Optimal Weights of Linear Combinations of the Independent Poisson Signals for Discrimination

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2 002 , V ol. 13, N o.2 p p . 307～315

Optim al W e ig h t s o f Lin e ar Com bin ation s o f t h e In de pe n de n t P oi s s on S ig n al s f or D i s c rim in ation

Jo o - H w an K im^{1 )} A b s tra c t

S u pp ose on e is g iv en a v ect or X of a finit e s et of qu ant it ies X i w h ich ar e in depen dent P ois s on sig n als . A nu ll hy pot h esis H₀ ab out E ( X ) is t o b e t est ed a g ain st an alt ern ativ e hy p ot h esis H1. A qu ant it y

i ix_i

is t o b e com put ed an d u s ed for th e t e st . T h e opt im al v alu e s of _i ar e calculat ed for t hr ee ca ses : (1) sig n al t o n ois e r at io is u sed in t h e t est , (2) n orm al appr ox im at ion s w it h un equ al v arian ce s t o th e P ois s on dist ribu tion s ar e u s ed in t h e t est , an d (3 ) th e P ois son distribut ion it self is u s ed. A com p arison is m ade of th e opt im al v alu es of _i in t h e t hr ee ca s es a s par am et er g oes t o infin it y .

K ey w or d s : P ois son sign al, H y pot h e sis t estin g , P ow er , Opt im al w eight s ,

1 . In tro du c ti on

A b eam of n eu t ral p articles can b e u sed t o est im at e th e den sit y or m a s s of an obj ect (F eller (1970)). A m et h od of dis crim in at ion h ere is t o u se a n eut r al part icle b eam (N P B ) aim ed at t h e obj ect , an d a sm all nu m b er of n eu t ron sign als ar e cou nt ed at t h e det ect or . Bey er an d Qu alls (1987 ) sh ow ed t h at t h e n um b er of r et u rn n eu tr on part icles fr om an obj ect in t err og at ion for a g iv en dw ell tim e follow P ois son distribut on . Kim (1995, 1996, 1997 an d 1998) st u died an applicat ion of sign al pr oces sin g of r et urn n eut r on sig n als fr om an obj ect irr adiat ed by a n eut r al part icle b eam in t h e ca se of on e pr ob e - on e obj ect - on e det ect or .

1. A s sociat e P r ofes sor , Dep artm en t of S t at istics an d In form at ion S cien ce , Don g gu k Un iv er sit y , Ky on gju , 780- 714. K orea .

E - m ail jhk @m ail.don g g uk .a c.kr

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T o ex t en d th e pr ev iou s stu dies , w e con sider t h e ca se of on e pr ob e - on e obj ect - k det ect or s . In depen dent P ois son coun t s in k b in s t h at h av e differ en t m ean s in each b in are con sider ed . T h e obj ectiv e is als o t o dis crim in at e b et w een a r e - en t ry v ehicle (RV ) an d a decoy u sin g t h e r et urn sign al. F igu re 1 sh ow s t h e sit u ation w e con sider in t his st u dy . T h is dis crim in at ion pr oblem is form u lat ed a s a t est of h y poth esis :

H0 : obj ect is an R V v s . H1 : obj ect is a decoy

(T h e t h eory of t estin g hy p ot h es es is giv en in Leh m ann (1986).) T h e ob serv ation s are form ed int o a v ect or of n eut r on coun t s in sev er al en er gy bin s :

x = ( x₁, x₂, , x_k) .

< F ig . 1> On e pr ob e - on e obj ect - k det ect or s ca se T h e sign al X _i are in dep en den t P ois son r an dom v ariables . Let

E ( X ) = ( t₁, t₂, , t_k) un der H₀ E ( X ) = ( d₁, d₂, , d_k) un der H₁ T h en

H₀ : E ( X ) = t

H₁ : E ( X ) = d d_i< t_i for all i

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T o dis crim in at e an obj ect u sin g k sig n als fr om k det ect or s , w e con sider a sum m ary st at ist ic th at is a lin ear com bin at ion of X _i

Y =

k

i = 0 iX _i

w h er e t h e w eigh t _i ar e t o b e ch osen lat er . S in ce t h e _i ar e t o b e ch osen p osit iv e (see (2.2), (3.3), an d (4.1) b elow ), w e r ej ect H0 if Y som e crit ical v alu e c . T hr ee m et h ods of ch oosin g th e _i ar e g iv en an d com par ed .

