Computational Intelligence for Modelling and Prediction

Saman K. Halgamuge, Lipo Wang (Eds.) Computational Intelligence for Modelling and Prediction

Studies in Computational Intelligence, Volume 2 Editor-in-chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: springeronline.com Vol. 1. Tetsuya Hoya Artificial Mind System – Kernel Memory Approach, 2005 ISBN 3-540-26072-2 Vol. 2. Saman K. Halgamuge, Lipo Wang (Eds.) Computational Intelligence for Modelling and Prediction, 2005 ISBN 3-540-26071-4

Saman K. Halgamuge Lipo Wang (Eds.)

Computational Intelligence for Modelling and Prediction

ABC

Dr. Saman K. Halgamuge

Dr. Lipo Wang

Associate Professor and Reader Mechatronics and Manufacturing Research Group Department of Mechanical and Manufacturing Engineering The University of Melbourne Victoria 3010 Australia E-mail: [email protected]

Associate Professor School of Electrical and Electronic Engineering Nanyang Technological University Block S1, 50 Nanyang Avenue Singapore 639798 Singapore E-mail: [email protected]

Library of Congress Control Number: 2005926347

ISSN print edition: 1860-949X ISSN electronic edition: 1860-9503 ISBN-10 3-540-26071-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-26071-4 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005
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Preface Understanding the decision-making processes of living systems, and the efforts to mimic them, are identified with research in Artificial Intelligence (AI). However, the recent popularity of neural networks, fuzzy systems and evolutionary computation, which are widely considered as areas related in AI, has created a need for a new definition to distinguish them from traditional AI techniques. Lotfi Zadeh, the inventor of fuzzy logic, has suggested the term “Soft Computing.” He created the Berkeley Initiative of Soft Computing (BISC) to connect researchers working in these new areas of AI. In contrast to hard computing, soft computing techniques account for the possibility of imprecision, uncertainty and partial truth. The first joint conference of neural networks, fuzzy systems, and evolutionary computation organized by the Institute of Electrical and Electronic Engineers (IEEE) in 1994 was the World Congress on Computational Intelligence. In his paper at this historic joint conference, James Bezdek defined three kinds of intelligence to distinguish Computational Intelligence from traditional AI and living systems: biological or organic, artificial or symbolic, and computational or numeric. All natural creatures belong to the first category, while traditional AI techniques remain in the second category. Computational Intelligence (CI) is the group of techniques based on numerical or sub-symbolic computation mimicking the products of nature. However a number of “new AI” methods have found a home in CI. The application of CI in emerging research areas, such as Granular Computing, Mechatronics, and Bioinformatics, show its maturity and usefulness. Recently IEEE changed the name of its Neural Network society to become the IEEE Computational Intelligence Society. The International Conference on Fuzzy Systems and Knowledge Discovery (FSKD) held in Singapore in 2002, in conjunction with two other conferences in CI, has led to the publication of these two edited volumes. This volume contains CI methods and applications in modeling, optimisation and prediction. The other volume entitled “Classification and Clustering for Knowledge Discovery”, from the same publisher, includes the papers on Clustering and Classification. Chapters 1-10 present theory and methods of Fuzzy and Hybrid Systems useful in modeling and prediction. In contrast to the unipolar scale of the unit interval [0,1] of a fuzzy set, a more psychologically justified interval [-1,1] is considered in Chapter 1. The qualities of metal containers are predicted using a fuzzy model coupled with a vision system in Chapter 2. An extended neuro-fuzzy system is proposed in Chapter 3 using the inclusion of fuzzy relations that enhance learning and rule flexibility. A new information retrieval system is presented in Chapter 4, which combines a fuzzy relevance model and a semantic based indexing. A novel algorithm on backward reasoning for deciding on the sequence of transition firing of a fuzzy-petri net is proposed in Chapter 5. The generalized rough approximation of a fuzzy set is investigated in Chapter 6 using a weak asymmetric relation. A new approach to the fuzzy shortest path problem including a new heuristic method to find the fuzzy shortest length is proposed in Chapter 7. A method to evaluate the degree of agreement in a group of experts and the application of the method in medical diagnosis is presented in Chapter 8. A method that represents and processes fuzzy information at the coarse granular level is proposed in Chapter 9. A study on intuitive fuzzy relational images and

