sci-kit 学习 PCA 和手动 PCA 的结果差异

Question

I'm really puzzled, hopefully someone can show me what I am missing. I'm trying to get principal components via two different methods:

import numpy as np
data = np.array([[ 2.1250045 , -0.17169867, -0.47799957],
               [ 0.7400025 , -0.07970344, -0.99600106],
               [ 0.15800177,  1.2993019 , -0.8030003 ],
               [ 0.3159989 ,  1.919297  ,  0.24300112],
               [-0.14800562, -1.0827019 , -0.2890004 ],
               [ 0.26900184, -1.3816979 ,  1.1239979 ],
               [-0.5040008 , -2.9066994 ,  1.6400006 ],
               [-1.2230027 , -2.415702  ,  3.1940014 ],
               [-0.54700005,  1.757302  , -1.825999  ],
               [-1.1860001 ,  3.0623024 , -1.8090007 ]]) # this should already be mean centered



# Method 1. Scikit-Learn
from sklearn.decomposition import PCA

pca = PCA(n_components=3).fit(data)
print(pca.components_)
[[-0.04209988 -0.79261507  0.60826717]
 [ 0.88594009 -0.31106375 -0.34401963]
 [ 0.46188501  0.52440508  0.71530521]]


# Method 2. Manually with numpy
cov = np.cov(data.T)

evals , evecs = np.linalg.eig(cov)

# The next three lines are just sorting by the largest eigenvalue
idx = np.argsort(evals)[::-1]
evecs = evecs[:,idx]
evals = evals[idx]

print(evecs.T)
[[ 0.04209988  0.79261507 -0.60826717]
 [ 0.88594009 -0.31106375 -0.34401963]
 [-0.46188501 -0.52440508 -0.71530521]]

The values for the eigenvectors are the same, but the signs are wrong. What I want is to get the output from sklearn PCA, but using only numpy. Thanks in advance for any suggestions.

Answer 1

That is expected because the eigenspace of a matrix (covariance matrix in your question) is unique but the specific set of eigenvectors is not. It is too much to explain here, so I would recommend the answer in math.se

PS: Notice that you're dealing with covariance matrix of 3x3 and you can imagine the eigenvectors as vectors in 3D with x-, y-, z-axis. Then you should notice your numpy answer vs sklearn answer are in exact opposite direction for 2 vectors and same direction for 1 vector.