一个关于PCA与eigenvector的问题

答案是1。

问题应该是“PCA之后得到的eigenvector里绝对值最大的值是多少”。

3个独立feature，求PCA时，会依次把方差第$i$大feature的单位向量作为第$i$个component， 最后PCA的eigenvector（component）是xyz轴的正交基（orthonormal）。所以x轴[+/-1 0 0]的最大绝对值为1. 

当复制一个feature后，xy轴不变，z轴扩展为一个平面，x轴扩展为[+/-1 0 0 0]，最大绝对值还是1.

做实验验证下：

import numpy as np
from sklearn.decomposition import PCA
np.random.seed(0)
n=100000
## Gaussian
A=np.random.multivariate_normal([0,0,0],np.diag([300,20,1]),n)
## Poisson
# A=np.zeros([n,3])
# for i in range(3):
#   A[:,i]=np.random.np.random.poisson((i+1)*100,size=n)
pca = PCA(3)
pca.fit(A)
print('---3 features PCA---')
print('eigenvectors:')
print(pca.components_)
print('eigenvalues:')
print(pca.explained_variance_)
print('max abs element of eignvectors is %f'%max(abs(pca.components_.flatten())))

B=np.concatenate((A, A[:,-1].reshape([n,1])),axis=1)
pca1 = PCA(4)
pca1.fit(B)
print('---3+1 features PCA---')
print('eigenvectors:')
print(pca1.components_)
print('eigenvalues:')
print(pca1.explained_variance_)
print('max abs element of eignvectors is %f'%max(abs(pca1.components_.flatten())))

结果：

---3 features PCA---
eigenvectors:
[[ 9.99999758e-01  6.61747537e-04 -2.16070381e-04]
 [ 6.61727522e-04 -9.99999777e-01 -9.26909242e-05]
 [-2.16131671e-04  9.25479220e-05 -9.99999972e-01]]
eigenvalues:
[299.24945095  19.99189396   0.99597203]
max abs element of eignvectors is 1.000000
---3+1 features PCA---
eigenvectors:
[[ 9.99999734e-01  6.61746150e-04 -2.16794328e-04 -2.16794328e-04]
 [ 6.61703761e-04 -9.99999772e-01 -9.78196862e-05 -9.78196862e-05]
 [-3.06684953e-04  1.38135016e-04 -7.07106741e-01 -7.07106741e-01]
 [-0.00000000e+00  5.26484535e-17  7.07106781e-01 -7.07106781e-01]]
eigenvalues:
[2.99249465e+02 1.99918941e+01 1.99194394e+00 2.49383447e-32]
max abs element of eignvectors is 1.000000