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reasonable theory?

3 messages · Spencer Graves, James Cloos

James Cloos
# 
Before coding this in C, I wanted to test the idea out in R.

But I'm unsure if the theory is well-founded.

I have a (user-supplied) black-box function which takes R^n -> R^3
and a defined domain for each of the input reals.

I want to send some samples through the box to determine an
approximation of the convex hull of the function's range.
(I'll use the library from http://www.qhull.org to compute
the convex hull from the function's outputs.)

My plan is to use the permutation of the min and max values
for the n inputs along with k-1 samples w/in [min,max], but
I want the adjust the k samples a bit to avoid sampling bias.

To make it simpler, let's set the domain to [0,1].

Then, K = { 1/k, 2/k, ... (k-1)/k }

One reasonably easy possibility is to add to each Kn
a linear RV in, say, [-1/k?,1/k?].

Would a normal RV be better?  Some other bell-shaped RV?

Does adding a bit (but not too much) of randomness to the
input values have reason at all?

-JimC
Spencer Graves
# 
If you know nothing about the black box except that its domain is 
bounded, then I would random sample uniformly from the domain.  If the 
function is monotonically increasing in all variables, then you only 
need to test the two extreme points.  If you know other things, you may 
be able to use the other logic.  Spencer
On 10/12/2011 1:42 PM, James Cloos wrote:

  
    

      
      
    

  


  


      

    

    
17 days later

    

      
      

        


  

        

      James Cloos
    

    

      
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SG> If you know nothing about the black box except that its domain is
SG> bounded, then I would random sample uniformly from the domain.  If the
SG> function is monotonically increasing in all variables, then you only
SG> need to test the two extreme points.  If you know other things, you
SG> may be able to use the other logic.

So I was overthinking it, then.

As it turns out, for most of the functions in question, the convex hull
of the range, as determined by uniform samples w/in the domain, was only
slightly bigger than the convex hull of the full set of extreme points
of the domain.

And for my needs, then, the latter is a good-enough approximation of the
former after all.

Thanks!

-JimC