T h e fir st m et h od is b a sed on a sig n al - t o - n oise r at io S/ N . M ax im izin g S/ N u su ally is int en ded t o m ax im ize t h e p ow er of th e st at istical t est , defin ed by th e crit erion for r ej ect in g H0. In t h e pr esen t sit u at ion t h e v arian ce of Y (t h e t est st atist ic ) is n ot th e sam e un der H₀ an d H₁; con s equ en tly , m ax im izin g S/ N an d m ax im izin g pow er ar e n ot equ iv alen t . T h e secon d m eth od of ch oosin g w eig ht is b a sed on m ax im izin g pow er for n orm al dist rib ut ion appr ox im at ion s w it h un equ al v arian ces un der H0 an d H1. T h e t hird m et h od m ax im izes pow er for th e ex act P ois son distribut ion s . N ot e t h at w e on ly con sider th e optim al w eigh tin g of th e lin ear com bin ation s of P ois son coun t s , bu t it h a s n ot b een v erified h er e a s b ein g able t o dis crim in at e t h e hy p ot h esis at r ea son able risk s an d .

2 . S ig n al - t o - n o i s e R atio M e th o d

S in ce t h e X _i ar e in dep en den t P ois son sign als , t h e m ean is E ( Y ) =

k

i = 1 it_i u n der H₀, b ut

E ( Y ) =

k

i = 1 idi

u n der H₁. T h e v arian ce V ( Y ) is V ( Y ) =

k

i = 1 2

i ti, u n der H0

V ( Y ) =

k i = 1

2

i d_i, un der H₁

In th e t h eory of t est in g hy poth eses con cern in g m ean s ₀ an d ₁ w it h com m on v arian ce ², th e pow er (probability of rej ectin g H₀: = ₀ w h en H₁: = ₁) is an in creasin g fun ct ion of t he sign al- t o- n oise r atio (i.e. m ean differ en ce b et w een H₀ an d H1 ov er comm on st an dard deviation )

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| 0- 1|

.

T his su g g est s t h at in th e pr esen t ca s e of t h e h y pot h e sis t estin g , on e m ig ht ch oose i t o m ax im ize

S

N =

k

i = 1 i( ti- di)

k

i = 1 2 i di

(2.1)

w h er e t h e den om in at or of (2.1) w a s ch osen t o t h e st an dar d dev iat ion of Y un der H₁. S in ce w e a s su m e th e st an dar d dev iat ion u n der H₀ is gr eat er t h an or equ al t o t h e on e u n der H1, th e m ax im u m of sign al- t o- n oise is obt ain ed w h en w e u se th e st an dar d dev iat ion of Y un der H₁ a s den om in at or .

By t h e Cau chy - S ch w ar z in equ alit y (Roh at g i 1976 p .165 ), w e h av e

k

i = 1( _i d_i) ti- di

d_i (^{i = 1}^k ²ⁱ^dⁱ) (^{i = 1}^k ^{( t}ⁱ^{- d}^di ⁱ⁾² )

w it h equ alit y h oldin g if an d only if

i di = K t_i- d_i

d_i , i = 1 , 2 , , k ,

for som e con st an t K . In ot h er w or d s , t h e sign al- t o- n oise r at io is m ax im ized by t h e ch oice

i= t_i- d_i

d_i (2.2)

S in ce an y con st an t m u lt iple of t h e i also m ax im izes S/ N , t h e i of (2.2) can b e r es caled s o t h at

k

i = 1 i= 1.

