Problem reconstructing retained principal components

Hi,

I think I am having an issue reconstructing retained principal components. I have a data set called data, size(40,101). So, I have 40 observations (subjects) and 101 variables (PCs). I would like to reconstruct the signal using all 101 pcs, then only PC1, PC2, and PC3. Here is my code:

[coeff, score, eigval] = princomp(data); subjects = 40;

pc_all = repmat(mean((data),1),subjects,1) + score(:,:)*coeff(:,:)';%all PC

pc1 = repmat(mean((data),1),subjects,1) + score(:,1)*coeff(:,1)';%PC1

pc2 = repmat(mean((data),1),subjects,1) + score(:,2)*coeff(:,2)';%PC2

pc3 = repmat(mean((data),1),subjects,1) + score(:,3)*coeff(:,3)';%PC3

When I take the mean(sd) for each PC, I end up with the same mean for each variable but a different sd.

ensem_pc_all(:,1) = mean(pc_all(1:subjects,:),1);

ensem_pc_all(:,2) = std(pc_all(1:subjects,:),0,1);

ensem_pc1(:,1) = mean(pc1(1:subjects,:),1);

ensem_pc1(:,2) = std(pc1(1:subjects,:),0,1); %...etc

Is there a reason the mean does not change depending on the number of components I retain?

Thanks, Eric

Answer 1

[Ilya]

PCA is applied to centered data. Observe that mean(score) is close to a vector of zeros. Rotating the scores back to the original coordinates should also produce vectors with means close to zero. For all pc's defined as they are above, the mean should be fairly close to mean(data).

[Eric]

I understand what you are saying, but in my case PC_all is identical to PC1, PC2, and PC3 to the fourth decimal place when I included all 40 observations in my mean ensemble calculations.

[Ilya]

Do you mean

all(PC_all(:)==PC1(:))

is true or

all(mean(PC_all)==mean(PC1))

is true? If the latter, this is due to centering, as I said.

[Eric]

The latter is true. It just seems odd that for the entire sample that regardless of how many PCs are retained and compared there is the same ensemble mean for PC3(:) as PC_all(:).

[Ilya]

Suppose you center your data. Or, to put it differently, move the coordinate origin to the computed center of the dataset. After you do that, the mean of observed values for any variable is zero. Equivalently, if you draw a vector from the new origin to each point in the dataset and add all the vectors, you will get a vector of zeros. The PCA transformation is an orthogonal rotation with respect to the new origin. An orthogonal rotation applied to a zero vector returns a zero vector. The mean of any variable, either in the (centered) space of original variables or in the rotated space, is zero.