🚀 go-pugleaf

In article <1992Aug16.234829.18925@ee.ubc.ca> victorw@ee.ubc.ca (Victor J. K. Wong) writes:

>For a 2-dimensional homogenous Poisson distribution, 
>p(i,A) = exp(-A*d)*(A*d)^i/i! where p(i,A) is the probability
>of i occurrences in an area A, d is the spatial density
>(number of occurrences per unit area). Similar to the
>one-dimensional Poisson distribution, 2-dimensional one
>also has the memoryless property. 

>To simulate a 2-dimensional Poisson environemnt, one can
>divide an area A into many small areas with equal size B.
>If B is small enough, p(0,B) + p(1,B) is very close to 1
>and hence, p(i,B), for i > 1, can be discarded. For each
>small area B, the probability of occurrence is p(1,B) and
>the probability of no occurrence is p(0,B), and this is
>a binominal process. If B is very small, one can assume
>that the location of the occurrence is the center of the 
>area B If this process repeats for each area B, the 
>distribution of of any area in A is a 2-dimensional
>Poisson. However, such a simulation takes long time, does
>anyone has better suggestion for a faster simulation of
>a 2-dimensional Poisson environment? If you have any
>suggestion, please email to me (email address victorw@ee.ubc.ca).

I am posting this response because I believe that there are
misconceptions which seem to be shared by many.  I believe
that what is asked is the simulation of a Poisson process, 
and that what is wanted is the entire set of sample points.
Now if the area of the entire region is A, there are roughly
A*d points to be computed, and there is no way to do the job
without this much computing.  But not much more is needed.

One way to do the job is to first compute the x-coordinates
of the points, which even with the greatest generality from
a one-dimensional Poisson process, with the measures of the
spacings being independent exponential random variables.  
Then the y-coordinates have the uniform (or other appropriate)
distribution given the x-coordinates.  This process can also
be modified to accommodate acceptance-rejection procedures
with little difficulty.

-- 
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hrubin@pop.stat.purdue.edu (Internet, bitnet)  
{purdue,pur-ee}!pop.stat!hrubin(UUCP)