3 . N orm al A pprox im ation w ith U n e qu al V ari an c e s

A s sum e t h at th e in dep en den t P ois s on dist ribu tion s P ( _i) of t h e X _i can b e appr ox im at ed by n orm al distribut ion s . T h en , appr ox im at ely ,

X _i N ( _i, ²_i)

w h ere _i= ²_i = t_i or _i= ²_i = d_i accor din g t o w h eth er H₀ or H₁ is tru e. Also, appr ox im at ely ,

Y N (

k

i = 1 i i,

k

i = 1 2 i i).

T h e det ection rat e is 1 - , w h er e

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= P ( r ej e ct H0 whe n H0 is tr u e)

= P ( Y c E ( X ) = t)

P Z c -

k

i = 1 iti k

i = 1 2 iti

an d Z N ( 0 , 1) . T h is is equ iv alen t t o c -

k

i = 1 iti k

i = 1 2 it_i

= ^{- 1}( ) = - ^{- 1}( 1 - ) , (3.1)

w h er e is th e cu m u lativ e distribut ion fu n ction of t h e st an dar d n orm al r an dom v ariable Z . T h e false alarm r at e is

= P ( a ccep t H₀ wh e n H₁ is tr u e)

= P ( Y > c E ( X ) = d)

W e a s su m e di>0 . T h erefor e, t h e di w ill r em ain in th e an aly sis . S o

1 - P Z

c -

k i = 1 id_i

k

i = 1 2 idi

or

c -

k

i = 1 id_i

k i = 1

2 i d_i

= ^{- 1}( 1 - ) (3.2)

T h e com bin at ion of (3.1) an d (3.2), for fix ed , sh ow s t h at satisfies

- 1( 1 - ) =

k

i = 1 i( ti- di) - ^{- 1}( 1 - )

k

i = 1 2 iti k

i = 1 2 i di

(3.3)

T h e qu ant it y

k

i = 1 i( t_i- d_i) is t h e "ex ces s " sign al u n der hy p ot h esis H₀ ov er t h at

u n der H₁. N ot e t h at t h e fir st t erm in (3.3 ):

k

i = 1 i( t_i- d_i)

k

i = 1 2 id_i is t h e sign al - t o - n oise r at io an d is ju st (2.1).

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It is r equir ed t o fin d { i} t h at m ax im ize t h e fun ct ion h ({ i}) defin ed by t h e righ t - h an d side of (3.3 ). On e can a s su m e t h at n ot all _i= 0 , sin ce ot h erw ise

= 1. On e fir st a s su m e s t h at 1= 0 an d inv est ig at es all m ax im a of t h e r esultin g fu n ct ion of k - 1 v ariables . T h en a s su m e ₁ 0 an d div ide nu m er at or an d den om in at or of (3.3) by ₁ an d put _i= _i/ ₁, i = 2 , 3 , , k . T h en h ({ _i}) b ecom es a fu n ct ion k - 1 of v ariables : h ( { i}) . A su fficient con dit ion (s ee Lu enb er g er (1973)) t h at th e p oint h ( { i

*}) b e a st rict local m ax im u m for h is t h at h ( { i

*}) = 0 an d t h at t h e m at rix ²h ( { i

*}) b e n eg at iv e defin it e. T his con dition can b e v erified b y ch eckin g t h at t h e eig en v alu es of t h e m atrix

2h ( { i

*}) ar e n eg ativ e. F in ally , h av in g ch eck ed all st rict local m ax im a , it is n eces sary t o in su r e t h at t h e fu n ction does n ot b ecom e elsew h er e g r eat er t h an it s v alu e at on e of t h e st rict local m ax im a . T h er e are ot h er pos sibilit ies t h at m u st b e ch eck ed su ch a s n on st rict m ax im a .

