Why pca() returns 1 less principal component?

    a = diag([1 2 3]);
    disp(pca(a));

I expected that it will display 3 principal components, since my matrix "a" is full-rank 3-by-3 one. However, only two PCs are displayed. Actually, when I do the same in R, it does give me 3 PCs, the first two of which are exactly the same as the two that matlab provides.

However, when the number of row exceeds the number of columns, pca() function seems to work well, for example:

a = [diag([1 2 3]); 0, 0, 4];
disp(pca(a));

Is it because I didn't use pca() properly, or do I need to specify the style for the result to be displayed? Thanks...

Answer 1

[KSSV]

See you need not to use inbuilt function for pca. Pca is straight forward with few steps. Check the below pca code with out using inbuilt pca function.

a = diag([1 2 3]);
% disp(pca(a));
A = cov(a) ;   % co-variance matrix 
[V,D] = eig(A) ;  % GEt Eigenvalues and Eigenvectors 
Eig = diag(D) ;
[val,idx] = sort(Eig,'descend') ;
PV = Eig(idx)
PC = V(:,idx)
disp(pca(a))

As you see one of the principal values (Eigenvalues) is zero. So pca is not displaying it.

[Zachary L]

Thank you for your answer!

So pca() doesn't use the raw input 'a' but the covariance 'cov(a)' instead, that makes sense. I see where my misunderstanding is, thank u!

[KSSV]

Yes..... pca acts on the covariance matrix. See this tutorial http://www.cs.otago.ac.nz/cosc453/student_tutorials/principal_components.pdf. It is amazing and very simple one to understand.

Answer 2

[Soma Mbadiwe]

   [coeff,score,latent] = pca(X, 'Economy',false);

The code above will tell MATLAB to return all the PCs and PVs even if some are zero.

No need re-inventing the wheel, especially because MATLAB's 'pca' function can give you so much more than just the PCs (coeff) and PVs (latent).

See documentation

PS: I know this is more of an answer to the title of this question than the question itself, but Google brings this page up quite often.

[Changoleon]

This is correct! Thank you for the clarification. If the number of observation is less than variables, pca function must be used with Economy as false.