Vanishing Decay Functions and Enlarged Deployment Regions to Facilitate the Design of Randomly Deployed Sensor Detection Systems Benedito J. B. Fonseca Jr. March 2013
Motivation Region of interest (city, park, stadium, campus)
Radioactive material being released One possible solution: restrict & control entry points Difficult when region has many entry points (e.g. city)
2
Sensor Detection System Region of interest
Sensors Sensorsdeployed deployedat at various variouspoint pointininthe theregion region Radioactive material
3
Sensor Detection System measurements noise
Fusion Center
H0 : Z = W
(Signal absent)
H1 : Z = A + W (Signal present) Z := ( Z1 ,..., Z K ) A := ( A1 ,..., AK )
Radioactive material
W := (W1 ,..., WK )
[Rao] (Oak Ridge National Lab) [Hills] (Lawrence Livermore National Lab) [Brennan],[Sundaresan],[Nemzek]
4
Sensor Detection System Fusion Center
H0 : Z = W
(Signal absent)
H1 : Z = A + W (Signal present) Z := ( Z1 ,..., Z K ) A := ( A1 ,..., AK )
Signal emitter
W := (W1 ,..., WK )
More generally, sensor detection systems can be used to... ●Detect radio transmissions (radio sensors) ●Detect the onset of a wildfire (temperature sensors) ●Detect intruders in a restricted area (seismic sensors) ●Submarines in the ocean (sonars) ●Aircrafts in an air space (radars) 5
Sensor Detection System H0 : Z = W
Fusion Center
(Signal absent)
H1 : Z = A + W (Signal present) Z := ( Z1 ,..., Z K ) A := ( A1 ,..., AK )
Signal emitter
Model Z1
q {0,1}
W := (W1 ,..., WK )
Sensor 1
U1
X1 Communication Subsystem
Measurement Process
ZK
Sensor K
UK
Fusion Center
XL
H0 H1 6
Designing a Sensor Detection System ●
●
Given: ●
Measurement process & Communication subsystem constraints
●
Prescribed minimum performance: –
Maximum probability of false alarm
–
Minimum probability of detection
Define: ●
Sensor functions
●
Fusion function
●
Number of sensors
Z1
q {0,1}
Sensor 1
U1
X1 Communication Subsystem
Measurement Process
ZK
Sensor K
UK
Fusion Center
XL
H0 H1 7
Designing Sensor Detection System under Conditionally Independent Measurements ●
Optimal sensor & fusion functions: [Tenney,Tsitsiklis,Willett,Varshney]
Li ( zi ) :=
f Zi |q =1 ( zi )
L0 ( u ) :=
f Zi |q =0 ( zi )
ì0, Li ( zi ) < ti ï fi ( z ) = íg i , Li ( zi ) = ti ï1, L ( z ) > t i i i î Z1
q {0,1}
Sensor 1
ì0, L0 ( u ) < t0 ï f0 ( z ) = íg 0 , L0 ( u ) = t0 ï1, L ( u ) > t 0 0 î
Communication Subsystem
ZK
Sensor K
P [ U = u | q = 0]
U1
Measurement Process conditionally independent
P [ U = u | q = 1]
UK
K dedicated error-free
Fusion Center
H0 H1 8
Designing Sensor Detection System under Conditionally Independent Measurements ●
Optimal sensor & fusion functions: [Tenney,Tsitsiklis,Willett,Varshney
Li ( zi ) :=
f Zi |q =1 ( zi )
f Zi |q =0 ( zi )
ì0, Li ( zi ) < ti ï fi ( z ) = íg i , Li ( zi ) = ti ï1, L ( z ) > t i i i î ●
L0 ( u ) :=
P [ U = u | q = 1]
P [ U = u | q = 0]
ì0, L0 ( u ) < t0 ï f0 ( z ) = íg 0 , L0 ( u ) = t0 ï1, L ( u ) > t 0 0 î
If measurements are conditionally dependent... – –
Sensor functions may no longer be based on the measurement likelihood [Chen,Willet] Fusion function depends on various correlation parameters [Kam] ● Difficult to know if sensors' locations are unknown 9
Conditional Dependence When signal depends on the distances between emitter and sensors, measurements become conditionally dependent ●
Sensors
Emitter
Much harder to determine optimal system
●
Harder to compute performance of any system
●
●
Requires integration over emitter location distribution
10
Conditional Dependence When signal depends on the distances between emitter and sensors, measurements become conditionally dependent ●
PLe Emitter
Much harder to determine optimal system
●
Harder to compute performance of any system
●
●
Requires integration over emitter location distribution
P1 éëf0 (U ) = 1ùû = ò P1 éëf0 (U ) = 1| Le = le ùû dPLe ( le ) Se ●
Numerical approaches often necessary
Emitter Emitterlocation locationcan canbe be considered consideredrandom random 11
Can Dependence be Disregarded? 1
Signal & Noise parameters as in [Rao'08] ● Signals decaying with d 2 ●
●
L-shaped region
Emitter & sensor locations uniformly distributed in region ●
●
Distributed detection system ● binary sensors
probability of detection
Example:
0.95
0.9
0.85 150 200
Designer Designerassumes assumes independence independence and anddecides decidesfor for 270 sensors 270 sensors
250 300 350 400 number of sensors
450
500
12
Can Dependence be Disregarded?
Signal & Noise parameters as in [Rao'08] ● Signals decaying with d2 ●
●
L-shaped region
Emitter & sensor locations uniformly distributed in region ●
●
Distributed detection system ● binary sensors
probability of detection
Example:
Measurements dependent Measurementsare areactually actually dependent 1 requirement! System does not meet System does not meet requirement!
0.95
0.9 considering dependence disregarding dependence 0.85 150 200
250 300 350 400 number of sensors
450
500
13
Can Dependence be Disregarded?
Signal & Noise parameters as in [Rao'08] ● Signals decaying with d2 ●
●
L-shaped region
Emitter & sensor locations uniformly distributed in region ●
●
Distributed detection system ● binary sensors
probability of detection
Example:
Number Numberofofsensor sensortotomeet meet requirement (95% PD, 5% requirement1(95% PD, 5%PFA) PFA) underestimated underestimatedby byaround around18% 18%
0.95
0.9 considering dependence disregarding dependence 0.85 150 200
250 300 350 400 number of sensors
450
500
14
More Difficulties to Design System ●
Emitter location is random ●
Requires integration over emitter location distribution
P1 éëf0 (U ) = 1ùû = ò P1 éëf0 (U ) = 1| Le = le ùû dPLe ( le ) Se ●
Emitter Location Distribution is generally unknown
PLe Emitter
Hypothesis H1 becomes composite
Sensors
H 0 : Z ~ PZ |q =0 H1 : Z ~ PZ |q =1, P(1) , PZ |q =1, P( 2) ,... Le
Le
●
Difficult to find uniformly most powerful designs
●
Difficult to ensure performance for all the possible distributions
15
Difficulties in Designing Distributed Detection Systems Tackle the model -Conditional dependence -Composite hypothesis
System designer options:
●
Difficult to make progress
Simplify the model -assume cond. independ. -assume an emitter location distribution
Design will be criticized Design may fail to satisfy requirement
If analytical treatment is difficult, it is not used. ●
Benefits & guidance of analytical treatment are lost
●
Increased risk in the design process 16
Research Research Question: Question: ●
Are there ways to circumvent the difficulties of ●
Conditional dependent measurements and
●
Composite hypothesis?
17
Key Assumptions
Le
Fusion Center
Li
H0 : Z = W H1 : Z = A + W
{Wi } iid Z := ( Z1 ,..., Z K ) A := ( A1 ,..., AK )
Amplitude function
x (d )
Ai = x ( Li - Le
Distance between sensor and emitter
)
W := (W1 ,..., WK )
Signal random variable depends on sensor and emitter locations through an Amplitude Function of the distance
18
Key Assumptions
Le
Fusion Center
Li
H0 : Z = W H1 : Z = A + W
{Wi } iid Z := ( Z1 ,..., Z K ) A := ( A1 ,..., AK )
Amplitude function
x (d )
Ai = x ( Li - Le
Distance between sensor and emitter
)
W := (W1 ,..., WK )
Sensors deployed at random i.i.d. locations Reasonable when... ● Sensors are mobile (spectrum sensing) ● Sensors are deployed from aircraft ● Sensor deployment may not be under the control of the system designer 19
Least Favorable Distributions for Emitter Location ●
Among other conditions,
Circular region
Sensor locations uniformly distr.
Z1
ZK
Le
P
Sensor 1
Sensor K
U1 Communication Subsystem
UK
· {Z i } conditionally i.i.d.
(
)
(
· "PLe , b PLe , f ³ b PL-e , f
K dedicated error-free
f0 ( u ) ì K ü 1 íÕFusion T0,i ( ui ) > t ý î i =1 Center þ
H0 H1
Cond. ●Cond.dependence dependenceproblem problemsolved solved ●Hypothesis H1 is now simple ●Hypothesis H1 is now simple ●Performance requirement met ●Performance requirement met ●
)
Probability of detection of f under PLe
20
Least Favorable Distributions for Emitter Location ●
Among other conditions,
Circular region
Sensor locations uniformly distr.
Z1
U1 Communication Subsystem
ZK
Le
P
UK
· {Z i } conditionally i.i.d.
(
)
(
· "PLe , b PLe , f ³ b PL-e , f
K dedicated error-free
f0 ( u )
ì KFusion ü 1 íå Zi > t ý Center î i =1 þ
H0 H1
Cond. ●Cond.dependence dependenceproblem problemsolved solved ●Hypothesis H1 is now simple ●Hypothesis H1 is now simple ●Performance requirement met ●Performance requirement met ●
)
Probability of detection of f under PLe 21
Least Favorable Distributions for Emitter Location ●
Among other conditions,
Circular region
Sensor locations uniformly distr.
Z1
ZK
Le
P
1{Sensor Z i > t s1}
1{Sensor Z i > tKs }
U1 Communication Subsystem
UK
· {Z i } conditionally i.i.d.
(
)
(
· "PLe , b PLe , f ³ b PL-e , f
K dedicated error-free
f0 ( u )
ì KFusion ü 1 íå Ui > t ý Center î i =1 þ
H0 H1
Cond. ●Cond.dependence dependenceproblem problemsolved solved ●Hypothesis H1 is now simple ●Hypothesis H1 is now simple ●Performance requirement met ●Performance requirement met ●
)
Probability of detection of f under PLe 22
Least Favorable Distributions for Emitter Location ●
Among other conditions,
Regular Polygon
Sensor locations uniformly distr.
Z1
ZK
Le
P
1{Sensor Z i > t s1}
1{Sensor Z i > tKs }
U1 Communication Subsystem
UK
· {Z i } conditionally i.i.d.
(
)
(
· "PLe , b PLe , f ³ b PL-e , f
K dedicated error-free
f0 ( u )
ì KFusion ü 1 íå Ui > t ý Center î i =1 þ
H0 H1
Cond. ●Cond.dependence dependenceproblem problemsolved solved ●Hypothesis H1 is now simple ●Hypothesis H1 is now simple ●Performance requirement met ●Performance requirement met ●
)
Probability of detection of f under PLe 23
Research Research Question: Question: ●
How to address the problems of conditional dependency and composite hypothesis when the region of interest is not circular or a regular convex polygon?
24
Proposition 1 Consider any region of interest...
IfIfthe theamplitude amplitudedecay decayfunction function vanishes vanishesatatsome somepoint point
Among other conditions... IfIfsensor sensorlocations locationsare are uniformly uniformlydistributed distributed ininan anenlarged enlargedregion region
Ss Ê
U B (l )
le ÎSe
d0
e
x (d ) Distance between sensor and emitter
d0
Then, the measurements become Then, the measurements become ● conditionally i.i.d.; and ● conditionally i.i.d.; and ● independent of the emitter location ● independent of the emitter location ► ►system systemdesigner designerno nolonger longerneeds needstoto know knowthe theemitter emitterlocation locationdistribution distribution 25
Can we use Proposition 1? Propose Proposethat thatsystem systemdesigner designer assume a modified amplitude assume a modified amplitude decay decayfunction function
ìx ( d ) if d < d 0 x0 ( d ) = í if d ³ d 0 î0
IfIfthe theamplitude amplitudedecay decayfunction function vanishes vanishesatatsome somepoint point
x (d )
Distance between sensor and emitter
d0
Proposition 1 can be used and design system for... ● conditionally i.i.d. measurements ● simple hypothesis
26
Can we use Proposition 1? Propose Proposethat thatsystem systemdesigner designer assume a modified amplitude assume a modified amplitude decay decayfunction function
IfIfthe theamplitude amplitudedecay decayfunction function vanishes vanishesatatsome somepoint point
ìx ( d ) if d < d 0 x0 ( d ) = í if d ³ d 0 î0 how does one choose d 0?
x (d )
Distance between sensor and emitter
d0
Proposition 1 can be used and design system for... ● conditionally i.i.d. measurements ● simple hypothesis
27
How to Modify the Decay Function? ●
High d 0
x (d )
Distance between sensor and emitter
d0 Design Designconsiders considersfarther farthersensors sensors Lower Lowersensor sensordensity density ●
Low d 0
x (d )
Difficult Difficulttotofind findoptimum optimumd 0 ●Decay function ●Decay function ●Shape & size of region ●Shape & size of region
Distance between sensor and emitter
d0 Design Designdisregards disregardssensors sensorswith withstrong strongsignal signal Higher Highersensor sensordensity density
28
How to Modify the Decay Function? Probability of detection
Probability of false alarm 0 ●
d0
Proposed method: increase d 0 in small steps till first local maximum is reached
29
How to Modify the Decay Function? Example: Signal & Noise parameters as in [Rao'08] ● Signals decaying with d2
1
●
L-shaped region
Emitter & sensor locations uniformly distributed in region ●
●
Distributed detection system ● binary sensors
probability of detection
●
0.8 0.6 D
0.4
D
0.2 0 0
0.5
1 d0
1.5
2
D 30
Actual Performance Propose Proposethat thatsystem systemdesigner designer assume a modified amplitude assume a modified amplitude decay decayfunction function
ìx ( d ) if d < d 0 x0 ( d ) = í if d ³ d 0 î0
IfIfthe theamplitude amplitudedecay decayfunction function vanishes vanishesatatsome somepoint point
x (d )
Distance between sensor and emitter
d0
Proposition 1 can be used and design system for... ● conditionally i.i.d. measurements ● simple hypothesis However, However,what whatwould wouldbe bethe thedetection detectionperformance performanceof ofthe the resulting resultingsystem systemunder underthe theactual actualamplitude amplitudedecay decayfunction? function? 31
Proposition 2 Propose Proposethat thatsystem systemdesigner designer assume assumeaamodified modifiedamplitude amplitude decay function decay function
IfIfthe theamplitude amplitudedecay decayfunction function vanishes vanishesatatsome somepoint point
ìx ( d ) if d < d 0 x0 ( d ) = í if d ³ d 0 î0
x (d )
Distance between sensor and emitter
d0
Under certain conditions and for a broad range of detection systems
f
The Theprobability probabilityofofdetection detectionunder underthe theactual actualdecay decayfunction function will willalways alwaysbe beequal equalororgreater greater
b min £ b (x 0 , S s , f ) £ b (x , S s , f )
prob. of detection requirement
prob. of detection of f under x0 and enlarged deployment region S s
32
How does a designer use this result? ●
Assume modified decay function, and deploy sensors in enlarged region
x (d )
Distance between sensor and emitter
d0 then the design is facilitated because Measurements conditionally i.i.d. Any emitter location distribution can be assumed ```` H1 simple Hypothesis ●
Detection performance ensured for actual decay function and actual emitter location distribution
b min £ b (x 0 , S s , f ) £ b (x , S s , f )
How Howconservative conservative isisthe thedesign? design? 33
How Conservative is the Design? 1
Signal & Noise parameters as in [Rao'08] ● Signals decaying with d2 ●
●
L-shaped region
Emitter & sensor locations uniformly distributed in region ●
●
Distributed detection system ● binary sensors
Probability of detection
Example:
0.9 0.8 0.7 baseline approach proposed approach
0.6 0
100
200 300 400 500 Number of sensors (K)
P1 éëf0 (U ) = 1ùû = ò P1 éëf0 (U ) = 1| Le = le ùû dPLe ( le ) Se
600
Assumes Assumesemitter emitter uniformly uniformlydistr. distr.
700
34
How Conservative is the Design? 1
Signal & Noise parameters as in [Rao'08] ● Signals decaying with d2 ●
●
L-shaped region
Emitter & sensor locations uniformly distributed in region ●
●
Distributed detection system ● binary sensors
Probability of detection
Example:
0.9 Proposed Proposedapproach approach (actual (actualdecay decayfunction) function)
0.8 0.7
baseline approach proposed approach
0.6 0
100
200 300 400 500 Number of sensors (K)
600
700
The Theprobability probabilityofofdetection detectionunder underthe theactual actualdecay decayfunction function will willalways alwaysbe beequal equalororgreater greater
b min £ b (x 0 , S s , f ) £ b (x , S s , f )
35
How Conservative is the Design? 1
Signal & Noise parameters as in [Rao'08] ● Signals decaying with d2 ●
●
L-shaped region
Emitter & sensor locations uniformly distributed in region ●
●
Distributed detection system ● binary sensors
Probability of detection
Example:
4.8x 4.8x
0.9 0.8 0.7
baseline approach proposed approach
0.6 0
100
200 300 400 500 Number of sensors (K)
600
700
36
How Conservative is the Design? 1
Signal & Noise parameters as in [Rao'08] ● Signals decaying with d2 ●
●
L-shaped region
Emitter & sensor locations uniformly distributed in region ●
●
Distributed detection system ● binary sensors
Probability of detection
Example:
0.9 0.8 0.7 baseline approach proposed approach
0.6 0
decay K ( prop )
K
( base )
1
100
2
200 300 400 500 Number of sensors (K)
3
4
5
600
700
15
10.3 4.8 3.8 3.5 3.3 3.1 37
Discussion Very Veryconservative. conservative. Design Designsystem systemunder under conditional conditionaldependency dependency
Probability of detection
1 4.8x 4.8x
0.9 0.8
0.7 Problem: Problem: Need Needemitter emitter location locationdistribution distribution 0.6 Assume Assumeaa“reasonable” “reasonable” emitter emitterlocation locationdistribution distribution
0
baseline approach proposed approach 100
200 300 400 500 Number of sensors (K)
600
700
Problem: Problem: design designmay mayfail failtoto meet meetspecifications specifications 38
Discussion Very Veryconservative. conservative.
Gather Gatherinformation informationabout about emitter emitterlocation locationdistribution distribution
Design Designsystem systemunder under conditional conditionaldependency dependency
Problem: Problem: Cost Costassociated associated with withinformation information
Problem: Problem: Need Needemitter emitter location locationdistribution distribution Assume Assumeaa“reasonable” “reasonable” emitter emitterlocation locationdistribution distribution Problem: Problem: design designmay mayfail failtoto meet meetspecifications specifications
PLe = $$
???
IsIsititbetter bettertotoinvest invest this thiscost costininadditional additionalsensors? sensors? proposed proposedapproach approachoffers offersthe theoption option ofofusing usingadditional additionalsensors sensorstoto compensate compensatefor forthe the lack lackofofinformation information 39
Summary & Conclusions Tackle the model -Conditional dependence
Difficult to make progress
-Composite hypothesis Simplify the model
system designer has more options:
-assume cond. independ. -assume an emitter location distribution
Design will be criticized Design may fail to satisfy requirement
Vanishing decay functions and enlarged deployment regions: -Simple hypothesis & cond iid
x (d )
-Design ensures requirement d0
40
