Southern Illinois University Carbondale
OpenSIUC
Conference Proceedings
Department of Electrical and Computer
Engineering
12-1987
Optimal Serial Distributed Decision Fusion
Ramanarayanan Viswanathan
Southern Illinois University Carbondale, viswa@engr.siu.edu
Stelios C. A. Thomopoulos
Southern Illinois University Carbondale
Ramakrishna Tumuluri
Southern Illinois University Carbondale
Follow this and additional works at: http://opensiuc.lib.siu.edu/ece_confs
Viswanathan, R., Thomopoulos, S.C.A., & Tumuluri, R. (1987). Optimal serial distributed decision
fusion. 26th IEEE Conference on Decision and Control, 1987, v.26, part 1, 1848 - 1849. DOI:
10.1109/CDC.1987.272831 ©1987 IEEE. Personal use of this material is permitted. However,
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Recommended Citation
Viswanathan, Ramanarayanan; Thomopoulos, Stelios C. A.; and Tumuluri, Ramakrishna, "Optimal Serial Distributed Decision
Fusion" (1987). Conference Proceedings. Paper 68.
http://opensiuc.lib.siu.edu/ece_confs/68
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FA8 12:30
Proceedings 01 the 26th Conference
on Decision and Control
Los Angeles, CA December 1087
OPTIMAL SERIAL DISTRIBUTED
DECISION
FUSION
RamanarayananViswanathan,
S t e l i o s C . A. Thomopoulos,RamakrishnaTumuluri
D e p a r t m e n to fE l e c t r i c a lE n g i n e e r i n g
S o u t h e r nI l l i n o i sU n i v e r s i t y
C a r b o n d a l e ,I l l i n o i s 62901
Abstract
H
The p r o b l e mo fd i s t r i b u t e dd e t e c t i o ni n v o l v i n g
N s e n s o r s is c o n s i d e r e d . The c o n f i g u r a t i o no fs e n s o r s i s s e r i a li nt h es e n s et h a tt h e
(j-l)th
sensor
p a s s e s i t s d e c i s i o n t o t h e j t h s e n s o ra n dt h a tt h e
j t h s e n s o rd e c i d e su s i n gt h ed e c i s i o n
i t receives
and i t s own o b s e r v a t i o n . When e a c hs e n s o re m p l o y s
t h e Neyman-Pearson t e s t , t h ep r o b a b i l i t yo fd e t e c t i o n is maximized f o r a g i v e n p r o b a b i l i t y o f f a l s e
1
N t h s t a g e . Withtwo
s e n s o r st h es e r i a l a r m ,a tt h e
a1scheme is b e t t e r t h a n t h e p a r a l l e l f u s i o n s c h e m e
a n a l y z e di nt h el i t e r a t u r e .
For c e r t a i nd i s t r i b u t i o n so fo b s e r v a t i o n s ,t h es e r i a ls c h e m ep e r f o r m s
N.
N * m e r i c a le x a m p l e si l l u s t r a t et h e
b e t t e rf o ra l l
g l o b a lo p t i m i z a t i o n
by t h e s e l e c t i o n o f o p e r a t i n g
t h r e s h o l d sa tt h es e n s o r s .
= o
Many t i m e s i t i s c o n v e n i e n tt ou s e
the log l i k e l i hood r a t i o , iln A ( Z j ) = A * ( Z j ) . Hence ,
A*(Zj)
9- [
Ho
T h i sr e s e a r c h i s s p o n s o r e d by SDI/ISDG and is managed by t h e O f f i c e of N a v a lR e s e a r c hu n d e rC o n t r a c t
NO0014k-0515.
if u.
J-1
=
1
t!'l
t.
J 90
if u.
J-1
=
0
and
Introduction
The t h e o r yo fd i s t r i b u t e dd e t e c t i o n
is r e c e i v [l-61.
i n g a l o t o fa t t e n t i o ni nt h el i t e r a t u r e
T y p i c a l l y , a number o fs e n s o r sp r o c e s s
t h e d a t at h e y
of one of t h e h y p o t h e s e s
r e c e i v ea n dd e c i d ei nf a v o r
a b o u tt h eo r i g i no ft h ed a t a .I n
a two c l a s sd e c i s i o np r o b l e m ,t h eh y p o t h e s e s
wouldbe
signalpresent
( H I ) o rt h es i g n a la b s e n t
(Ho).
T h e s ed e c i s i o n sa r e
For t h e f i r s t s t a g e ,
*
t l , l - t l*, O '
A t t h e j t h s t a g e ,t h ef a l s e
is given by
alarm p r o b a b i l i t y
a finaldecision
t h e ns e n tt o
a f u s i o nc e n t e rw h e r e
is made. T h i s
r e g a r d i n gt h ep r e s e n c eo ft h es i g n a l
scheme
can
be
termed
p a r a l l e ld e c i s i o nm a k i n g .I n
t h i s p a p e r , we c o n s i d e r a s e r i a l d i s t r i b u t e dd e c i 1).
