[R] R: generating independent vectors

[Clark Allan]

hi all

i would like to generate independent vectors. i have included the code
below. i display the correlation matrix of the n*p (n=the number of
samples, p= the number of variables) matrix. 
what i find is that as n increases, the correlation matrix tends to an
identity matrix. i.e. independence. for small samples (see examples
below, for n=20, p=5 ; n=50,p=5, n=100,p=5; n=10000,p=5) the variables
does not appear to be independent. note that i have not tested this
statement statistically.

ARE these variables independent? by setting the seed for each variable
run i HOPE that the variables are now independent. IS this true??? if
not does anyone know how to generate these independent variables?

how does R generate its random variables? does it use the box muller
technique? if so how does it generate the random uniform variables?

thanking you in advance!!!
***
allan



IND<-function(rows.=10,col.=3)
{
a<-matrix(nrow=rows.,ncol=col.)
for (j in 1:col.)
{
	set.seed(j)
	r<-rnorm(rows.)
	a[,j]<-r
}
#list(a=a,cor=cor(a))
cor(a)
}
IND(rows.=10,col.=3)



> IND(rows.=20,col.=5)
            [,1]        [,2]        [,3]        [,4]       [,5]
[1,]  1.00000000  0.25677165 -0.14882130 -0.09190797  0.1562481
[2,]  0.25677165  1.00000000  0.02585515  0.13735712 -0.1443301
[3,] -0.14882130  0.02585515  1.00000000 -0.11311416  0.1437001
[4,] -0.09190797  0.13735712 -0.11311416  1.00000000  0.1833647
[5,]  0.15624807 -0.14433011  0.14370006  0.18336467  1.0000000


> IND(rows.=50,col.=5)
              [,1]        [,2]          [,3]        [,4]        [,5]
[1,]  1.0000000000  0.07915025  0.0009239851 -0.14102117 -0.07335342
[2,]  0.0791502463  1.00000000 -0.1764530631  0.10021081  0.19742285
[3,]  0.0009239851 -0.17645306  1.0000000000  0.02968062  0.14543350
[4,] -0.1410211698  0.10021081  0.0296806188  1.00000000  0.07234953
[5,] -0.0733534183  0.19742285  0.1454335014  0.07234953  1.00000000
> 

> IND(rows.=100,col.=5)
            [,1]       [,2]         [,3]         [,4]       [,5]
[1,]  1.00000000 -0.1537208 -0.023741715 -0.135245915 0.01961224
[2,] -0.15372076  1.0000000 -0.141796984  0.157219334 0.15518443
[3,] -0.02374171 -0.1417970  1.000000000  0.005865698 0.19118563
[4,] -0.13524592  0.1572193  0.005865698  1.000000000 0.07345299
[5,]  0.01961224  0.1551844  0.191185627  0.073452993 1.00000000
> 

> IND(rows.=10000,col.=5)
             [,1]         [,2]         [,3]         [,4]         [,5]
[1,]  1.000000000  0.015928444 -0.008288940 -0.005646904  0.006936662
[2,]  0.015928444  1.000000000 -0.005444611  0.005242395 -0.008246009
[3,] -0.008288940 -0.005444611  1.000000000  0.007277489  0.012299247
[4,] -0.005646904  0.005242395  0.007277489  1.000000000  0.001918704
[5,]  0.006936662 -0.008246009  0.012299247  0.001918704  1.000000000