Generating correlated numbers

Question

Here is a fun one: I need to generate random x/y pairs that are correlated at a given value of Pearson product moment correlation coefficient, or Pearson r. You can imagine this as two arrays, array X and array Y, where the values of array X and array Y must be re-generated, re-ordered or transformed until they are correlated with each other at a given level of Pearson r. Here is the kicker: Array X and Array Y must be uniform distributions.

I can do this with a normal distribution, but transforming the values without skewing the distribution has me stumped. I tried re-ordering the values in the arrays to increase the correlation, but I will never get arrays correlated at 1.00 or -1.00 just by sorting.

Any ideas?

--

here is the AS3 code for random correlated gaussians, to get the wheels turning:

public static function nextCorrelatedGaussians(r:Number):Array{             
         var d1:Number;
         var d2:Number;
         var n1:Number;
         var n2:Number;
         var lambda:Number;
         var r:Number;
         var arr:Array = new Array();
         var isNeg:Boolean; 

        if (r<0){
            r *= -1;
              isNeg=true;
        }            
        lambda= (   (r*r)  -  Math.sqrt(  (r*r) - (r*r*r*r)  )     )   /   ((  2*r*r ) - 1  );

        n1 = nextGaussian();
        n2 = nextGaussian();           
        d1 = n1;            
        d2 = ((lambda*n1) + ((1-lambda)*n2)) / Math.sqrt( (lambda*lambda) + (1-lambda)*(1-lambda));

        if (isNeg) {d2*= -1}           
        arr.push(d1);
        arr.push(d2);
        return arr;
    }

Answer 1

Here is an implementation of of twolfe18's algorithm written in Actionscript 3:

for (var j:int=0; j < size; j++) {
 xValues[i]=Math.random());
}
var varX:Number = Util.variance(xValues);
var varianceE:Number = 1/(r*varX) - varX;

for (var i:int=0; i < size; i++) {
 yValues[i] = xValues[i] + boxMuller(0, Math.sqrt(varianceE));
}

boxMuller is just a method that generates a random Gaussian with the arguments (mean, stdDev). size is the size of the distribution.

Sample output

Target p: 0.8
Generated p: 0.04846346291280387
variance of x distribution: 0.0707786253165176
varianceE: 17.589920412141158

As you can see I'm still a ways off. Any suggestions?

Answer 2

I ended up writing a short paper on this

It doesn't include your sorting method (although in practice I think it's similar to my first method, in a roundabout way), but does describe two ways that don't require iteration.