Least Favorable Distributions to Facilitate the Design of Detection Systems with Sensors at Deterministic Locations Benedito J. B. Fonseca Jr. September 2014

Motivation Region of interest (city, park, stadium)

2

Motivation Region of interest (city, park, stadium)

Radioactive material being released

3

Sensor Detection System Region of interest (city, park, stadium)

Sensors Sensorsdeployed deployedat at various variouspoint pointininthe theregion region Radioactive material being released

4

Sensor Detection System measurement noise

Fusion Center

H 0 : Z i , j = Wi , j

(Signal absent)

H1 : Z i , j = Ai + Wi , j (Signal present)

Each sensor i obtains M measurements

5

Sensor Detection System Fusion Center

H 0 : Z i , j = Wi , j

(Signal absent)

H1 : Z i , j = Ai + Wi , j (Signal present)

Signal emitter

More generally, sensor detection systems can be used to... ●Detect radio transmissions ●Detect the onset of a wildfire ●Military applications (radar, sonar) 6

Key Assumptions Fusion Center

H 0 : Z i , j = Wi , j H1 : Z i , j = Ai + Wi , j

noise component

{W } i.i.d. i, j

7

Key Assumptions Le

Li

Fusion Center

H 0 : Z i , j = Wi , j

Ai = x ( Li - Le Amplitude function

x (d )

Distance between sensor and emitter

{W } i.i.d.

H1 : Z i , j = Ai + Wi , j

i, j

)

sensor location

emitter location Signal random variable depends on sensor and emitter locations through an Amplitude Function of the distance

More generally, Ai may have a distribution with parameter depending on x ( Li - Le

)

8

Key Assumptions Le

Li

Fusion Center

H 0 : Z i , j = Wi , j

{W } i.i.d.

H1 : Z i , j = Ai + Wi , j

Ai = x ( Li - Le

i, j

)

Amplitude function

x (d )

Distance between sensor and emitter

Emitter Location is random with unknown distribution

9

Problem Le

Li

Fusion Center

H 0 : Z i , j = Wi , j

{W } i.i.d.

H1 : Z i , j = Ai + Wi , j

Ai = x ( Li - Le

i, j

)

Amplitude function

x (d )

Distance between sensor and emitter

Emitter Location is random with unknown distribution

H1 is composite hypothesis

10

Problem Le

Li

Fusion Center

H 0 : Z i , j = Wi , j

{W } i.i.d.

H1 : Z i , j = Ai + Wi , j

Ai = x ( Li - Le

i, j

)

Signal depends on the distance to a common emitter location

Emitter Location is random with unknown distribution

Measurements conditionally dependent

H1 is composite hypothesis

11

Problem Le

Li

Fusion Center

H 0 : Z i , j = Wi , j

{W } i.i.d.

H1 : Z i , j = Ai + Wi , j

Ai = x ( Li - Le

i, j

)

Signal depends on the distance to a common emitter location

Emitter Location is random with unknown distribution

Measurements conditionally dependent

H1 is composite hypothesis

Difficult to design sensor detection system 12

Research Question Le

Li

Fusion Center

H 0 : Z i , j = Wi , j H1 : Z i , j = Ai + Wi , j

How Howcan canaasystem systemdesigner designercircumvent circumventthe thedifficulties difficultiesof... of... ● Conditional dependent measurements and ● Conditional dependent measurements and ● Composite hypothesis ● Composite hypothesis while whilesatisfying satisfyingaagiven givenperformance performancerequirement? requirement?

13

Research Question Le

Li

Fusion Center

H 0 : Z i , j = Wi , j H1 : Z i , j = Ai + Wi , j

How Howcan canaasystem systemdesigner designercircumvent circumventthe thedifficulties difficultiesof... of... ● Conditional dependent measurements and ● Conditional dependent measurements and ● Composite hypothesis ● Composite hypothesis while whilesatisfying satisfyingaagiven givenperformance performancerequirement? requirement? Proposal: choose a Least Favorable Distribution for the emitter location 14

Least Favorable Distributions for a System X := ( X 1 ,..., X N ) with unknown distributions under H 0 and under H1 Detection system f : X ® { H 0 , H1} ìï H 0 : X ~ PX | H 0 Î W0 composite hypothesis test: í ïî H1 : X ~ PX |H1 Î W1 Possible distributions for X