T h is pr ocedur e w ill b e illu st r at ed in only on e ca se : k = 2 . A s sum e ₁ 0 . P u t x = ₂ = ₂/ ₁. T h en

h ( x ) = x ( t₂- d₂) + ( t₁- d₁) - ^{- 1}( 1 - ) x²t₂+ t₁

x²d₂+ d₁ (3.4)

A ft er calcu latin g h ' ( x ) = 0 , t ran sp osin g a squ ar e r oot an d s qu arin g b oth sides , a qu adratic equ at ion in x is obt ain ed :

[ d₂( t₂- d₁)x - d₁( t₂- d₂) ]²( t₂x²+ t₁) - ( d₁t₂- t₁d₂)² ^{- 1}( 1 - )x²= 0 . (3.5) In t h e sp ecial ca se t h at d₁= 1, d₂= 2 , t₁= 3 , t₂= 4 , ^{- 1}( 1 - ) = 4 , equ at ion (3.5 ) r edu ces t o 16x⁴- 16x³- 12x + 3 = 0 w hich h a s t w o r eal r oot s , on ly on e of w h ich giv es a v alu e zer o t o h ' : x = 1 . 3998 : an d h ' ' (1 . 3998 ) <0 . T h e con dit ion

1+ ₂= 1 fin ally giv es ₁= .4274 , ₂= . 5726 . S in ce h ( 0) <0 an d h ( ) = - 2 , t h er e is n o ot h er m ax im um v alu e of h ( x ) t o t h e sp ecial ca s e.

F or t h e ca se k = 3 , h in (3.4 ) b ecom es biv ariat e fun ction of ₂^* an d ₃^* an d w e m ay fin d t h e m ax im u m poin t for s om e specified v alu es of t h e p ar am et er s . F or g en er al k , h b ecom es k - 1 v ariat e fun ct ion of ₂^*, ^*₃, , ^*_k.

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4 . Optim al W e ig h t s f or P oi s s on D i s trib uti on s

F or P ois s on coun t s in on e en er gy bin , Bey er an d Qu alls (1987) g iv e a r at h er com plet e an aly t ical an aly sis . F or t w o en er gy bin s , t h e discrim in at ion surface, an alog ou s in t h e dis crim in at ion cu rv e in Kim (1997 ) is a m appin g of R² t o R². F or th is an aly sis , on e n eed s t o dev elop on e or m or e t est st at istics . In t his sect ion w e giv e an opt im al t est st at istic b a s ed on t h e ob s erv at ion t h at t o m in im ize is t o m ax im ize t h e p ow er 1 - . W e seek th e m ost p ow erful (M P ) t est of t h e h y poth esis H0. T h e N ey m an - P ear s on lem m a [Lehm an n , p .74] com put es th e M P t est in t erm s of a

r ej e ction r eg ion = {^x ^{p ( x ; d)}p ( x ; t) >K }^,

w h er e

p ( x ; d) p ( x ; t) =

k

i = 1 d_i^xⁱe^{- d}ⁱ/ x_i!

k

i = 1 t_i^xⁱe^{- t}ⁱ/ x_i!

= ex p{^- ^{i = 1}^k ^xⁱ ^log ^d^tⁱⁱ }^{i = 1}^k ^e ^{( t}ⁱ^{- d}ⁱ⁾

is t h e lik elih ood r at io. T h e M P t est h a s t h e form R ej e ct H₀ if y =

k

i = 1 ix_i c,

w h er e

i= log t_i

d_i (4.1)

A g ain , w e can re scale th e _i s o t h at

k

i = 1 i= 1.

5 . Com p ari s on an d R em ark

W e n ow h av e t hr ee m et h ods of calcu lat in g w eigh t s : (1) S/ N , (2) n orm al appr ox im ation s , an d (3 ) P ois s on M P t est . In t his sect ion w e com p ar e t h e optim al w eig ht s for th ree m eth od s .

Com p arison is m ade in t h e lim it for lar g e P ois son coun t s . Let ti= pidi w it h p_i> 1 for all i. T h en con sider t h e lim it of t h e _i for t h e t hr ee m et h od s of t his p aper a s t h e decoy cou nt s b ecom e infin it e, i.e . a s m in (d_i) .

