plume tracking in sensor networks glenn nofsinger phd thesis defense august 22, 2006

Plume Tracking in Sensor Networks

Glenn NofsingerPhD Thesis Defense

August 22, 2006

Outline

1. Motivation and Problem Statement2. Other Work3. Theoretical Background4. 2-Step Algorithm5. Experiments6. Results and Conclusions

Motivation Current monitoring lacks information

sharing and high sampling density

Method needed for estimating highly unpredictable events: chemical, biological, radioactive agents

Many current sensors for such agents are binary

Problem Statement (1)

Gedanken-experiment: city with fixed, binary sensors of harmful agent

At an unexpected time a series of sensors activated, cause of release unknown

Where was the release? How many release sources? How are observations correlated?


t=o


t=1


t=2


t=3


t=4 What is the best estimate of the true source locations given these observations?


t=1 True initial state: two source locations

Thesis work estimates this truth state


This problem is hard!

Having an unknown number of sources and only binary detections at a large number of nodes is a new type of problem


Problem Summary:

Use a sensor network capable of only binary detection to estimate source locations

Evaluate performance of this estimation As a function of wind As a function of sensor density

Other work

ModelComplexity

Mobility

Mobile scout robots

Traditionalenvironmental techniqueswith high resolution sensors,low sensor density

Swarm robots

Static sensor networkswith high densitycheap fixed sensors

(Our approach)

III

III IV

Graphical Conventions:

Source Sensor Sensor with detection Track

A collection of sensors with detections believed to originate from the same event

Each track has different color

+

Theoretical Background (1)

Agent concentration for some area, A

Likelihood map given sensor observations

Plot Conventions:



Fick’s Law for diffusion and linear wind

First order approximation to process

Standard Gaussian solution

y

c

x

c

y

cD

x

cD

t

cyx

2

2

2

2

)4

)(

4

)((

22

4),,( tD

ty

tD

tx

yx

yxeDDt

AtyxC

Advection-diffusion Model


Solution of differential equations for advection-diffusion lead to a superposition of Gaussians

Peclet number measures relative strengths of diffusion to wind. A typical Peclet number is 10. This ratio determines plume width in our model

Plume Model


Assume a spatially uniform wind over the matrix A

Concentration state matrix A is designed to simulate an area of size 25mi x 25mi

Decorrelation length scale in wind data indicates the distances over which spatially uniform assumption holds

Typical values are on the order of 50-200 miles, therefore to first order we can assume spatially uniform wind

Wind Model

•Unique response per source location

•Relative differences of Tmax unique

•3 sensors for 2D location

•Can solve for (X0,Y0)

)4

)(

4

)((

21

22

4),,( tD

ty

tD

tx

yx

yxeDDt

AAtyxC

Classic Analytical Approaches


Analytical Approach Can solve differential equations for advection-

diffusion Solution of the source (X0,Y0) based on

measurements of C(t)

Method breaks down: No continuous time series available Very noisy, possibly binary data



t

C(x

,y,t

)

A B

A

B

Sensors

Radii of location for rising and falling edges of agent detection – one for each edge is possible in binary sensor

Analytical approach no longer useful, need statistical methods.

Leads to Bayesian formulation

Typical Sensor Response Curve

Bayesian Estimation Goal to obtain good estimate of target state Xt

based on measurement history Zt p(x) – a priori probability distribution function of

state x (plume concentration) – assumed uniform

p(z|x) – the likelihood function of z given x

p(x|z) – the a posteriori distribution of x given measurement z, also called the current belief


Bayesian Formulation Relationship between a posteriori distribution, a

priori distribution, and the likelihood function

Our state estimate

True state

•Want our state estimate to be as close to true state as possible

•Given observation set, what is:


Estimators

MMSE –minimum-mean-squared error. It is the mean posterior density. Equal weight to obs.

