Calibration in Sensor Systems based on Statistical Error Models

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Calibration in Sensor Systems based on
Statistical Error Models
Jessica Feng, Gang Qu, and Miodrag Potkonjak
Computer Science Dept. University of California,
Los Angeles
Electrical and Computer Engineering Dept.
University of Maryland
jessicaf_at_cs.ucla.edu
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Why Calibration?
Process of mapping raw sensor readings to the
corrected values (golden standard, consistency
among sensors)

• 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

• Systematic bias vs. random error

Objective
Identify and correct the systematic bias in the
sensor reading so it is as close as possible to
the correct values
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Why Statistical Error Modeling?
Location Discovery

• L1 norm 1.927m

• L2 norm 5.737m

• Max. likelihood with Gaussian 1.028m

• Statistical error modeling 1.662x10-3m

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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

• Assumptions

• Preliminaries
• Light intensity measurements (point-light model)
• Acoustic signal-based distance measurements

• Statistical model construction

• Actuator-based calibration

• Experimental Results

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State-of-the-art

• Hightower, J., Vakili, C., and Borriello, G.
• Design and Calibration of the SpotON Ad-Hoc
  Location Sensing System, Univ. of Washington,
  2001.

• Whitehouse, K. and Culler, D.
• Calibration as Parameter Estimation in Sensor
  Networks, ACM WSNA, 2002.

• Bychkovskiy, V., Megerian, S., Estrin, D., and
  Potkonjak, M.
• Colibration A Collaborative Approach to
  In-Place Sensor calibration, IPSN, 2003.

• Ihler, A., Fisher, J., Moses, R., and Willsky, A.
• Nonparametric Belief Propagation for
  Self-Calibration in Sensor Networks, IPSN, 2004

• Elson, J., Girod, L., and Estrin, D.
• Fine-Grained Network Time Synchronization using
  Reference Broadcasts, OSDI, 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
Point-Light Model

• K light sources

• Photocell

• Miniature silicon solar cell

• Photovoltaic detector

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Courtesy to Seapahn Megerian
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Preliminaries
Acoustic Signal-based Distance Measurements

• Deployed in the Fort Leonard Wood Self Healing
  Minefield Test Facility (size 200m x 50m)

• 90 sensor nodes

• Sh4 processor running at 200MHz

• 64MB RAM

• 2.4GHz TDMA frequency hopping radio

• 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

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Courtesy to Lewis Girod
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Statistical Model Construction I
Suitability Evaluation

• Acoustic signal-based distance measurements

• Correct distances calculated off-line as the
  golden standard

• 3 Observations

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Statistical Model Construction II
Technical Details

• Kernel weight estimation function

• Sliding window

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Statistical Model Construction III
3-dimensional PDF
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Statistical Model Construction III
3-dimensional PDF function
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Statistical Model Construction III
3-dimensional PDF function
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Statistical Analysis of Consistency
Consistency Predictability

• Interval of confidence,
• 80 of the confidence ?
• modeling error 5.5 1.5

• Prediction capability

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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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Calibration Model Piece-wise Polynomials

• Why piece-wise polynomials?

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Light Intensity Measurements
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Application of the Statistical Error Model
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 I
Static Stimuli and Environment

• Probability of sensors being stable

• Length of Stability

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Actuator-based Calibration II
Formulation

• M deployed light sensors
• Aware of its own position and orientation
• Light intensity measurement rt at time moment t

• A single point light source S
• Intensity It at time moment t

• Environmental impact function Bt (It) at time
  moment t

• Sensor is calibration function Ci (rt) , i
  1,,M t 1,,T

• T time moments

Ci (rit ) Bt (It ) t 1,,T
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Actuator-based Calibration III
Optimization
Ci (rit ) Bt (It ) t 1,,T

• Optimization objective function

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Actuator-based Calibration IV
Solvability
et 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

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ILP-based Broadcasting Tree I
Variables
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ILP-based Broadcasting Tree II
Constraints

• 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 gt 0

• All variables must hold value 0

• Only sensor nodes in communication range can
  exchange messages

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Experimental Results I
Pairs 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
Sensors Broadcast (U V 2, 15 snapshots)

• Calibration error vs. of broadcasting sensor
  nodes

• Communication cost vs. of broadcasting sensor
  nodes

• 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

• 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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