Hi Sneha,
I understand that you want an explanation of the MATLAB code implementing Principal Component Analysis (PCA) as defined in the function "pca1".
The function "pca1" takes a matrix "A" as input and computes the PCA, returning the reconstructed matrix "Z", the compressed principal components "PCred", the eigenvectors "Vp", and the eigenvalues "Di".
The code begins by defining and initializing some variables and matrices. The input matrix "A" is defined with specific values, and its size is determined.
A = [3 2 1; 4 5 6; 7 8 9];
[n, m] = size(A);
Amean = mean(A);
Astd = std(A);
The matrix "A" is standardized by subtracting the mean and dividing by the standard deviation for each column. This results in the standardized matrix "B".
B = (A - repmat(Amean, [n 1])) ./ repmat(Astd, [n 1]);
% "repmat(Amean, [n 1])" replicates the mean vector to match the size of "A".
% "repmat(Astd, [n 1])" replicates the standard deviation vector to match the size of "A".
The eigenvectors "Vp" and eigenvalues "Di" of the covariance matrix of "B" are computed. These are used to determine the principal components.
[Vp, Di] = eig(cov(B));
The cumulative sum of the eigenvalues is calculated to determine the percentage of variance explained by each principal component.
C = cumsum(flipud(diag(Di))) / sum(diag(Di));
% "flipud(diag(Di))" flips the diagonal elements of "Di" upside down.
% "cumsum(...)" computes the cumulative sum.
The original principal components are computed by multiplying the standardized matrix "B" with the eigenvectors "Vp".
PC = B * Vp;
The third column of the eigenvectors "Vp" is selected for compression, and the compressed principal components "PCred" are computed.
Vpred = Vp(:, 3);
PCred = B * Vpred;
The decompressed matrix "PCdecomp" is computed by projecting back the compressed components, and the reconstructed matrix "Z" is obtained by reversing the standardization process.
PCdecomp = PCred * Vpred';
Z = (PCdecomp .* repmat(Astd, [n 1])) + repmat(Amean, [n 1]);
Hope this helps!
