1. Using Prior Information in Bayesian Inference - with Application to Fault Diagnosis Anna Pernestål and Mattias Nyberg Department of Electrical Engineering, Linköping University, Sweden Scania CV AB, Sweden MaxEnt 2007, Saratoga Springs 8 – 13 July

2. Outline Motivation: The Fault Diagnosis Problem Problem Formulation Our Approach Small Example Conclusions

3. Motivation: Automotive Fault Diagnosis Go to workshop? Stop immediately? Ignore and go on?

4. Motivation: Automotive Fault Diagnosis Why Fault Diagnosis? Safety Uptime Fuel Consumption Environmental Issues (Emissions) Guidance at the workshop

5. The Diagnosis Problem Probability of faults, c System under diagnosis Diagnosis system Observations, x Pre- processing

6. System under diagnosis Diagnosis system Probability of faults, c Observations, x Pre- processing Complex system several hundreds of faults observations. Uncertainty due to noise, missing information, lack of understanding of the system under diagnosis. No model of the probabilistic relations between observations and faults available. Training data from some, but not all faults. Training data collected by implementing faults and run the system. Prior probabilities of faults are known. Engineering skills and prior knowledge may be available. Observations have different characteristics: sensor readings, model based diagnostic tests, ”ad hoc tests” constructed by engineers. The Diagnosis Problem

7. Example: catalyst diagnosis Gas flow in a catalyst Two tests:

8. Prior Response Knowledge C = c 1 No Fault C = c 2 T s1 C = c 3 T s2 C = c 4 Catalyst x 1 =-10√00 x 1 =0√√√√ x 1 =10√00 x 2 =0√√√√ x 2 =10√√√ Means that some values of the observation is impossible under some faults Assume that the thresholds are such that the probabilities for false alarms are zero (in practice) Simple in Bayesian framework!

9. Prior Causality Knowledge C = c 1 No Fault C = c 2 T s1 C = c 3 T s2 C = c 4 catalyst x1x1 000 x2x2 0 Know that some observations are not affected under certain faults.

10. Summary of Probelm Formulation We have –data from some faults –prior knowledge about causality Determine the probability of different faults Previous works use either prior information or data. Now: Be Bayesian and combine training data and prior knowledge!

11. Notation Observation vector X= x, with x = (x 1, x 2,… x m ) State of the system C, with values c 1, c 2,… Z = (X,C), Z d ={1, 2, 3...K} Training data, D State of knowledge I

12. Training Data Only Assume that Then Dirichlet distribution

13. Response Knowledge Then

14. Prior Causality Knowledge x1x1 cP(x 1,c|I) 01θ1θ1 02θ2θ2 11θ3θ3 12θ4θ4 Let: Example

15. Causality knowledge, cont. Dirichlet Can be solved e.g. using variable substitution.

16. Application to Fault Diagnosis

17. Example: Two classes and two binary observations. Training data from fault c 2 only. Do inference about c 1 x1x1 x2x2 Have reused training data, and learned that x 1 is far probable under c 1 also! 01 1 0

18. Conclusion Formulated the fault isolation problem in the Bayesian framework Emphasized the use of prior information Data and prior knowledge solves different parts of the diagnosis problems, the optimal solution is when both are used together! Future work General solution of the integral Compare to MaxEnt

19. Thank you!

20. Some Previous Work Determine the faults that are logically consistent with the observations, using prior information only. Ignores noise. –DeKleer & Williams (1992), Reiter (1992), …. Use response information and fault models. –Gertler (1998), Blanke et. al. (2003), … Qualitative information about signs, magnitudes etc. –Pulido et. al. (2005), Daigle et. al. (2006), … Fuzzy logic. –Fagarasan et. al. (2001) Construct a Baysian network from expert knowledge. –Schwall(2002), Lerner et. al.(2000), … Use Training data only. Classification methods, SVM. –Pernestål et. Al. (2007), Gareth et. Al. (2007) Now: Be Bayesian and combine training data and prior knowledge!

21. References 1.DeKleer & Williams (1992), Diagnosis with Behavioral Modes, Readings in Model Based Diagnosis. 2.Reiter (1992), A Theory of Diagnosis From First Principles, Readnings in Model Based Diagnosis. 3.Gertler (1998), Fault Detection and Diagnosis in Engineering Systems, Marcel & Decker. 4.Blanke, Kinnaert, Lunze, Staroswiecki and Schröder, (2003) Diagnosis and Fault Tolerant Control, Springer. 5.Pulido, Puig, Escobet, and Quevedo (2005), A new Fault Localization Algorithm that Improves the Integration Between Fault Detection and Localization in Dynamic Systems. 16th International Workshop on Principles of Diagnosis, DX05. 6.Daigle, Koutsoukos and Biswas (2006), Multiple Fault Diagnosis in Complex Systems, 17th International Workshop on Principles of Diagnosis, DX06. 7.Sala (2006), Fuzzy Logic Diagnostic Rules – a Constraint Optimization Viewpoint, Proceedings of ECC 2006 8.Schwall and Gerdes (2002), A probabilistic Approach to Residual Processing for Vehicle Fault Detection, Proceedings of ACC. 9.Lerner, Parr, Koller, and Biswas (2000), Bayesian Fault Detection and Diagnosis in Dynamic Systems, AAAI/IAAI 10.Pernestål and Nyberg, (2007), Probabilistic Fault Diagnosis Based on Incomplete Data with Application to an Automotive Engine, Proceedings of ECC. 11.Lee, Bahri, Shastri, and Zaknich (2007) A Multi-Category Decission Support for the Tennesse Eastman Problem, Proceedings of ECC.