Transcript Slide 1

Graphical Technique of Inference
Graphical Technique of Inference
Using max-product (or correlation product) implication
technique, aggregated output for r rules would be:




 B  y   max A inputi   A input j 
k
~
k
k
k
~1
~2


k  1,2,, r


 B  y   max A inputi    A input j 
k
~
k
k  1,2,, r
k
k
~1
~2
Graphical Technique of Inference
Case 3: input(i) and input(j) are fuzzy variables










 
 
 B  y   maxminmax A x    x1, max A x    x2 
k
~


k
~1






k
~2
 
 
 B  y   maxminmax A x    x1, max A x    x2 
k
~


k
~1
k
~2
Graphical Technique of Inference
Case 4: input(I) and input(j) are fuzzy, inference using
correlation product








 
 
 B  y   maxminmax A x    x1  max A x    x2 
k
~
k


k
~1
k
~2
Graphical Technique of Inference
Example:
1
1
1
B
A
A
Rule 1: if x1 is ~ 1 and x2 is ~ 2 , then y is ~
2
2
2
B
A
A
Rule 2: if x1 is ~ 1 or x2 is ~ 2 , then y is ~
input(i) = 0.35
input(j) = 55
Fuzzy Nonlinear Simulation
Virtually all physical processes in the real world are
nonlinear.
Input
X
Nonlinear
System
Input vector and output vector

X  x1, x2 , xn  in Rn space

Y   y1, y2 ,, ym  in Rm space
Output
Y
Approximate Reasoning or Interpolative
Reasoning
1. The space of possible
conditions or inputs, a collection
k
of fuzzy subsets, A~ for k = 1,2,…
A y
k
~
p
2. The space of possible outputs B~ p = 1,2,…
B y
p
~
3. The space of possible mapping relations, fuzzy relations
q
R q = 1,2,…
~
 R  x, y 
q
~
Fuzzy Relation Equations
B  A R
~
~
We may use different ways to find R~
a. look up table
b. linguistic rule of the form
IF A THEN B
~
~
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:
1
2
r
y  x  R AND x  R AND  AND x  R

~


~


~

Fuzzy Relation Equations
Fuzzy Relation Equations
Equivalently

y  x  R AND R AND AND R
1
2
~
~
~
r

y  x R
~
R  R  R  R
1
~
2
~
~
r
~
R: fuzzy system transfer for a single input x.
If a system has n non-interactive fuzzy inputs xi and a
single output y
y  x1  x2   xn  R
~
If the fuzzy system is described by a system of disjunctive
rules:
1
2
r

 

