I am trying to perform PCA on some data. From my knowledge, the correlation matrix should have entries of 1 along the main diagonal. This is not what I am seeing for .get_covariance() in sklearn PCA. I am wondering why this is the case?
For my own purposes, I can just perform a scaling of the matrix to obtain a matrix with diagonal entries of 1, but I was just wondering, since I have already standardized my data, why are the diagonal entries still not 1?
In [1]: import pandas as pd
In [2]: import numpy as np
In [3]: from sklearn.decomposition import PCA
In [4]: df = pd.read_csv('myTable.csv')
In [5]: df
Out[5]:
a1 a2 a3 a4 a5
0 -0.559104 0.185914 -2.331367 0.231150 0.357008
1 0.769835 -0.408685 0.375754 0.051397 -0.075885
2 -1.376530 -0.764808 -2.383611 -0.327153 1.746765
3 -0.830105 -0.197574 1.835807 -0.695089 0.881297
4 -0.991861 1.089319 -0.164139 -0.335003 0.795937
5 -1.132968 -2.240598 -0.101935 0.680038 -0.033921
6 -1.205631 -1.492009 -0.602400 -0.065256 -0.494267
7 -1.210978 -1.220986 -0.017062 0.024422 -0.224585
8 -0.332957 2.114870 0.818108 0.612831 -1.879758
9 -0.350612 -0.563872 0.869303 -0.325626 -0.372874
In [6]: df = (df-df.mean())/df.std()
In [7]: pca = PCA()
In [8]: pca.fit(df)
Out[8]: PCA(copy=True, n_components=None, whiten=False)
In [10]: pca.explained_variance_, pca.components_, pca.get_covariance()
Out[10]:
(array([ 1.8780651 , 1.1526052 , 0.78052872, 0.55167761, 0.13712337]),
array([[-0.47790108, -0.36036503, -0.38619941, -0.35716396, 0.60417838],
[ 0.25426743, 0.32305024, 0.47784502, -0.72831952, 0.26870322],
[-0.17613902, -0.7303121 , 0.6250759 , -0.05118019, -0.20562097],
[ 0.82132736, -0.45982165, -0.21938834, 0.03274499, 0.25452296],
[ 0.03681087, -0.14485808, -0.42855924, -0.58162955, -0.67505936]]),
array([[ 0.9 , 0.30943895, 0.29916112, 0.12605405, -0.32333097],
[ 0.30943895, 0.9 , 0.14715469, 0.00295615, -0.24279645],
[ 0.29916112, 0.14715469, 0.9 , -0.13683409, -0.38167791],
[ 0.12605405, 0.00295615, -0.13683409, 0.9 , -0.56418468],
[-0.32333097, -0.24279645, -0.38167791, -0.56418468, 0.9 ]]))
Closed
The problem was with my standardization. I was supposed to use df.std(ddof=0) as suggested by Tonechas
You need to normalize the standard deviation by N rather than by N-1 (which is the default value). This can be changed using the ddof parameter in the call to pandas.DataFrame.std() like this:
In [146]: from sklearn.decomposition import PCA
In [147]: df
Out[147]:
a1 a2 a3 a4 a5
0 -0.559104 0.185914 -2.331367 0.231150 -0.559104
1 0.769835 -0.408685 0.375754 0.051397 0.769835
2 -1.376530 -0.764808 -2.383611 -0.327153 -1.376530
3 -0.830105 -0.197574 1.835807 -0.695089 -0.830105
4 -0.991861 1.089319 -0.164139 -0.335003 -0.991861
5 -1.132968 -2.240598 -0.101935 0.680038 -1.132968
6 -1.205631 -1.492009 -0.602400 -0.065256 -1.205631
7 -1.210978 -1.220986 -0.017062 0.024422 -1.210978
8 -0.332957 2.114870 0.818108 0.612831 -0.332957
9 -0.350612 -0.563872 0.869303 -0.325626 -0.350612
In [148]: df = (df-df.mean())/df.std(ddof=0)
In [149]: pca = PCA()
In [150]: pca.fit(df)
Out[150]:
PCA(copy=True, iterated_power='auto', n_components=None, random_state=None,
svd_solver='auto', tol=0.0, whiten=False)
In [151]: pca.get_covariance()
Out[151]:
array([[ 1. , 0.34, 0.33, 0.14, 1. ],
[ 0.34, 1. , 0.16, 0. , 0.34],
[ 0.33, 0.16, 1. , -0.15, 0.33],
[ 0.14, 0. , -0.15, 1. , 0.14],
[ 1. , 0.34, 0.33, 0.14, 1. ]])