calibration in sensor systems based on statistical error models computer science dept. university of...
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Calibration in Sensor Systems based on Statistical Error Models
Computer Science Dept. University of California, Los Angeles
Jessica Feng, Gang Qu*, and Miodrag Potkonjak
*Electrical and Computer Engineering Dept. University of Maryland
CENS
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Why Calibration?
Inevitable due to the natural process of device aging and imperfections
Particularly important in wireless distributed sensor networks
Manual calibration is either infeasible or expensive
Process of mapping raw sensor readings to the corrected values (golden standard, consistency among sensors)
Systematic bias vs. random error
Identify and correct the systematic bias in the sensor reading so it is as close as possible to the correct values
Objective:
CorrectSensorReading
Systematic Bias
Random Noise
RawSensorReading
Time
SensorReading
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Why Statistical Error Modeling?
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Location Discovery
L1 norm: 1.927m
Statistical error modeling: 1.662x10-3m
Max. likelihood with Gaussian: 1.028m
L2 norm: 5.737m
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Our Approach
Nonparametric statistical model construction For each measured value, provide probabilities for all possible
real/correct values 4 calibration alternatives based on different objectives Statistical validation: resubstitution and prediction Demonstrative example: acoustic signal-based distance
measurements
Actuator-based On-line Calibration Intrinsically localized Energy (communication cost) efficient Arbitrary forms of calibration model Demonstrative example: light intensity measurements
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Presentation Organization
State-of-the-art calibration techniques
Statistical model construction
Actuator-based calibration
Preliminaries: Light intensity measurements (point-light model) Acoustic signal-based distance measurements
Experimental Results
Assumptions
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State-of-the-art
Bychkovskiy, V., Megerian, S., Estrin, D., and Potkonjak, M. “Colibration: A Collaborative Approach to In-Place
Sensor calibration”, IPSN, 2003.
Elson, J., Girod, L., and Estrin, D. “Fine-Grained Network Time Synchronization using
Reference Broadcasts”, OSDI, 2002.
Hightower, J., Vakili, C., and Borriello, G. “Design and Calibration of the SpotON Ad-Hoc Location
Sensing System”, Univ. of Washington, 2001.
Ihler, A., Fisher, J., Moses, R., and Willsky, A. “Nonparametric Belief Propagation for Self-Calibration
in Sensor Networks”, IPSN, 2004
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Whitehouse, K. and Culler, D. “Calibration as Parameter Estimation in Sensor
Networks”, ACM WSNA, 2002.
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Assumptions
Nonparametric statistical model construction Golden standard available (only off-line model construction) On-line model construction: solutions proposed by the solver
Actuator-based On-line Calibration Static stimuli Static environment Correct Point-light model Independence of errors (only when max. likelihood is used)
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Preliminaries
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Courtesy to: Seapahn Megerian
Point-Light Model
Photocell Miniature silicon solar cell
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Photovoltaic detector
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Preliminaries
Courtesy to: Lewis Girod
Acoustic Signal-based Distance Measurements
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Merrill, W., Girod, L., Elson, J., Sohrabi, K., Newberg, F., and Kaiser, W. “Autonomous Position Location in Distributed Embedded Wireless
Systems”, IEEE CAS Workshop on Wireless Communications and Networking, 2002
Merrill, W., Newberg, F., Girod, L., and Sohrabi, K. “Battlefield Ad-Hoc LANs: A Distributed Processing Perspective”,
GOMACTech, 2004
Sh4 processor running at 200MHz
64MB RAM
Deployed in the Fort Leonard Wood Self Healing Minefield Test Facility (size 200m x 50m)
2.4GHz TDMA frequency hopping radio
~90 sensor nodes
Statistical Model Construction I
Acoustic signal-based distance measurements Correct distances calculated off-line as the golden standard
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Suitability Evaluation
3 Observations
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Kernel weight estimation function Sliding window
Statistical Model Construction IITechnical Details
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Statistical Analysis of Consistency
Interval of confidence,
80% of the confidence
modeling error = [5.5% ± 1.5%]
Prediction capability
Consistency & Predictability
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Correct Value Selection Alternatives
Peak: select the real distance that has the highest PDF value.
