Soft Sensor for Faulty Measurements Detection and Reconstruction in Urban Traffic

1. Soft Sensor for Faulty Measurements Detection andReconstruction in Urban Traffic Department of Adaptive systems, Institute of Information Theory and Automation, June 2010, Prague

2. Outline • Problem description • Soft sensors • Gaussian Process models • Soft sensor for faulty measurement detection and reconstruction • Conclusions

3. Outline • Problem description • Soft sensors • Gaussian Process models • Soft sensor for faulty measurement detection and reconstruction • Conclusions

4. 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

5. Example of controlled network (Zličin shopping centre, Prague) • Sensors on crossroads • Failure:control system has no means to react • Possible solution: soft sensor for failure detection and signal reconstruction

6. 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.

7. Outline • Problem description • Soft sensors • Gaussian Process models • Soft sensor for faulty measurement detection and reconstruction • Conclusions

8. GP model • Probabilistic (Bayes) nonparametric model. • GP model determined by: • Input/output data (data points, not signals) (learning data – identification data): • Covariance matrix:

9. Covariance function • 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

10. Hyperparameters • Identification of GP model = optimisation of covariance function parameters • Cost function: maximum likelihood of data for learning

11. GP model prediction • Prediction of the output based on similarity test input – training inputs • Output: normal distribution • Predicted mean • Prediction variance

12. Nonlinear fuctionand GP model 10 8 Nonlinear function to be modelled from learning points 8 y=f(x) 6 Learning points 6 4 y 2 4 0 y 2 -2 0 -4 Learning points m ± s 2 -2 -6 m -1.5 -1 -0.5 0 0.5 1 1.5 2 f(x) x -4 Prediction error and double standard deviation of prediction -1.5 -1 -0.5 0 0.5 1 1.5 2 x s 2 6 |e| 4 e 2 0 -1.5 -1 -0.5 0 0.5 1 1.5 2 x Static illustrative example • Static example: • 9 learning points: • Prediction • Rare data density  increased variance (higher uncertainty).

13. 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)

14. Outline • Problem description • Soft sensors • Gaussian Process models • Soft sensor for faulty measurement detection and reconstruction • Conclusions

15. The profile of vehicle arrival data

16. 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 3s band

17. Estimation

18. Validation

19. Proposed algorithm for detecting irregularities and for reconstruction the data with prediction Sensor fault: longer lasting outliers.

20. 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)

21. Soft sensor applied on faulty data

22. Conclusions • Soft sensors: promising for FD and signal reconstruction. • GP models: excessive noise, outliers, no delay in prediction, measure of prediction confidence. • The excessive noise limits the possibility to develop better predictor. • Traffic sensor problem successfully solved for working days.