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- 1. Fuzzy Rule Based NetworksAlexander GegovUniversity of Portsmouth, UKalexander.gegov@port.ac.uk
- 2. Presentation OutlinePart 1 - Theory: Introduction Types of Fuzzy Systems Formal Models for Fuzzy Networks Basic Operations in Fuzzy Networks Structural Properties of Basic Operations Advanced Operations in Fuzzy NetworksPart 2 - Applications: Feedforward Fuzzy Networks Feedback Fuzzy Networks Evaluation of Fuzzy Networks Fuzzy Network Toolbox Conclusion References 2
- 3. IntroductionModelling Aspects of Systemic Complexity • non-linearity (input-output functional relationships) • uncertainty (incomplete and imprecise data) • dimensionality (large number of inputs and outputs) • structure (interacting subsystems)Complexity Management by Fuzzy Systems • model feasibility (achievable in the case of non-linearity) • model accuracy (achievable in the case of uncertainty) • model efficiency (problematic in the case of dimensionality) • model transparency (problematic in the case of structure) 3
- 4. Types of Fuzzy SystemsFuzzification • triangular membership functions • trapezoidal membership functionsInference • truncation of rule membership functions by firing strengths • scaling of rule membership functions by firing strengthsDefuzzification • centre of area of rule base membership function • weighted average of rule base membership function 4
- 5. Types of Fuzzy SystemsLogical Connections • disjunctive antecedents and conjunctive rules • conjunctive antecedents and conjunctive rules • disjunctive antecedents and disjunctive rules • conjunctive antecedents and disjunctive rulesInputs and Outputs • single-input-single-output • single-input-multiple-output • multiple-input-single-output • multiple-input-multiple-output 5
- 6. Types of Fuzzy SystemsOperation Mode • Mamdani (conventional fuzzy system) • Sugeno (modern fuzzy system) • Tsukamoto (hybrid fuzzy system)Rule Base • single rule base (standard fuzzy system) • multiple rule bases (hierarchical fuzzy system) • networked rule bases (fuzzy network) 6
- 7. Types of Fuzzy SystemsFigure 1: Standard fuzzy system 7
- 8. Types of Fuzzy SystemsFigure 2: Hierarchical fuzzy system 8
- 9. Types of Fuzzy SystemsFigure 3: Fuzzy network 9
- 10. Formal Models for Fuzzy NetworksNode Modelling by If-then RulesRule 1: If x is low, then y is smallRule 2: If x is average, then y is mediumRule 3: If x is high, then y is bigNode Modelling by Integer TablesLinguistic terms for x Linguistic terms for y1 (low) 1 (small)2 (average) 2 (medium)3 (high) 3 (big) 10
- 11. Formal Models for Fuzzy NetworksNode Modelling by Boolean Matricesx/y 1 (small) 2 (medium) 3 (big)1 (low) 1 0 02 (average) 0 1 03 (high) 0 0 1Node Modelling by Binary Relations{(1, 1), (2, 2), (3, 3)} 11
- 12. Formal Models for Fuzzy NetworksNetwork Modelling by Grid Structures Layer 1Level 1 N11(x, y)Network Modelling by Interconnection Structures Layer 1Level 1 y 12
- 13. Formal Models for Fuzzy NetworksNetwork Modelling by Block Schemesx N11 yNetwork Modelling by Topological Expressions[N11] (x | y) 13
- 14. Basic Operations in Fuzzy NetworksHorizontal Merging of Nodes[N11] (x11 | z11,12) * [N12] (z11,12 | y12) = [N11*12] (x11 | y12)* symbol for horizontal mergingHorizontal Splitting of Nodes[N11/12] (x11 | y12) = [N11] (x11 | z11,12) / [N12] (z11,12 | y12)/ symbol for horizontal splitting 14
- 15. Basic Operations in Fuzzy NetworksVertical Merging of Nodes[N11] (x11 | y11) + [N21] (x21 | y21) = [N11+21] (x11, x21 | y11, y21)+ symbol for vertical mergingVertical Splitting of Nodes[N11–21] (x11, x21 | y11, y21) = [N11] (x11 | y11) – [N21] (x21 | y21)– symbol for vertical splitting 15