W1 W0

15

Least Favorable Distributions for a System X := ( X 1 ,..., X N ) with unknown distributions under H 0 and under H1 Detection system f : X ® { H 0 , H1} ìï H 0 : X ~ PX | H 0 Î W0 composite hypothesis test: í ïî H1 : X ~ PX |H1 Î W1 Possible distributions for X

X |H 0

P

W0

W1

PX-|H1

ìï H 0 : X ~ PX-|H 0 simple hypothesis test: í H : X ~ P ïî 1 X | H1

16

Least Favorable Distributions for a System X := ( X 1 ,..., X N ) with unknown distributions under H 0 and under H1 Detection system f : X ® { H 0 , H1} ìï H 0 : X ~ PX | H 0 Î W0 composite hypothesis test: í ïî H1 : X ~ PX |H1 Î W1 Possible distributions for X

X |H 0

P

W0

W1

PX-|H1

ìï H 0 : X ~ PX-|H 0 simple hypothesis test: í H : X ~ P ïî 1 X | H1 Prob. of False Alarm of system f

(

)

(

"PX |H 0 Î W0 , a PX |H 0 , f £ a PX-|H 0 , f

Prob. of Detection of system f

)

(

)

(

"PX |H1 Î W1 , b PX |H1 , f ³ b PX-|H1 , f

) 17

Least Favorable Distributions for Emitter Location Z := ( Z1 ,..., Z K ) under H1 depends on Le with unknown distribution ìï H 0 : Z ~ PZ |H 0 composite hypothesis test: í ïî H1 : Z ~ PZ |H1 induced by PLe Î W Le Possible distributions for Z

W0

PZ |H 0

W1

W Le Possible distributions for emitter location

18

Least Favorable Distributions for Emitter Location Z := ( Z1 ,..., Z K ) under H1 depends on Le with unknown distribution ìï H 0 : Z ~ PZ |H 0 composite hypothesis test: í ïî H1 : Z ~ PZ |H1 induced by PLe Î W Le

W0

PZ |H 0

W1

Z | H1

P

PL-e

W Le

ìï H 0 : Z ~ PZ |H 0 simple hypothesis test: í H : Z ~ P ïî 1 Z | H1 induced by PLe

19

Least Favorable Distributions for Emitter Location Z := ( Z1 ,..., Z K ) under H1 depends on Le with unknown distribution ìï H 0 : Z ~ PZ |H 0 composite hypothesis test: í ïî H1 : Z ~ PZ |H1 induced by PLe Î W Le

W0

PZ |H 0

W1

Z | H1

P

PL-e

W Le

ìï H 0 : Z ~ PZ |H 0 simple hypothesis test: í H : Z ~ P ïî 1 Z | H1 induced by PLe Prob. of Detection of system f when measurements induced by

(

)

(

PLe

"PLe Î W Le , b PLe , f ³ b PL-e , f

) 20

Least Favorable Distributions for Emitter Location Z := ( Z1 ,..., Z K ) under H1 depends on Le with unknown distribution ìï H 0 : Z ~ PZ |H 0 composite hypothesis test: í ïî H1 : Z ~ PZ |H1 induced by PLe Î W Le

W0

PZ |H 0

W1

Z | H1

P

PL-e

W Le

ìï H 0 : Z ~ PZ |H 0 simple hypothesis test: í H : Z ~ P ïî 1 Z | H1 induced by PLe If f satisfies a detection requirement when assuming PL , then f will satisfy e for the actual, unknown, distribution

Prob. of Detection of system f when measurements induced by

(

)

(

PLe

"PLe Î W Le , b PLe , f ³ b PL-e , f

) 21

When Sensors are at Random Locations... ●

Among other conditions,

Sensor locations uniformly distributed

P éë Le Î ¶BR ( 0 ) ùû = 1

22

When Sensors are at Random Locations... ●

Among other conditions,

Sensor locations uniformly distributed

P éë Le Î ¶BR ( 0 ) ùû = 1 P éë Le Î ¶BR ( 0 ) ùû = 1 isisLeast LeastFavorable FavorableDistribution Distribution and Allerton'2010; IEEE Trans. IT'2014

{Zi } are conditionally i.i.d. 23

When Sensors are at Deterministic Locations... Sensor at fixed deterministic locations

Emitter Location still random!!

