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Terrell, G. R.

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A W E M T# Gc!4ERA T ION OF aJ I SSM-0I STR IBUTED
R.Jr.1Q" *MRS WITH LARGE W-M
By
George R. Terrell
Approved By:	 J - C.
. C. Minter, Supervisor
Techniques Developrr*nt Section
(230 -10224)	 A hLTE ON :HE GEeEnAIiCN GF
POISSGN-OI5 BlBU°ED HANSUM NU!!PERS LIIH
LARGE MEAN (Lockheed Engineering and
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April 1980
	
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Exact methods for commuter generatiar. of Foisscn ;se;.do- C ando^ variables tend
to be expensive for large -alves of the par`r7eter.	 :nis memorandu rn suggests
an approximate s+etsod far ms re accurate In those cases than the usual
normal approxi-mation.
17. Key Words (Supgnfed by Author ftll 18	 Distribution Statement
Pseudc-random numbers,
	
simuiatl:-in.
Poissrn distribution,	 goodness of fit,
discrete random variables, computer
simulation
19. Security Oata,f lof this report) 20	 secvroy Ciassif	 (of this pagel 21. No. of POW 22. Pt-'
Unclassified Unclassified
'For We by the Natiomol Technical Information Se ict. Sptng f ield. Veripma 22161
AC Far. 142 4 INw qo. 751	 NASA — JSC
i 1
CON TEN'TS
Section	 Page
I-	 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. APPROXIMATELY P HISS!'*! PHL' I _	 -.=': D"ir : ,:,R	 i- rS . . . . . 
_ . _ . . . 2
3. KjWER TRANSEORtiV.
	 S . . . . . . . . . . . . . . . . . . . . . . . 3
4. A GENERAi I?ti ALGOR:ins . . . . . .	 . _ . . . _ . . . . . . . . . . 4
5. EXTENSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
6. REFERENCE,	 . . . . .	 . _ . . . . . . . . . . . . . 7
i i i
TABLES
Table
	
P aye
1 Comparison of Maximum Er rers in Two Approximations
To The Poisson 	 . . . • . . . . . . . . . . . . . . . . 5
iv
I . INTRODUCTION
The computer s i mulation of picture-elerr-ent (pixel) data sets in remote
sensing problems is an important method fer testing proposed methods of
data analysis. In such si.-nulatior.s the distribution of the number of pixels
to be assigned to a rare signatu,-e or cr y:; type can be modeled by a Poisson
process. Unfortunately, Poisson pseudo-rando:i generators for moderate mean
rates (^-50) tend to to either slcw zr ii:3 prat:. This technical memorandum
proposes an alternative method of generation to partially overcome these
problems.
0; APrrt0?; ILMATELY POISSON PSEUDO-RANDOM VARIABLES
A Poisson-distributed random variable x with probability e-Xax/x
has mean and variance -^. Because of the iisportance of Poisson processes,
computat i onal procedures for generatin g pseudo-random Poisson-distributed
numbers have received some attention in the literature (Snow 1968, Schaffer
1910). The methods ocSCribed there „cork :suite satisfactorily for r small,
but involves a sequence of calculai ion'- ­ d:.­ atien order X. Thus, as a
becomes large, the time required to genera`-- a single Poisson variate may be
prohibitive. The IMSL program literary (IMSL 1919) which implements the
above proced!:rr•_ ZTPIKCS them w. th a normally distributed random variable
with A mean anu	 ince for A , :50. ns wi i • be seen later, this approximation
is not particularly good.
The criterion for a good approximation to a particular distribution will be
the --metric
max IFdPProx(x)-F(x)Iwhere F(x) is the cummuIative distribution
function of the rar^om variable to be approximated. Since satisfactory
procedures exist t;, generate normal pseudo-random variables, we will use
approximations of the following form:
I' t 
dist. 
Normal (0;1), then x approx = T(t:+uZ)
where T is an appropriate one-one transformation and u and o are constants
depending on T and A. Thus, if ^ is the ncanal (0,1) cumulative distribution
function,
t approx - SGT-i(x)-1!)/']
The Edgeworth ex:;ansion (Abramovitz and Stegun 1910) of the function T-1(x)
(where x is Poisson distributed) shows that the highest order correction
term to the normality of T I (x) i% the skewness Y I = " 3 /o 3 (where u 3 is the
third central moment) t.im,2,, a `unction of x independent of T. Thus, we wish
to find a T such that T -1 (x) has sk(2wn(,%s as small as possible.
2
3. POWER TRANSFORMU;TIONS
The family of functions 
T-1 
which we will explore will be the powers
x a , a>o, which are all defined since x Poisson > 0. This
choice of family is sugges t ed by the well-knor;n "variance-stabilizing"
transformation Vx whose variarce is constant to ^^rder a -1 . This has been
useful in analyziny Poisson-cent data since ,ifter transformation the
separate counts ma y he treated as ',omosredastic.
This corresponds to our case a = 112. To es`imate the skewness of xa,
x Poisson, we write
a
E(x a ) _ ?dEX)d)
= a a E	 1 +	
) a)
and using the binomial theorem
A E ( 1 + ax- 1 + a a- I^ / x-__)2 + a a-1'
6 
a^ (X-X)3tJ
a a-li l y- > ^^ -3)	 X_',)4 }
	
