svd()函数获取矩阵主成分

X_centered = X - X.mean(axis=0)
U, s, V = np.linalg.svd(X_centered)
c1 = V.T[:, 0]
c2 = V.T[:, 1]

在获取主成分之前, 应当将训练集集中(第一行代码).

得到的V即为主成分矩阵, c1和c2分别对应第一, 第二主成分.

问题: 为什么要调用T()方法对主成分矩阵进行转置???

import numpy as np

X = np.random.random([6, 6])
X_centered = X - X.mean(axis=0)
U, s, V = np.linalg.svd(X_centered)
c1 = V.T[:, 0]
c2 = V.T[:, 1]

print("原始矩阵(训练集)")
print(X)
print("="*8 + "集中后的训练集")
print(X_centered)
print("="*8 + "主成分矩阵V")
print(V)
print("="*8 + "未转置的主成分1")
print(V[:, 0])
print("="*8 + "转置后的主成分1")
print(c1)
print("="*8 + "转置后的主成分2")
print(c2)

上述代码输出:

原始矩阵(训练集)
[[0.48925597 0.10007899 0.24868701 0.24582956 0.01189903 0.92280551]
 [0.58452584 0.08522259 0.4727072  0.16183505 0.37137143 0.28484798]
 [0.83314739 0.6253953  0.5853588  0.943809   0.39430325 0.73108571]
 [0.26039373 0.52392961 0.29923633 0.99798253 0.39681654 0.76359518]
 [0.34860457 0.61275214 0.89554728 0.03253871 0.41150234 0.92642534]
 [0.61715827 0.18374784 0.02390839 0.69360349 0.40653224 0.86420085]]
========集中后的训练集
[[-0.03292499 -0.25510875 -0.17222049 -0.26677016 -0.32017178  0.17397875]
 [ 0.06234488 -0.26996515  0.0517997  -0.35076467  0.03930063 -0.46397878]
 [ 0.31096643  0.27020755  0.1644513   0.43120928  0.06223245 -0.01774105]
 [-0.26178723  0.16874187 -0.12167117  0.48538281  0.06474574  0.01476842]
 [-0.17357639  0.2575644   0.47463978 -0.48006101  0.07943153  0.17759858]
 [ 0.09497731 -0.17143991 -0.39699912  0.18100377  0.07446144  0.11537409]]
========主成分矩阵V
[[ 0.10367429  0.17534132 -0.33069463  0.91277858  0.09734234  0.08067125]
 [-0.07961523  0.66591982  0.68886836  0.09517681  0.21607987  0.14115045]
 [-0.42666386  0.08883363 -0.20928781 -0.08933653 -0.26604371  0.82915914]
 [ 0.87535833  0.0054328   0.10825683 -0.06744275 -0.26173744  0.38593229]
 [-0.15981178  0.00365394  0.27606895  0.23488069 -0.87780784 -0.26929025]
 [ 0.09528378  0.71963319 -0.53326021 -0.30013023 -0.18439526 -0.25417092]]
========未转置的主成分1
[ 0.10367429 -0.07961523 -0.42666386  0.87535833 -0.15981178  0.09528378]
========转置后的主成分1
[ 0.10367429  0.17534132 -0.33069463  0.91277858  0.09734234  0.08067125]
========转置后的主成分2
[-0.07961523  0.66591982  0.68886836  0.09517681  0.21607987  0.14115045]

可以看出, 一行为一个主成分, 如果不转置, 实际上得到的是一列(非主成分向量).

实际上, c1 = V.T[:, 0]等价于c1 = V[0, :].

证明: 矩阵截取实验

import numpy as np

x = np.random.random([3, 3])
print(x)

print("="*8 + "\nx[:, 0]:")
print(x[:, 0])

print("="*8 + "\nx[0, :]:")
print(x[0, :])

print("="*8 + "\nx.T[:, 0]:")
print(x.T[:, 0])

输出:

[[0.75288091 0.75186556 0.70652022]
 [0.71672155 0.36091306 0.1308125 ]
 [0.71698569 0.95277093 0.25093763]]
========
x[:, 0]:
[0.75288091 0.71672155 0.71698569]
========
x[0, :]:
[0.75288091 0.75186556 0.70652022]
========
x.T[:, 0]:
[0.75288091 0.75186556 0.70652022]