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Do anyone know what is the algorithm that MATLAB used in its built-in function "pca"?

I have the following data set:

148.9820 55.8438 210.2150

149.3030 56.8891 208.4280

151.4400 55.8180 208.9240

146.5530 55.9677 211.5800

146.5770 57.2682 209.3680

145.5330 58.4735 207.6970

153.9680 55.8386 207.9600

143.6960 57.2371 211.1020

152.5960 57.3995 206.2770

144.1070 56.1439 212.9730

149.6670 58.8746 205.1560

142.8440 58.1240 209.7220

143.2190 59.3990 207.0410

146.3050 60.2445 204.1980

156.7100 55.9361 207.4610

141.0470 57.3240 212.4660

where the number of rows are number of observations and each observation is of dimension 3.

I want to perform principal component analysis on this data set so I wrote

 P = pca(A)

where A is the above matrix. The answer I got is

    0.9480    0.2104    0.2387

   -0.0980   -0.5204    0.8483

   -0.3027    0.8276    0.4727

However, when I use the following program:

function [evects,evals] = pca_test(dataset)

if (size(dataset,1)>size(dataset,2))

    dataset = dataset';

end

N = size(dataset,2);

mm = mean(dataset,2);

dataset = dataset - mm*ones(1,N);

cc = cov(dataset',1);

[cvv,cdd] = eig(cc);

[~,ii] = sort(diag(cdd));

ii = flip(ii,1);

evects = cvv(:,ii);

cdd = diag(cdd);

evals = cdd(ii);

it gives

evects = 

-0.9480    0.2104    0.2387
0.0980   -0.5204    0.8483
0.3027    0.8276    0.4727

The first column is of opposite sign to the result generated by the built-in pca. Why is there such a change?

I ask this because I think the matlab built-in pca is really slow. The pca_test above is around 3 times faster than the built-in function. But I want it to have exactly the same result as the built-in one. Can anyone help?

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    $\begingroup$ You can read about MATLAB's implementation on their website. PCA usually involves SVD, and singular vectors are only known up to sign (for real-valued matrices). This means that the choice of sign is arbitrary. $\endgroup$
    – Bill Barth
    Sep 8, 2014 at 14:24
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    $\begingroup$ You can use the command 'edit pca' to open up the pca.m file in Matlab's editor. From there you can look to see exactly how it works. $\endgroup$
    – Nick Alger
    Sep 8, 2014 at 16:11

1 Answer 1

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Principal Components Analysis (PCA) is conducted using a Singular Value Decomposition (SVD) algorithm. As Bill Barth says above, the choice of sign of the principal component vectors is entirely arbitrary.

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