Fuzzy Sets and Systems

Lecture 5(Fuzzy Inference)

Bu- Ali Sina UniversityComputer Engineering Dep.

Spring 2010

Fuzzy Inference

OutlineFuzzy Inference

– Fuzzy inference rules• Modus ponens• Modus tollens• Hypothetical Syllogism

Fuzzy inference rules

• Inference rules in classical logic � based on the various tautologies.• Inference rules can be generalized within the framework of fuzzy logic

– To facilitate approximate reasoning.

Here we describe generalizations for three classical inference rules:

• modus ponens,• modus tollens,• hypothetical syllogism.

Fuzzy Inference rule

For a given fuzzy relation R on X x Y, and a givenfuzzy set A’ on X, a fuzzy set B’ on Y can bederived for all y ɛ Y, so that

In matrix form, compositional rule of inference is

Fuzzy inference rulesThe fuzzy relation R is given by (one or more) conditionalfuzzy propositions.

For a given fuzzy proposition:

p : If X is A; then Y is B

Is determined for all x ɛ X and y ɛ Y by

where J stands for a fuzzy implication.

Generalized Modus ponens

Using relation R obtained from given proposition p (previous slide)and given another proposition q of the form

q : x is A'

we may conclude that y is B' by the compositional rule of inference.B’ is calculated by

This procedure is called a generalized modus ponens.

Example

Generalized modus tollens,Another inference rule in fuzzy logic, which is a generalized

modus tollens, is expressed by the following schema:

the compositional rule of inference has the form

Example

Generalization of hypothetical syllogism

The generalized hypothetical syllogism isexpressed by the following schema:

In this case, X , v , :Z are variables taking values in sets X, Y, Z,respectively, and A, B , C are fuzzy sets on sets X, Y, Z, respectively.In this case, x,y,z are variables taking values in sets X, Y, Z,respectively, and A, B , C are fuzzy sets on sets X, Y, Z, respectively.

For each conditional fuzzy proposition in (8.43), there is a fuzzy relation

Generalization of hypothetical syllogism,

Given R1, R2, R3, obtained by these equations, wesay that the generalized hypothetical syllogismholds if

This equation may also be written in the matrix form

Example

Inference from conditional and qualified propositions

Given a conditional and qualified fuzzy proposition pof the form

where S is a fuzzy truth qualifier, and a fact is in the form "X is A'," wewant to make an inference in the form “y is B'."

method of truth-value restrictions, is based on a manipulation oflinguistic truth values. The method involves the following four steps.

Example

Homework8-2, 8-4, 8-6, 8-9,

Outline

Fuzzy inference system– Fuzzifiers– Defuzzifiers

Fuzzy Systems with Fuzzifier and Defuzzifier (Fuzzy inference)

RV⊂nn RUUU ⊂××= L1

Fuzzy Rule BaseFuzzy Rule Base

Fuzzy InferenceFuzzy InferenceEngineEngine

x in U y in V

FuzzifierFuzzifier DefuzzifierDefuzzifier

Fuzzy Sets in U Fuzzy Sets in V

Fuzzy inference systems

Fuzzifiers (construction of fuzzy sets)With experts knowledge (Direct and indirect

methods)

– Direct methods: Experts give answers to questions thatexplicitly pertain to the constructed membership function.

– Indirect methods: Experts answer simpler questions, easier toanswer, which pertain to the constructed membership functiononly implicitly.

Direct method with one expert and multiple experts

An expert is expected to assign to each given element x ɛX a membership grade A(x) that, according to his or heropinion, best captures the meaning of the linguistic termrepresented by the fuzzy set A.

When a direct method is extended from one expert tomultiple experts, the opinions of individual experts mustbe appropriately aggregated. One of the most commonmethods is based on a probabilistic interpretation ofmembership functions.

Or where

Indirect methods with one and multiple experts

Let xl, x2, . . . , xn, be elements of the universal set X for whichwe want to estimate the grades of membership in A.our problem is to determine the values ai = A (x,) for all i ɛ N.Instead of asking the expert to estimate values ai directly, weask him or her to compare elements x1, x2, . . . , xn, in pairsaccording to their relative weights of belonging to A.

The pairwise comparisons is

And with simplication we have

Construction from sample dataLagrange interpolationLeast square curve fittingConstruction by neural networkConstruction by genetic algorithm

In each of the discussed methods, we assume thatn sample data:

Lagrange curve fitting

ExampleFor this data samples:

We have

Least square curve fittingin this method we chose f

where E is minimized:

One of the best choice isthe bell function:

Therefore the membershipfunction is:

Another choice is thetrapezoidal function:

Example

By NNIn general, constructions by neural networks are

based on learning patterns from sample data.

Defuzzifiers� The defuzzifier is defined as a mapping from fuzzy

set B' in V R to crisp point y* V.� Conceptually, the task of the defuzzifier is to

specify a point in V that best represents the fuzzyset B'.

� This is similar to the mean value of a randomvariable. However, since the B' is constructed insome special ways, we have a number of choicesin determining this representing point.

⊂ ∈

DefuzzificationDefuzzifiers

• Mean of maximum (MOM)• Center of area (COA)• The height method

Mean of maximum (MOM)Calculates the average of those output values

that have the highest possibility degrees

Center of area (COA)Calculate the center-of-gravity (the weighted

sum of the results)

Center of Gravity Defuzzifier

The height methodConvert the consequent membership function

Ci into crisp consequent y = ci

Apply the centroid defuzzification

wi is the degree to which the ithrule matches the input data