VI

Preface

their application in the representation of linguistic hedges and in the development of a meaningful concept of a fuzzy rough set is presented in Chapter 10. A book on Modelling and Prediction must have the essence of the various applications. We capture the range of applications varying from image processing (Chapters 11-15), audio processing (Chapters 16-17), commerce and finance (Chapters 18-20), communication networks (Chapters 21-22), and other applications (Chapters 23-28). The prediction of weed dispersal using Geographic Information Systems (GIS) spatial images, optimisation of image compression, multi-layer image transmission with inverse pyramidal decomposition, multiple feature relevance feedback in content-based image retrieval, and non-iterative independent component analysis (ICA) for detecting motion in image sequences are topics discussed in the five chapters in image processing. A practical approach to chord analysis in acoustic recognition and the concepts and applications of audio fingerprinting are presented in the two audio processing chapters. Mining patterns in the US stock market, a fuzzy rule-based trading agent, and a case study with the Brazilian Central Bank are presented in the three chapters on finance. Congestion control in packet networks and prediction of the timeout and retransmission of Transmission Control Protocol (TCP) are the subjects of the two chapters on communication networks. Chapters on other applications include user satisfaction in web searching (Chapter 23), population studies (Chapter 24), information retrieval (Chapter 25), learning Boolean formulas from examples (Chapter 26), fuzzy linear programming (Chapter 27) and controlling hybrid power systems (Chapter 28). Our thanks go to the Department of Mechanical and Manufacturing Engineering, University of Melbourne, Australia, and the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, which supported our project in various ways. We thank the many people who supported the International Conference on Fuzzy Systems and Knowledge Discovery (FSKD) held in Singapore in 2002. We especially thank its honorary chair Prof Lotfi Zadeh, who suggested the conference's name and motivated us throughout the process. Most of the papers in this book reflect the extended work from this conference. Our special thanks go to Mr. Chon Kit (Kenneth) Chan and Mr. Siddeswara Mayura Guru for managing the paper submission and formatting process, and Mr. Sean Hegarty for proofreading. We also acknowledge the partial support of the Australian Research Council. We are grateful to all the authors for enthusiastically submitting high quality work to this publication, and Prof Janusz Kacprzyk and Springer Verlag for realizing this book project. Saman K. Halgamuge and Lipo Wang March 8, 2005

Contents Chapter 1

Symmetrization Of Fuzzy Operators: Notes On Data Aggregation Wladyslaw Homenda and Witold Pedrycz

1

Chapter 2

An Integrity Estimation Using Fuzzy Logic P. Mariño, C. A. Sigüenza, V. Pastoriza, M. Santamaría, E. Martínez and F. Machado

19

Chapter 3

Connectionist Fuzzy Relational Systems Rafa Scherer and Leszek Rutkowski

35

Chapter 4

Semantic Indexing and Fuzzy Relevance Model in Information Retrieval Bo-Yeong Kang, Dae-Won Kim and Sang-Jo Lee

49

Chapter 5

Backward Reasoning on Rule-Based Systems Modeled by Fuzzy Petri Nets Through Backward Tree Rong Yang, Pheng-Ann Heng and Kwong-Sak Leung

61

Chapter 6

On The Generalization of Fuzzy Rough Approximation Based on Asymmetric Relation Rolly Intan and Masao Mukaidono

73

Chapter 7

A new approach for the fuzzy shortest path problem Tzung-Nan Chuang and Jung-Yuan Kung

89

Chapter 8

Distances Between Intuitionistic Fuzzy Sets and their Applications in Reasoning Eulalia Szmidt and Janusz Kacprzyk

