Transcript Plume Tracking and Process Detection in Sensor Networks
Plume Tracking in Sensor Networks
Glenn Nofsinger PhD Thesis Defense August 22, 2006
Outline
1. Motivation and Problem Statement 2. Other Work 3. Theoretical Background 4. 2-Step Algorithm 5. Experiments 6. 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?
Problem Statement (1)
t=o
Problem Statement (1)
t=1
Problem Statement (1)
t=2
Problem Statement (1)
t=3
Problem Statement (1)
t=4
What is the best estimate of the true source locations given these observations ?
Problem Statement (1)
t=1
True initial state: two source locations
Thesis work estimates this truth state
Problem Statement (2)
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 Statement (3)
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
Swarm robots II Static sensor networks with high density cheap fixed sensors III ( Our approach ) Mobile scout robots I IV Model Complexity Traditional environmental techniques with high resolution sensors, low sensor density Mobility
Theoretical Background (1) 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 (2) Plot Conventions:
Agent concentration for some area, A
Likelihood map given sensor observations
Theoretical Background (3)
Advection-diffusion Model
Fick’s Law for diffusion and linear wind
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First order approximation to process
Standard Gaussian solution
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Theoretical Background (4)
Plume 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
Theoretical Background (5)
Wind 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
Theoretical Background (6) Classic Analytical Approaches
C
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D x D y e
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) •Unique response per source location •Relative differences of Tmax unique •3 sensors for 2D location •Can solve for (X0,Y0)
Theoretical Background (7)
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
Theoretical Background (8)
A
B
Typical Sensor Response Curve A
B
t 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
Theoretical Background (9)
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
Theoretical Background (10)
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:
Theoretical Background (11) 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
Theoretical Background (12) 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
Theoretical Background (13) Uniform State Estimation
Theoretical Background (14)
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
Theoretical Background (15)
MHT 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
observations Tracks
2-Step Algorithm (1) 2-Step track-estimate algorithm 1. Step 1 is track building 2. 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 groups 2. Overlapping tracks, mixing sensor groups
End of Background (20 minutes)
2-Step Algorithm (2)
Input: Sensor “Hits” (x,y,t) Step 1: Track estimation Output: N Tracks Step 2: State Estimation For each track 300 Aggregated Belief Probability With N= 57, 15 hits. 250 150 M(Track2) … 50 M(TrackN) 0 -50 0 50 100 X 150 200 250 300 4.5
4 3.5
3 2.5
2 1.5
1 0.5
0
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
2-Step Algorithm (4)
Step 1: Track Formation
1.2 Data Association – All sensors with new observations calculate a likelihood function based on wind history. Function evaluated at all leader nodes
2-Step Algorithm (5)
Step 1: Track Formation
1.3
Track extension –
observations that were associated in step 1.2 become the new leader nodes.
2-Step Algorithm (6)
Step 1: Track Formation
1.4
Track termination –
The track is terminated once simulation ends or no new associations within cutoff parameter. Track outputs sent to Step 2
2-Step Algorithm (7)
Detail of likelihood function for track association Track A Leader nodes New Observation Track B Likelihood Function
300 250 200 150 100 50 0 -50 Aggregated Belief Probability With N= 57, 15 hits.
2-Step Algorithm (8)
4.5
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Step 2:
Each track sequence produces an individual likelihood map
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In this case only 4 sensor observations used to form belief map
0 0 50 100 X 150 200 250 300
300 Aggregated Belief Probability With N= 57, 15 hits.
2-Step Algorithm (9)
250 200 150 100 50
Step 2:
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Each track sequence produces an individual likelihood map.
Only subset of observations applied to belief
0 -50 0 50 100 X 150 200 250 300 0
2-Step Algorithm (10) Track Assisted State Estimation
Gradual update of estimated source position, as sensor data is aggregated along the path ABCD.
2-Step Algorithm (11) Track Assisted State Estimation
Final update of estimated source position, as sensor data is aggregated along the path ABCD. Final estimated likelihood map after integration of ABCD, and renormalization for easier viewing.
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
Experiments (2) LabVIEW simulation design
Initialize, setup scenarios And control batch runs 2-Step Alg.
Main loop – heavy computation Result Outputs, statistical calculations
Experiments (3) 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 (4) 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 (5)
Wind Data Example, heavy processing needed
Data Imported from web http://www.ndbc.noaa.gov/ 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 9999 2004 12 31 23 30 136 8.2 999 99.0 9999
Experiments (6)
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
Results and Conclusions Likelihood performance metrics
Maximum likelihood, ML(M), in each belief map compared to likelihood value at true source M(i,j)
Source A(i,j) Belief M(i,j) ML(M)
Results and Conclusions 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
Results and Conclusions (2) Typical M for same data
2-Step predictor Z T
Observation set
MMSE predictor
Results and Conclusions – KEY RESULT Wind Experimental Results Summary
Results and Conclusions – KEY RESULT Sensor Node Density Results Summary
Results and Conclusions
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
Results and Conclusions
Summary of Wind performance zones
Best performance zone
Mean wind Speed scaled into 4 groups Standard deviation of direction divided into 4 groups This produced 16 total wind categories
high Intermediate performance 1 2 3 4 5 …
ZONE 1 ZONE 3
low 16 high Worst performance Zone: low wind Speed, with Frequent shifts Wind direction Std. deviation
Results and Conclusions 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?
Results and Conclusions (3) Sensor Density Study N=50 Sensors
P(M)=1 E-4
Results and Conclusions (4) N=100
P(M)=1 E-4
Results and Conclusions (5) N=200
P(M)=1.3 E-4
Results and Conclusions (6) N=300
P(M)=1E-4
Results and Conclusions (7) N=400 Sensors
P(M)=9E-5
Results - Belief Map From Uniform Predictor
Aggregated Belief Probability With N= 57, 15 hits. 300 14 12 250 10 200 8 150 6 100 4 50 2 0 -50 0 50 100 X 150 200 250 300
Results - Belief Map, Track 1
Aggregated Belief Probability With N= 57, 15 hits. 300 250 200 150 100 50 0 -50 0 50 100 X 150 200 250 300 4.5
4 1.5
1 0.5
0 3.5
3 2.5
2
Results - Belief Map Track2
Aggregated Belief Probability With N= 57, 15 hits. 300 250 200 150 100 50 0 -50 0 50 100 X 150 200 250 300 3 2 1 0
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 can we differentiate two sources as a function of sensor density?
In this example, two sources in constant wind can superimpose to create a 3 rd 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
We want to construct a belief map after each trial, and look at the value of the cell where the actual source was released.
Once we introduce tracking, we get sharper regions with higher Values per cell. This allows us to compare the predicted map with ground truth on any selected trial.
Forward simulation Likelihood Map M
Belief No Tracks
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 that The source was released from that particular cell.
Typical values for a single cell are between 10E-3 and 10E-5
Forward Probability, P(B|A) Known release event : A 60 50 40 30 100 90 80 70 20 10 10 No Wind Constant Wind 20 Inverse Probability, P(A|B) 100 90 30 No Wind Bayes Rule: 40 50 80 70 Known detection event : B 60 50 40 30 20 10 10 20 30 40 Constant Wind 50 60 70 100 60
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