/* This Gauss program 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). */ 

/* Step 1: generate nxk design matrix X with two near dependencies */
n = 100; k = 5;
rndseed 1;              /* Reset generator's seed to original state  
                        ** to facilitate replication. */
x=rndn(n,k); 
x[.,1]=ones(n,1);
x[.,3]=x[.,1]+x[.,2]+rndn(n,1)*0.05; 
x[.,5]=x[.,4]+rndn(n,1)*0.05;
x2 = x.^2;
length = sumc(x2)^.5;
lm = ones(n,1)*length';
scaledx = x./lm; 

/* Step 2:  analyze design matrix */
print on;
{U,D,T} = svd2(scaledX); /* Singular value decomposition of the design matrix */
D;                       /* Print the diagonal matrix D */ 
T;                       /* Print eigenvectors of X'X */
mu = diag(D);            /* Put the singular values of X in vector mu */
mumax = mu[1];           /* Assign maximum singular value to mumax */
mumaxvec = mumax*ones(k,1);
eta = mumaxvec./mu;      /* Condition indices 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 = sumc(tom);         /* Variance inflation factors */
vifmat = ones(k,1)*vif';
vdp = tom./vifmat;       /* Variance decomposition proportions */
sumc(vdp);               /* Check that proportions sum to one */
table = eta~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 */