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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

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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