graphical technique of inference. using max-product (or correlation product) implication technique,...

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


,,2,1

max2~1~~


2~1~~

max rk ,,2,1


Case 3: input(i) and input(j) are fuzzy variables

2max,1maxminmax

2~1~~

xxxxy kkk AAB

2max,1maxminmax

2~1~~

xxxxy kkk AAB


2max1maxminmax

2~1~~

xxxxy kkk AAkB

Case 4: input(I) and input(j) are fuzzy, inference using correlation product


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

~


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

Nonlinear Simulation using Fuzzy Rule-Based System

: 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~


This model may also involve Spline functions to represent the output instead of crisp singletons.

riforxfyBTHEN

xxxAIF

ii

iii

,,2,1:

:

~

1~


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~

::


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)

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


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


Example:

Non-linear membership function: y = 10 sin x


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


Graphical Inference Method showing membership propagation and defuzzification:


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


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}


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


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.

graphical technique of inference. using max-product (or correlation product) implication technique,...

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