[R] generate two sets of random numbers that are correlated

[Duncan Murdoch]

On 11/08/2011 1:33 PM, Duncan Murdoch wrote:
> On 11/08/2011 12:01 PM, Kathie wrote:
> >  almost forgot. In fact, I want to generate correlated Poisson random vectors.
>
> Saying you want two random variables to be correlated doesn't specify
> the joint distribution, so there will be a lot of solutions.  Here's
> one, for the case where both variables have the same mean mu, and you
> want a positive correlation.
>
> We know that the sum of independent Poissons is Poisson, so we'll
> generate 3 variables: X with mean nu, and Y&  Z with mean mu-nu, and return
> A = X+Y and B = X+Z.  If nu=0 then A and B are independent, and if
> nu=mu, they have correlation 1, so you must be able to solve for a value
> where they have any desired correlation in between.
>
> If the means aren't the same, this method will still work up to a point,
> but you won't be able to get really high correlations.
>
> If you want negative correlations it's harder, but you could use the
> following trick:  Generate U ~ Unif(0, 1).  Calculate A by the inverse
> CDF method from U.  Compute V to be equal to U if U<  a or U>  1-a, and
> equal to 1-U otherwise.  Calculate B by the inverse CDF method on V.
>
> Then both U and V will have Poisson distributions (and you can choose

I meant A and B in the line above...

> the means as you like), and there will be some range of achievable
> correlations which will be quite close to [-1, 1].  The joint
> distribution will be very weird, but you didn't say that was a problem...
>
> Some R code:
>
> U<- runif(10000)
> A<- qpois(U, 5)
> a<- 0.115
> V<- ifelse(U<  a | U>  1-a, U, 1-U)
> B<- qpois(V, 5)
> cor(A, B)
>
> This gives a correlation around 0.4.
>
>
> Duncan Murdoch