%Candace Metoyer
%Stat 135
%Chapter 9 Demos

%EXAMPLE 9.3
%Factor Analysis of Consumer Preference Data

clear;
clc;
format compact;
format short g;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Example 9.3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

p = 5;

%Enter the correlation matrix

R1 = [1 .02 .96 .42 .01];
R2 = [.02 1 .13 .71 .85];
R3 = [.96 .13 1 .5 .11];
R4 = [.42 .71 .50 1 .79];
R5 = [.01 .85 .11 .79 1];

R = [R1;R2;R3;R4;R5];

%Obtain the eigenvalues and eigenvectors

[eigVEC, eigVAL] = eig(R);

lambda = zeros(p,1);
ReigVEC = zeros(size(eigVEC));
j = 1;

for i = p:-1:1      %Decrease from p to 1 in steps of 1
    lambda(i) = eigVAL(j,j);
    ReigVEC(:,i) = eigVEC(:,j);        %re-ordered eigenvector matrix
    j = j+1;
end

lambda;
ReigVEC;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Compute the cumulative percentage of the total variance
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

for i = 1:p
    cpercent(i) = sum(lambda(1:i))/sum(lambda);
end

cpercent

%Compute the estimated factor loadings using m = 2

elltilde = [sqrt(lambda(1))*ReigVEC(:,1) sqrt(lambda(2))*ReigVEC(:,2)]

%Compute the communalities and specific variances

htilde2 = zeros(p,1);
psitilde = zeros(p,p);

for i = 1:p
    htilde2(i) = (norm(elltilde(i,:)))^2;
    psitilde(i,i) = 1 - htilde2(i);
end

lambda'     %the eigenvalues
htilde2
psitilde

%Compare the estimate of R with the original matrix

R

Rest = elltilde*elltilde' + psitilde