Calibration in Sensor Systems based on Statistical Error Models Computer Science Dept. University of...

Calibration in Sensor Systems based on Statistical Error Models

Computer Science Dept. University of California, Los Angeles

Jessica Feng, Gang Qu*, and Miodrag Potkonjak

jessicaf@cs.ucla.edu

*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

K light sources:

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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

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Acoustic signal-based distance measurements Correct distances calculated off-line as the golden standard

CENS

Suitability Evaluation

3 Observations

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Kernel weight estimation function Sliding window

Statistical Model Construction IITechnical Details

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Statistical Model Construction III3-dimensional PDF

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Statistical Model Construction III3-dimensional PDF function

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Statistical Model Construction III3-dimensional PDF function

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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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Calibration Model ~Piece-wise Polynomials

Why piece-wise polynomials?

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Light Intensity Measurements16

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Application of the Statistical Error Model17

Location Discovery

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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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Simulation Results

U = V = 2 U = V = 1

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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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