●

How to solve the conditional dependent and the composite hypothesis problems?

24

Least Favorable Distributions for Emitter Location Prob. of Detection of system f when measurements induced by

(

Prob. of Detection of system f conditioned on each emitter location

PLe

)

(

)

"PLe , b PLe , f = ò b PLe , f dPLe ( le ) Se

( le )

P [ Le = le ] = 1

25

Least Favorable Distributions for Emitter Location Prob. of Detection of system f when measurements induced by

(

Prob. of Detection of system f conditioned on each emitter location

PLe

)

(

)

"PLe , b PLe , f = ò b PLe , f dPLe ( le ) Se

( le )

P [ Le = le ] = 1 ●

Find

e

l that lower bounds the conditional prob. detection l ) ( (l ) æ ö "le Î Se , b PL , f ³ b ç PL , f ÷

(

e

e

)

e

è

ø

e

l ) ( æ "PL , b ( PL , f ) ³ b ç PL , f ö÷ e

e

e

è

e

ø

é P ë Le = le ùû = 1 isisLFD LFD 26

Least Favorable Distributions for Emitter Location Prob. of Detection of system f when measurements induced by

(

Prob. of Detection of system f conditioned on each emitter location

PLe

)

(

)

"PLe , b PLe , f = ò b PLe , f dPLe ( le ) Se

( le )

P [ Le = le ] = 1 ●

Find

e

l that lower bounds the conditional prob. detection l ) ( (l ) æ ö "le Î Se , b PL , f ³ b ç PL , f ÷

(

e

e

)

e

è

ø

e

l ) ( æ "PL , b ( PL , f ) ³ b ç PL , f ö÷ e

e

H1 is simple hypothesis

e

è

e

ø

é P ë Le = le ùû = 1 isisLFD LFD 27

Least Favorable Distributions for Emitter Location Prob. of Detection of system f when measurements induced by

(

Prob. of Detection of system f conditioned on each emitter location

PLe

)

(

)

"PLe , b PLe , f = ò b PLe , f dPLe ( le ) Se

( le )

P [ Le = le ] = 1 ●

Find

e

l that lower bounds the conditional prob. detection l ) ( (l ) æ ö "le Î Se , b PL , f ³ b ç PL , f ÷

(

e

e

)

e

è

e

ø

(

Ai = x li - leH1 is simple hypothesis

é P ë Le = le ùû = 1 isisLFD LFD

)

Measurements cond. indep. 28

Least Favorable Distributions for Emitter Location ●

Need to find

le- such that

(

)

l ) ( æ "le Î Se , b P , f ³ b ç PL , f ö÷

( le ) Le

e

è

e

ø

P éë Le = le- ùû = 1

29

Least Favorable Distributions for Emitter Location ●

Need to find

le- such that

(

)

l ) ( æ "le Î Se , b P , f ³ b ç PL , f ö÷

( le ) Le

e

è

e

ø

P éë Le = le- ùû = 1 Uncountably Uncountablymany manylocations locations ●Global min methods often need convexity ●Global min methods often need convexity ● ●

30

Least Favorable Distributions for Emitter Location ●

Need to find

le- such that

(

)

l ) ( æ "le Î Se , b P , f ³ b ç PL , f ö÷

( le ) Le

e

è

e

ø

P éë Le = le- ùû = 1 Uncountably Uncountablymany manylocations locations ●Global min methods often need convexity ●Global min methods often need convexity ● ●

Idea: Idea:Use Usetechniques techniquesfrom fromthe thefield fieldof of Operations OperationsResearch Research 31

Obnoxious Facility Location Problem ●

Where to place undesired facility? city B

city A

city F

?? chemical factory

city E city C city D

32

Obnoxious Facility Location Problem ●

Where to place undesired facility? city A

city B

city F

?? chemical factory

city E city C city D

Find...

K

l := arg min å gi ( li - le * e

le ÎSe

i =1

)

cost function 33

Obnoxious Facility Location Problem ●

Where to place undesired facility?

●

Where to place emitter?

city A

city B

sensor 1 sensor 2 city F

sensor 6

??

emitter

chemical factory

city E

sensor 3

city C

K

l := arg min å gi ( li - le * e

le ÎSe

i =1

sensor 5

sensor 4

city D

Find...

