[R] Generation of correlated variables

[(Ted Harding)]

On 15-Mar-2012 Filoche wrote:
> Hi everyone.
> 
> Based on a dependent variable (y), I'm trying to generate some
> independent variables with a specified correlation. For this
> there's no problems.
> However, I would like that have all my "regressors" to be
> orthogonal (i.e. no correlation among them).
> 
> For example, 
> 
> y = x1 + x2 + x3 where the correlation between y x1 = 0.7,
> x2 = 0.4 and x3 = 0.8.  However, x1, x2 and x3 should not be
> correlated to each other.
> 
> Anyone can help me?
> 
> Regards,
> Phil

Your fundamental problem here (with the correlations you specify)
is the following.

Your desired correlation matrix can be constructed by

  C <- cbind( c(1.0,0.7,0.4,0.8),c(0.7,1.0,0.0,0.0),
              c(0.4,0.0,1.0,0.0),c(0.8,0.0,0.0,1.0) )
  rownames(C) <- c("y","x1","x2","x3")
  colnames(C) <- c("y","x1","x2","x3")

  C
  #      y  x1  x2  x3
  # y  1.0 0.7 0.4 0.8
  # x1 0.7 1.0 0.0 0.0
  # x2 0.4 0.0 1.0 0.0
  # x3 0.8 0.0 0.0 1.0

And now:

  det(C)
  # [1] -0.29

and it is impossible for the determinant of a correlation
matrix to have a negative determinant: a correlation matyrix
must be positive-semidefinite, and therefore have a non-negative
determinant.

An alternative check is to look at the eigen-structure of C:

  eigen(C)
  # $values
  # [1]  2.1357817  1.0000000  1.0000000 -0.1357817
  # 
  # $vectors
  #           [,1]          [,2]       [,3]       [,4]
  # [1,] 0.7071068  0.000000e+00  0.0000000  0.7071068
  # [2,] 0.4358010 -1.172802e-16  0.7874992 -0.4358010
  # [3,] 0.2490291 -8.944272e-01 -0.2756247 -0.2490291
  # [4,] 0.4980582  4.472136e-01 -0.5512495 -0.4980582

so one of the eigenvalues (-0.1357817) is negative, again
impossible for a correlation matrix.

The suggestions made by the other respondents amount to
messing with the correlations so as to achieve a possible
situation; but then this does not achieve you goal (which
is impossible) but a different one, whose relationship
with your question may or may not suit what you hope to
achieve.

Ted.

-------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at wlandres.net>
Date: 15-Mar-2012  Time: 23:23:24
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