1. Fuzzy Rule Based Networks
Alexander Gegov
University of Portsmouth, UK
alexander.gegov@port.ac.uk

2. Presentation Outline
Part 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 Networks
Part 2 - Applications: Feedforward Fuzzy Networks
Feedback Fuzzy Networks
Evaluation of Fuzzy Networks
Fuzzy Network Toolbox
Conclusion
References
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3. 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)
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4. 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
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5. 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
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6. 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)
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7. Types of Fuzzy Systems
Figure 1: Standard fuzzy system
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8. Types of Fuzzy Systems
Figure 2: Hierarchical fuzzy system
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9. Types of Fuzzy Systems
Figure 3: Fuzzy network
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10. 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)
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11. 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)}
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12. Formal Models for Fuzzy Networks
Network Modelling by Grid Structures
Layer 1
Level 1 N11(x, y)
Network Modelling by Interconnection Structures
Layer 1
Level 1 y
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13. Formal Models for Fuzzy Networks
Network Modelling by Block Schemes
x N11 y
Network Modelling by Topological Expressions
[N11] (x | y)
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14. Basic Operations in Fuzzy Networks
Horizontal Merging of Nodes
[N11] (x11 | z11,12) * [N12] (z11,12 | y12) = [N11*12] (x11 | y12)
* symbol for horizontal merging
Horizontal Splitting of Nodes
[N11/12] (x11 | y12) = [N11] (x11 | z11,12) / [N12] (z11,12 | y12)
/ symbol for horizontal splitting
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15. Basic Operations in Fuzzy Networks
Vertical Merging of Nodes
[N11] (x11 | y11) + [N21] (x21 | y21) = [N11+21] (x11, x21 | y11, y21)
+ symbol for vertical merging
Vertical Splitting of Nodes
[N11–21] (x11, x21 | y11, y21) = [N11] (x11 | y11) – [N21] (x21 | y21)
– symbol for vertical splitting
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16. Basic Operations in Fuzzy Networks
Output Merging of Nodes
[N11] (x11,21 | y11) ; [N21] (x11,21 | y21) = [N11;21] (x11,21 | y11, y21)
; symbol for output merging
Output Splitting of Nodes
[N11:21] (x11, 21 | y11, y21) = [N11] (x11,21 | y11) : [N21] (x11,21 | y21)
: symbol for output splitting
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17. Structural Properties of Basic Operations
Associativity 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)
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18. Structural Properties of Basic Operations
Variability 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)
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19. Structural Properties of Basic Operations
Associativity 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)
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20. Structural Properties of Basic Operations
Variability 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)
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21. Structural Properties of Basic Operations
Associativity 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)
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22. Structural Properties of Basic Operations
Variability 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)
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23. Advanced Operations in Fuzzy Networks
Node 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)
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24. 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
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25. 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
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26. 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
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27. 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
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28. 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
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29. 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
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30. Feedforward Fuzzy Networks
Network with Single Level and Single Layer
• fuzzy system
1,1
[N11] (x11 | y11)
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31. Feedforward Fuzzy Networks
Network with Single Level and Multiple Layers
• queue of ‘n’ fuzzy systems
1,1 … 1,n
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32. Feedforward Fuzzy Networks
Network with Multiple Levels and Single Layer
• stack of ‘m’ fuzzy systems
1,1
…
m,1
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33. 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
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34. Feedback Fuzzy Networks
Network with Single Local Feedback
• one node embraced by feedback
N(F) N N N(F)
N N N N
N N N N
N(F) N N N(F)
N(F) – node embraced by feedback
N – nodes with no feedback
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35. 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 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) N
N(F1), N(F2) – nodes embraced by separate feedback
N – nodes with no feedback
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36. 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 N N1(F)
N2(F) N N N2(F)
{N1(F), N2(F)} – set of nodes embraced by feedback
N – nodes with no feedback
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37. 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(F1) N1(F2)
N1(F2) N2(F2) N2(F1) N2(F2)
{N1(F1), N2(F1)}, {N1(F2), N2(F2)} – sets of nodes embraced
by separate feedback
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38. Evaluation of Fuzzy Networks
Linguistic Evaluation Metrics
• composition of hierarchical into standard fuzzy systems
• decomposition of standard into hierarchical fuzzy systems
Composition Formula
x–1 x–1
NE,x = *p=1 (N1,p + +q=p+1 Iq,p)
NE,x – equivalent node for a fuzzy network with x inputs
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39. Evaluation of Fuzzy Networks
Decomposition Algorithm
1. 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
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40. Evaluation of Fuzzy Networks
Functional Evaluation Metrics
• model performance indicators
• applications to case studies
Feasibility Index (FI)
n
FI = sum i=1 (pi / n)
n – number of non-identity nodes
p i – number of inputs to the i-th non-identity node
lower FI ⇒ better feasibility
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41. Evaluation of Fuzzy Networks
Accuracy Index (AI)
AI = sum i=1nl sum j=1qil sum k=1vji (|yjik – djik| / 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
yjik , djik – 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
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42. Evaluation of Fuzzy Networks
Efficiency Index (EI)
EI = sum i=1n (qiFID . riFID)
n – number of non-identity nodes
FID – fuzzification-inference-defuzzification
FID
qi – number of outputs from the i-th non-identity node with an
associated FID sequence
riFID – number of rules for the i-th non-identity node with an
associated FID sequence
lower EI ⇒ better efficiency
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43. 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
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44. Evaluation of Fuzzy Networks
Case Study 1 (Ore Flotation)
Inputs:
x1 – copper concentration in ore pulp
x2 – iron concentration in ore pulp
x3 – debit of ore pulp
Output:
y – new copper concentration in ore pulp
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45. Evaluation of Fuzzy Networks
Figure 4: Standard fuzzy system (SFS) for case study 1
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46. Evaluation of Fuzzy Networks
Figure 5: Hierarchical fuzzy system (HFS) for case study 1
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47. Evaluation of Fuzzy Networks
Figure 6: Fuzzy network (FN) for case study 1
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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
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49. Evaluation of Fuzzy Networks
Case Study 2 (Retail Pricing)
Inputs:
x1 – expected selling price for retail product
x2 – difference between selling price and cost for retail product
x3 – expected percentage to be sold from retail product
Output:
y – maximum cost for retail product
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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
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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
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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
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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
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54. 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
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55. 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
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56. 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
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57. 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)
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58. References
A.Gegov, Complexity Management in Fuzzy Systems,
Springer, Berlin, 2007
A.Gegov, Fuzzy Networks for Complex Systems,
Springer, Berlin, 2010
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