??

)

cost function

Find...

(

( le )

l := arg min b PLe , f e

le ÎSe

)

depends on li - le 34

Obnoxious Facility Location Problem ●

Where to place undesired facility?

●

Where to place emitter?

city A

city B

sensor 1 sensor 2 city F

sensor 6

??

emitter

chemical factory

city E

sensor 3

city C

K

l := arg min å gi ( li - le * e

le ÎSe

sensor 5

sensor 4

city D

Find...

??

i =1

) Û

K sensors

Find...

(

( le )

l := arg min b PLe , f e

le ÎSe

g1 ,...., g K } such Condition: exist that { Condition:ififthere there exist such thatK ( le ) b PLe , f isisnondecreasing nondecreasingwith with å i =1 gi ( li - le

(

)

)

) 35

Obnoxious Facility Location Problem Find...

K

l := arg min å gi ( li - le * e

le ÎSe

i =1

Û )

Find...

(

( le )

l := arg min b PLe , f e

le ÎSe

g1 ,...., g K } such Condition: exist { Condition:ififthere there exist suchthat thatK ( le ) b PLe , f isisnondecreasing nondecreasingwith with å i =1 gi ( li - le

(

)

depends on f and distribution of ●

)

)

{Z } i, j

When is condition satisfied?

36

Obnoxious Facility Location Problem Find...

K

l := arg min å gi ( li - le * e

le ÎSe

i =1

Find...

Û )

(

( le )

l := arg min b PLe , f e

le ÎSe

g1 ,...., g K } such Condition: exist { Condition:ififthere there exist suchthat thatK ( le ) b PLe , f isisnondecreasing nondecreasingwith with å i =1 gi ( li - le

(

)

depends on f and distribution of

ìK M ü f ( z ) = 1 íåå zi , j > t ý î i =1 j =1 þ

(

( le )

b PLe , f

)

increasing with

)

{Z } i, j

~ N ( 0, s 2 )

When is condition satisfied?

●

)

H 0 : Z i , j = Wi , j

gi

å

K i =1

H1 : Z i , j = Ai + Wi , j

x ( li - le

)

Ai = x ( li - Le

) 37

Obnoxious Facility Location Problem Find...

K

l := arg min å gi ( li - le * e

le ÎSe

i =1

Find...

Û )

(

( le )

l := arg min b PLe , f e

le ÎSe

g1 ,...., g K } such Condition: exist { Condition:ififthere there exist suchthat thatK ( le ) b PLe , f isisnondecreasing nondecreasingwith with å i =1 gi ( li - le

(

)

depends on f and distribution of

ìK M ü f ( z ) = 1 íåå zi , j > t ý î i =1 j =1 þ

(

( le )

b PLe , f

)

increasing with

)

{Z } i, j

When is condition satisfied?

●

)

~ Poisson ( D ) H 0 : Z i , j = Wi , j

gi

å

K i =1

H1 : Z i , j = Ai + Wi , j

x ( li - le

)

(

~ Poisson x ( li - Le

)) 38

Obnoxious Facility Location Problem g1 ,...., g K } such Condition: exist that { Condition:there there exist such that K l ( e) b PLe , f isisnondecreasing nondecreasingwith with å i =1 gi ( li - le

(

●

)

~ N ( 0, s 2 )

When is condition satisfied?

H 0 : Z i , j = Wi , j

ì ìK ü ï f0 ( u ) = 1 íå ui > 0 ý î i =1 þ ï ( or ) f := í M ì ü ïf ( z ) = 1 zi , j > t ý í å i ï î j =1 þ î

(

( le )

b PLe , f

)

increasing with

K

H1 : Z i , j = Ai + Wi , j Ai = x ( li - Le

)

gi

(

(

2 log 1 Q t , M x l L , M s ( ) å i e i =1

)

))

(similar result for Poisson case) 39

Obnoxious Facility Location Problem g1 ,...., g K } such Condition: exist that { Condition:there there exist such that K l ( e) b PLe , f isisnondecreasing nondecreasingwith with å i =1 gi ( li - le

(

●

)

~ Poisson ( D )

When is condition satisfied?

