21 jun. 2015

The loading matrix “P” : a good example of orthogonal matrix

We know that for an orthogonal matrix A:

At.A=A.At=I

When we calculate the loading matrix during the PCA process, each loading is orthogonal (perpendicular) to all others. So we can check for fun in R, Excel,…., this condition with the loading matrix.

P is a very large matrix, so we will check it with just a few columns (6 loadings or terms) and the same number of files (6 wavelengths):
 
> round(gas.loadings[1:6,1:6],digits=4)
          PC1   PC2   PC3    PC4   PC5    PC6
900 nm -0.011 0.022 0.034 -0.039 0.042 -0.020
902 nm -0.010 0.022 0.031 -0.041 0.039 -0.022
904 nm -0.011 0.022 0.030 -0.042 0.036 -0.021
906 nm -0.012 0.024 0.027 -0.045 0.031 -0.012
908 nm -0.013 0.021 0.025 -0.045 0.035 -0.013
910 nm -0.014 0.023 0.023 -0.046 0.036 -0.018
 
Pt is the transpose, so the columns are the wavelengths and the files the loadings:
 
> round(t(gas.loadings[1:6,1:6]),digits=4)
    900 nm 902 nm 904 nm 906 nm 908 nm 910 nm
PC1 -0.011 -0.010 -0.011 -0.012 -0.013 -0.014
PC2  0.022  0.022  0.022  0.024  0.021  0.023
PC3  0.034  0.031  0.030  0.027  0.025  0.023
PC4 -0.039 -0.041 -0.042 -0.045 -0.045 -0.046
PC5  0.042  0.039  0.036  0.031  0.035  0.036
PC6 -0.020 -0.022 -0.021 -0.012 -0.013 -0.018

Now we multiply the two matrix:

> round((gas.loadings[1:6,1:6])%*% solve((gas.loadings[1:6,1:6])),digits=4)
       900 nm 902 nm 904 nm 906 nm 908 nm 910 nm
900 nm      1      0      0      0      0      0
902 nm      0      1      0      0      0      0
904 nm      0      0      1      0      0      0
906 nm      0      0      0      1      0      0
908 nm      0      0      0      0      1      0
910 nm      0      0      0      0      0      1
 

> round(((solve(gas.loadings[1:6,1:6]))%*%(gas.loadings[1:6,1:6])),digits=4)
    PC1 PC2 PC3 PC4 PC5 PC6
PC1   1   0   0   0   0   0
PC2   0   1   0   0   0   0
PC3   0   0   1   0   0   0
PC4   0   0   0   1   0   0
PC5   0   0   0   0   1   0
PC6   0   0   0   0   0   1

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