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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 GFPOISSGN-OI5 BlBU°ED HANSUM NU!!PERS LIIHLARGE MEAN (Lockheed Engineering andsdnageaent) 11 U HC A0..(/'!F AJ1 CSCL 111
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April 1980
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Exact methods for commuter generatiar. of Foisscn ;se;.do- C ando^ variables tendto be expensive for large -alves of the par`r7eter. :nis memorandu rn suggestsan approximate s+etsod far ms re accurate In those cases than the usualnormal approxi-mation.
17. Key Words (Supgnfed by Author ftll 18 Distribution StatementPseudc-random numbers, simuiatl:-in.Poissrn distribution, goodness of fit,
discrete random variables, computersimulation
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 ApproximationsTo 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,
tapprox - 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(&-21) ^a-1 + a a -_i
2a4-2)(3a-5) ^a-2 +
Manipulation of this expansion gives us
u a as + 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 = 2/3X1/6
+ 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 ISOF POOR QUALITY
A
4
TABLE l.--COMPARISON OF MAXIMUM ERRORS IN IWO APPROXIMATIONS
TO THE POISSON DISTRIBUTION
-Max Error In
NormaI (Normal)3/2P.nrox. Approx.
2 .044 .00775 .0%9:.I 00293
10 .0208 .00'4415 .0170 .0009520 .0148 .00069525 .0132 .00055630 .0121 .00045740 .13105 .00034150 .000938 .00027175 .00766 .000178100 .00664 .000132150 .00495 .000072
r.ii:l\ 1f. i'.1^^1. ltiti! 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
(02/3 "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 Pearson2 ^
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 Satisfyingthe Poisson Distribution. Communications of the ACM, 11 (12), p. 819,1968.
.. .,m