Average: find the smallest (Min) and the largest (Max) correct distance that have PDF values greater than zero or a threshold; calculated the average of the two values.
50%: select the real distance that has the highest PDF value.
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Weighted-Error: for each real distance Y, calculate the summation of weighted error defined as , where yi, i=1,…,n are the correct
distances; and PDF(X, yi) is the PDF value of a specific real distance yi given the measured distance X.
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Our Approach
Nonparametric statistical model construction For each measured value, provide probabilities for all possible
real/correct values 4 calibration alternatives based on different objectives Statistical validation: resubstitution and prediction Demonstrative example: acoustic-based distance
measurements
Actuator-based On-line Calibration Intrinsically localized Energy (communication cost) efficient Arbitrary forms of calibration model and environmental impact model Optimal broadcasting tree formulated as ILP instance Demonstrative example: light intensity measurements
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Actuation-based Calibration IStatic Stimuli and Environment
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Probability of sensors being stable
Length of Stability
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Actuator-based Calibration IIFormulation
M deployed light sensors Aware of its own position and orientation Light intensity measurement rt at time moment t
Environmental impact function Bt (It) at time moment t
A single point light source S Intensity It at time moment t
T time moments
Sensor i’s calibration function Ci (rt) , i =1,…,M; t =1,…,T
Ci (rit ) = Bt (It ) t = 1,…,T
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Linear environmental impact factor, constant calibration model: rit + Ci = It Bt t = 1,…,T.
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Actuator-based Calibration IIIOptimization
Ci (rit ) = Bt (It ) t = 1,…,T
εt = Ci (rit ) – Bt (It ) t = 1,…,T
Optimization objective function
L1 Norm: L2 Norm: L∞ Norm:
Gaussian: Statistical Model:
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Linear environmental impact factor, constant calibration model: εt = rit + Ci – It Bt t = 1,…,T.
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Actuator-based Calibration IVSolvability
εt = Ci (rit ) – Bt (It ) t = 1,…,T
Each of the T environmental impact function Bt has U parameters Each of the M sensors has calibration function that has V parameters,
i =1,…M # of equations:
M T.
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M T ≥ V M + U T. ..
# of unknown variables: V M + U T. .
# of Sensors # of Time Moments 1 –
2 –
3 3
4 2
5 2
6 2
7 2
8 2
# of Time Moments # of Sensors 1 –
2 4
3 3
4 3
5 3
6 3
7 3
8 3
U=2, V=1
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ILP-based Broadcasting Tree I
Eij = {1 node i & j are within communication range 0 o/w
xi = {k node i has at 1 neighbor belongs to level k–1 0 o/w
xij = {1 node i sends message to node j 0 o/w
yi = {1 node i belongs to the broadcasting tree 0 o/w
wik = {1 node i belongs to level k 0 o/w
Variables
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ILP-based Broadcasting Tree IIConstraints
Each sensor node I, i =1,…M must receive the broadcasting message
Sensor node I belongs to level k in the broadcasting tree iff neighboring sensor node j has level (k-1) and xij = 1
Sensor node I must be in the broadcasting tree if the neighboring sensor node j receives message from i
Root node has level 1
All sensor nodes must be assigned with level > 0
All variables must hold value ≥ 0
Only sensor nodes in communication range can exchange messages
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Experimental Results IPairs of sensors
Calibration Error: difference between the correct value and the calibration model (polynomial function estimate) of the calibrated value
Calibration error vs. # of time
moments(U = V = 2)
Interval of confidence 92% of the confidence calibration error = [7.3% ± 0.5%]
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Experimental Results II
Calibration error vs. # of broadcasting sensor nodes
Communication cost vs. # of broadcasting sensor nodes
Sensors Broadcast (U = V = 2, 15 snapshots)
Interval of confidence 82% of the confidence calibration error = [7.5% ± 0.5%]
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Conclusion
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Nonparametric statistical model construction Complete PDF for all possible values 4 calibration alternatives Off-line and on-line model construction
Actuator-based On-line Calibration Energy (communication cost) efficient Arbitrary forms of calibration model and environmental
impact model
Statistical Validation measured by the Interval of Confidence
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