101

Chapter 9

Efficient Reasoning With Fuzzy Words Martin Spott

117

Chapter 10

Intuitionistic Fuzzy Relational Images Martine De Cock, Chris Cornelis and Etienne E. Kerre

129

Chapter 11

Thematic Fuzzy Prediction of Weed Dispersal Using Spatial Dataset Andrew Chiou and Xinghuo Yu

147

Chapter 12

Optimization of Image Compression Method Based on Fuzzy Relational Equations by Overlap Level of Fuzzy Sets Hajime Nobuhara, Eduardo Masato Iyoda, Kaoru Hirota and Witold Pedrycz

163

VIII

Contents

Chapter 13

Multi-layer Image Transmission with Inverse Pyramidal Decomposition Roumen Kountchev, Mariofanna Milanova, Charles Ford and Roumiana Kountcheva

179

Chapter 14

Multiple Feature Relevance Feedback in Content-Based Image Retrieval using Probabilistic Inference Networks Campbell Wilson and Bala Srinivasan

197

Chapter 15

Non-Iterative ICA for Detecting Motion in Image Sequences Yu Wei and Charayaphan Charoensak

209

Chapter 16

A Practical Approach to the Chord Analysis in the Acoustical Recognition Process Marcin Szlenk and Wladyslaw Homenda

221

Chapter 17

Audio Fingerprinting Concepts And Applications Pedro Cano, Eloi Batlle, Emilia Gómez, Leandro de C.T.Gomes and Madeleine Bonnet

233

Chapter 18

Mining Technical Patterns in The U. S. Stock Market through Soft Computing Ming Dong and Xu-Shen Zhou

247

Chapter 19

A Fuzzy Rule-Based Trading Agent: Analysis and Knowledge Extraction Tomoharu Nakashima, Takanobu Ariyama, Hiroko Kitano and Hisao Ishibuchi

265

Chapter 20

An Algorithm for Determining the Controllers of Supervised Entities: A Case Study with the Brazilian Central Bank Vinícius Guilherme Fracari Branco, Li Weigang and Maria Pilar Estrela Abad

279

Chapter 21

Fuzzy Congestion Control In Packet Networks Tapio Frantti

291

Chapter 22

Fuzzy Logic Strategy of Prognosticating TCP's Timeout and Retransmission Zhongwei Zhang, Zhi Li and Shan Suthaharan

309

Chapter 23

Measuring User Satisfaction in Web Searching M. M. Sufyan Beg and Nesar Ahmad

321

Contents

IX

Chapter 24

Long-Range Prediction of Population by Sex, Age and District Based on Fuzzy Theories Pyong Sik Pak and Gwan Kim

337

Chapter 25

An Efficient Information Retrieval Method for Brownfield Assessment with Sparse Data Linet Ozdamar and Melek Basak Demirhan

355

Chapter 26

A Fuzzy Method for Learning Simple Boolean Formulas from Examples Bruno Apolloni, Andrea Brega, Dario Malchiodi, Christos Orovas and Anna Maria Zanaboni

367

Chapter 27

Fuzzy Linear Programming: A Modern Tool For Decision Making Pandian Vasant, R. Nagarajan and Sazali Yaacob

383

Chapter 28

Fuzzy Logic Control in Hybrid Power Systems Josiah Munda, Sadao Asato and Hayao Miyagi

403

CHAPTER 1 Symmetrization Of Fuzzy Operators: Notes On Data Aggregation 1

Wladyslaw Homenda and Witold Pedrycz

2

1

Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-661 Warsaw, Poland, [email protected] 2 Dept. of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada T6G 2G7 and Systems Research Institute, Polish Academy of Sciences, 01-447 Warsaw, Poland, [email protected]

Abstract: Fundamental operators of set theory, fuzzy sets and uncertain information processing explore a unipolar scale of the unit interval [0,1] . However, it is known that human beings also handle other types of scale including a bipolar one that can be expressed on the interval [− 1,1] . This scale is of particular interest since it supports a representation of symmetric phenomena encountered in human behavior. In the paper, we investigate fundamental aspects of information aggregation operators from the standpoint of their symmetric behavior. Likewise, classical set operators are analyzed within the same setting. New type of aggregation operators - balanced operators - are defined and discussed. Keywords: unipolar and bipolar scale, information aggregation, t-norm and t-conorm, uninorm, balanced fuzzy operators

