Qualitative Modelling of Complex Systems with Fuzzy Inductive Reasoning

1. UNIVERSITAT POLITÈCNICA DE CATALUNYAPrograma de doctorat: Tecnologies avançades de la produccióQualitative Modelling of Complex Systems by Means of Fuzzy Inductive Reasoning. Variable Selection and Search Space Reduction. Josep Maria Mirats Tur Directors: Rafael Huber François E. Cellier (Univ. Politècnica de Catalunya) (Univ. of Arizona) Barcelona, Novembre 2001

2. Context and motivation • To model and simulate the output or outputs of a system in order to control it • To solve the modelling and simulation problem we can use Deductive and/or Inductive modelling approaches

3. Context and motivation • FIR, is a modelling and simulation methodology capable of generating an input-output model • Unfortunately FIR, in its previous state, could not deal with large-scale systems • It qualitatively learns the behaviour of a system from its real past data. Interesting for ill-defined systems

4. Objectives • Problem: To obtain a qualitative model of a (ill-defined) system with a large number of measurable variables • Objective: To reduce the FIR model search space in order to solve the aforementioned problem

5. Index of subjects • About the Fuzzy Inductive Reasoning methodology • Main lines of research in the dissertation • Developed methods • Sub-optimal mask search algorithm • Method based on spectral coherence functions • Subsystem decomposition methods • Applications • Conclusions and future work

6. About the FIR methodology Prediction error real-valued trajectories from the system variables Quantitative predictions FIR Fuzzification Module (recoding) Defuzzification Module (regeneration) Qualitative data Qualitative predictions Model (mask + behaviour) Qualitative Modelling Engine Qualitative Simulation Engine

7. About the FIR methodologyThe Qualitative modelling engine

8. Index of subjects • About the Fuzzy Inductive Reasoning methodology • Main lines of research in the dissertation • Developed methods • Sub-optimal mask search algorithm • Method based on spectral coherence functions • Subsystem decomposition methods • Applications • Conclusions

9. Sub-optimal mask search algorithms Methods that directly simplify the FIR mask search space Reducing the number of possible m-input variables to the FIR model Sets of variables containing maximum information about the system Methods that obtain a decomposition of the system into subsystems Sets of variables maximally related between them Main lines of research

10. Index of subjects • About the Fuzzy Inductive Reasoning methodology • Main lines of research in the dissertation • Developed methods • Sub-optimal mask search algorithm • Method based on spectral coherence functions • Subsystem decomposition methods • Applications • Conclusions

11. Developed methodsNew sub-optimal search algorithm • A new approach for reducing the model search space of FIR is proposed • The algorithm is based on proposing to FIR mask candidates of increasing depth • It can deal with previously unmanageable large-scale MISO systems

12. Propose a new candidate matrix Does the overall quality increases? c < 5 Developed methodsNew sub-optimal search algorithm Complexity, c Depth, d = 1, Mcan = (-1 -1 …. -1 -1 +1) Exhaustive search keeping information about Q of each mask c = 2 d = d+1 Find the highest quality mask Qbest, and compute relative quality of all masks Qrel=Q/Qbest Elaborate matrix M where the rows r  d are filled by -1 at the locations of significant m-inputs and 0 for insignificant m-inputs Determine the good masks, with quality Qrel > s c = c+1 Yes Yes END FIR sub-optimal models together with a value for parameter d are obtained Determine significant inputs as those used at least in the t% of all good masks No No

13. Developed methodsNew sub-optimal search algorithmApplication to a garbage incinerator plant y(t) = f{y(t-1),y(t-4),x2(t-8)} Qopt=0.6548 Q=0.6312 Qloss = 3.60%

14. Index of subjects • About the Fuzzy Inductive Reasoning methodology • Main lines of research in the dissertation • Developed methods • Sub-optimal mask search algorithm • Method based on spectral coherence functions • Subsystem decomposition methods • Applications • Conclusions

15. Developed methodsMethod based on spectral coherence functions • To propose FIR a unique sparse candidate matrix to obtain a dynamic qualitative model of a large-scale system • Computing the energy of the signals it can be determined at which delays each input variable has more energy related to the output • Each variable trajectory is seen as the collection of values measuring a desired physical characteristic

16. Start Decide a mask depth, d Compute the cross-coherence function, Cxy and significant peaks for the pair <xi,y> Form a mask candidate matrix with information from delays 2 up to d. Obtain the significant delays for which each xi is most related to the output in energy terms. Fill rows 0 and 1 of the candidate matrix, for example using the suboptimal algorithm of Section 2.4.1 Last input variable i = n ? no Compute the corresponding FIR models yes End Developed methodsMethod based on spectral coherence functions

17. Developed methodsMethod based on spectral coherence functions Application to a garbage incinerator plant y(t) = f{y(t-1),y(t-8),x2(t-9), x7(t)} Qopt=0.6548 Q=0.6274 Qloss = 4.18%

18. Index of subjects • About the Fuzzy Inductive Reasoning methodology • Main lines of research in the dissertation • Developed methods • Sub-optimal mask search algorithm • Method based on spectral coherence functions • Subsystem decomposition methods • Applications • Conclusions

