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PCA scaling and centering documentation wrong?

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Ari Paul
Ari Paul on 24 Mar 2015
Answered: the cyclist on 26 Jun 2019
The pca() documentation says that the raw data is automatically centered at the start of the process. If true, then pca(X) should be equal to pca(Y), where Y = centered data. But they're not (specific data below). Additionally, when I use either eig() or svd() to compute the principal components, I can only get them to match the pca output when I first manually center the data before using pca(). Ultimately my question is simply how do I correctly calculate the principal components of raw data? I.e. do I need to manually center and scale it first? Only manually center? Only manually scale?
Sample data: X =
1.0000 -3.0000 -1.0000; 2.0000 -2.0000 -0.5000; 3.0000 -0.5000 0.2500; 4.0000 2.0000 1.0000; 5.0000 5.0000 2.5000;
Centering X -> Y= -2.0000 -3.3000 -1.4500; -1.0000 -2.3000 -0.9500; 0 -0.8000 -0.2000; 1.0000 1.7000 0.5500; 2.0000 4.7000 2.0500;
pca(X) = -0.7360 -0.6037 -0.3062; -0.6688 0.7186 0.1907; -0.1049 -0.3452 0.9327;
pca(Y) =
0.4058 0.8414 0.3569
0.9124 -0.3960 -0.1036
0.0542 0.3676 -0.9284
svd(Y) = 0.4058 0.9124 0.0542; 0.8414 -0.3960 0.3676; 0.3569 -0.1036 -0.9284;
eig(cov(Y)) = 0.0542 0.9124 0.4058; 0.3676 -0.3960 0.8414; -0.9284 -0.1036 0.3569; ^this is the same output just in a different order.

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Answers (2)

Sagar
Sagar on 9 Aug 2015
You got it little wrong. When you do PCA(Y), by default, PCA again centers the data. So if you want to get the same values as PCA(X), use 'centered', 'off' name-value pair option: PCA_of_Y = PCA (Y, 'centered', 'off'); Now it will definitely be equal to PCA(X).

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the cyclist
the cyclist on 26 Jun 2019
Answering a gazillion years after-the-fact, because I just turned this up in my own search.
X = [1.0000 -3.0000 -1.0000; 2.0000 -2.0000 -0.5000; 3.0000 -0.5000 0.2500; 4.0000 2.0000 1.0000; 5.0000 5.0000 2.5000];
Y = X - mean(X);
pca(X)
pca(Y)
both give the same PCA results for me, in MATLAB Online (as of when I answered this).
So, either something got fixed, or you made a mistake.

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