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 inputi A input j
k
~
k
k
k
~1
~2
k 1,2,, r
B y max A inputi 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 maxminmax A x x1, max A x x2
k
~
k
~1
k
~2
B y maxminmax 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 maxminmax 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 :yR
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
max0, A 53.1 0.41
y 6 max0, A 36.9 0.59
y 4 max0, A 23.6 0.74
y 2 max0, A 11.5 0.87
y 0 max0,1,1 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
IJ
T J I heightI 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
axm in
I1 x1, I 2 x 2
I1 I 2 m
x1 x 2
~
~
~
~
m axm in
I1 7 , I 2 7 , m in I1 7 , I 2 6
~
~
~
~
m axm in0.8,0.7 , m in0.8,11
m ax0.7,0.8 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 axm in1,0.8, m in0.8,0.8, m in0.8,1 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.