## scivdp.R 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
set.seed(676,kind="default")  # Set seed to facilitate replication.
xvec <- rnorm(n*k)
X <- array(xvec, c(n,k))
X[,1] <- 1
X[,3] <- X[,1]+X[,2]+rnorm(n)*0.05
X[,5] <- X[,4]+rnorm(n)*0.05
## Step 2: analyze design matrix
X2 <- X^2
length <- (t(rep(1,n))%*%X2)^0.5  # Lengths of columns of design matrix
lm <- rep(1, n)%*%length
scaledX <- X/lm  # Scaled design matrix has columns of unit length.
svdsX <- svd(scaledX)  # Singular value decompositon
mu <- svdsX$d  #  The singular values of X
etatilde <- mu[1]/mu  # Scaled condition indices of X.  Each large condition
                      # index corresponds to a near linear dependence.
T <- svdsX$v  # Eigenvectors of X'X
t2 <- t(T)*t(T)
mu2 <- mu^2
mu2mat <- mu2%*%t(rep(1,k))
tom <- t2/mu2mat
vif <- t(rep(1,k))%*%tom  # variance inflation factors
vifmat <- rep(1,k)%*%vif
vdp <- tom/vifmat  # variance decomposition proportions
rep(1,k)%*%vdp  # Check that proportions sum to one.
table <- cbind(etatilde, vdp)
table 
## 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.