- 16. Basic Operations in Fuzzy NetworksOutput Merging of Nodes[N11] (x11,21 | y11) ; [N21] (x11,21 | y21) = [N11;21] (x11,21 | y11, y21); symbol for output mergingOutput Splitting of Nodes[N11:21] (x11, 21 | y11, y21) = [N11] (x11,21 | y11) : [N21] (x11,21 | y21): symbol for output splitting 16
- 17. Structural Properties of Basic OperationsAssociativity of Horizontal Merging[N11] (x11 | z11,12) * [N12] (z11,12 | z12,13) * [N13] (z12,13 | y13) =[N11*12] (x11 | z12,13) * [N13] (z12,13 | y13) =[N11] (x11 | z11,12) * [N12*13] (z11,12 | y13) =[N11*12*13] (x11 | y13) 17
- 18. Structural Properties of Basic OperationsVariability of Horizontal Splitting[N11/12/13] (x11 | y13) =[N11] (x11 | z11,12) / [N12/13] (z11,12 | y13) =[N11/12] (x11 | z12,13) / [N13] (z12,13 | y13) =[N11] (x11 | z11,12) / [N12] (z11,12 | z12,13) / [N13] (z12,13 | y13) 18
- 19. Structural Properties of Basic OperationsAssociativity of Vertical Merging[N11] (x11 | y11) + [N21] (x21 | y21) + [N31] (x31 | y31) =[N11+21] (x11, x21 | y11, y21) + [N31] (x31 | y31) =[N11] (x11 | y11) + [N21+31] (x21, x31 | y21, y31) =[N11+21+31] (x11, x21, x31 | y11, y21, y31) 19
- 20. Structural Properties of Basic OperationsVariability of Vertical Splitting[N11–21–31] (x11, x21, x31 | y11, y21, y31) =[N11] (x11 | y11) – [N21–31] (x21, x31 | y21, y31) =[N11–21] (x11, x21 | y11, y21) – [N31] (x31 | y31) =[N11] (x11 | y11) – [N21] (x21 | y21) – [N31] (x31 | y31) 20
- 21. Structural Properties of Basic OperationsAssociativity of Output Merging[N11] (x11,21,31 | y11) ; [N21] (x11,21,31 | y21) ; [N31] (x11,21,31 | y31) =[N11;21] (x11,21,31 | y11, y21) ; [N31] (x11,21,31 | y31) =[N11] (x11,21,31 | y11) ; [N21;31] (x11,21,31 | y21, y31) =[N11;21;31] (x11,21,31 | y11, y21, y31) 21
- 22. Structural Properties of Basic OperationsVariability of Output Splitting[N11:21:31] (x11,21,31 | y11, y21, y31) =[N11] (x11,21,31 | y11) : [N21:31] (x11,21,31 | y21, y31) =[N11:21] (x11,21,31 | y11, y21) : [N31] (x11,21,31 | y31) =[N11] (x11,21,31 | y11) : [N21] (x11,21,31 | y21) : [N31] (x11,21,31 | y31) 22
- 23. Advanced Operations in Fuzzy NetworksNode Transformation in Input Augmentation[N] (x | y) ⇒ [NAI] (x , xAI | y)Node Transformation in Output Permutation PO[N] (x | y1, y2) ⇒ [N ] (x | y2, y1)Node Transformation in Feedback Equivalence[N] (x, z | y, z) ⇒ [NEF] (x, xEF | y, yEF) 23
- 24. Advanced Operations in Fuzzy NetworksNode Identification in Horizontal MergingA*U =CA, C – known nodes, U – non-unique unknown nodeU*B=CB, C – known nodes, U – non-unique unknown node 24
- 25. Advanced Operations in Fuzzy NetworksA*U*B=CA, B, C – known nodes, U – non-unique unknown nodeA*U=DD*B=CD – non-unique unknown nodeU*B=EA*E=CE – non-unique unknown node 25
- 26. Advanced Operations in Fuzzy NetworksNode Identification in Vertical MergingA+U=CA, C – known nodes, U – unique unknown nodeU+B=CB, C – known nodes, U – unique unknown node 26
- 27. Advanced Operations in Fuzzy NetworksA+U+B=CA, B, C – known nodes, U – unique unknown nodeA+U=DD+B=CD – unique unknown nodeU+B=EA+E=CE – unique unknown node 27
- 28. Advanced Operations in Fuzzy NetworksNode Identification in Output MergingA;U =CA, C – known nodes, U – unique unknown nodeU;B=CB, C – known nodes, U – unique unknown node 28
- 29. Advanced Operations in Fuzzy NetworksA;U;B=CA, B, C – known nodes, U – unique unknown nodeA;U=DD;B=CD – unique unknown nodeU;B=EA;E=CE – unique unknown node 29
- 30. Feedforward Fuzzy NetworksNetwork with Single Level and Single Layer • fuzzy system1,1[N11] (x11 | y11) 30
- 31. Feedforward Fuzzy NetworksNetwork with Single Level and Multiple Layers • queue of ‘n’ fuzzy systems1,1 … 1,n 31
