[R] Create correlated data with skew

[bbolker]

Mike Lawrence wrote:
> 
> Hi all,
> 
> I understand that it is simple to create data with a specific  
> correlation (say, .5) using mvrnorm from the MASS library:
> 
>  > library(MASS)
>  > set.seed(1)
>  >
>  > a=mvrnorm(
> + 	n=10
> + 	,mu=rep(0,2)
> + 	,Sigma=matrix(c(1,.5,.5,1),2,2)
> + 	,empirical=T
> + )
>  > a
>              [,1]         [,2]
> [1,] -1.0008380 -1.233467875
> [2,] -0.1588633 -0.003410001
> [3,]  1.2054727 -0.620558768
> [4,]  1.9580971  2.389495155
> [5,] -0.9447473 -0.141852055
> [6,]  0.6236799 -0.826952659
> [7,]  0.1421782  0.452217611
> [8,] -0.9050954  0.330991444
> [9,] -0.7261632  0.217740460
> [10,] -0.1937206 -0.564203311
>  > cor(a)
>       [,1] [,2]
> [1,]  1.0  0.5
> [2,]  0.5  1.0
> 
> 
> But I'm looking to create data where the variables are non-normally  
> distributed (i.e. somewhat skewed). Any suggestions?
> 
> Mike
> 
> --
> Mike Lawrence
> Graduate Student, Department of Psychology, Dalhousie University
> 
> Website: http://memetic.ca
> 
> Public calendar: http://icalx.com/public/informavore/Public
> 
> "The road to wisdom? Well, it's plain and simple to express:
> Err and err and err again, but less and less and less."
> 	- Piet Hein
> 
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> 

The simplest (?) solution is probably to exponentiate your MVN data,
leading to a bivariate log-normal distribution.  The hard part is
specifying the parameters of the lognormal in terms of the desired
variance-covariance matrix.   Variances are not too bad, but correlation
may not be solvable.  (Of course, if you don't care much about the
precise characteristics of the simulated data and/or are willing to
use some trial and error to get the desired variance/correlation you
don't have to deal with this.)
See e.g.

http://www.stuart.iit.edu/faculty/workingpapers/thomopoulos/SomeMeasuresontheStandardBivariateLognormalDistribution.doc

 for some of the relevant formulas.

  good luck
    Ben Bolker

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