Need Normalization Before PCA

Normalization is important in PCA since it is a variance maximizing exercise. It projects your original data onto directions which maximize the variance. The first plot below shows the amount of total variance explained in the different principal components wher we have not normalized the data. As you can see it seems like only component one explains all the variance in the data.

If you look at the second picture we have normalized the data first. Here it is clear that the other components contribute as well. The reason for this is because PCA seeks to maximize the variance of each component. And since the covariance matrix of this particular dataset is:

             Murder   Assault   UrbanPop      Rape
Murder    18.970465  291.0624   4.386204  22.99141
Assault  291.062367 6945.1657 312.275102 519.26906
UrbanPop   4.386204  312.2751 209.518776  55.76808
Rape      22.991412  519.2691  55.768082  87.72916

From this structure the PCA will of course select to project as much as possible in the direction of Assault since that variance is much greater. So for finding features usable for any kind of model a PCA without normalization would perform badly.

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