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. Graphical Technique of Inference
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

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

7. Approximate Reasoning or Interpolative Reasoning
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,…

8. Fuzzy Relation Equations
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:

9. Fuzzy Relation Equations

10. Fuzzy Relation Equations
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:

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. Nonlinear Simulation using Fuzzy Rule-Based System
: 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.

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 A1 A2 A3 A4 A5 A6 A7 B1 C1 C4 C3 B2 C2 B3 B4 B5
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 x1
N B
N S z
P S
P B y

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. x1
-135
-45
45
135 y -7 7

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 y x1
-10
-90 -8
-126.9
-53.1 -6
-143.1
-36.9 -4
-156.4
-23.6 -2
-168.5
-11.5
-180
180 2
11.5
168.5 4
23.6
156.4 6
36.9
143.1 8
53.1
126.9

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

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. Fuzzy Ordering 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:

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.