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Graphical Technique of Inference
Graphical Technique of InferenceUsing max-product (or correlation product) implication technique, aggregated output for r rules would be:
rk
jinputiinputy kkk AAkB
,,2,1
max2~1~~
rk
jinputiinputy kkk AAkB
,,2,1
max2~1~~
jinputiinputy kkk AAkB
2~1~~
max rk ,,2,1
Graphical Technique of Inference
Case 3: input(i) and input(j) are fuzzy variables
2max,1maxminmax
2~1~~
xxxxy kkk AAB
2max,1maxminmax
2~1~~
xxxxy kkk AAB
Graphical Technique of Inference
2max1maxminmax
2~1~~
xxxxy kkk AAkB
Case 4: input(I) and input(j) are fuzzy, inference using correlation product
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
1
1~A 1
2~A
~
1B2
1~A 2
2~A
~
2B
Fuzzy Nonlinear Simulation
Virtually all physical processes in the real world are nonlinear.
NonlinearSystem
Input Output
X Y
Input vector and output vector
in Rn space
in Rm space
nxxxX ,, 21
myyyY ,,, 21
Approximate Reasoning or Interpolative Reasoning
1. The space of possible conditions or inputs, a collection of fuzzy subsets, for k = 1,2,…
2. The space of possible outputs p = 1,2,…
3. The space of possible mapping relations, fuzzy relationsq = 1,2,…
kA~
pB~
ykA~
ypB~
qR~
yxqR,
~
Fuzzy Relation Equations
~~RAB
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:
~R
~A
~B
rRxANDANDRxANDRxy~
2
~
1
~
Fuzzy Relation Equations
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:
r
r
RRRR
Rxy
RANDANDRANDRxy
~
2
~
1
~~
~
~
2
~
1
~
~21 Rxxxy n
r
r
r
RRRR
RxRORORRORRxy
RxORORRxORRxy
~
2
~
1
~~
~~
2
~
1
~
~
2
~
1
~
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
: 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”
1. 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.
1
~R 1
~A 1
~B
2
~R 2
~A 2
~B
rR~
rA~
rB~
iA~
iB~
riforyyBTHEN
xxxAIF
ii
iii
,,2,1:
:
~
1~
Nonlinear Simulation using Fuzzy Rule-Based System
This model may also involve Spline functions to represent the output instead of crisp singletons.
riforxfyBTHEN
xxxAIF
ii
iii
,,2,1:
:
~
1~
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.
iii
i ByTHENxxxAIF~
1~:
iiii
i RyBTHENxxxAIF~~
1~
::
Nonlinear Simulation using Fuzzy Rule-Based System
4. Input: fuzzy Output: singleton or functions.
xfy
BTHENAxIFa
or
yyBTHENAxIF
i
ii
iii
~~
~~
.
:
If fi is linearQuasi-linear fuzzy model (QLFM)
constp
xpxpxppy
BTHENAxIFb
ji
nin
iii
ii
:
.
22110
~~
linearnonfxfy
BTHENAxIFc
ii
ii
:
.~~
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 A2 A3 A4 A5 A6 A7
B1 C1 C4 C4 C3 C3
B2 C1 C2
B3 C4 C1 C1 C2
B4 C3 C3 C1 C1 C2
B5 C3 C4 C4 C1 C3
A A1 A7
B B1 B5
Output: C C1 C4
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 z4. IF x1 is NS, THEN y is NB
FAM for the four simple rulesx1 N B N S z P S P B
y z N B z P B z
Fuzzy Associative Memories (FAMs)
Graphical Inference Method showing membership propagation and defuzzification:
Fuzzy Associative Memories (FAMs)
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 0 0 0 0
-7 0 0 7
-7 7
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 tabley 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
09010
41.08
59.01.146,9.36max6
74.04.156,6.23max4
87.05.168,5.11max2
11,1,0max0
87.05.11,0max2
74.06.23,0max4
59.09.36,0max6
41.01.53,0max
1.53,9.126max8
09010
1
11
11
11
1
1
1
1
11
1
Ay
y
AAy
AAy
AAy
y
Ay
Ay
Ay
A
AAy
y A
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 , for a given situation , solving~R
~
~~~Re
Fuzzy Ordering
Given two fuzzy numbers I and J
yxJIT JI
yx ~~
,minsup~~
dd
JIheightIJT
JIiffJII
JI~~
~~~~
~~1
Fuzzy OrderingIt can be extended to the more general case of many fuzzy sets
~~2
~1 ,,, kIII
8.08.0,7.0max
11,8.0min,7.0,8.0minmax
6,7min,7,7minmax
2,1minmax
8/5.04/12/8.0
6/14/7.0
7/8.03/1
:
,,,
~2
~1
~2
~1
~2
~121~
2~1
~3
~2
~1
~~~2
~~1
~
~~2
~1
~
IIII
IIxx
k
k
xxIIT
I
I
I
Example
IITIITandIIT
IIIIT
Fuzzy Ordering
8.0,
7.0
0.1
0.1
0.1
,
8.01,8.0min,8.0,8.0min,8.0,1minmax
4,7min
2,7min,2,3min
max
~3
~2
~1
~2
~3
~1
~3
~3
~2
~1
~2
~3
~1
~2
~1
~2
~1
~2
~1
IIIT
Then
IIT
IIT
IIT
IIT
Similarly
IIT
II
IIII
Fuzzy Ordering
7.0,
1,
~2
~1
~3
~3
~1
~2
IIIT
IIIT
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:
~3
~1
~2 ,, III
yfxf
xfyxf
xy
y
,max|
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.