MAP- maximum a posteriori, maximizes the posterior distribution

ML- maximum likelihood, considers information in measurement only


Estimator Example: Source Localization

Each sensor measurement produces independent likelihood function

Cone shaped likelihood function Localization based on sequential

Bayesian estimation Measurements combined, assuming

independence of likelihood


Uniform State Estimation


MHT The previous uniform estimator can be improved

with advanced data association (DA) techniques such as multiple hypothesis tracking (MHT)

By maintaining multiple “tracks” observations partitioned into subsets which correspond to unique “targets” – in this case unique plume sources



MHT handles the combinatorial growth of possible track assignments via accurate pruning

Once tracks are built in the plume problem, assume 1 target per track, therefore focusing the custom estimation on one exclusive source

MHT

observations

Tracks

2-Step Algorithm (1) 2-Step track-estimate algorithm

1. Step 1 is track building2. Step 2 is state estimation of tracks

Custom Estimator based on tracks, ignoring observations not associated to a track

Able to work in two scenarios:1. Sources distant, distributed sensor groups2. Overlapping tracks, mixing sensor groups

End of Background (20 minutes)

2-Step Algorithm (2)

Step 1:Track estimation

Input: Sensor “Hits” (x,y,t)

Step 2:State EstimationFor each track

Output: N Tracks

M(Track1)

M(Track2)

M(TrackN) 0

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Aggregated Belief Probability With N= 57, 15 hits.

…

2-Step Algorithm (3) Step 1: Track Formation

1.1 Track Initialization –All new observations potentially create tracks. The terminal node on track is designated leader node


1.2 Data Association – All sensors with new observations calculate a likelihood function based on wind history. Function evaluated at all leader nodes


1.3 Track extension – observations that were associated in step 1.2 become the new leader nodes.


1.4 Track termination – The track is terminated once simulation ends or no new associations within cutoff parameter. Track outputs sent to Step 2


Track A

Track B

Likelihood Function

New Observation

Leader nodes

Detail of likelihood function for track association

2-Step Algorithm (8) Step 2: State Estimation

Each track sequence produces an individual likelihood map

In this case only 4 sensor observations used to form belief map

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2-Step Algorithm (9) Step 2: State Estimation

Each track sequence produces an individual likelihood map.

Only subset of observations applied to belief

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Gradual update of estimated source position, as sensor data is aggregated along the path ABCD.

Track Assisted State Estimation

Track Assisted State Estimation


Final estimated likelihood map after integration of ABCD, and renormalization for easier viewing.

Final update of estimated source position, as sensor data is aggregated along the path ABCD.

End of 2-Step Algorithm (40 minutes)

Experiments (1)

Experimental Setup Originally intended on collecting data

from a field of physical sensors, however this hardware component distracted from analytical purpose of thesis

Forward data generated based on real wind data, numerical approximation to diffusion, on a grid size m=n=250

All code implemented in LabVIEW graphical programming language, allows for easy future hardware integration

LabVIEW simulation design

Experiments (2)

Initialize, setup scenariosAnd control batch runs

Main loop – heavy computation

Result Outputs, statistical calculations

2-Step Alg.

DiffuseNumerical implementation

Fick’s law for diffusion implemented numerically using standard 2D centered difference scheme

Concentration of Agent assumed=0 at boundaries, agent “floats off screen”

Same code used for forward diffusion and backward belief state propagation

Experiments (3)

Large Batch Study #1: Wind Study Likelihood as a function of wind direction

standard deviation

As wind variability increases tracks become critical and perform dramatically better, operating in regions of high wind shift

A dataset containing 40,000 samples of real wind data are used to generate samples of length 200 spanning 5 degrees to 90 degrees

Experiments (4)

Wind Data Example, heavy processing needed

Experiments (5)

YYYY MM DD hh mm DIR SPD GDR GSP GTIME 2004 12 31 23 00 116 7.5 999 99.0 9999 2004 12 31 23 10 115 6.7 999 99.0 9999 2004 12 31 23 20 134 7.2 999 99.0 99992004 12 31 23 30 136 8.2 999 99.0 9999

Data Imported from webhttp://www.ndbc.noaa.gov/

Large Batch Study #2: Sensor Density Study

Increase number of sensors from N=50, 100, 150, 200, 250, 300 for a 250x250 grid. Random addition of new sensors to existing set.

