t2 fs talk

Fuzzy Set Type-2 Theory, Applications and Examples

Mahmoud Alish Noha El-Prince Nouf Al Alawi

Agenda •  Introduction to type-2 fuzzy set.

o  Interval Type-2 & General Type-2 o  Type-2 Terminologies

•  Type-2 fuzzy set Theory & Systems o  Theory & operation o  Inference System o  Example (Image Edge Detection)

•  Application (Computing With Words) o  Computing with Words (CWW) o  Perceptual Computing (Per-C) o  Example of Per-C: Journal Publishing Judgment Advisor (JPJA) o  Analysis of Per-C o  Conclusion

Introduction •  Introduced by Zadeh as theoretical concept in 1975 to

handles the uncertainty that Type-1 cannot handle.

•  Three sources of uncertainties –  The meanings of the words used in the antecedents & consequents

•  words mean different things to different people

–  The noise associated with Measurements of activating in the fuzzy system.

–  The noise associated with data that are used to tune the parameters in the fuzzy system.

Introduction

•  Able to model and minimize the effects of uncertainties in the fuzzy logic systems.

•  The grades of membership are fuzzy not crisp like Type-1 –  Called a “fuzzy-fuzzy set.”

•  Fuzzy set of type-2 denoted by ( ) •  If all uncertainty disappears è Type-2 reduces to a Type-1

Interval & General Type-2 Fuzzy Set

•  The membership function is three-dimensional (model uncertainty). –  More difficult to use and understand. –  Third dimension difficult to draw. –  Not easy to collect simple well-defined terms for the third dimension –  The use of type-2 is computationally more complicated.

•  There are two types of fuzzy set type-2 –  General Type-2 Fuzzy Set –  Interval Type-2 Fuzzy set

Interval & General Type-2 Fuzzy Set

General Type-2 IntervalType-2

•  Primary Membership (2D Domain) •  Secondary Membership (3D Domain)

•  The third dimension value is constant everywhere è ignored

The Terminologies

•  FOU: Footprint of uncertainty. •  UMF & LMF: Upper and lower membership functions

Type-2 Theory

•  Two representations: –  The vertical-slice

•  The basis for most computations

–  The wavy-slice •  The basis for most theoretical

derivations

Type-2 Theory

•  Both representations called covering theorems –  Union of all vertical slices

–  Union of all embedded Type-1 that

cover the entire FOU.

•  known as the Mendel-John Representation Theorem (RT)

Operations

Union Intersection Complement

Type-2 Fuzzy System

•  Type Reduction

Type-2 Fuzzy System

Type-2 Fuzzy System

•  Singleton Type-2 Fuzzy Logic Systems –  Based on Mandani –  Uncertainties in the antecedents or consequents

•  Non-singleton Fuzzy Logic Systems –  Based on Mandani –  Uncertainties in both the antecedents, consequents and input

measurement –  More complicated.

•  Sugeno Type-2 Fuzzy Systems –  Based on Takagi and Sugeno

The Use of Interval Type-2 Fuzzy Logic as a Method for Edge Detection using Gaussian as MF •  Detecting the edges allows feature extraction & construction of input

vectors for neural networks with aims of image recognition

•  Traditional Methods –  The gradient methods like Roberts, Prewitt and Sobel for detect edges.

•  Looking for maximum and minimum in first derived –  The Laplacian methods like Marrs-Hildreth

•  Finding the zeros of second derived

The Use of Interval Type-2 Fuzzy Logic as a Method for Edge Detection using Gaussian as MF

Inputs Inputs equation Rules

DH: Derivative Horizontal

If (DH is LOW) and (DV is LOW) then (EDGES is LOW) If (DH is MEDIUM) and (DV is MEDIUM) then (EDGES is HIGH) If (DH is HIGH) and (DV is HIGH) then (EDGES is HIGH) If (DH is MEDIUM) and (HP is LOW) then (EDGES is HIGH) If (DV is MEDIUM) and (HP is LOW) then (EDGES is HIGH) If (M is LOW) and (DV is MEDIUM) then (EDGES is LOW) If (M is LOW) and (DH is MEDIUM) then (EDGES is LOW)

DV: Derivative

Vertical

HP: high-pass

filter M:

low-pass filter


•  By using Type-2 fuzzy more than half of the pixels were cleared without depreciating the image

–  reduce in drastic form the cost of training in a neural network.

Computing with Words (CWW) CWW is :

“A methodology in which words are used

in place of numbers for compu7ng and

reasoning..”

Ann lives near Mary Distance =

500

Example: consider the proposi<ons: p1 = Ann lives near Mary p2 = Mary lives near Clara. Query : “How far is Ann from Clara?,” Answer: Ann lives not far from Clara.

(Zadeh, 1996- Ref[4])

Fig. Compute with numbers vs. compute with words

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Perceptual Computer (Per-C)

•  Words mean different things to different People = uncertainty in words = FL •  Uncertainty a person has about the meaning of a word (intra-personal

uncertainty)=> T1 FL •  Uncertainties that a group of people have about the meaning of the word

(inter-personal uncertainty) => T2 FL

Encoder

Decoder

CWW Engine (T2 FLS)

Words (perceptions)

Words (word, rank, class)

IT2 FS

IT2 FS

Fig. Architecture of a Per-C (Mendel Ref[5],2002)

[Ref 6]

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Perceptual Computer (Per-C) cont. Encoder: Words --------------- > IT2 FSs (Code Book = Words + their FOUs) q  Encoding Approaches:

Ø  Person FOU

Ø  Interval End-points

Ø  IA Approach (mostly practically used)

Encoding Approach

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q  IA Approach: (1) Collect interval end point data for a word

from a group of persons. (2) After removing outliers, intervals

are classified as either : interior, left-shoulder, right-shoulder IT1FS.