T h o u g ht h pe e r f o r m a n c oe f
s i o ns c h e m e( F i g .
t h i s c o n f i g u r a t i o n i s s u s c e p t i b l et ol i n kf a i l u r e s ,
t h e p e r f o r m a n c eo ft h es e r i a ls c h e m ec a ne x c e e dt h a t
A l s o , t h eg e o g r a p h i c a l
o tf h ep a r a l l e sl c h e m e .
c l o s e n e s s o f some of t h e s e n s o r sm i g h t make a s e r i a l
or s e r i a l - p a r a l l e l c o n f i g u r a t i o n d e s i r a b l e .
Developmentof
Key Equations
C o n s i d e rt h es e r i a lc o n f i g u r a t i o no f
distrib u t e ds e n s o r s h o w ni nF i g .
1.
D e n o t et h es e n s o r
d e c i s i o n sa s
u 1 , u2
,...,
c e i v e s t h e decisionuj-1and
t o make i t s d e c i s i o nu j .
UN.
The j t h s e n s o r r e -
i t s own o b s e r v a t i o n Z j
The d e c i s i o n
UN
a tt h e
U s i n g ( 3 ) , (4) a n dt h ec o n d i t i o n a li n d e p e n d e n c ea s sumption, we have
Nth
s e n s o r is t h e f u s e d d e c i s i o n a b o u t t h e h y p o t h e s e s .
We a s s u m et h a t t h e d a t a a t t h es e n s o r s ,c o n d i t i o n e d
on e a c hh y p o t h e s i s ,a r es t a t i s t i c a l l yi n d e p e n d e n t .
T h i s i m p l i e s t h a t Z j and u j - 1a r ea l s oc o n d i t i o n a l l y
Similarly,
j t h s e n s o re m p l o y sa n
N-P t e s t
i n d e p e n d e n tT. h e
u s i n g t h e d a t a ( Z ~ ~ u j -)1.
The o p t i m a l i t yo f t h i s
Knowing t h e d i s t r i b u t i o n o f t h e o b s e r v a t i o n s
assumption is shown by Theorem 1 , d i s c u s s e d l a t e r .
D e n o t i n gt h ed i s t r i b u t i o no f
p ( Z j l H o ) ,t h el i k e l i h o o dr a t i ot e s t
Z j asp(ZjlH1)and
becomes
CH2505-6187/0000-1848$1.00 0 1987 lEEE
Zj
and
u s i n g ( 2 ) and (4 through 6 ) , i t i s p o s s i b l e t o comP ~ , j ' sa r e
p u t e t h e P ~ , j ' sr e c u r s i v e l yp r o v i d e dt h e
1848
Authorized licensed use limited to: Southern Illinois University Carbondale. Downloaded on May 30, 2009 at 16:13 from IEEE Xplore. Restrictions apply.
s p e c i f i e d .i f
tnFi P ~ , j ' sa r ex e p t h es a n e t, h e
S r i n i v a s a n , R . , " D i s t r i b u t e dR a d a rD e t e c t i o n
No. 1 ,
Theory," IZE P r o c e e d i n g s , Vol. 1 3 3P, t F ,
February 1 9 6 6 , p p . 55-60.
s e r i a lc o n f i g u r a t i o ne x h i b i t ss o m en i c ep r o p e r t i e s
~ 7 ; . H o w e v e rf, o r
a g i v e n PO,% a tt h e
~ t sht a g s ,
T i o m o p o u l o s , S. C . A . , Viswanathan, R. and Boug o u l i a s , D . P . , "ComputableOptimal
Distributed
DecisionFilsion,"underpreparation.
a naximum P c , ~ j .
t h i s p r o c e a u r ed o e sn o tg d a r a n t e e
I no r d e r t o g l o b a l l yo p t i m i z et h e? e r f o r m a n c e ,t h a t
is t oz a x i m i z e P ~ , Nf o r a g i v e n ? F , N , we need a nul-
Thomopouios, S. C . A . , Viswanathan, R . and Boug o u l i a s , 0 . P . , ' l O p t i m a lD e c i s i o nF u s i o ni n
IEEE
M u l t i p l eS e n s o rS y s t e m s , "t oa p p e a ri n
T r a n s a c t i o n s on A e r o s p a c ea n dE l e c t r o n i cS y s tems.
t i a i n e n s i o n a i s e a r c h w i t h r e s p e c tt ot h ev a r i a b l e s
P F , ~ ' sj ~ = 1 ,2,. ( X - 1 ) . The r e s u l t s o b t a i n e d
u s i n gt h en m e r i c a ls e a r c hp r o c e d u r e
will De p r e s e n t ed i nt h en e x tS e c t i o n .
Tne Theorem 1 s t a t e db e l o w
shows : h a tt n e
I;-? t e s t 3 a t t h e s e n s o r s
i s optimum
f o rt h as e r i a i3 i s t - i b u t e ad e c i s i o np r o b l e m .
The
b e found i n [6j.