Source fixed Same wind series for each trial Compared performance of belief maps

generated by sensor network using tracks Vs. No Tracks

Experiments (6)

Results and Conclusions

Maximum likelihood, ML(M), in each belief map compared to likelihood value at true source M(i,j)

Likelihood performance metrics

Source A(i,j) Belief M(i,j)

ML(M)

Performance Metric Definition, For a Single Source:

[M(i,j) / ML(M) ] = P(M), the performance of M

For P(M)=1, sensor occurs at the position (i,j) within M of maximum likelihood. 1 is considered a perfect score, while 0 is considered the lowest score

This is the metric used in wind study and sensor node density study


Typical M for same data

ZT

Observation set

MMSE predictor

2-Step predictor

Results and Conclusions (2)

Wind Experimental Results Summary

Results and Conclusions – KEY RESULT

Sensor Node Density Results Summary

Results and Conclusions – KEY RESULT

Density Conclusion

Identical network with tracks can achieve sharper maps with lower densities of sensors

Major advantage of using tracks is the ability to establish number of unique sources

Theoretical information content of a sensor network grows as log(N), therefore diminishing returns as N gets large. Both estimators approach this limit but at different rates


Summary of Wind performance zones


Wind direction Std. deviation

Mean w

ind speed

1 2

16

4

5

3

…

…

Worst performanceZone: low windSpeed, with Frequent shifts

Best performancezone

low high

high Intermediate performance

ZONE 1 ZONE 2 ZONE 3

Mean wind Speed scaled into 4 groupsStandard deviation of direction divided into 4 groupsThis produced 16 total wind categories

The 2-Step tracking based algorithm allows provides enhanced performance compared to uniform estimator

Sensor density – on average the tracker based maps received a likelihood metric better by a factor of 2

High Wind variability – in conditions of high wind direction variability, the tracking based estimator performs much better than uniform estimator. Maintaining tracks and therefore estimates up to 30 degrees Std. deviation higher.


Future Plans Application of sensor network physical process tracking to extreme

remote environments

The computationally intensive data association portion of the 2-Step algorithm method could be exported to existing MHT/PQS infrastructures and improved (pruning, track maintenance, hypothesis management).

Questions?

Sensor Density Study N=50 Sensors

P(M)=1 E-4


N=100P(M)=1 E-4


N=200

P(M)=1.3 E-4


N=300

P(M)=1E-4


N=400 Sensors

P(M)=9E-5


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Results - Belief Map From Uniform Predictor

Results - Belief Map, Track 1

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Results - Belief Map Track2

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Future Plans Application of sensor network physical process tracking to extreme

remote environments

The computationally intensive data association portion of the 2-Step algorithm method could be exported to existing MHT/PQS infrastructures and improved (pruning, track maintenance, hypothesis management).

Questions?

My papers

SPIE 2004 MILCOM 2005 SPIE 2006

Backup Slides

Source Separation Problem

To what extent canwe differentiate two sourcesas a function of sensordensity?

In this example, two sources in constant wind can superimpose to create a 3rd peak

The goal of this sensor network is to correctly identify exactly 2 sources, not 3

Belief Map Without Tracks

Inverse Belief Map of Sensor Network

Forward simulation Likelihood Map M

We want to construct a belief map after each trial, and look atthe value of the cell where the actual source was released.Once we introduce tracking, we get sharper regions with higherValues per cell. This allows us to compare the predicted mapwith ground truth on any selected trial.

Belief No Tracks

Track Formation

3 sources M=N=250 300 sensors Constant Wind

Example Likelihood (Belief) Map

The inverse scale here is E-5, which is likelihood thatThe source was released from that particular cell.Typical values for a single cell are between 10E-3 and 10E-5

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Forward Probability, P(B|A)

Inverse Probability, P(A|B)

Known release event : A

Known detection event : B

No Wind

Constant Wind

Variable WindConstant Wind

No Wind

Bayes Rule:

Variable Wind

)(

)()|()|(

BP

APABPBAP

plume tracking in sensor networks glenn nofsinger phd thesis defense august 22, 2006

Documents

binary slide

problem statement

o slide

new type of problem

uniform wind wind model

model plume model slide

truth state slide

source sensor sensor