% scivdp.m generates and analyzes nearly collinear data.
% The key concepts--singular value decomposition and variance decomposition
% proportions--are explained in D. Belsley, E. Kuh, and R. Welsch, Regression 
% Diagnostics (1980) and D. Belsley, Conditioning Diagnostics (1991).  Some 
% of the Matlab code used here is taken from bkw.m, written by J. LeSage.

% Step 1: generate nxk design matrix X with two near dependencies
n=100; k=5;
randn('state',0)         % Return random number generator to 
                         % original state to facilitate
                         % replication.
X=randn(n,k);
X(:,1)=ones(n,1);
X(:,3)=X(:,1)+X(:,2)+randn(n,1)*0.05; 
X(:,5)=X(:,4)+randn(n,1)*0.05;
% Step 2:  analyze design matrix
X2 = X.^2;
length = sum(X2).^.5;     % Lengths of columns of design matrix
lm = ones(n,1)*length;
scaledX = X./lm;          % Scaled design matrix has columns of unit length.
[U,D,T] = svd(scaledX,0); % Singular value decomposition
D                         % Print the diagonal matrix D.
T                         % Print eigenvectors of X'X.
mu = diag(D)              % Put the singular values of X in vector. 
mumax = mu(1)           
mumaxvec = mumax*ones(k,1)
etatilde = mumaxvec./mu % Scaled condition indexes of X.  Each large condition
                        % index corresponds to a near linear dependence.
t2 = T'.*T'
mu2 = mu.^2             % Eigenvalues of X'X.  
mu2mat = mu2*ones(1,k)
tom = t2./mu2mat
vif = sum(tom)'         % Variance inflation factors.
vifmat = ones(k,1)*vif'
vdp = tom./vifmat       % Variance decomposition proportions.
sum(vdp)                % Check that proportions sum to one.
table = [etatilde vdp]       
% A large  element in the first column indicates a near linear dependence.
% Large elements to its right indicate which variables are involved in this
% near dependence.