- 32. Feedforward Fuzzy NetworksNetwork with Multiple Levels and Single Layer • stack of ‘m’ fuzzy systems1,1…m,1 32
- 33. Feedforward Fuzzy NetworksNetwork with Multiple Levels and Multiple Layers • grid of ‘m by n’ fuzzy systems1,1 … 1,n… … …m,1 … m,n 33
- 34. Feedback Fuzzy NetworksNetwork with Single Local Feedback • one node embraced by feedbackN(F) N N N(F)N N N NN N N NN(F) N N N(F)N(F) – node embraced by feedbackN – nodes with no feedback 34
- 35. Feedback Fuzzy NetworksNetwork with Multiple Local Feedback • at least two nodes embraced by separate feedbackN(F1) N(F2) N N N(F1) NN N N(F1) N(F2) N N(F2)N(F1) N N N(F1) N N(F1)N(F2) N N N(F2) N(F2) NN(F1), N(F2) – nodes embraced by separate feedbackN – nodes with no feedback 35
- 36. Feedback Fuzzy NetworksNetwork with Single Global Feedback • one set of at least two adjacent nodes embraced by feedbackN1(F) N2(F) N NN N N1(F) N2(F)N1(F) N N N1(F)N2(F) N N N2(F){N1(F), N2(F)} – set of nodes embraced by feedbackN – nodes with no feedback 36
- 37. Feedback Fuzzy NetworksNetwork with Multiple Global Feedback • at least two sets of nodes embraced by separate feedback with at least two adjacent nodes in each setN1(F1) N2(F1) N1(F1) N1(F2)N1(F2) N2(F2) N2(F1) N2(F2){N1(F1), N2(F1)}, {N1(F2), N2(F2)} – sets of nodes embracedby separate feedback 37
- 38. Evaluation of Fuzzy NetworksLinguistic Evaluation Metrics • composition of hierarchical into standard fuzzy systems • decomposition of standard into hierarchical fuzzy systemsComposition Formula x–1 x–1NE,x = *p=1 (N1,p + +q=p+1 Iq,p)NE,x – equivalent node for a fuzzy network with x inputs 38
- 39. Evaluation of Fuzzy NetworksDecomposition Algorithm1. Find N1,1 from the first two inputs and the output.2. If x=2, go to end.3. Set k=3.4. While k≤x, do steps 5-7.5. Find NE,k from the first k inputs and the output.6. Derive N1,k–1 from the formula for NE,k, if possible.7. Set k=k+1.8. Endwhile.NE,k – equivalent node for a fuzzy subnetwork with first k inputs 39
- 40. Evaluation of Fuzzy NetworksFunctional Evaluation Metrics • model performance indicators • applications to case studiesFeasibility Index (FI) nFI = sum i=1 (pi / n)n – number of non-identity nodesp i – number of inputs to the i-th non-identity nodelower FI ⇒ better feasibility 40
- 41. Evaluation of Fuzzy NetworksAccuracy Index (AI)AI = sum i=1nl sum j=1qil sum k=1vji (|yjik – djik| / vij)nl – number of nodes in the last layerqil – number of outputs from the i-th node in the last layervji – number of discrete values for the j-th output from the i-thnode in the last layeryjik , djik – simulated and measured k-th discrete value for the j-thoutput from the i-th node in the last layerlower AI ⇒ better accuracy 41
- 42. Evaluation of Fuzzy NetworksEfficiency Index (EI)EI = sum i=1n (qiFID . riFID)n – number of non-identity nodesFID – fuzzification-inference-defuzzification FIDqi – number of outputs from the i-th non-identity node with anassociated FID sequenceriFID – number of rules for the i-th non-identity node with anassociated FID sequencelower EI ⇒ better efficiency 42
- 43. Evaluation of Fuzzy NetworksTransparency Index (TI)TI = (p + q) / (n + m)p – overall number of inputsq – overall number of outputsn – number of non-identity nodesm – number of non-identity connectionslower TI ⇒ better transparency 43
- 44. Evaluation of Fuzzy NetworksCase Study 1 (Ore Flotation)Inputs:x1 – copper concentration in ore pulpx2 – iron concentration in ore pulpx3 – debit of ore pulpOutput:y – new copper concentration in ore pulp 44
- 45. Evaluation of Fuzzy Networks Figure 4: Standard fuzzy system (SFS) for case study 1 45