?roofcan
..,
Theorem 1
a DistriS a d j a d i , F. A . , " H y p o t h e s e sT e s t i n gi n
butedEnvironment,"
I E E E T r a n s a c t i o n s o n AeroVol. AES-22, No.
s p a c ea n dE l e c t r o n i cS y s t e m s ,
2, March 1966, p p . 134-137.
G i v e nt h a tt h eo b s e r v a t i o n sa te a c hs t a g ei n
a
w i t h N sens e r i a ld i s t r i b a t e dd e t e c t i s ne n v i r o n m e n t
is
s o r sa r ei . i . d . ,t n ep r o b a b i l i t yo fd e t e c t i o n
maximized f o r a g i v e n p r o b a b i l i t y o f f a l s e a l a r m , a t
Viswanathan, R., Thomopoulos, S. C . A. and Tumul u r i , R . , " S e r i a lD e c i s i o ni nM u l t i p l eS e n s o r
F u s i o n , ' !t oa p p e a ri nt h eP r o c e e d i n g so ft h e
1987CISS Conference,JohnHopkinsUniversity.
t h e N t n s t a g e , when e a c hs t a g e
? e a r s o nt e s t .
employsthe
Neyman-
Viswanathan, R . , Thomopoulos, S. C . A . and Tumul u r i , B . , " O p t i m a lS e r i a lD i s t r i b u t e dD e c i s i o n
F u s i o n , "s u b m i t t e dt o
ISEE T r a n s a c t i o n s onAerospaceandElectronicSystems.
PerformanceEvaluation
Llsing s t a n d a r dn u m e r i c a l? r o c e d u r e ,
we e v a l u of a s e r i a l scheme f o r t h e c a s e
a t e dt h ep e r f o r m a n e e
i n additive
o ft h eo e t e c t i o no f
a c o n s t a n ts i g n a l
it v i t ht h ep a r a l white'Saussiannoiseandcompared
i c l s c h e m e .T h er e s u l tf o r
two s e n s o r s is shown i n
2 s e n s o r s ,t h es e r i a l
F i g . 2 . I n g e n e r a l f, o r
scheme. The
schema i s n o ti n f e r i o rt ot h ep a r a l l e l
proofof
t n i sf o l l o w sf r o m
Theorem 2 [ e ] .
Theorem 2
I ft h es w i t c h i n gf u n c t i o nc o r r e s p o n d i n gt ot h e
o p t i m a lp a r a l l e lf u s i o nc a nb er e a l i z e di nt e r m so f
'with s i n g l e
a s e q u e n c eo ft w ov a r i a c l ef u n c t i o n s
t h e o p t i m a ls e r i a ls c h e m e
is b e t t e r
o u t p u t ,t h e n
t h a nt h eo p t i m a lp a r a l l e ls c h e m e .
Conclusion
A s e r i a ld i s t r i a u t e dn e t w o r ko f
N s e n s o r sd e t e c t i n gt h ep r e s e n c eo ra b s e n c eo f
a signal is analyzed
i nt h i sp a p e r .
When t h es e n s o ro b s e r v a t i o n sc o n d i t i o n e d on t h e h y p o t h e s i s , a r e s t a t i c a l l y i n d e p e n d e n t ,t h es e n s o r s
employNeyman-Pearson
t e s t f o r maxim i z i n gt h ed e t e c t i o np r o b a b i l i t yf o r
a g i v e nf a l s e
Nth s t a g e (Theorem 1 ) .
For
a l a r mp r o b a S i l i t ya tt h e
c e r t a i nn o i s ed i s t r i b u t i o n s ,t h ep a r a l l e ls t r u c t u r e
r e q u i r i n g i t s f u s i o ns c h e m et ob e l o n gt oc e r t a i n
i s i n f e r i o rt ot h e
- l a s so fs w i t c h i n gf u n c t i o n s ,
2).
As a drawback,anyseris e r i a l scheme(Theorerr.
i s v u l n e r a b l et ol i n kf a i l u r e s .
Some
a nl e t w o r k
n u m e r i c a ls x a a 7 l e si l l u s t r a t et h ep e r f o r m a n c eo ft h e
o p t i m a ls e r i a ld e c i s i o ns z h e m s .
\
References
-3.
11
T e n n e y , R . R . a n dS i n d e l l ,
N . R., J r .",D e t e c IEEE Transact i o nw i t hD i s t r i b u t e dS e n s o r s , "
t i o n so nA e r o s p a c ea n dE l e c t r o n i cS y s t e m s ,V o l .
A E S - 1 7 , ; U ~ Y 1981,pp.501-510.
21
C n a i r , Z . a nV
d arshney,
P . K . , '!Optimal
Data
F u s i o ni nM u l t i p l eS e n s o rD e t e c t i o nS y s t e m s , "
I E E E T r a n s a c t i o n s o nA e r o s p a c ea n dE l e c t r o n i c
Systems,Vol.
AES-22, No. 1 , J a n u a r y1 9 6 6 p, p .
c
J
0
1849
Authorized licensed use limited to: Southern Illinois University Carbondale. Downloaded on May 30, 2009 at 16:13 from IEEE Xplore. Restrictions apply.