Interval Type-2 Fuzzy Logic System versus ... - Semantic Scholar

Interval Type-2 Fuzzy Logic System versus Perceptual Computer: Similarities and Differences Jerry M. Mendel University of Southern California Los Angeles, CA

This Talk • Is about the similarities and differences between an IT2 FLS and a Per-C • Approach: Focus on their generic architectures and six associated issues

IT2 FLS vs Per-C: Issues

•Inputs •Fuzzifier •Rules •Inference •Output Processing •Outputs

IT2 FLS vs Per-C: Issues

•Inputs

•Inputs

•Fuzzifier

•Encoder

•Rules

•CWW Engine

•Inference

•Output of CWW Engine

•Output Processing

•Decoder

•Outputs

•Recommendation + Data

IT2 FLS vs Per-C: Applications

•“Function Approximation” •Fuzzy logic control •Signal processing •Rule-based classification

IT2 FLS vs Per-C: Applications

•“Function Approximation”

•“Computing With Words”

•Fuzzy logic control

•Investment advising

•Signal processing

•Social judgments

•Rule-based classification

•Decision making

IT2 FLS vs Per-C

•Inputs •Numbers first, then the Membership Functions (MFs) •Doesn’t matter what you call the fuzzy sets

IT2 FLS vs Per-C

•Inputs •Numbers first, then the Membership Functions (MFs) •Doesn’t matter what you call the fuzzy sets

•Inputs •Words first, then the MFs •Words that mean something to end-user label the fuzzy sets

IT2 FLS vs Per-C

•Fuzzifier—Different kinds (choices) • Singleton • T1 FS—a fuzzy number • IT2 FS—a fuzzy-fuzzy number

IT2 FLS vs Per-C

•Fuzzifier—Different kinds (choices) • Singleton • T1 FS—a fuzzy number • IT2 FS—a fuzzy-fuzzy number

•Encoder • Words mean different things to different people • IT2 FS—No choice • Data from group of subjects • IA maps data into an FOU • Three canonical FOUs • Codebook {Wi, FOU(Wi)}

IT2 FLS vs Per-C

•Rules—IF-THEN • From experts • From data • Independent of kind of FSs used • Words in antecedents and consequents modeled as IT2 FSs

IT2 FLS vs Per-C

•Rules—IF-THEN

•CWW Engine

• From experts

• IF-THEN rules

• From data

• LWA

• Independent of kind of FSs used

• Others under development

• Words in antecedents and consequents modeled as IT2 FSs

• All words used by the CWW Engine must be in a Codebook

IT2 FLS vs Per-C

•Inference • Mamdani • Extended sup-star composition: firing interval • Computations only involve LMFs and UMFs • Fired rule outputs may be combined or not, depending on kind of output processing

IT2 FLS vs Per-C

•Inference • TSK • Firing interval • Computations only involve LMFs and UMFs • Fired rule outputs combined using TSK formula

IT2 FLS vs Per-C

•Inference • Mamdani • TSK

•Output of CWW Engine • IF-THEN rules • Similarity used to compute firing level • Perceptual Reasoning used to aggregate fired rules • Resulting output IT2 FS resembles word FOUs—new requirement for CWW

IT2 FLS vs Per-C

•Perceptual Reasoning


i=1 fi G
i M

Y
PR =




M j=1

fj

=
i=1 M




• IF-THEN rules

fi M j=1

•Output of CWW Engine

fj

G
i

• Similarity used to compute firing level • Perceptual Reasoning used to aggregate fired rules • Resulting output IT2 FS resembles word FOUs—new requirement for CWW

IT2 FLS vs Per-C

•Inference • Mamdani • TSK

•Output of CWW Engine • LWA (words, T1 FSs, intervals and numbers) • Extension Principle • Alpha-cuts function decomposition theorem • Two FWAs • IWAs • FOU(LWA) resembles word FOUs

IT2 FLS vs Per-C

•Output of CWW Engine

•LWA


i=1 X
iW
i M

Y
LWA =





j W j=1

M

• LWA (words, T1 FSs, intervals and numbers) • Extension Principle • Alpha-cuts function decomposition theorem • Two FWAs • IWAs • FOU(LWA) resembles word FOUs

IT2 FLS vs Per-C

•Output Processing • Type-reduction • Different kinds • KM algorithms • TR set is an IVFS—uncertainty measure

• Defuzzification • Average of TR FS

IT2 FLS vs Per-C

•Output Processing • Type-reduction

•Decoder • Three kinds (so far)

• Different kinds

• Similarity

• KM algorithms

• Rank (Centroid/ ranking band)

• TR set is an IVFS—uncertainty measure

• Subsethood+classification

• Defuzzification • Average of TR FS

IT2 FLS vs Per-C

•Outputs • Crisp outputs that are used in an action • TR IVFS that can be used as a measure of uncertainties that have flowed through the IT2FLS (analogous to a confidence interval)

IT2 FLS vs Per-C

•Outputs • Crisp outputs that are used in an action • TR IVFS that can be used as a measure of uncertainties that have flowed through the IT2FLS (analogous to a confidence interval)

•Recommendation + Data • People want to know “Why?” • Linguistic and numerical outputs • Centroid and ranking bands can be used as measures of uncertainties that have flowed through the Per-C

IT2 FLS vs Per-C: Recapitulation

• Words before FSs • Words mean different things to different people • Words mean similar things to different people • Words or a mixture of words and numbers always excite the Per-C • CWW Engines are constrained so that their outputs resemble the FOUs in the Codebook • Computations developed for IT2 FLSs are used in Perceptual Computing • Similarity, rank and subsethood are important in Per-C

Conclusions • There are many differences between an IT2 FLS and a Perceptual Computer • By comparing their architectures, blockby-block, it is easy to enumerate those differences • They are used for very different kinds of problems

IT2 FLS vs Per-C

•One Reference J. M. Mendel, Uncertain RuleBased Fuzzy Logic Systems: Introduction and New Directions, Prentice-Hall, 2001 • There are now a multitude of references for IT2 FLSs

IT2 FLS vs Per-C

•Reference

•Reference

J. M. Mendel, Uncertain RuleBased Fuzzy Logic Systems: Introduction and New Directions, Prentice-Hall, 2001

J. M. Mendel and D. Wu, Perceptual Computing: Aiding People in Making Subjective Judgments, Wiley and IEEE Press, 2010

• There are now a multitude of references for IT2 FLSs

• There is not yet a multitude of references for Perceptual Computing

Thanks