Fuzzy Networks for Complex Systems Alexander Gegov University of Portsmouth, UK alexander.gegov@port.ac.uk
Presentation Outline Introduction Types of Fuzzy Systems Formal Models for Fuzzy Networks Basic Operations in Fuzzy Networks Structural Properties of Basic Operations Advanced Operations in Fuzzy Networks Feedforward Fuzzy Networks Feedback Fuzzy Networks Evaluation of Fuzzy Networks Fuzzy Network Toolbox Conclusion References 2
Introduction Modelling 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
Types of Fuzzy Systems Fuzzification triangular membership functions trapezoidal membership functions Inference truncation of rule membership functions by firing strengths scaling of rule membership functions by firing strengths Defuzzification centre of area of rule base membership function weighted average of rule base membership function 4
Types of Fuzzy Systems Logical Connections disjunctive antecedents and conjunctive rules conjunctive antecedents and conjunctive rules disjunctive antecedents and disjunctive rules conjunctive antecedents and disjunctive rules Inputs and Outputs single-input-single-output single-input-multiple-output multiple-input-single-output multiple-input-multiple-output 5
Types of Fuzzy Systems Operation 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
Types of Fuzzy Systems Figure 1: Standard fuzzy system 7
Types of Fuzzy Systems Figure 2: Hierarchical fuzzy system 8
Types of Fuzzy Systems Figure 3: Fuzzy network 9
Formal Models for Fuzzy Networks Node Modelling by If-then Rules Rule 1: If x is low, then y is small Rule 2: If x is average, then y is medium Rule 3: If x is high, then y is big Node Modelling by Integer Tables Linguistic terms for x Linguistic terms for y 1 (low) 1 (small) 2 (average) 2 (medium) 3 (high) 3 (big) 10
Formal Models for Fuzzy Networks Node Modelling by Boolean Matrices x / y 1 (small) 2 (medium) 3 (big) 1 (low) 1 0 0 2 (average) 0 1 0 3 (high) 0 0 1 Node Modelling by Binary Relations {(1, 1), (2, 2), (3, 3)} 11
Formal Models for Fuzzy Networks Network Modelling by Grid Structures Layer 1 Level 1 N 11 (x, y) Network Modelling by Interconnection Structures Layer 1 Level 1 y 12
Formal Models for Fuzzy Networks Network Modelling by Block Schemes x N 11 y Network Modelling by Topological Expressions [N 11 ] (x y) 13
Basic Operations in Fuzzy Networks Horizontal Merging of Nodes [N 11 ] (x 11 z 11,12 ) * [N 12 ] (z 11,12 y 12 ) = [N 11*12 ] (x 11 y 12 ) * symbol for horizontal merging Horizontal Splitting of Nodes [N 11/12 ] (x 11 y 12 ) = [N 11 ] (x 11 z 11,12 ) / [N 12 ] (z 11,12 y 12 ) / symbol for horizontal splitting 14
Basic Operations in Fuzzy Networks Vertical Merging of Nodes [N 11 ] (x 11 y 11 ) + [N 21 ] (x 21 y 21 ) = [N 11+21 ] (x 11, x 21 y 11, y 21 ) + symbol for vertical merging Vertical Splitting of Nodes [N 11 21 ] (x 11, x 21 y 11, y 21 ) = [N 11 ] (x 11 y 11 ) [N 21 ] (x 21 y 21 ) symbol for vertical splitting 15
Basic Operations in Fuzzy Networks Output Merging of Nodes [N 11 ] (x 11,21 y 11 ) ; [N 21 ] (x 11,21 y 21 ) = [N 11;21 ] (x 11,21 y 11, y 21 ) ; symbol for output merging Output Splitting of Nodes [N 11:21 ] (x 11, 21 y 11, y 21 ) = [N 11 ] (x 11,21 y 11 ) : [N 21 ] (x 11,21 y 21 ) : symbol for output splitting 16
Structural Properties of Basic Operations Associativity of Horizontal Merging [N 11 ] (x 11 z 11,12 ) * [N 12 ] (z 11,12 z 12,13 ) * [N 13 ] (z 12,13 y 13 ) = [N 11*12 ] (x 11 z 12,13 ) * [N 13 ] (z 12,13 y 13 ) = [N 11 ] (x 11 z 11,12 ) * [N 12*13 ] (z 11,12 y 13 ) = [N 11*12*13 ] (x 11 y 13 ) 17
Structural Properties of Basic Operations Variability of Horizontal Splitting [N 11/12/13 ] (x 11 y 13 ) = [N 11 ] (x 11 z 11,12 ) / [N 12/13 ] (z 11,12 y 13 ) = [N 11/12 ] (x 11 z 12,13 ) / [N 13 ] (z 12,13 y 13 ) = [N 11 ] (x 11 z 11,12 ) / [N 12 ] (z 11,12 z 12,13 ) / [N 13 ] (z 12,13 y 13 ) 18
