Non-rectangular region integration

Roofit and RooStats
1

Hi experts,
I would like to do a 2D functional integration with 2D bounds. That is, I would like to integrate over a non-rectangular region, with things like AdaptiveIntegratorMultiDim or GSLMCIntegrator. Is this possible? It seems that only rectangular regions are supported, but I want to make sure.

Many thanks,
Mike

2

some ideas: Boundary integral
some warnings: Creating TF1 from TGraph and Convolution of two TF1

3

That is very helpful. Since I have a slightly different problem, I run it past you:

The boundaries are different: I need x>1-0.5*y-0.5*sqrt(y^2-a^2). This is easily done by changing the example a bit. See below:

double f2(const double *X)
{
//BRUTAL METHOD:
//if(x<=1.0 - 0.5*y - 0.5*sqrt(y*y-a*a)) return 0;

//GOOD METHOD
  double f = 0.0;
  double t = X[0]; // "t" (i.e. the original "x")
  double y = X[1]; // "y"
  double x = (0.5*y + 0.5*sqrt(y*y-a*a)) * t + (1.0 - 0.5*y - 0.5*sqrt(y*y-a*a)); // calculate new "x"
  //when t=0, x = 1.0 - 0.5*y - 0.5*sqrt(y*y-a*a), i.e. the lower bound.
  //when t=1, x=1
  
  // calculate the original function "f(x,y)"
  f = 1.0/(x+y-1.0-a*a/4);
  
  f *= (0.5*y + 0.5*sqrt(y*y-a*a)); // multiply the result by "dx/dt"
  return f;
}

The real follow-up question is whether this is also applicable to TFoam densities. I see that it is applicable to all cases of simple integration, but TFoam is sampling from the distribution also. I see no reason why the method should not work (it is just calculus I guess). But do the experts see some problem with shifting basis in this way for the density function in TFoam? Explicitly, could I set the function above via TFoam->SetRho(f2), and expect better sampling than the brutal method?

Of course I will experiment with it, but any expert comments are welcome.

Many Thanks!

EDIT. I added part of the function I am sampling/integrating to show that there is at least a potential problem with divergence.

4

It seems that since my function is singular if you get too far out of the region, that the change of variables does not work.

The constraint x>1 - 0.5*y - 0.5*sqrt(y*y-a*a) results in x+y>1+a*a/4+O(a^4). So the constraint keeps the denominator from becoming singular. When I change variables, the formula is evaluated at a singular value.

I don’t immediately see how to avoid this. I keep looking. Any thoughts?

5

I was wrong about the singular values and solved it. (Was due to y values leaving valid regions).

I checked TFoam, and it seems I must apply the original transformation in the sampling phase if I expect to get sampled data in the original x,y.