ì ì ü ïf0 ( u ) = 1 íå ui = K ý î i =1 þ ï ( and ) f := í M ì ü ïf ( z ) = 1 zi , j > t ý í å i ï î j =1 þ î K

(

( le )

b PLe , f

)

increasing with

)

H 0 : Z i , j = Wi , j H1 : Z i , j = Ai + Wi , j

(

~ Poisson x ( li - Le

))

gi

å log ( Q ( t , M × éëD + x ( l - L )ùû ) ) K

i =1

i

e

(similar result for Gaussian case) 40

Obnoxious Facility Location Problem g1 ,...., g K } such Condition: exist that { Condition:ififthere there exist such thatK l ( e) b PLe , f isisnondecreasing nondecreasingwith with å i =1 gi ( li - le

(

●

)

)

Borrow tools and results from Operations Research ●

Analytical solutions difficult to find

●

Several specific numerical methods proposed to find K

l := arg min å gi ( li - le * e

le ÎSe

i =1

)

41

Obnoxious Facility Location Problem ●

Big Triangle Small Triangle (BTST) method ●

Under certain conditions, finds lower bound for K * g l l å i =1 i ( i e ) that converges to global minimum

function to minimize

[Drezner and Suzuki'2004]

42

Obnoxious Facility Location Problem ●

Big Triangle Small Triangle (BTST) method ●

●

Under certain conditions, finds lower bound for K * g l l å i =1 i ( i e ) that converges to global minimum Steps overview: –

1) Partition region in Δ

[Drezner and Suzuki'2004]

43

Obnoxious Facility Location Problem ●

Big Triangle Small Triangle (BTST) method ●

●

Under certain conditions, finds lower bound for K * g l l å i =1 i ( i e ) that converges to global minimum Steps overview: – –

1) Partition region in Δ 2) Compute UB and LB for K * g l l å i =1 i ( i e ) within each Δ

UB2 LB2 UB3 LB3

UB1 LB1

function to minimize

UB4 LB4

[Drezner and Suzuki'2004]

44

Obnoxious Facility Location Problem ●

Big Triangle Small Triangle (BTST) method ●

●

Under certain conditions, finds lower bound for K * g l l å i =1 i ( i e ) that converges to global minimum Steps overview: – – –

1) Partition region in Δ 2) Compute UB and LB for K * g l l å i =1 i ( i e ) within each Δ 3) Eliminate all Δ with LB > lowest UB

[Drezner and Suzuki'2004]

UB3 LB3

45

Obnoxious Facility Location Problem ●

Big Triangle Small Triangle (BTST) method ●

●

Under certain conditions, finds lower bound for K * g l l å i =1 i ( i e ) that converges to global minimum Steps overview: – – – –

1) Partition region in Δ 2) Compute UB and LB for K * g l l å i =1 i ( i e ) within each Δ 3) Eliminate all Δ with LB > lowest UB 4) Partition Δ with lowest LB into 4 smaller Δ and return to 2)

[Drezner and Suzuki'2004]

46

Obnoxious Facility Location Problem ●

Big Triangle Small Triangle (BTST) method ●

●

Under certain conditions, finds lower bound for K * g l l å i =1 i ( i e ) that converges to global minimum Steps overview: – – – –

1) Partition region in Δ 2) Compute UB and LB for K * g l l å i =1 i ( i e ) within each Δ 3) Eliminate all Δ with LB > lowest UB 4) Partition Δ with lowest LB into 4 smaller Δ and return to 2)

[Drezner and Suzuki'2004]

Εasy Εasyto toobtain obtain UBs UBsand andLBs LBs for foreach eachΔΔ for forsuch suchfunction! function!

47

Example ●

Region of Interest: equilateral triangle

●

3 Sensors, one at each vertex ●

●

Similar to [Rao'2008] in system to detect radiation sources

Model parameters ●

Resemble [Rao'2008] Wi , j ~ Poisson ( 8 )

(

Ai ~ Poisson x ( li - le

x (d ) = 1

))

d2

(dimensions in km) 48

Example ●

Region of Interest: equilateral triangle

●

3 Sensors, one at each vertex ●

●

Similar to [Rao'2008] in system to detect radiation sources

●

For the centralized system...

