Subject: 3D probability

From: Krzysztof Radacki <[email-masked]>
Date: Tue, 07 Oct 1997 09:06:11 +0100

Dear Colleagues,

sorry for my unknowledge and maybe basic question (but if I knew 
the answer, I wouldn't ask).

Let me describe my question.

I have 10 000 cartessian points (X,Y,Z - known) in space.
	1. How can I calculate 3D function describing probability
	   of finding point in this space?
	2. How can I plot 3D shape closing space with probability
	   bigger than f.e. 50%.
	3. What disciplin describe this question (I know generaly
	   it's probability caclulation, but something more 
	   preciss - where can I try to look for it)?
	4. Is there free available algorithm to make one of this 
	   two steps?

If someone could answer I would apreciate.

           Krzys Radacki

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From: "Peter Shenkin" <[email-masked]>
Date: Tue, 7 Oct 1997 11:36:48 -0400

On Oct 7,  9:06am, Krzysztof Radacki wrote:
> Subject: CCL:3D probability

> 	   of finding point in this space?

You want to convert a point sampling to a density distribution.
One way to do this is to construct the Voronoi polyhedron about
each point and ascribe a density of 1/its_volume to each polyhedral
region.  I think there are programs available the net that can determine
the polyhedrons given a PDB file (others might be able to respond
with detail).  This doesn't work for the outermost points, which
will have infinite volume (and give zero density).  Anyway, if
you normalize the density, you can view it as a probability 
distribution. 

Note that this density distribution is discrete:  some region of
space is tiled with these polhedra, and each polyhedron has
a uniform density.  You can ascribe each density to the location
of the "central" point, and if you want a continuous distribution,
use some method to interpolate density.  This sort of thing is
done by finite-element methods.  You use different interpolation
methods depending on whether the subsequent use of the distribution
requires continuous function values only, or continuous function
plus first derivative, etc.
 
> 	2. How can I plot 3D shape closing space with probability

This problem is not well defined.  For instance, you can draw a
plane parallel to the X axis and position it so that 50% of the
points are on either side of it, then select either side of the
plane as your solution.  Or you can use the Y or Z axis.  

This is probably not what you have in mind.  You probably have
in mind a picture in which the density is highest in the middle
and tails of gradually in all directions, and you want to draw
your boundary along an isodensity countour so as to enclose the
densest region.  But your data may not look like this.  Of
course, if your data look *approximately* like this, you can
fit them to some function that *does* look like this (for
instance, a 3D gaussian), then do what I described.  But you
had better be sure this is a reasonable model for your data.

> 	   preciss - where can I try to look for it)?

Finite element analysis; statistical pattern recognition.

Hope this helps,
-P.

-- 
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