Structural Properties of Basic Operations Associativity of Vertical Merging [N 11 ] (x 11 y 11 ) + [N 21 ] (x 21 y 21 ) + [N 31 ] (x 31 y 31 ) = [N 11+21 ] (x 11, x 21 y 11, y 21 ) + [N 31 ] (x 31 y 31 ) = [N 11 ] (x 11 y 11 ) + [N 21+31 ] (x 21, x 31 y 21, y 31 ) = [N 11+21+31 ] (x 11, x 21, x 31 y 11, y 21, y 31 ) 19
Structural Properties of Basic Operations Variability of Vertical Splitting [N 11 21 31 ] (x 11, x 21, x 31 y 11, y 21, y 31 ) = [N 11 ] (x 11 y 11 ) [N 21 31 ] (x 21, x 31 y 21, y 31 ) = [N 11 21 ] (x 11, x 21 y 11, y 21 ) [N 31 ] (x 31 y 31 ) = [N 11 ] (x 11 y 11 ) [N 21 ] (x 21 y 21 ) [N 31 ] (x 31 y 31 ) 20
Structural Properties of Basic Operations Associativity of Output Merging [N 11 ] (x 11,21,31 y 11 ) ; [N 21 ] (x 11,21,31 y 21 ) ; [N 31 ] (x 11,21,31 y 31 ) = [N 11;21 ] (x 11,21,31 y 11, y 21 ) ; [N 31 ] (x 11,21,31 y 31 ) = [N 11 ] (x 11,21,31 y 11 ) ; [N 21;31 ] (x 11,21,31 y 21, y 31 ) = [N 11;21;31 ] (x 11,21,31 y 11, y 21, y 31 ) 21
Structural Properties of Basic Operations Variability of Output Splitting [N 11:21:31 ] (x 11,21,31 y 11, y 21, y 31 ) = [N 11 ] (x 11,21,31 y 11 ) : [N 21:31 ] (x 11,21,31 y 21, y 31 ) = [N 11:21 ] (x 11,21,31 y 11, y 21 ) : [N 31 ] (x 11,21,31 y 31 ) = [N 11 ] (x 11,21,31 y 11 ) : [N 21 ] (x 11,21,31 y 21 ) : [N 31 ] (x 11,21,31 y 31 ) 22
Advanced Operations in Fuzzy Networks Node Transformation in Input Augmentation [N] (x y) [N AI ] (x, x AI y) Node Transformation in Output Permutation [N] (x y 1, y 2 ) [N PO ] (x y 2, y 1 ) Node Transformation in Feedback Equivalence [N] (x, z y, z) [N EF ] (x, x EF y, y EF ) 23
Advanced Operations in Fuzzy Networks Node Identification in Horizontal Merging A * U = C A, C known nodes, U non-unique unknown node U * B = C B, C known nodes, U non-unique unknown node 24
Advanced Operations in Fuzzy Networks A * U * B = C A, B, C known nodes, U non-unique unknown node A * U = D D * B = C D non-unique unknown node U * B = E A * E = C E non-unique unknown node 25
Advanced Operations in Fuzzy Networks Node Identification in Vertical Merging A + U = C A, C known nodes, U unique unknown node U + B = C B, C known nodes, U unique unknown node 26
Advanced Operations in Fuzzy Networks A + U + B = C A, B, C known nodes, U unique unknown node A + U = D D + B = C D unique unknown node U + B = E A + E = C E unique unknown node 27
Advanced Operations in Fuzzy Networks Node Identification in Output Merging A ; U = C A, C known nodes, U unique unknown node U ; B = C B, C known nodes, U unique unknown node 28
Advanced Operations in Fuzzy Networks A ; U ; B = C A, B, C known nodes, U unique unknown node A ; U = D D ; B = C D unique unknown node U ; B = E A ; E = C E unique unknown node 29
Feedforward Fuzzy Networks Network with Single Level and Single Layer fuzzy system 1,1 [N 11 ] (x 11 y 11 ) 30
Feedforward Fuzzy Networks Network with Single Level and Multiple Layers queue of n fuzzy systems 1,1 1,n 31
Feedforward Fuzzy Networks Network with Multiple Levels and Single Layer stack of m fuzzy systems 1,1 m,1 32
Feedforward Fuzzy Networks Network with Multiple Levels and Multiple Layers grid of m by n fuzzy systems 1,1 1,n m,1 m,n 33
Feedback Fuzzy Networks Network with Single Local Feedback one node embraced by feedback N(F) N N N N N N(F) N N N(F) N N N N N N(F) N(F) node embraced by feedback N nodes with no feedback 34
Feedback Fuzzy Networks Network with Multiple Local Feedback at least two nodes embraced by separate feedback N(F1) N(F2) N N N(F1) N N(F2) N N N N(F1) N(F2) N N(F1) N N(F2) N(F1) N N N(F2) N N(F1) N(F2) N N(F1), N(F2) nodes embraced by separate feedback N nodes with no feedback 35
Feedback Fuzzy Networks Network with Single Global Feedback one set of at least two adjacent nodes embraced by feedback N1(F) N2(F) N N N N N1(F) N2(F) N1(F) N N2(F) N N N N1(F) N2(F) {N1(F), N2(F)} set of nodes embraced by feedback N nodes with no feedback 36
Feedback Fuzzy Networks Network with Multiple Global Feedback at least two sets of nodes embraced by separate feedback with at least two adjacent nodes in each set N1(F1) N2(F1) N1(F2) N2(F2) N1(F1) N1(F2) N2(F1) N2(F2) {N1(F1), N2(F1)}, {N1(F2), N2(F2)} sets of nodes embraced by separate feedback 37
