nd component")
plt.subplot(2,2,4,aspect='equal')
plt.scatter(X_back[reds,0],X_back[reds,1],c="red",s=20,edgecolor='k')
plt.scatter(X_back[blues,0],X_back[blues,1],c="blue",s=20,edgecolor='k')
plt.title("Original space after inverse transform")
plt.xlabel("$x_1$")
plt.ylabel("$x_2$")
plt.subplots_adjust(0.02, 0.10, 0.98, 0.94, 0.04, 0.35)
plt.show()
算法：核主成分(Kernel Principal Component Analysis, Kernel PCA)是主成分推广，使用了核函数，将原始主成分线性变换转换到核希尔伯特空间。