19. Developed methodsSubsystem decomposition methods • Reconstruction analysis based method • Using FIR to find the structure of a system • Statistical method combined with FIR

20. Developed methodsStatistical method combined with FIR • Inclusion of time in the analysis • Variable selection to eliminate redundancy • Linear relationship search • Non-linear relationship search • FIR modelling from formed subsystems

21. Developed methodsStatistical method combined with FIRInclusion of time

22. Developed methodsStatistical method combined with FIRVariable selection. Linear relationship search • Forming subsets of linearly related variables - Singular value decomposition of the remaining correlation matrix - Projection of the eigen-vectors onto the principal axes -Groups of variables are formed • A cheap variable selection by means of a correlation analysis is performed to eliminate information redundancy

23. Developed methodsStatistical method combined with FIRLinear relationship search. Application to a garbage incinerator plant

24. Developed methodsStatistical method combined with FIRNon-linear relationship search • Complete subsets of linearly related variables with possible non-linear relations between them - The correlation among (Xi*,ξm) is calculated, where: Xi*=spline(Xi), is a non-linear transformation of variable Xi ξm = linear(Xm1… Xmj) is a linear combination of the j variables from m-th subset.

25. Number of variables >5 ? Last cluster ? Developed methodsStatistical method combined with FIRFIR modelling Yes No Use cluster variables as mask candidate matrix Use cluster variables as FIR model Calculate optimal FIR model of complexity 5 Add to list of good FIR models No Yes Use FIR simulation to determine best model

26. Developed methodsStatistical method combined with FIR Application to a garbage incinerator plant S2 --> 30 Models S4 --> 561 Models Classical FIR --> 428.812.560 Models y(t) = f{y(t-1), x1(t-9), x2(t),x7(t-14) }

27. Index of subjects • About the Fuzzy Inductive Reasoning methodology • Main lines of research in the dissertation • Developed methods • Sub-optimal mask search algorithm • Method based on spectral coherence functions • Subsystem decomposition methods • Applications • Conclusions

28. Liquid Fuel system Gas Fuel system Ql Qg Electric power to the grid P2 Combustion Chamber P1 T2 T1 P0 IGV air filter T0 Gearbox Generator Compressor Turbine section Q0 Exhaust to atmosphere T3 P3 ApplicationGas turbine for electric power generation 215 variables reduced to 64 using prior knowledge of the system

29. ApplicationGas turbine for electric power generation

30. Conclusions • Since the FIR modelling engine is of exponential complexity, new techniques had to be devised that would reduce the number of masks to be visited • This can be accomplished either by reducing the number of ‘-1’ elements in the mask candidate matrix, or by decomposing the system into subsystems. • The so enhanced FIR toolbox can now easily cope with large-scale systems comprising of dozens if not hundreds of variables. • The new tools were built in a modular fashion so that they can be combined to form a variety of search-space reduction algorithms.

31. Main contributionsReducing FIR model search space • New sub-optimal mask search algorithm • Spectral coherence functions based method • Subsystem decomposition: • Using Fuzzy Reconstruction Analysis • Re-implementation of the FRA module • Using FIR to find the structure of a system • Using statistical techniques

32. Other resultsFIR methodology • Improvement to the FIR simulation engine - Corrected five-neighbours prediction formula • New use of the unreconstructed variance methodology • The concept of variable acceptability, ‘envelopes’ • A variable similarity measure based on a modified Hr value

33. Future research • Investigate alternative algorithms to include time in the analysis • Parameters intrinsic to the sub-optimal search algorithm and the energy method • Is a subsystem decomposition preferable to a whole model? • Parameters intrinsic to FIR • More thorough validation of the search-space reduction algorithms

34. UNIVERSITAT POLITÈCNICA DE CATALUNYAPrograma de doctorat: Tecnologies avançades de la produccióQualitative Modelling of Complex Systems by Means of Fuzzy Inductive Reasoning. Variable Selection and Search Space Reduction. Josep Maria Mirats Tur Directors: Rafael Huber François E. Cellier (Univ. Politècnica de Catalunya) (Univ. of Arizona) Barcelona, Novembre 2001

36. Qualitative data Distance Computation 5-nearest neighbours Input patterns matching Output Forecast Computation Forecasted value About the FIR methodologyQualitative Simulation Engine

37. Perform a cheap variable selection Calculate optimal FIR model of complexity 5 X20 = f1 (X4, X5, X6, X12, X19) Q = 0.2089 Determine the relative importance of the inputs used by the model X20 = f2 (X4, X5, X6, X19)Q = 0.1752 X20 = f3 (X4, X5, X12, X19)Q = 0.1735 X20 = f4 (X5, X6, X12, X19)Q = 0.1521 X20 = f5 (X4, X6, X12, X19)Q = 0.1339 X20 = f6 (X4, X5, X6, X12)Q = 0.1165 For each input, starting by the less important one, obtain a FIR model x7, x8,x11,x14 f11 x19 x2 x7 f9 x4 x8 x2,x9,x13,x14 x9 f1 x20 x10 f7 x12 x7,x9,x13 x11 x13 f8 x6 x14 x7, x8, x13 x18 f10 x8,x10,x11,x14,x18 x5 Developed methodsUsing FIR to find the structure of a system