Aleksander gegov

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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 OperationsAdvanced 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 function4 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 system7 8. Types of Fuzzy SystemsFigure 2: Hierarchical fuzzy system8 9. Types of Fuzzy SystemsFigure 3: Fuzzy network9 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 1N11(x, y)Network Modelling by Interconnection Structures Layer 1Level 1y12 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[N1121] (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 splitting16 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[N112131] (x11, x21, x31 | y11, y21, y31) =[N11] (x11 | y11) [N2131] (x21, x31 | y21, y31) =[N1121] (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 PermutationPO[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 node24 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 node26 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 node28 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,n31 32. Feedforward Fuzzy NetworksNetwork with Multiple Levels and Single Layer stack of m fuzzy systems1,1m,132 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)NN N NNN 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 NN(F1) NN N N(F1) N(F2)N N(F2)N(F1) N NN(F1) N N(F1)N(F2) N NN(F2) N(F2) NN(F1), N(F2) nodes embraced by separate feedbackN nodes with no feedback35 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 x1 x1NE,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 kx, do steps 5-7.5. Find NE,k from the first k inputs and the output.6. Derive N1,k1 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 inputs39 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 accuracy41 42. Evaluation of Fuzzy NetworksEfficiency Index (EI)EI = sum i=1n (qiFID . riFID)n number o