Skip to content

simulate zero-truncated Poisson distribution

5 messages · Glazko, Galina, (Ted Harding), Peter Dalgaard +1 more

Glazko, Galina
# 
Dear All

I would like to know whether it is possible with R to 
generate random numbers from zero-truncated Poisson 
distribution.

Thanks in advance,
Galina
(Ted Harding)
#
On 01-May-05 Glazko, Galina wrote:
Maybe someone has written an efficient function for this.

If not, you anyway have at least two options with basic R.

(A) "Rejection sampling": OK if the probability of 0 is small.
In that case, suppose you want n zero-truncated Poisson values.
Sample n values from rpois. Reject any which are 0, leaving
you with say r to find. Sample r from rpois, and continue
in the same way until you have all n.

This could be done by something on the following lines:

  n<-1000; T<-3.5
  Y<-rpois(n,T); Y0<-Y[Y>0]; r<-(n - length(Y0))
  while(r>0){
    Y<-rpois(r,T); Y0<-c(Y0,Y[Y>0]); r<-(n - length(Y0))
  }

and Y is then the required sample of n=1000 from a Poisson
distribution with mean T=3.5, after rejecting all zeros.

(B) This has a deeper theoretical base.

Suppose the mean of the original Poisson distribution (i.e.
before the 0's are cut out) is T (notation chosen for intuitive
convenience).

The number of events in a Poisson process of rate 1 in the
interval (0,T) has a Poisson distribution with mean T.
The time to the first event of a Poisson process of rate 1
has the negative exponential distribution with density exp(-t).

Conditional on the first event lying in (0,T), the time
to it has the conditional distribution with density

  exp(-t)/(1 - exp(-T)) (0 <= t <= T)

and the PDF (cumulative distribution) is

  F(t) = (1 - exp(-t))/(1 - exp(-T))

If t is randomly sampled from this distribution, then
U = F(t) has a uniform distribution on (0,1). So, if
you sample U from runif(1), and then

  t = -log(1 - U*(1 - exp(-T)))

you will have a random variable which is the time to
the first event, conditional on it being in (0,T).

Next, the number of Poisson-process events in (t,T),
conditional on t, simply has a Poisson distribution
with mean (T-t).

So sample from rpois(1,(T-t)), and add 1 (corresponding to
the "first" event whose time you sampled as above) to this
value.

The result is a single value from a zero-truncated Poisson
distribution with (pre-truncation) mean T.

Something like the following code will do the job vectorially:

  n<-1000       # desired size of sample
  T<-3.5        # pre-truncation mean of Poisson
  U<-runif(n)   # the uniform sample
  t = -log(1 - U*(1 - exp(-T))) # the "first" event-times
  T1<-(T - t)   # the set of (T-t)

  X <- rpois(n,T1)+1 # the final truncated Poisson sample

The expected value of your truncated distribution is of course
related to the mean of the pre-truncated Poisson by

  E(X) = T/(1 - exp(-T))

Hoping this helps,
Ted.


--------------------------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at nessie.mcc.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 01-May-05                                       Time: 17:20:09
------------------------------ XFMail ------------------------------
Peter Dalgaard
# 
(Ted Harding) <Ted.Harding at nessie.mcc.ac.uk> writes:
.....
There must be an easier way... Anything wrong with

 rtpois <- function(N, lambda)
     qpois(runif(N, dpois(0, lambda), 1), lambda)

 rtpois(100,5)

?
(Ted Harding)
#
On 01-May-05 Peter Dalgaard wrote:
Well, that's neat! (When I saw that, I had to switch in
backup-brain to think clearly about qpois, but there you are ... )

Best wishes,
Ted.


--------------------------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at nessie.mcc.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 01-May-05                                       Time: 21:20:43
------------------------------ XFMail ------------------------------
Spencer Graves
# 
Hi, Peter:

Your solution is simple, elegant, very general yet sufficiently subtle 
that I (and I suspect many others) might not have thought of it until 
you mentioned it.  It is worth remembering.

Thanks.  spencer graves
Peter Dalgaard wrote: