Improving the Scan Statistic to Design Sensor Detection Systems

Benedito J. B. Fonseca Jr.July 2016

Motivation

Region of Interest(city, stadium, park)

Motivation

Radioactive materialbeing released

Region of Interest(city, stadium, park)

Motivation

Possible solution: restrict & control entry points(Difficult when region has many entry points)

Radioactive materialbeing released

Region of Interest(city, stadium, park)

Sensor Detection Systems

Sensors deployed at various points in the region

Sensor Detection Systems

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

measurement sensor at position (I,j)

noise

signal

(emitter absent)

(emitter present)

Sensor Detection Systems

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

measurement sensor at position (I,j)

noise

signal

(emitter absent)

(emitter present)

• [Rao] (Oak Ridge National Lab – USA) • [Hills] (Lawrence Livermore National Lab – USA)• [Drukier],[Qian],... IEEE Conferences on Homeland Security (annual)• [Liu],[Stella],[Sundaresan],... • Safecast.org

Sensor Detection Systems

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

measurement sensor at position (I,j)

noise

signal

(emitter absent)

(emitter present)

More generally, sensor detection systems can be used to… • Detect radio transmissions• Detect the onset of a wildfire • Detect intruders in a restricted area (seismic sensors)• Submarines in the ocean (sonars)• Aircraft in an air space (radars)

Sensor Detection Systems

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

measurement sensor at position (I,j)

signal

(emitter absent)

(emitter present)

IIDnoise

Sensor Detection Systems

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

measurement sensor at position (I,j)

noise

(emitter absent)

(emitter present)

Signal random variable depends on sensor and emitter locations through an amplitude function of the distance

,i j eL L

Distance between sensor and emitter

dAmplitude Function

Sensor location

Emitter location

IID

How to combine measurements??

Problem: How to design the Fusion Center?

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

measurement sensor at position (I,j)

noise

(emitter absent)

(emitter present)

Signal random variable depends on sensor and emitter locations through an amplitude function of the distance

,i j eL L

Distance between sensor and emitter

dAmplitude Function

Sensor location

Emitter location

??

IID

How to combine measurements??

Problem: How to design the Fusion Center?

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

measurement sensor at position (I,j)

noise

(emitter absent)

(emitter present)

Signal random variable depends on sensor and emitter locations through an amplitude function of the distance

,i j eL L

Distance between sensor and emitter

dAmplitude Function

Sensor location

??

Which sensors are close to the emitter?

IID

How to combine measurements??

Problem: How to design the Fusion Center?

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

measurement sensor at position (I,j)

noise

(emitter absent)

(emitter present)

,i j eL L

Distance between sensor and emitter

dAmplitude Function

Sensor location

??

Which sensors are close to the emitter?

Emitter location is unknown

Emitter may be close toany of the sensors!

IID

Possible Fusion Rules

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

Distance between sensor and emitter

dAmplitude Function

, 1

, 0

max

max

i j

or

i j

Z t H

Z t H

z

Gives same important to all sensorsNeed to increase t to keep PFA low ( low PD)

(emitter absent)

(emitter present)

Fusion Center

Possible Fusion Rules

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

, 1

, 0

i j

sum

i j

Z t H

Z t H

z

Gives same important to all sensorsAverages out noiseCombines weak and strong measurements( low PD)

(emitter absent)

(emitter present)

strong signal weak signal

Distance between sensor and emitter

d

Scan Statistic

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

(emitter absent)

(emitter present)

• Combine measurements from sensors within a cluster C

[Guerriero, Willett, Glaz; 2009]

Scan Statistic

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

(emitter absent)

(emitter present)

• Combine measurements from sensors within a cluster C

• Scan over several clusters

[Guerriero, Willett, Glaz; 2009]

Scan Statistic

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

(emitter absent)

(emitter present)

• Combine measurements from sensors within a cluster C

• Scan over several clusters

[Guerriero, Willett, Glaz; 2009]

Scan Statistic

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

(emitter absent)

(emitter present)

• Combine measurements from sensors within a cluster C

• Scan over several clusters

[Guerriero, Willett, Glaz; 2009]

Scan Statistic

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

(emitter absent)

(emitter present)

• Combine measurements from sensors within a cluster C

• Scan over several clusters• Decide for H1 if any cluster has

high enough statistic

, 1,

, 0,

max

max

i ji j CC

scan

i ji j CC

Z t H

Z t H

z

Scan Statistic

Fusion Center

0 , ,

1 , , ,

:

:

i j i j

i j i j i j

H Z W

H Z A W

(emitter absent)

(emitter present)

• Combine measurements from sensors within a cluster C

• Scan over several clusters• Decide for H1 if any cluster has

high enough statistic

, 1,

, 0,

max

max

i ji j CC

scan

i ji j CC

Z t H

Z t H

z

Gives same important to all sensorsAverages out noiseCombines measurements only from sensors within cluster

Example

Intensity of emitter signal (I0)P

rob

abili

ty o

f d

etec

tio

n

• 5km x 5km square region• 64 sensors uniformly placed in 8x8 grid• Assume emitter in center of region

0 ,

01 , 2

: ~ , 8

: ~ ,

i j

i j i e

H Z Poisson

IH Z Poisson l l d

d

(d in km) le in center

Example

Intensity of emitter signal (I0)P

rob

abili

ty o

f d

etec

tio

n

• 5km x 5km square region• 64 sensors uniformly placed in 8x8 grid• Assume emitter in center of region

0 ,

01 , 2

: ~ , 8

: ~ ,

i j

i j i e

H Z Poisson

IH Z Poisson l l d

d

(d in km) le in center

Example

Intensity of emitter signal (I0)P

rob

abili

ty o

f d

etec

tio

n

• 5km x 5km square region• 64 sensors uniformly placed in 8x8 grid• Assume emitter in center of region

0 ,

01 , 2

: ~ , 8

: ~ ,

i j

i j i e

H Z Poisson

IH Z Poisson l l d

d

(d in km)

Clusters to scan:• All 2x2 clusters• Clusters with 2 sensors

in boundary• Clusters with 1 sensor

in corners

le in center

Example

Intensity of emitter signal (I0)P

rob

abili

ty o

f d

etec

tio

n

• 5km x 5km square region• 64 sensors uniformly placed in 8x8 grid• Assume emitter in center of region

0 ,

01 , 2

: ~ , 8

: ~ ,

i j

i j i e

H Z Poisson

IH Z Poisson l l d

d

(d in km)

Clusters to scan:• All 2x2 clusters• Clusters with 2 sensors

in boundary• Clusters with 1 sensor

in corners

le in center

P[detect]=0.95 for emitters up to 33%

weaker

Example:Emitter at Corner

• 5km x 5km square region• 64 sensors uniformly placed in 8x8 grid

0 ,

01 , 2

: ~ , 8

: ~ ,

i j

i j i e

H Z Poisson

IH Z Poisson l l d

d

emitter at corner

Example:Emitter at Corner

Intensity of emitter signal (I0)P

rob

abili

ty o

f d

etec

tio

n

• 5km x 5km square region• 64 sensors uniformly placed in 8x8 grid

(d in km)

le in corner

worse performance of Scan Statistics

emitter at corner

0 ,

01 , 2

: ~ , 8

: ~ ,

i j

i j i e

H Z Poisson

IH Z Poisson l l d

d

Example:Emitter at Corner

Intensity of emitter signal (I0)P

rob

abili

ty o

f d

etec

tio

n

Why? Because just 1 sensor has high SNR

le in corner

worse performance of Scan Statistics

, 1,

, 0,

max

max

i ji j CC

scan

i ji j CC

Z t H

Z t H

z

Example:Emitter at Corner

Intensity of emitter signal (I0)P

rob

abili

ty o

f d

etec

tio

n

Why? Because just 1 sensor has high SNR

le in corner

Should we worry about it?• Yes, if need to ensure performance

for all conditions

worse performance of Scan Statistics

, 1,

, 0,

max

max

i ji j CC

scan

i ji j CC

Z t H

Z t H

z

Missing sensor

Example:Missing Sensor

• 5km x 5km square region• 63 sensors uniformly placed in 8x8 grid

0 ,

01 , 2

: ~ , 8

: ~ ,

i j

i j i e

H Z Poisson

IH Z Poisson l l d

d

Example:Missing Sensor

• 5km x 5km square region• 63 sensors uniformly placed in 8x8 grid

Worst case foremitter location

0 ,

01 , 2

: ~ , 8

: ~ ,

i j

i j i e

H Z Poisson

IH Z Poisson l l d

d

Example:Missing Sensor

• 5km x 5km square region• 63 sensors uniformly placed in 8x8 grid

(d in km)

Worst case foremitter location Intensity of emitter signal (I0)

Pro

bab

ility

of

det

ecti

on

Missing sensor

Degraded performance of Scan Statistics

0 ,

01 , 2

: ~ , 8

: ~ ,

i j

i j i e

H Z Poisson

IH Z Poisson l l d

d

Example:Missing Sensor

Intensity of emitter signal (I0)P

rob

abili

ty o

f d

etec

tio

n

Missing sensor

Degraded performance of Scan Statistics

Why? • clusters close to emitter have only 3 sensors

, 1,

, 0,

max

max

i ji j CC

scan

i ji j CC

Z t H

Z t H

z

Worst case foremitter location

Example:Missing Sensor

Intensity of emitter signal (I0)P

rob

abili

ty o

f d

etec

tio

n

Missing sensor

Degraded performance of Scan Statistics

Is this a problem?• Yes, it may be difficult to deploy

sensors in a regular pattern

, 1,

, 0,

max

max

i ji j CC

scan

i ji j CC

Z t H

Z t H

z

Why? • clusters close to emitter have only 3 sensors

Research Question:

Scan Statistics is able to• Combine measurements to average noise• Combine measurements only from sensors near the emitter

However, detection performance suffers when the cluster that detects emitter has a small number of sensors.

Research Question:

Scan Statistics is able to• Combine measurements to average noise• Combine measurements only from sensors near the emitter

However, detection performance suffers when the cluster that detects emitter has a small number of sensors.

Research Question:• How to improve the performance of the scan statistic

when the number of sensors in each cluster vary?

Normalized Scan Statistic

• Original Scan Statistic:

• We propose that the Scan Statistic be normalized as follows:

g(|C|): a function of the number of sensors in the cluster C

, 1,

, 0,

max

max

i ji j CC

scan

i ji j CC

Z t H

Z t H

z

, 1,

, 0,

1max

1max

i ji j CC

new

i ji j CC

Z t Hg C

Z t Hg C

z

Normalized Scan Statistic

• What if we just normalize by the number of sensors (g(|C|)=|C|) ?

, 1,

, 0,

1max

1max

i ji j CC

new

i ji j CC

Z t Hg C

Z t Hg C

z

le in corner

Intensity of emitter signal (I0)

Pro

bab

ility

of

det

ecti

on

Normalized Scan Statistic

• What if we just normalize by the number of sensors (g(|C|)=|C|) ?• Does not solve the problem in all situations

, 1,

, 0,

1max

1max

i ji j CC

new

i ji j CC

Z t Hg C

Z t Hg C

z

le in corner le in center Missing sensor

Intensity of emitter signal (I0)

Pro

bab

ility

of

det

ecti

on

Normalized Scan Statistic

• What if we just normalize by the number of sensors (g(|C|)=|C|) ?• Does not solve the problem in all situations

• Why?• All cluster statistics have the same expected value

• High |C| cluster statistic with low varianceLow |C| cluster statistic with high variance

• Higher variance higher tail probabilities

• Probabilities of Detection and False Alarm depends on the tail probabilities!

, 1,

, 0,

1max

1max

i ji j CC

new

i ji j CC

Z t Hg C

Z t Hg C

z

Normalized Scan Statistic

• What if we just normalize by the number of sensors (g(|C|)=|C|) ?• Does not solve the problem in all situations

• Why?• All cluster statistics have the same expected value

• High |C| cluster statistic with low varianceLow |C| cluster statistic with high variance

• Higher variance higher tail probabilities

• Probabilities of Detection and False Alarm depends on the tail probabilities!

• Need normalization that normalizes the tail statistics for various |C|

, 1,

, 0,

1max

1max

i ji j CC

new

i ji j CC

Z t Hg C

Z t Hg C

z

Normalized Scan Statistic

• We propose that the Scan Statistic be normalized with g(|C|) that satisfies…

, 1,

, 0,

1max

1max

i ji j CC

new

i ji j CC

Z t Hg C

Z t Hg C

z

0 , 0 1,1,

1i ji j C

P Z t P Z tg C

Probability of False Alarmof cluster with |C| sensors

Probability of False Alarmof cluster with 1 sensor

Normalized Scan Statistic

• We propose that the Scan Statistic be normalized with g(|C|) that satisfies…

, 1,

, 0,

1max

1max

i ji j CC

new

i ji j CC

Z t Hg C

Z t Hg C

z

0 , 0 1,1,

1i ji j C

P Z t P Z tg C

Probability of False Alarmof cluster with |C| sensors

Probability of False Alarmof cluster with 1 sensor CFAR scan statistic

Normalized Scan Statistic

• We propose that the Scan Statistic be normalized with g(|C|) that satisfies…

• Easy to compute g(|C|) numerically for |C|=2,3,4,… up to the maximum number of sensors in a cluster

, 1,

, 0,

1max

1max

i ji j CC

new

i ji j CC

Z t Hg C

Z t Hg C

z

0 , 0 1,1,

1i ji j C

P Z t P Z tg C

Probability of False Alarmof cluster with |C| sensors

Probability of False Alarmof cluster with 1 sensor CFAR scan statistic

Example

• Performance of proposed Normalized Scan StatisticP

rob

abili

ty o

f d

etec

tio

n

Intensity of emitter signal (I0)

Conclusions

Scan Statistics is able to• Combine measurements to average noise• Combine measurements only from sensors near the emitter

• However, detection performance suffers when the cluster that detects emitter has a small number of sensors.

• Important when designer needs to ensure performance even at the worst possible emitter location

Conclusions

Scan Statistics is able to• Combine measurements to average noise• Combine measurements only from sensors near the emitter

• However, detection performance suffers when the cluster that detects emitter has a small number of sensors.

• Important when designer needs to ensure performance even at the worst possible emitter location

• Proposed normalized Scan Statistic that normalizes PFA in each cluster (CFAR scan statistic) and improves detection performance when emitter is in clusters with small number of sensors

• Maintains detection performance at clusters with designed # of sensors