ìK M ü f ( z ) = 1 íåå zi , j > t ý î i =1 j =1 þ

Model parameters ●

Resemble [Rao'2008] Wi , j ~ Poisson ( 8 )

(

Ai ~ Poisson x ( li - le

x (d ) = 1

))

d2

(dimensions in km) 49

Example ●

Region of Interest: equilateral triangle

●

3 Sensors, one at each vertex ●

●

Similar to [Rao'2008] in system to detect radiation sources

Model parameters ●

Resemble [Rao'2008]

●

For the centralized system...

ìK M ü f ( z ) = 1 íåå zi , j > t ý î i =1 j =1 þ Least Favorable Distribution places emitter at...

Wi , j ~ Poisson ( 8 )

(

Ai ~ Poisson x ( li - le

x (d ) = 1

))

d2

(dimensions in km) 50

Example ●

Region of Interest: equilateral triangle

●

3 Sensors, one at each vertex

probability of detection

●

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

●

Similar to [Rao'2008] in system to detect radiation sources

For the centralized system...

ìK M ü f ( z ) = 1 íåå zi , j > t ý î i =1 j =1 þ Least Favorable Distribution places emitter at...

Easy Easytotocompute computePD PD when whenconsider consideremitter emitteratatLFD LFD 5 10 15 20 25 30 35 M (number of measurements/sensor)

40 51

Example ●

Region of Interest: equilateral triangle

●

3 Sensors, one at each vertex

probability of detection

●

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

●

Similar to [Rao'2008] in system to detect radiation sources

For the centralized system...

ìK M ü f ( z ) = 1 íåå zi , j > t ý î i =1 j =1 þ Least Favorable Distribution places emitter at...

M=30 M=30totosatisfy satisfy requirement of requirement ofPD>0.95 PD>0.95

5 10 15 20 25 30 35 M (number of measurements/sensor)

40 52

Example ●

Region of Interest: equilateral triangle

●

3 Sensors, one at each vertex

probability of detection

●

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

●

Similar to [Rao'2008] in system to detect radiation sources

For the centralized system...

ìK M ü f ( z ) = 1 íåå zi , j > t ý î i =1 j =1 þ Least Favorable Distribution places emitter at...

Design considering emitter at LFD Design considering emitter at uniform distribution

5 10 15 20 25 30 35 M (number of measurements/sensor)

40 53

Example ●

Region of Interest: equilateral triangle

●

3 Sensors, one at each vertex

probability of detection

●

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

●

Similar to [Rao'2008] in system to detect radiation sources

For the centralized system...

ìK M ü f ( z ) = 1 íåå zi , j > t ý î i =1 j =1 þ

Least Favorable Distribution LFD 50% places LFDneeds needs 50% emitter at...

more moremeasurements measurements Design considering emitter at LFD Design considering emitter at uniform distribution

5 10 15 20 25 30 35 M (number of measurements/sensor)

40 54

Example ●

Region of Interest: equilateral triangle

●

3 Sensors, one at each vertex ●

●

Similar to [Rao'2008] in system to detect radiation sources

Model parameters ●

Resemble [Rao'2008]

(

x (d ) = 1

For the distributed system...

ì ìK ü ï f0 ( u ) = 1 íå ui > 0 ý î i =1 þ ï ( or ) f := í M ì ü ïf ( z ) = 1 zi , j > t ý í å i ï î j =1 þ î LFD places emitter at...

Wi , j ~ Poisson ( 8 ) Ai ~ Poisson x ( li - le

●

))

d2

(dimensions in km) 55

Example ●

Region of Interest: equilateral triangle

●

3 Sensors, one at each vertex ●

Similar to [Rao'2008] in system to detect radiation sources

probability of detection

1 0.9

~66% ~66% more more meas. meas.

0.8 0.7 0.6

●

For the distributed system...

ì ìK ü ï f0 ( u ) = 1 íå ui > 0 ý î i =1 þ ï ( or ) f := í M ì ü ïf ( z ) = 1 zi , j > t ý í å i ï î j =1 þ î LFD places emitter at...

Design considering emitter at LFD

0.5 0.4

Design considering emitter at uniform distribution

0.3 0.2 0.1

10

20 30 40 50 60 M (number of measurements/sensor)

56

Example ●

Region of Interest: equilateral triangle

●

3 Sensors, one at each vertex ●

●

Similar to [Rao'2008] in system to detect radiation sources

Model parameters ●

Resemble [Rao'2008]

(

x (d ) = 1

For the distributed system...