(3) Each word’s data intervals is mapped into its respective T1 interior, left shoulder or right shoulder MF (4) Interpret the MFs as an embedded T1 FS. Using Representation theorem and take their union.

Perceptual Computer (Per-C) cont.

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(5) Result is a FOU for an IT2FS model of the word. (6) The words + their FOUs constitute a codebook.

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Perceptual Computer (Per-C) cont. v  Code Book Sample:

v  FOUs of 4 words

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Perceptual Computer (Per-C) cont. q  CWW Engine

(a) Linguistic Weighted Average (LWA)

= data features

= The weights

Encoder

Decoder

CWW Engine (T2 FLS)

Words (perceptions)


IT2 FS

IT2 FS

There are different kinds of CWW engines:

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(b) Perceptual Reasoning (PR) Given : A rule base with K rules, each of the form:

Perceptual Computer (Per-C) cont.

Where and are words modeled by IT2 FSs

new input = (j = 1,...,p) are also words modeled by IT2 FSs

Output: where Firing level

of Rk

q  CWW Engine

Jaccard similarity for IT2 FSs

and are upper and lower MFs of

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Perceptual Computer (Per-C) cont. q Decoder

Recomm. Type

Decoder Type

Word

Similarity is calculated between the CWW o/p and all the words in the codebook. The o/p is the word with max. similarity.

Rank

A centroid-based ranking method for IT2 FSs can be used to select the best alternative.

Class

A decision category is the o/p made by a classifier e.g. Average subsethood

Encoder

Decoder

CWW Engine (T2 FLS)

Words (perceptions)


IT2 FS

IT2 FS

•  Three Types of decoders according to three types of recommendations:

Application of Perceptual Computing

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q The Journal Publishing Judgment Advisor (JPJA)

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Ø Step#1: Modify Paper review form


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Ø Step#2: Design the Encoder Two code books are needed : ( words (used by the reviewer), weights )

Fig. FOUs for the five-word Sub-codebook R1

Fig. FOUs for the three-word Sub-codebook R2

I. Codebook for the reviewer


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Ø Step#2: Design the Encoder – cont.

Fig. FOUs for the three-word Sub-codebook W1

Fig. FOUs for the four-word Sub-codebook W2

II. Codebook for the weights Weights correspond to Importance (W ̃I), Content (W ̃Co) and Depth (W ̃D).

Weights correspond to Weights correspond to Style (W ̃S), Organization (W ̃O), Clarity (W ̃Cl) and Reference (W ̃R).

Weights correspond to Weights correspond to Weights correspond to Technical Merit (W ̃T ) and Presentation ( W ̃P ) .

Fig. FOUs for the two-word Sub-codebook W3


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LWAs are used : all assessments and weights are words (or FOUs)

q The Journal Publishing Judgment Advisor (JPJA) Ø  Step#3: Design the CWW Engine


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•  The decoder for the JPJA is a classifier: it classifies the overall paper quality

into 3 classes: Accept, Rewrite, or reject.

•  A decoding codebook is needed to store the FOUs for these 3 words.


•  2 approaches for constructing such a codebook: a.) using a survey (used in JPJA) b.) using training examples.


Ø  Step#4: Design the decoder

•  O/p:

Analysis of Per-C

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q  Advantages

o  Very good tool for hierarchal decision making.

o  Diverse i/ps can be aggregated across different hierarchies

o  Uncertainties associated with these i/p s are preserved and

propagate into the final evaluation. q  Limitations

o  Not data adaptive o  Inputs (Interval end points) needs human interaction.

Conclusion

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•  Type-2 fuzzy set enables us to model the effect of uncertainty in (FLS)

better than Type-1 fuzzy set as it combines uncertainty about the

membership function into fuzzy set theory.

•  Type-2 fuzzy set is used when there is uncertainty about the

membership grades or difficult to determine the membership

functions of fuzzy logic system.

•  Per-C is a very useful tool for hierarchal and distributed decision making

and has great potential in solving complex decision-making problems.

Questions ?

References •  [1] Oscar Castillo and Patricia Melin. Type-2 Fuzzy Logic: Theory and Applications. 223, 2008

Springer-Verlag Berlin Heidelberg. July 2007. •  [2] Jerry M. Mendel and Robert I. Bob John. Type-2 Fuzzy Sets Made Simple. IEEE

TRANSACTIONS ON FUZZY SYSTEMS, VOL. 10, NO. 2, APRIL 2002. •  [3] Jerry M Mendel. Type-2 fuzzy sets and system: an overview. IEEE computational intelligent

magazine, vol 2, no. 1, pp. 20-29. February 2007.

•  [4] L.A. Zadeh, Fuzzy logic = computing with words, IEEE Trans. on Fuzzy Systems 4 (1996) 103-111.

•  [5] J.M. Mendel, An architecture for making judgments using computing with words, Int. J. Appl. Math. Comput. Sci. 12 (3) (2002) 325–335.

•  [6] J.M.Mendel, Perceptual Computing, Willey, 2010.

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