[EM] Spatial models -- Polytopes vs Sampling

[Daniel Carrera]

On Tue, Feb 1, 2022 at 5:43 AM Kristofer Munsterhjelm <km_elmet at t-online.de>
wrote:

> I'm hardly an expert on these matters, but I found a paper suggesting a
> way to transform a simplex into a hypercube (by, I think, using extended
> barycentric coordinates as a change of bases). If that's a workable
> approach, then you could decompose the polytope into simplices,
> transform these simplices into cubes, and do QMC on the cubes without
> throwing any points out.
>
>
> https://istam.iitkgp.ac.in/resources/2014/proceedings/59-istam-nm-fp-268.pdf
>
>
So I've been thinking about this, and I think I've come to the conclusion
that at least in this instance it is perfectly fine to throw out QMC
points. Here's why: Suppose I want to integrate a function `f()` over some
polytope `P`.

Voters = Integral_P { f( x ) }

Let `C` be some cube that contains the polytope `P`. I can always define a
function `g()` that is defined on `C` but has compact support in `P`:

g(x) = { f(x) if x in P; or 0 otherwise }

There's no reason why I shouldn't be allowed to integrate `g(x)` over the
cube `C` and technically I'm not throwing away any points. But that is
exactly equivalent to the original plan of generating QMC points inside the
`C`, selecting those in `P`, and evaluating `f()` on those.

Wikipedia says that one of the key advantages of quasi-MC over regular MC
is that a low-dispersion sequence gives you more even coverage of the space
without making a lattice. Random sampling will naturally create clusters of
points and voids with few points. It is the nature of randomness that
clustering happens. Low-dispersion sequences avoid clustering. Now... I
don't know why we're told to not throw away points. But if the reason is
that in general throwing away points will create gaps, then... well...
slicing the space would be an exception to the rule. If the points
uniformly cover `C` then they uniformly cover any subset of `C`. You just
need to be aware that you are effectively integrating over that subset of
`C` and not all of `C`, but in our case that is exactly what we want.

Cheers,
-- 
Dr. Daniel Carrera
Postdoctoral Research Associate
Iowa State University
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