Using Prior Information in Bayesian Inference - with Application to Fault Diagnosis

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 Stop immediately? Ignore and go on? Go to workshop?

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

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

6. The Diagnosis Problem Observations, x Probability of faults, c System under diagnosis Pre-processing Diagnosis system Complex system several hundreds of faults observations. Uncertaintydue 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.

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

8. Prior Response Knowledge Assume that the thresholds are such that the probabilities for false alarms are zero (in practice) Means that some values of the observation is impossible under some faults Simple in Bayesian framework!

9. Prior Causality Knowledge 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 = (x1, x2,… xm) • State of the system C, with values c1, c2,… • Z = (X,C), Zd ={1, 2, 3...K} • Training data, D • State of knowledge I

12. Training Data Only Assume that Dirichlet distribution Then

13. Prior Causality Knowledge Example Let:

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

15. Example: Two classes and two binary observations. Training data from fault c2only. Do inference about c1 1 x2 0 0 1 x1 Have reused training data, and learned that x1 is far probable under c1also!

16. 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

17. Thank you!

18. Some Previous Work • Determine the faults that are logically consistent with the observations, using prior informationonly. 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!

19. References • DeKleer & Williams (1992), Diagnosis with Behavioral Modes, Readings in Model Based Diagnosis. • Reiter (1992), A Theory of Diagnosis From First Principles, Readnings in Model Based Diagnosis. • Gertler (1998), Fault Detection and Diagnosis in Engineering Systems, Marcel & Decker. • Blanke, Kinnaert , Lunze , Staroswiecki and Schröder, (2003) Diagnosis and Fault Tolerant Control, Springer. • 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. • Daigle, Koutsoukos and Biswas (2006), Multiple Fault Diagnosis in Complex Systems, 17th International Workshop on Principles of Diagnosis, DX06. • Sala (2006), Fuzzy Logic Diagnostic Rules – a Constraint Optimization Viewpoint, Proceedings of ECC 2006 • Schwall and Gerdes (2002), A probabilistic Approach to Residual Processing for Vehicle Fault Detection, Proceedings of ACC. • Lerner, Parr, Koller, and Biswas (2000), Bayesian Fault Detection and Diagnosis in Dynamic Systems, AAAI/IAAI • Pernestål and Nyberg, (2007), Probabilistic Fault Diagnosis Based on Incomplete Data with Application to an Automotive Engine, Proceedings of ECC. • Lee, Bahri, Shastri, and Zaknich (2007) A Multi-Category Decission Support for the Tennesse Eastman Problem, Proceedings of ECC.