/+	 y2 4	 (1 a /	
. .
then using standard results about the central moments of a Poisson variable
a s + a(&-
2
1) ^a-1 + a a -_i 
2
a
4
-2)(3a-5) ^a-2 +
Manipulation of this expansion gives us
u a	 a
s + j -11 A a-1 + ...
X	 2.
o x a - aa a-1/2 + 3/4a(a-1)2^a-3/2 + ...
-,
Y lx'I
 = a(3a-2)^	 /` i ...
Thus, by choosing a - 2/3, x  is asymptotically unskewed as a becomes large.
3
4. P. GENERATION ALGORITHM
The random number generation scheme suggested by this result is as follows:
Generate a normal (0,1) pseudorandom variate J. Muitiply it by
o = 2/3X 1/6 + 1/18A - ' /~ and add u = 
X2/3 _ 1/9X-1/3
then raise the result to the 3/2-power. Th. Poisson variate x is taken to be
the non-negative integer nearest to this value.
In table i, the --metric error for this approximate scheme is tabulated
for various values of X, along with the error for the usual untransformed
(a = 1) normal approximation. It is apparent that the 3/2-scheme is
enormously more accurate; for X as small as 2 it is already better than
the usual procedure at A - 50. Also, 	 • ^n, i:_ases the 3/2-scheme becomes
better as X -1 , whereas the old scheme becomes better as X -1/2 . Prichard
(1980) has implemented this scheme on a programmable calculator with very
little difficulty.
The disadvantages of the scheme include the extra computation involved in
calculating ti and U, which may be negligible if a number of variates are
to be calculated, and the overhead involver) in taking a 312 power for each
variate. The higher accuracy may well compensate for this.	 In mixed
exact-and-asymptotic for small and large X schemes, speed may actually be
enhanced because the asymptotic approximation can be tolerated for much
smaller Vs.
ORIGI\AI. PAGE IS
OF POOR QUALITY
A
4
TABLE l.--COMPARISON OF MAXIMUM ERRORS IN IWO APPROXIMATIONS
TO THE POISSON DISTRIBUTION
-Max Error In
NormaI	 (Normal)3/2
P.nrox.	 Approx.
2 .044 .0077
5 .0%9:.I 00293
10 .0208 .00'44
15 .0170 .00095
20 .0148 .000695
25 .0132 .000556
30 .0121 .000457
40 .13105 .000341
50 .000938 .000271
75 .00766 .000178
100 .00664 .000132
150 .00495 .000072
r.ii:l\ 1f. i'.1^^1.
	
lti
ti! it	 CY'
5
5.	 EXILNSIONS
As a further application of the results of this memorandum, we may construct
a goodness-of-fit test for categories t.hii are believed to have independent
Poisson processes as generators. Let Oil i - 1, ..,, n be the observed
counts and E  be the expected cou-its (A's). 'Then
(0	2/3 "ij'	 2
2
i
	
n	 213
=1	 ^J2/3(E	
?^
i)	 n
where the approximation is mu ,-Ii Moser ^ (:,^ sma.l l 1. .	 than is the Pearson
2	 ^
X . Linear constraints, however, are no lunget linear ^.o this test warrants
further investigation.
6
6. REFERENCES
1. Abramovitz, M.; and Stegun, I. A.. 	 Handbook of Mathematical Functions.
Dover p. 935, 1970.
2. Prichard, H. M.:	 Per ,,on-,l rormunicat ioi, 1930.
3. Schaffer, H. E.: Algorithm 169, Generilor of Random Numbers Satisfying
The Poisson Distribution.	 Coudnunic,ation ,  of the ACM 13 (1), p. 49, 1970.
4. Snow, R. H.: Algorithm 342, Generator of kaiIdom Numbers Satisfying
the Poisson Distribution.
	
Communications of the ACM, 11 (12), p. 819,
1968.
.. .,m


A note on the generation of Poisson-distributed random numbers with large mean

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