1. Soft Sensor for Faulty Measurements Detection and Reconstruction 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.  Probabilistic (Bayes) nonparametric model.  GP model determined by: Input/output data (data points, not signals) (learning data – identification data): Covariance matrix: GP model

9.  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

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

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

12. Static illustrative example  Static example:  9 learning points:  Prediction  Rare data density  increased variance (higher uncertainty). -1.5-0.500.511.52 -4 -2 0 2 4 6 8 x y Nonlinear function to be modelled from learning points y=f(x) Learning points -1.5-0.500.511.52 -6 -4 -2 0 2 4 6 8 10 x y Nonlinear fuction and GP model -1.5-0.500.511.52 0 2 4 6 x e Prediction error and double standard deviation of prediction 2  |e| Learning points   2   f(x)

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 3  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.