- 46. Evaluation of Fuzzy Networks Figure 5: Hierarchical fuzzy system (HFS) for case study 1 46
- 47. Evaluation of Fuzzy Networks Figure 6: Fuzzy network (FN) for case study 1 47
- 48. Evaluation of Fuzzy Networks Table 1: Models performance for case study 1 Index / Model Standard Hierarchical Fuzzy Fuzzy System Fuzzy System Network Feasibility 3 2 2 Accuracy 4.35 4.76 4.60 Efficiency 343 126 343 Transparency 4 1.33 1.33 FI – FN better than SFS and equal to HFS AI – FN worse than SFS and better than HFS EI – FN equal to SFS and worse than HFS TI – FN better than SFS and equal to HFS 48
- 49. Evaluation of Fuzzy NetworksCase Study 2 (Retail Pricing)Inputs:x1 – expected selling price for retail productx2 – difference between selling price and cost for retail productx3 – expected percentage to be sold from retail productOutput:y – maximum cost for retail product 49
- 50. Evaluation of Fuzzy Networks 100 90 80 70 60 output 50 40 30 20 10 0 0 20 40 60 80 100 120 140 input permutations Figure 7: Standard fuzzy system (SFS) for case study 2 50
- 51. Evaluation of Fuzzy Networks 100 80 60 output 40 20 0 0 20 40 60 80 100 120 140 input permutations Figure 8: Hierarchical fuzzy system (HFS) for case study 2 51
- 52. Evaluation of Fuzzy Networks 100 80 60 output 40 20 0 0 20 40 60 80 100 120 140 input permutations Figure 9: Fuzzy network (FN) for case study 2 52
- 53. Evaluation of Fuzzy Networks Table 2: Models performance for case study 2 Index / Model Standard Hierarchical Fuzzy Fuzzy System Fuzzy System Network Feasibility 3 2 2 Accuracy 2.86 5.57 3.64 Efficiency 125 80 125 Transparency 4 1.33 1.33 FI – FN better than SFS and equal to HFS AI – FN worse than SFS and better than HFS EI – FN equal to SFS and worse than HFS TI – FN better than SFS and equal to HFS 53
- 54. Evaluation of Fuzzy NetworksComposition of Hierarchical into Standard Fuzzy System • full preservation of model feasibility • maximal improvement of model accuracy • fixed loss of model efficiency • full preservation of model transparencyDecomposition of Standard into Hierarchical Fuzzy System • no change of model feasibility • minimal loss of model accuracy • fixed improvement of model efficiency • no change of model transparency 54
- 55. Fuzzy Network ToolboxToolbox Details • developed by the PhD student Nedyalko Petrov • implemented in the Matlab environment • to be uploaded on the Mathworks web site • to be uploaded on the Springer web siteToolbox Structure • Matlab files for basic operations on nodes • Matlab files for advanced operations on nodes • Matlab files for auxiliary operations • Word files with illustration examples 55
- 56. ConclusionTheoretical Significance of Fuzzy Networks • novel application of discrete mathematics and control theory • detailed validation by test examples and case studiesMethodological Impact of Fuzzy Networks • extension of standard and hierarchical fuzzy systems • bridge between standard and hierarchical fuzzy systems • novel framework for non-fuzzy rule based systemsApplication Areas of Fuzzy Networks • modelling and simulation in the mining industry • modelling and simulation in the retail industry 56
- 57. ConclusionOther Applications of Fuzzy Networks • finance (mortgage evaluation) • healthcare (radiotherapy margins) • robotics (path planning) • games (obstacle avoidance) • sports (boxer training) • security (terrorist threat) • environment (natural preservation) 57
- 58. ReferencesA.Gegov, Complexity Management in Fuzzy Systems,Springer, Berlin, 2007A.Gegov, Fuzzy Networks for Complex Systems,Springer, Berlin, 2010 58

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