Question about different results using PCA and SVD

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HpW 2022-8-16

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https://ww2.mathworks.cn/matlabcentral/answers/1780365-question-about-different-results-using-pca-and-svd

评论： HpW 2022-8-16

采纳的回答： Steven Lord

在 MATLAB Online 中打开

Hello

I'm in the process of larning about SVD and PCA, and have been experimenting by using Matlab

I have a 3D set of points as a nx3 column vector called XYZ_orig

I subtracted the mean from X, Y, and Z and perform SVD and PCA on the data:

cent=[mean(XYZ_orig(:,1)), mean(XYZ_orig(:,2)), mean(XYZ_orig(:,3))];
XYZ = [XYZ_orig(:,1)-cent(1),XYZ_orig(:,2)-cent(2),XYZ_orig(:,3)-cent(3)];

[U,S,V] = svd(XYZ);
[coeff,~,~,~,~,~] = pca(XYZ);

my understanding is the basis vectors/coefficients for SVD in 'V' and PCA in 'coeff' should be the same, however, although the 1st column is always the same, I am finding that the second and third columns are always of different sign with the same coefficients.

V =
0.5926   -0.2694   -0.7591
0.7193   -0.2470    0.6493
-0.3624   -0.9308    0.0474

coeff =
0.5926    0.2694    0.7591
0.7193    0.2470   -0.6493
-0.3624    0.9308   -0.0474

Can someone explain why there is this sign difference for columns 2 and 3?

thanks!

hpw

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Answer 1

Steven Lord 2022-8-16

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https://ww2.mathworks.cn/matlabcentral/answers/1780365-question-about-different-results-using-pca-and-svd#answer_1027635

From the Algorithms section on the pca documentation page: "The pca function imposes a sign convention, forcing the element with the largest magnitude in each column of coefs to be positive. Changing the sign of a coefficient vector does not change its meaning."