 
OR OR R   x  R
y  x  R OR x  R OR OR x  R

~
y  x  R OR R
1
~
~
~
2
r
~
R  R  R  R
1
~
~
2
~
~
~
r
~
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)
Partitioning
Partitioning
Nonlinear Simulation using Fuzzy Rule-Based
System
1
1
1
2
2
2
r
r
r
R : If x is A , then y is B
~
~
~
R : If x is A , then y is B
~
~
~
R : If x is A , then y is B
~
~
~
Rules can be connected by “AND” or “OR” or “ELSE”
i
i
1. IF A : x = xi THEN B : 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.
IF
A : xi 1  x  xi
i
~
THEN B : y  yi for i  1,2, , r
i
~
Nonlinear Simulation using Fuzzy Rule-Based
System
This model may also involve Spline functions to represent
the output instead of crisp singletons.
IF
A : xi 1  x  xi
i
~
THEN B : y  f i x  for i  1,2,, r
i
~
Nonlinear Simulation using Fuzzy Rule-Based
System
3. Input conditions are crisp sets and output is fuzzy set or
fuzzy relation
IF
A : xi 1  x  xi THEN
i
~
The output can be
defuzzied.
i
IF
A : xi 1  x  xi THEN
~
y B
i
~
B :yR
i
~
~
i
Nonlinear Simulation using Fuzzy Rule-Based
System
4. Input: fuzzy
IF
Output: singleton or functions.
x  A THEN
i
~
B : y  yi
i
~
or
a.
IF x  A
i
~
y  f i x 
b.
THEN
B
i
~
If fi is linear
Quasi-linear fuzzy model (QLFM)
IF x  A
i
~
THEN
B
i
~
y  p0i  p1i x1  p2i x2    pni xn
pij : const
c.
IF x  A
i
~
y  f i x 
THEN
B
i
~
f i : non  linear
Quasi-nonlinear fuzzy model (QNFM)
Nonlinear Simulation using Fuzzy Rule-Based
System
Nonlinear Simulation using Fuzzy Rule-Based
System
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
Fuzzy Associative Memories (FAMs)
Example: for n = 2
A1
B1
A2
C1
B2
A3
A4
C4
C4
A5
C4
B4
C3 C3
B5
C3
A  A1  A7
B  B1  B5
Output: C  C1  C4
A7
C3 C3
C1
B3
A6
C2
C1
C1 C2
C1
C4
C4
C1 C2
C1
C3
Fuzzy Associative Memories (FAMs)
Example:
Non-linear membership function: y = 10 sin x
Fuzzy Associative Memories (FAMs)
Few simple rules for y = 10 sin x
1. IF x1 is Z or P B, THEN y is z.
2. IF x1 is PS, THEN y is PB.
3. IF x1 is z or N B, THEN y is z
4. IF x1 is NS, THEN y is NB
FAM for the four simple rules
x1
NB
NS
z
PS
PB
y
z
NB
z
PB
z
Fuzzy Associative Memories (FAMs)
Graphical Inference Method
propagation and defuzzification:
showing
membership
Fuzzy Associative Memories (FAMs)
Fuzzy Associative Memories (FAMs)
Defuzzified results for simulation of y = 10 sin x1
x1
-135
-45
45
135
y
0
0
0
0
-7
0
0
7
-7
7
select value with maximum absolute value in each column.
Fuzzy Associative Memories (FAMs)
More rules would result in a close fit to the function.
Comparing with results using extension principle:
Let
1. x1 = Z or PB
2. x1 = PS
3. x1 = Z or NB
4. x1 = NS
Let B = {-10,-8,-6,-4,-2,0,2,4,6,8,10}
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
0
-180
180
2
11.5
168.5
4
23.6
156.4
6
36.9
143.1
8
53.1
126.9
Fuzzy Associative Memories (FAMs)
For rule1, x1 = Z or PB
 y  10  
 90  0
 y  8  max A  126.9,  A  53.1
 max0,  A  53.1  0.41
 y  6  max0,  A  36.9  0.59
 y  4  max0,  A  23.6   0.74
 y  2  max0,  A  11.5  0.87
 y 0   max0,1,1  1
 y 2  max A 11.5,  A 168.5  0.87
 y 4  max A 23.6,  A 156.4  0.74
 y 6   max A 36.9,  A 146.1  0.59
 y 8  0.41
 y 10   A 90  0
A1
1
1
1
1
1
1
1
1
1
1
1
1
1
Graphical approach can give
solutions very close to those
using extension principle
Fuzzy Decision Making
Fuzzy Synthetic Evaluation
An evaluation of an object, especially ill-defined one, is
often vague and ambiguous.
First, finding R~ , for a given situation ~ , solving
e   R
~
~
~
Fuzzy Ordering
Given two fuzzy numbers I and J
 
T I  J  sup min  I x ,  J  y 
~
~
~
 ~

x y
 
IJ
T J  I   heightI  J 
I I  J  1iff
~
~
~
~
  I d    J d 
~
~
~
~
Fuzzy Ordering
It can be extended to the more general case of many fuzzy
sets I1 , I 2 ,, I k
~
~
~


T  I  I1 , I 2 ,  , I k 
~
~
~ 
~

 T
 I~  I1  and
~ 

Exam ple:



T
 I~  I 2  T  I~  I k 
~ 

~ 

I1  1 / 3  0.8 / 7
~
I 2  0.7 / 4  1 / 6
~
I 3  0.8 / 2  1 / 4  0.5 / 8
~



T
axm in
 I1  x1,  I 2  x 2 
 I1  I 2   m

x1 x 2
~
 ~

~ 
 ~





 m axm in
  I1 7 ,  I 2 7 , m in  I1 7 ,  I 2 6  
~
~
 ~

 ~


 m axm in0.8,0.7 , m in0.8,11
 m ax0.7,0.8  0.8
Fuzzy Ordering

  3,  2 , m in  7 ,  2 
m
in
 I1

 I1

I2
I2

~
~

 ~

 ~



T  I1  I 3   m ax

~
~


m in

 I1 7 ,  I 2 4 



~
 ~



 m axm in1,0.8, m in0.8,0.8, m in0.8,1  0.8
Sim ilarly,
T
 I2 
 ~

T  I2 
 ~

T  I3 
 ~

T  I3 
 ~
Then
I1 
  1.0
~ 

I 3   1.0
~ 

I 1   1 .0
~ 

I 2   0 .7
~ 


T  I1  I 2 , I 3   0.8
~
~ 
 ~
Fuzzy Ordering


T  I 2  I1 , I 3   1
~
~ 
 ~


T  I 3  I1 , I 2   0.7
~
~ 
 ~
Then the ordering is:
I 2 , I1 , I 3
~
~
~
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:
f x | y  

f y x 

max f y x , f x  y 
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.