ì ìK ü ïf0 ( u ) = 1 íå ui = K ý î i =1 þ ï ( and ) f := í M ì ü ïf ( z ) = 1 zi , j > t ý í å i ï î j =1 þ î LFD places emitter at...

Wi , j ~ Poisson ( 8 ) Ai ~ Poisson x ( li - le

●

))

d2

(dimensions in km) 57

Example ●

Region of Interest: equilateral triangle

●

3 Sensors, one at each vertex ●

Similar to [Rao'2008] in system to detect radiation sources

probability of detection

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

~160% ~160% more more meas. meas.

●

For the distributed system...

ì ìK ü ïf0 ( u ) = 1 íå ui = K ý î i =1 þ ï ( and ) f := í M ì ü ïf ( z ) = 1 zi , j > t ý í å i ï î j =1 þ î LFD places emitter at...

Design considering emitter at LFD Design considering emitter at uniform distribution

0.2 0.1

50 100 150 200 250 300 350 400 450 M (number of measurements/sensor)

58

Discussion probability of detection

Very Veryconservative. conservative.

1 0.9 0.8 0.7 0.6 0.5

~160% ~160%more more measurements measurements

0.4 0.3 0.2 0.1

50 100 150 200 250 300 350 400 450 M (number of measurements/sensor)

59

Discussion

Design Designsystem systemunder under conditional conditionaldependency dependency

probability of detection

Very Veryconservative. conservative.

1 0.9 0.8 0.7 0.6 0.5

~160% ~160%more more measurements measurements

0.4 Problem: Problem: Need 0.3 Needemitter emitter location locationdistribution distribution 0.2 0.1

50 100 150 200 250 300 350 400 450 M (number of measurements/sensor)

60

Discussion

Design Designsystem systemunder under conditional conditionaldependency dependency

probability of detection

Very Veryconservative. conservative.

1 0.9 0.8 0.7 0.6 0.5

PD<0.8 PD<0.8 requirement requirementnot notsatisfied! satisfied!

0.4 Problem: Problem: Need 0.3 Needemitter emitter location locationdistribution distribution 0.2

Assume Assumeaa“reasonable” “reasonable” emitter emitterlocation locationdistribution distribution

0.1

50 100 150 200 250 300 350 400 450 M (number of measurements/sensor)

Problem: Problem: design designmay mayfail failtoto meet meetspecifications specifications 61

Discussion

Design Designsystem systemunder under conditional conditionaldependency dependency

probability of detection

Very Veryconservative. conservative.

1 0.9 0.8 0.7 0.6 0.5

PD<0.8 PD<0.8 requirement requirementnot notsatisfied! satisfied!

0.4 Problem: Problem: Need 0.3 Needemitter emitter location locationdistribution distribution 0.2

Assume Assumeaa“reasonable” “reasonable” emitter emitterlocation locationdistribution distribution Problem: Problem: design designmay mayfail failtoto meet meetspecifications specifications

0.1

50 100 150 200 250 300 350 400 450 M (number of measurements/sensor)

LFD LFDapproach approachoffers offersthe theoption option ofofcollecting collectingadditional additional measurements to measurements tocompensate compensate for forthe thelack lackofofinformation information 62

Summary Sensor Detection System Design: H1 is composite hypothesis

Measurements conditionally dependent

63

Summary Sensor Detection System Design: H1 is composite hypothesis

Measurements conditionally dependent

Least LeastFavorable FavorableDistribution Distribution for forEmitter EmitterLocation Location

64

Summary Sensor Detection System Design: H1 is composite hypothesis

Measurements conditionally dependent results results&&tools toolsfrom from Obnoxious ObnoxiousFacility FacilityLocation LocationProblem Problem

Least LeastFavorable FavorableDistribution Distribution for forEmitter EmitterLocation Location

65

Summary Sensor Detection System Design: H1 is composite hypothesis

Measurements conditionally dependent results results&&tools toolsfrom from Obnoxious ObnoxiousFacility FacilityLocation LocationProblem Problem

Least LeastFavorable FavorableDistribution Distribution for forEmitter EmitterLocation Location H1 is simple hypothesis

Measurements conditionally independent

Resulting design satisfies detection requirement for unknown emitter location distribution 66