1

Introduction

Fundamentals of fuzzy sets were formulated on the basis of max and min operators applied to membership functions [27]. These operators were, then, generalized to triangular norms. In both theory and application the concept of triangular norms borrowed from [21] play an important role. They are widely utilized in many areas, e.g. logic, set theory, reasoning, data aggregation, etc. To satisfy practical requirements new operators were proposed and developed. The study is structured into 6 sections. In Section 2 we start with a review of fuzzy operators with special attention paid to triangular norms and uninorms. In Section 3 the motivation of this work is given. In Section 4 we introduce balanced operators based on triangular norms, then we study relations between balanced t-conorms and uninorms and finally we define balanced uninorms, the operators that satisfy intuitive constraints put on aggregation operators. The main sections of the paper are supplemented by conclusions and a bibliography.

W. Homenda and W. Pedrycz: Symmetrization Of Fuzzy Operators: Notes On Data Aggregation, Studies in Computational Intelligence (SCI) 2, 1–17 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


2

W. Homenda and W. Pedrycz

2

PRELIMINARIES

So far most of studies of aggregation operators have been conducted on the unit interval 0,1 . The list of such operators includes triangular norms [21] and uninorms [18,26] that are of special interest of this study.

[ ]

2.1

Triangular norms

Triangular norms were introduced in probabilistic metric spaces, c.f. [21] and then they were adopted as generalization of union and intersection of fuzzy sets. Triangular norms are mappings from the unit square into the unit interval satisfying the following axioms: Definition: Triangular norms (t-norms and t-conorms) are mappings p : 0,1 × 0,1 → 0,1 , where p stands for both t-norm t and t-conorm s , satisfying the following axioms: 1. p ( a, p (b, c )) = p ( p ( a, b), c ) associativity

[ ] [ ] [ ]

p (a, b) = p (b, c) commutativity monotonicity 3. p ( a, b) ≤ p (c, d ) if a ≤ c and b ≤ d boundary conditions 4. t (1, a ) = a for a ∈ [0,1] , s (0, a ) = a for a ∈ [0,1] Any t-norm t satisfies the following boundary condition t (0, a ) = 0 for all 0 ≤ a ≤ 1 . For any s-norm we have s (1, a ) = 1 for 0 ≤ a ≤ 1 . 2.

A t-norms (s-norm respectively) is called to be strong if it is continuous and strictly decreasing (increasing, respectively). Strong t-norm (t-conorm, respectively) satisfies the following condition:

t ( x , x ) < x ( s ( x, x ) > x )

( )

x ∈ 0,1 . for all Example The following mappings are t-norms and t-conorms: 1. minimum and maximum, i.e. t ( a, b) = min(a, b) , s ( a, b) =

max(a, b) 2. product and probabilistic-sum, i.e. t (a, b) = a * b , s ( a, b) = a + b − a * b 3. every strict t-norm t is generated by an additive generator f t : t (x, y ) = g t ( f t (x ) + f t ( y )) where: • ft : [0,1] → [0,+∞ ] , f t (1) = 0 , f t ( x )   + → +∞ x →0

•

ft is continuous, strictly decreasing, • gt denotes the inverse of ft . 4. every strict t-conorm s is generated by an additive generator f s : s ( x, y ) = gs ( fs ( x) + fs ( y )) where: • f s : [0,1] → [0,+∞ ] f s : (0 ) = 0 , f s ( x )  − → +∞ x →1

•

fs is continuous, strictly increasing, • gs denotes the inverse of fs .

