soft sensor for faulty measurements detection and reconstruction in urban traffic department of...
TRANSCRIPT
Soft Sensor for Faulty Measurements Detection and
Reconstruction in Urban Traffic
Department of Adaptive systems, Institute of Information Theory and Automation, June 2010, Prague
Outline
Problem description Soft sensors Gaussian Process models Soft sensor for faulty measurement
detection and reconstruction Conclusions
Outline
Problem description Soft sensors Gaussian Process models Soft sensor for faulty measurement
detection and reconstruction Conclusions
Problem description
Traffic crossroad - count of vehicles Inductive loop is a popular choice Devastating for traffic control
system Failure detection and recovery of
sensor signal
Example of controlled network (Zličin shopping centre, Prague)
Sensors on crossroadsFailure: control system has no means to reactPossible solution: soft sensor for failure detection and signal reconstruction
Soft sensors
Models that provide estimation of another variable
`Soft sensor’: process engineering mainly
Applications in various engineering fields Model-driven, data-driven soft sensors Issues:
missing data, data outliers, drifting data, data co-linearity, different sampling rates, measurement delays.
Outline
Problem description Soft sensors Gaussian Process models Soft sensor for faulty measurement
detection and reconstruction Conclusions
Probabilistic (Bayes) nonparametric model.
GP model determined by: • Input/output data (data points, not
signals)
(learning data – identification data):• Covariance matrix:
GP model
Covariance function:• functional part and noise part• stationary/unstationary, periodic/nonperiodic,
etc.• Expreses prior knowledge about system
properties, • frequently: Gaussian covariance function
» smooth function» stationary function
Covariance function
Identification of GP model = optimisation of covariance function parameters • Cost function: maximum likelihood of
data for learning
Hyperparameters
GP model prediction
Prediction of the output based on similarity test input – training inputsOutput: normal distribution
•Predicted mean •Prediction variance
-2 +2
Static illustrative example
Static example: 9 learning points: Prediction
Rare data density increased variance (higher uncertainty).
-1.5 -1 -0.5 0 0.5 1 1.5 2-4
-2
0
2
4
6
8
xy
Nonlinear function to be modelled from learning points
y=f(x)
Learning points
-1.5 -1 -0.5 0 0.5 1 1.5 2-6
-4
-2
0
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4
6
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10
x
y
Nonlinear fuction and GP model
-1.5 -1 -0.5 0 0.5 1 1.5 20
2
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x
e
Prediction error and double standard deviation of prediction
2|e|
Learning points
2f(x)
GP model attributes (vs. e.g. ANN) Smaller number of parameters Measure of confidence in prediction, depending on data Data smoothing Incorporation of prior knowledge * Easy to use (engineering practice)
Computational cost increases with amount of data Recent method, still in development Nonparametrical model
* (also possible in some other models)
Outline
Problem description Soft sensors Gaussian Process models Soft sensor for faulty measurement
detection and reconstruction Conclusions
Modelling
One working day for estimation data Different working day for validation
data Validation based regressor selection the fourth order AR model (four delayed output values as regressors) Gaussian+constant covariance function Residuals of predictions with 3 band
50 100 150 200 250 300 350 400 450-10
-5
0
5
10
15
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Samples
Que
ue le
ngth
3measurements
50 100 150 200 250 300 350 400 4500
5
10
Samples
e
3
|e|
-8 -6 -4 -2 0 2 4 6 80
20
40
60
80
100
120
Value of residual
Residuals distribution
Estimation
50 100 150 200 250 300 350 400 450-10
-5
0
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Samples
Que
ue le
ngth
3measurements
50 100 150 200 250 300 350 400 4500
5
10
Samples
e
3
|e|
-8 -6 -4 -2 0 2 4 6 80
10
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100
Value of residual
Residuals' distribution
Validation
Proposed algorithm for detecting irregularities and for reconstruction the data with prediction
Sensor fault: longer lasting outliers.
The comparison of MRSE for k-step-ahead predictions
Purposiveness of the obtained model(the measure of measurement validity, close-enough
prediction, fast calculation, model robustness)
0 5 10 15 20 250.46
0.48
0.5
0.52
0.54
0.56
0.58
0.6
0.62
Prediction steps
MR
SE
Soft sensor applied on faulty data
50 100 150 200 250 300 350 400 450-20
0
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80
Samples
Que
ue le
ngth
3measurements
50 100 150 200 250 300 350 400 4500
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15
Samples
Que
ue le
ngth
corr. measurements