Python：使用 SVD 实现 PCA

Question

I am trying to figure out the differences between PCA using Singular Value Decomposition as oppossed to PCA using Eigenvector-Decomposition.

Picture the following matrix:

 B = np.array([          [1, 2],
                         [3, 4],
                         [5, 6] ])

When computing the PCA of this matrix B using eigenvector-Decomposition, we follow these steps:

1. Center the data (entries of B) by substracting the column-mean from each column
2. Compute the covariance matrix C = Cov(B) = B^T * B / (m -1) , where m = # rows of B
3. Find eigenvectors of C
4. PCs = X * eigen_vecs

When computing the PCA of matrix B using SVD, we follow these steps:

1. Compute SVD of B: B = U * Sigma * VT
2. PCs = U * Sigma

I have done both for the given matrix.

With eigenvector-Decomposition I obtain this result:

[[-2.82842712  0.        ]
 [ 0.          0.        ]
 [ 2.82842712  0.        ]]

With SVD I obtain this result:

[[-2.18941839  0.45436451]
 [-4.99846626  0.12383458]
 [-7.80751414 -0.20669536]]

The result obtained with eigenvector-Decomposition is the result given as solution. So, why is the result obtained with the SVD different?

I know that: C = Cov(B) = V * (Sigma^2)/(m-1)) * VT and I have a feeling this might be related to why the two results are different. Still. Can anyone help me understand better?

Answer 1

Please see below a comparision for your matrix with sklearn.decomposition.PCA and numpy.linalg.svd. Can you compare or post how you derived SVD results.

Code for sklearn.decomposition.PCA:

from sklearn.decomposition import PCA
import numpy as np 
np.set_printoptions(precision=3)

B = np.array([[1.0,2], [3,4], [5,6]])

B1 = B.copy() 
B1 -= np.mean(B1, axis=0) 
n_samples = B1.shape[0]
print("B1 is B after centering:")
print(B1)

cov_mat = np.cov(B1.T)
pca = PCA(n_components=2) 
X = pca.fit_transform(B1)
print("X")
print(X)

eigenvecmat =   []
print("Eigenvectors:")
for eigenvector in pca.components_:
   if eigenvecmat == []:
        eigenvecmat = eigenvector
   else:
        eigenvecmat = np.vstack((eigenvecmat, eigenvector))
   print(eigenvector)
print("eigenvector-matrix")
print(eigenvecmat)

print("CHECK FOR PCA:")
print("X * eigenvector-matrix (=B1)")
print(np.dot(PCs, eigenvecmat))

Output for PCA:

B1 is B after centering:
[[-2. -2.]
 [ 0.  0.]
 [ 2.  2.]]
X
[[-2.828  0.   ]
 [ 0.     0.   ]
 [ 2.828  0.   ]]
Eigenvectors:
[0.707 0.707]
[-0.707  0.707]
eigenvector-matrix
[[ 0.707  0.707]
 [-0.707  0.707]]
CHECK FOR PCA:
X * eigenvector-matrix (=B1)
[[-2. -2.]
 [ 0.  0.]
 [ 2.  2.]]

numpy.linalg.svd:

print("B1 is B after centering:")
print(B1)

from numpy.linalg import svd 
U, S, Vt = svd(X1, full_matrices=True)

print("U:")
print(U)
print("S used for building Sigma:")
print(S)
Sigma = np.zeros((3, 2), dtype=float)
Sigma[:2, :2] = np.diag(S)
print("Sigma:")
print(Sigma)
print("V already transposed:")
print(Vt)
print("CHECK FOR SVD:")
print("U * Sigma * Vt (=B1)")
print(np.dot(U, np.dot(Sigma, Vt)))

Output for SVD:

B1 is B after centering:
[[-2. -2.]
 [ 0.  0.]
 [ 2.  2.]]
U:
[[-0.707  0.     0.707]
 [ 0.     1.     0.   ]
 [ 0.707  0.     0.707]]
S used for building Sigma:
[4. 0.]
Sigma:
[[4. 0.]
 [0. 0.]
 [0. 0.]]
V already transposed:
[[ 0.707  0.707]
 [-0.707  0.707]]
CHECK FOR SVD:
U * Sigma * Vt (=B1)
[[-2. -2.]
 [ 0.  0.]
 [ 2.  2.]]