Evaluation of Fuzzy Networks Linguistic Evaluation Metrics composition of hierarchical into standard fuzzy systems decomposition of standard into hierarchical fuzzy systems Composition Formula N E,x = * p=1 x 1 (N 1,p + + q=p+1 x 1 I q,p ) N E,x equivalent node for a fuzzy network with x inputs 38
Evaluation of Fuzzy Networks Decomposition Algorithm 1. Find N 1,1 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 N E,k from the first k inputs and the output. 6. Derive N 1,k 1 from the formula for N E,k, if possible. 7. Set k=k+1. 8. Endwhile. N E,k equivalent node for a fuzzy subnetwork with first k inputs 39
Evaluation of Fuzzy Networks Functional Evaluation Metrics model performance indicators applications to case studies Feasibility Index (FI) FI = (sum i=1 n p i ) / n n number of non-identity nodes p i number of inputs to the i-th non-identity node lower FI better feasibility 40
Evaluation of Fuzzy Networks Accuracy Index (AI) AI = sum i=1 nl sum j=1 qil sum k=1 vji ( y ji k d ji k / vij) nl number of nodes in the last layer qil number of outputs from the i-th node in the last layer vji number of discrete values for the j-th output from the i-th node in the last layer y ji k, d ji k simulated and measured k-th discrete value for the j-th output from the i-th node in the last layer lower AI better accuracy 41
Evaluation of Fuzzy Networks Efficiency Index (EI) EI = sum i=1 n (q i FID. r i FID ) n number of non-identity nodes FID fuzzification-inference-defuzzification q i FID number of outputs from the i-th non-identity node with an associated FID sequence r i FID number of rules for the i-th non-identity node with an associated FID sequence lower EI better efficiency 42
Evaluation of Fuzzy Networks Transparency Index (TI) TI = (p + q) / (n + m) p overall number of inputs q overall number of outputs n number of non-identity nodes m number of non-identity connections lower TI better transparency 43
Evaluation of Fuzzy Networks Case Study 1 (Ore Flotation) Inputs: x 1 copper concentration in ore pulp x 2 iron concentration in ore pulp x 3 debit of ore pulp Output: y new copper concentration in ore pulp 44
Evaluation of Fuzzy Networks Figure 4: Standard fuzzy system (SFS) for case study 1 45
Evaluation of Fuzzy Networks Figure 5: Hierarchical fuzzy system (HFS) for case study 1 46
Evaluation of Fuzzy Networks Figure 6: Fuzzy network (FN) for case study 1 47
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
Evaluation of Fuzzy Networks Case Study 2 (Retail Pricing) Inputs: x 1 expected selling price for retail product x 2 difference between selling price and cost for retail product x 3 expected percentage to be sold from retail product Output: y maximum cost for retail product 49
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
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
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
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
Evaluation of Fuzzy Networks Composition 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 transparency Decomposition 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
Fuzzy Network Toolbox Toolbox 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 site Toolbox 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
Conclusion Theoretical Significance of Fuzzy Networks novel application of discrete mathematics and control theory detailed validation by test examples and case studies Methodological 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 systems Application Areas of Fuzzy Networks modelling and simulation in the mining industry modelling and simulation in the retail industry 56
Conclusion Other 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
References A.Gegov, Complexity Management in Fuzzy Systems, Springer, Berlin, 2007 A.Gegov, Fuzzy Networks for Complex Systems, Springer, Berlin, 2010 58