Fuzzy Systems Michael J. Watts .

Fuzzy Systems

Michael J. Watts

http://mike.watts.net.nz

Lecture Outline

• Fuzzy systems• Developing fuzzy systems• Advantages of fuzzy systems• Disadvantages of fuzzy systems

Fuzzy Systems

• Three major componentso Fuzzy membership functionso Fuzzy ruleso Fuzzy inference engine

Fuzzy Systems

• Membership Functionso numerous types available

Gaussian, Triangular, Singleton, Trapezoidalo each type has different parameters

e.g. centre, spreado parameters effect the output of the function

Fuzzy Systems

• Fuzzy Ruleso map fuzzy facts to fuzzy conclusionso different antecedent operators

• AND, OR, NOTo consequents

• Zadeh-Mamdani• Takagi-Sugeno

Fuzzy Systems

• Fuzzy inference engineo final result depends on

inference composition defuzzification

Fuzzy Systems

• Used in fuzzy expert systems• Like other expert systems, but knowledge

base is a fuzzy logic system

Fuzzy Systems

Developing a Fuzzy System

• Stepso identify the problemo define the input and output variableso define the membership functionso define the fuzzy ruleso select the inference / composition methodso select the defuzzification methodo validate the system


• Identify the problemo most essential step of solving any problemo if you don’t know what the problem is, how can

you solve it?o is the problem clearly defined?o what are the goals of the system?


• Identify the problemo is the problem suited to a fuzzy system?o will one fuzzy system do, or are several needed?

• modular approach• easier to optimise


• Define the input and output variableso are all available input variables needed?o are all available output variables needed?o data analysiso ranges of the variables

• universe of discourseo variation of the variables

• implications for MF design


• Define the membership functionso what type to use?

Gaussian smoothly tends towards zero Triangular has a more constant slope Singelton useful for binary values Rectangular good for clear, non-overlapping groups Trapezoidal combines triangular and rectangular


• Define the membership functionso what parameters?o spread of each MFo distribution of the MFo labels attached to each MF

meaningful


• Define the fuzzy ruleso main problem in developing fuzzy systemso how to get them?

expert extract from data

clustering extract from learning algorithm

ANN EA


• Rules from expertso does the developer understand the problem?

Can the developer define the rules themselves? Can the developer understand the expert well enough

to transcribe accurate rules?o does the expert understand fuzzy logic?

Can the expert define the rules directly? Can the expert verify the rules created by the

developer?


• Rules from datao use a clustering methodo clusters of a class indicate regions to define with

rules a rule defines a region

Rules from Clusters


• Rules from learning systemso Artificial neural networks (ANN)

train an artificial neural network analyse the connection weights

o devise fuzzy rules from those weightso Evolutionary Algorithms (EA)

use performance of system to drive evolution


• Completeness of the ruleso do the rules cover the entire input space?o what happens if they don’t?

is "no response" acceptable?o contradictions in the rules?o combine or split OR'd rules?


• Select the inference / composition methodso different methods give very different resultso how to choose them?


• Select the defuzzification methodo many to choose fromo each gives different crisp resultso issues

speed accuracy


• Validate the systemo involves

gathering test data evaluating performance of system adjusting as necessary

o testing system in situ does the system give results the experts are happy

with? does it fail gracefully?


• Making adjustmentso improve performanceo correct errors

• Adding / deleting ruleso modularity of rules helps here

• Altering inference / composition / defuzzification methods


• Modifying MF o alter

types parameters

• Adding / removing variableso modularity of rules helps again here


• Adjustment gotchaso a change in the MF may require a change in the

ruleso a change in the rules may require a change in the

MFo a change in the inference process will effect

everything else• optimisation is difficult

Advantages of Fuzzy Systems

• Universal function approximatorso given enough rules, a fuzzy system can

approximate any function to any degree of precision

o number of rules required smaller than crisp rule based function approximator


• Comprehensibilityo well crafted fuzzy rules are easy to understand

requires meaningful labels for MFo makes a fuzzy expert system a "white box"

See workings of the system


• Modularityo rules can be added and removed as neededo eases development

start with a small number of ruleso add as necessary to improve performanceo remove redundant rules to improve execution

speedo optimise individual rules


• Explainabilityo execution traceo which rules firedo explains how system reached conclusion


• Uncertaintyo rules can fire even if all antecedents don't matcho can deal with inexact concepts

smaller, faster etc.o each rule corresponds to a wider range of input

values


• Parallel execution of ruleso output calculated once at end of cycle o rules are evaluated in parallelo order does not mattero no need for execution selection methods

• Compare to crisp ruleso order of rule execution can alter output of system

Disadvantages of Fuzzy Systems

• Computational costo more computations involved

fuzzification fuzzy operators composition of output fuzzy set defuzzification

o complex MF can aggravate this problem simple triangular vs.. S or Pi functions


• Defining the ruleso where do the rules come from?o major problem with rule-based systemso need to get enough rules to be accurateo rules need to be expressive

comprehensibilityo rules need to be accurate


• Optimisationo a change in the MF can require a change in the

ruleso a change in the rules can require a change in the

MFo each parameter / choice effects the otherso multi-parameter optimisation problem

Summary

• Identifying the problem is the most important step of developing a fuzzy system

• Data analysis can help in determining the variables to include in the system

• Care must be taken in defining the membership functionso data analysis can again help with this

Summary

• Biggest problem with fuzzy systems in defining the rules

• Rules can come from several sources• Fuzzy rules more expressive than crisp rules

o need fewer of them• Modular nature of the rules make

development easier

Summary

• Fuzzy rule based systems overcome most of the problems with crisp rule based systems

• It can be difficult to optimise fuzzy systemso MF <-> Rules <->Inference methods

Fuzzy Systems Michael J. Watts .

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