Graphical Technique of Inference

1. Graphical Technique of Inference

2. Graphical Technique of Inference Using max-product (or correlation product) implication technique, aggregated output for r rules would be:

3. Graphical Technique of Inference Case 3: input(i) and input(j) are fuzzy variables

4. Graphical Technique of Inference Case 4: input(I) and input(j) are fuzzy, inference using correlation product

5. Example: Rule 1: if x1 is and x2 is , then y is Rule 2: if x1 is or x2 is , then y is input(i) = 0.35 input(j) = 55 Graphical Technique of Inference

6. Input Output X Y Nonlinear System Input vector and output vector in Rn space in Rm space Fuzzy Nonlinear Simulation Virtually all physical processes in the real world are nonlinear.

7. The space of possible conditions or inputs, a collection of fuzzy subsets, for k = 1,2,… • The space of possible outputs p = 1,2,… • 3. The space of possible mapping relations, fuzzy relations • q = 1,2,… Approximate Reasoning or Interpolative Reasoning

8. We may use different ways to find a. look up table b. linguistic rule of the form IF THEN If the fuzzy system is described by a system of conjunctive rules, we could decompose the rules into a single aggregated fuzzy relational equation for each input, x, as follows: Fuzzy Relation Equations

9. Fuzzy Relation Equations

10. Equivalently R: fuzzy system transfer for a single input x. If a system has n non-interactive fuzzy inputs xi and a single output y If the fuzzy system is described by a system of disjunctive rules: Fuzzy Relation Equations

11. Partitioning How to partition the input and output spaces (universes of discourse) into fuzzy sets? 1. prototype categorization 2. degree of similarity 3. degree similarity as distance Case 1: derive a class of membership functions for each variable. Case 2: create partitions that are fuzzy singletons (fuzzy sets with only one element having a nonzero membership)

12. Partitioning

13. Partitioning

14. : If x is , then y is • : If x is , then y is • : If x is , then y is • Rules can be connected by “AND” or “OR” or “ELSE” • IF : x = xi THEN : y = yi • It is a simple lookup table for the system description • 2. Inputs are crisp sets, Outputs are singletons • This is also a lookup table. Nonlinear Simulation using Fuzzy Rule-Based System

15. Nonlinear Simulation using Fuzzy Rule-Based System This model may also involve Spline functions to represent the output instead of crisp singletons.

16. Nonlinear Simulation using Fuzzy Rule-Based System 3. Input conditions are crisp sets and output is fuzzy set or fuzzy relation The output can be defuzzied.

17. Nonlinear Simulation using Fuzzy Rule-Based System 4. Input: fuzzy Output: singleton or functions. If fi is linear Quasi-linear fuzzy model (QLFM) Quasi-nonlinear fuzzy model (QNFM)

18. Nonlinear Simulation using Fuzzy Rule-Based System

19. Nonlinear Simulation using Fuzzy Rule-Based System

20. Fuzzy Associative Memories (FAMs) A fuzzy system with n non-interactive inputs and a single output. Each input universe of discourse, x1, x2, …, xn is partitioned into k fuzzy partitions The total # of possible rules governing this system is given by: l = kn or l = (k+1)n Actual number r << 1. r: actual # of rules If x1 is partitioned into k1 partitions x2 is partitioned into k2 partitions : . xn is partitioned into kn partitions l = k1 k2  …  kn

21. Fuzzy Associative Memories (FAMs) Example: for n = 2 A  A1  A7 B  B1  B5 Output: C  C1  C4

22. Fuzzy Associative Memories (FAMs) Example: Non-linear membership function: y = 10 sin x

23. Fuzzy Associative Memories (FAMs) • Few simple rules for y = 10 sin x • IF x1 is Z or P B, THEN y is z. • IF x1 is PS, THEN y is PB. • IF x1 is z or N B, THEN y is z • IF x1 is NS, THEN y is NB • FAM for the four simple rules

24. Fuzzy Associative Memories (FAMs) Graphical Inference Method showing membership propagation and defuzzification:

25. Fuzzy Associative Memories (FAMs)

26. Fuzzy Associative Memories (FAMs) Defuzzified results for simulation of y = 10 sin x1 select value with maximum absolute value in each column.

27. Fuzzy Associative Memories (FAMs) • More rules would result in a close fit to the function. • Comparing with results using extension principle: • Let • x1 = Z or PB • x1 = PS • x1 = Z or NB • x1 = NS • Let B = {-10,-8,-6,-4,-2,0,2,4,6,8,10}

28. Fuzzy Associative Memories (FAMs) To determine the mapping, we look at the inverse of y = f(x1) i.e. x1 = f-1(y) in the table

29. Fuzzy Associative Memories (FAMs) For rule1, x1 = Z or PB Graphical approach can give solutions very close to those using extension principle

30. Fuzzy Synthetic Evaluation An evaluation of an object, especially ill-defined one, is often vague and ambiguous. First, finding , for a given situation , solving Fuzzy Decision Making

31. Fuzzy Ordering Given two fuzzy numbers I and J

32. Fuzzy Ordering It can be extended to the more general case of many fuzzy sets

33. Fuzzy Ordering

34. Then the ordering is: Sometimes the transitivity in ordering does not hold. We use relativity to rank. fy(x): membership function of x with respect to y fx(y): membership function of y with respect to x The relationship function is: Fuzzy Ordering

35. Fuzzy Ordering This function is a measurement of membership value of choosing x over y. If set A contains more variables A = {x1,x2,…,xn} A’ = {x1,x2,…,xi-1,xi+1,…,xn} Note: here, A’ is not complement. f(xi | A’) = min{f(xi | x1),f(xi | x2),…,f(xi | xi-1),f(xi | xi+1),…,f(xi | xn)} Note: f(xi|xi) = 1 then f(xi|A’) = f(xi|A) We can form a matrix C to rank many fuzzy sets. To determine overall ranking, find the smallest value in each row.