Comments on: Random Correlation Matrices

By: Jorge Durango

Jorge Durango — Mon, 11 Oct 2010 05:23:37 +0000

Can you help me to improve this algorithm did in R

# Descriptive measures of multivariate scatter and linear dependence
hp = function(p){
#x<-seq(0.01,2.99,length=30)
#malla<-expand.grid(x,x) ### Malla (0,3)^2
### Parametros de la distribución beta (ai,Bi)
alfa1<-runif(900,0,3)
beta2<-runif(900,0,3)
malla<-cbind(alfa1,beta2) ### (0,3)^2
bet<-function(x) rbeta(p,x[1],x[2]) ### Generación aleatoria de la distribución Beta
vp<-apply(malla,1,bet) ### Valores propios
sumas<-apply(vp,2,sum)
vpN<-p*vp/matrix(sumas,ncol=900,nrow=p,byrow=T) ### Valores propios normalizados
dete<-apply(vpN,2,prod)
De<- 1-(dete)^(1/p)
f1=0.7)/p ### escogencia del numero de componentes principales
propor<-apply(vpN,2,f1)
list("De" = De, "hsp" = propor)
}
hsp<-De<-numeric(0)
for(i in seq(40,440,40)){
hspi<-hp(i)
hsp<-c(hsp,hspi$hsp)
De<-c(De,hspi$De)
}
hsp; De
plot(hsp~De, pch=20,ylab="h/p",xlab="De(X)") ### se observa que la relacion emtre h/p y De es un Sigmoideo
### La función sigmoide: Su gráfica tiene una típica forma de "S".
### A menudo la función sigmoide se refiere al caso particular de la función logística
### Aproximacion del modelo lineal
AML<-aov(hsp~De)
AML$coefficients
summary(AML)
plot(De,hsp,pch=20, ylab="h/p",xlab="De(X)",xlim=c(0.1,0.9))
abline(lm(hsp~De),v=c(0.1,0.9))

By: Kornelius Rohmeyer

Kornelius Rohmeyer — Thu, 23 Sep 2010 12:20:14 +0000

Take a look at HOLMES or BENDEL AND MICKEY which are proposing the following:

Start with a diagonal matrix with the eigenvalues you are looking for:

D <- diag(rbeta(p,a,b))

Then take a random orthogonal matrix M and create B=MDM’. This matrix will be unfortunately no correlation matrix. Therefore determine a product of elementary orthogonal rotation matrices P such that PBP’ has only ‘ones’ along the diagonal. This is then a random correlation matrix with the wanted spectrum.

By: Jorge Durango Borja

Jorge Durango Borja — Sun, 19 Sep 2010 04:51:21 +0000

Excuse my Englishman I am of colombia believe that you can help me with this:
We want to study the relationship between the De of the sample and the proportion of components, h/p; needed to explain 90% of the total variability. We carried out a simulation study by generating random correlation matrices of dimension p as follows:

1. the eigenvalues of the correlation matrix are drawn from a Beta(a, b) distribution, with a and b chosen froma grid in the interval (0,3)^2; obtaining 900 pairs of parameters a, b.

2. The values are normalized so that their sum is p: For each fixed value p; we generated 900 matrices. This process was performed for p=40, 80…440; so that 9900 correlation matrices were generated in total.

For each one of these matrices, we calculate (h/p)e[0,1] and the De(X): We observed that the relation between (h/p) and De(X) is a sigmoid, but in the interval De(X)e[0.1; 0.9] we can approximate it by the linear relation
h/p= 0.8230 -0.492De(X)