F or th e S/ N m et h od, w e ob t ain

i= t_i- d_i

di

= t_i di

- 1= pi- 1 pi- 1 as m in ( di) (5.1)

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F or th e P ois son M P t est m et h od , w e obt ain

i= log ( t_i

d_i ) = log ( p_i) log ( p_i) as m in ( d_i) . (5.2) F or t h e n orm al approx im at ion m et h od , w e furt h er sp ecialize t h e lim itin g pr oces s s o t h at t h e decoy cou nt s d_i= d⁰_i for d⁰_i 0 for all i w it h . Ex pr es sion (3.3 ) b ecom es

ip_id⁰_i

2 id⁰i

- ^{- 1}( 1 - )

2 ip_id⁰_i

2 id⁰i

(5.3) T h e s econ d t erm of (5.3 ) for v ariou s ch oices of th e _i is b oun ded ab ov e (an d b elow ), sin ce t h e m in (d⁰i) >0 . Con sequ ent ly , a s , th e fir st t erm dom in at es an d t h e _i th at m ax im ize (5.3 ) conv er g e t o t h e S/ N w eig ht s ; i.e.

i ( pi- 1) as .

It is int er est in g t h at t h e n orm al appr ox im ation s t o t h e P ois son dist ribu tion s im plicitly in clu ded in (3.3) an d (5.3 ) b ecom e b et t er a s ti an d di b ecom e lar g e ( ) but t h e _i th at m ax im ize S/ N do n ot con v er g e t o th e P ois s on M P t est w eig ht s .

R e f e re n c e s

1. Bey er , W . A . an d Qu alls , C. R . (1987 ), Discrim in at ion w it h N eu tr al P art icle B eam s an d N eut r on s , L os A lam os N a tional L ab ora tory , LA - 8 7- 3140.

2. F eller , W . (1970). A n I n trod uction to P robability T he ory and I ts A pp lica t ions : V olum e I I , T h ir d E d., J oh n W iley & S on s .

3. Grav es, R. E . (1986), Sign al- t o- back groun d noise ratio optimization for an N P B n eut r on det ect or w ith en er g y m ea sur em en t capabilit y , L os A lam os N a t ional L ab., LA - UR - 86 - 3101.

4.“Kim , J oo - H w an (1995), P r op erties of t h e P ois son - pow er F un ct ion

Dist ribu t ion , T he K or ean Com m un ica t ions in S ta t is t ics , V ol. 2, N o. 2, pp . 166 - 175

5. Kim , J oo - H w an (1996), Err or Rat e for th e Lim it in g P ois son - pow er F u n ct ion Dist ribu t ion , T he K or ean Com m un ica t ions in S ta t is t ics, V ol. 3, N o. 1, pp . 243 - 255, 1996.

6. Kim , J oo - H w an (1997), T h e M inim um Dw ell T im e A lg orith m for t h e P ois son Dist ribu tion an d th e P ois s on - p ow er F u n ction Distribut ion , T he K orean Com m un ica tions in S ta tis tics , V ol. 4, N o. 1. pp . 229 - 241.

7. Kim , Joo- H w an (1998), M on ot on e Likelih ood Rat io P r operty of th e P ois son S ign al Dist ribution w it h T h ree S ou rces of Err or s in t h e P ar am et er , T he

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K orean Com m un ica tions in S ta tis tics , V ol. 5, N o. 2, pp . 503 - 515.

8. Lehm an n , E . L . (1986 ), T es ting S ta tis t ical H yp othes es , 2n d E dit ion , J ohn W iley & S on s .

9. Lu en b er g er , D . G. (1973 ), I n trod uct ion to L in ear an d N onlin ear P rog ram m ing , A ddison - W esley P u blishin g Com pan y , 1973.

10. Lohat gi, V. K. (1976), An Introduction to Probability T heory and M at h em at ical S t at istics , J oh n W iley & S on s .

[ 2002년 9월 접수, 2002년 10월 채택 ]