Symmetrization Of Fuzzy Operators: Notes On Data Aggregation

3

f t : [0,1] → [0,+∞] f (x ) = (1 − x) / x and f (x ) = x /(1 − x) are applied as additive generators of t-norms

Note: the following two mapping:

f s : [0,1] → [0,+∞]

and t-conorms ongoing. In the Figures 8 and 9 the right upper quarter of the plots of balanced t-norm and balanced t-conorm present plots of the above t-norm and t-conorm. t-norm t and t-conorm s are said to be dual if they satisfy the following condition with respect to strong negation operator

n( x ) = 1 − x :

s (a, b ) = 1 − t (1 − a,1 − b ) Note: cf., for instance, [18] for weaker conditions on representation of triangular norms by additive generators, [17,18] for studies on continuous triangular norms, [18] for discussion on fuzzy negation operators. t-conorm s and t-norm t could be replaced each by other in this formula in order to define t-norm dual to a given t-conorm. Duality of norms implies that properties of dual norms are dual. Note that the max-min and product-probabilistic-sum are dual pairs of tnorms and t-conorms.

2.2

Uninorms

Recently, unification, extensions and generalizations of triangular norms were introduced, c.f. [4,18,26]. Both the neutral element 1 of a t-norm and the neutral element 0 of a tconorm are boundary points of the unit interval. However, there are many important operations whose neutral element is an interior point of the underlying set. Employing this information we can replace the boundary condition of the definition of triangular norms by any value of the unit interval 0,1 . The fact that the first three axioms of the definition of triangular norms coincide, i.e. the only axiomatic difference lies in the location of neutral element, leads to close relations between triangular norms and uninorms and nullnorms. Definition: Uninorm u and nullnorm v are mappings, where p stands for both u

( )

and v : p : 1. 2. 3. 4.

[0,1]× [0,1] → [0,1] satisfying the following axioms: p

is associative, commutative and monotone,

there exists an identity element for all

x ∈ [0,1] u ( x, e) = x

there exists a neutral element

e ∈ [0,1] such that

a ∈ [0,1] such that

x ∈ [a,1] v( x,1) = x boundary condition for uninorm u and nullnorm v Obviously, a t-norm t is a special uninorm with identity element e = 1 and a t-conorm s is a special uninorm with identity element e = 0 . Assuming that u is a uninorm with identity e and if v is defined as v(x, y ) = 1 − u (1 − x, 1 − y ) , then v is a uninorm with identity 1 − e . v is called the for all

x ∈ [0, a ] v( x,0) = x

and for all

dual uninorm of u . This means that uninorm and its dual analogue differ quantitatively while they are similar from the perspective of global properties discussed in this paper. So that duality is not discussed here. Assuming that u is a uninorm with identity element e, we have:

4

W. Homenda and W. Pedrycz

u (a,0 ) = 0 for all a ≤ e , u (a,1) = 1 for all a ≥ e 2. x ≤ u ( x, y ) ≤ y for all x ≤ e and e ≤ y 3. either u (0,1) = 0 or u (0,1) = 1 Assuming that u is a uninorm with identity e ∈ (0,1) , the mappings tu and su are t1.

norm and t-conorm respectively, cf. [7], we obtain

tu ( x , y ) = su (x, y ) =

u (ex, ey ) e

and

u (e + (1 − e )x, e + (1 − e ) y ) 1− e

or equivalently:

x y uu (x, y ) = etu  ,  e e

if x, y ∈ [0, e]

and

 x−e y −e uu (x, y ) = e + (1 − e )su  ,  if x, y ∈ [e,1]  1− e 1− e  The above results show that uninorm domain is split into four areas: two squares determined by left-bottom and right-top vertexes 0,0 , e, e and e,e , 1,1 respectively, on one hand, and two remaining rectangles fitting up to the unit square. Uninorm restricted to the first square remains a squeezed t-norm and restricted to the second square remains a squeezed t-conorm, cf. Figure 3.

{( ) ( )}

2.3

{( ) ( )}

Brief survey of other aggregation operators

In [23] OWA (ordered weighted averaging aggregation operators) were introduced. This family of new type operators were concerned with the problem of aggregating multicriteria information to form an overall decision function. In particular, OWA operators have property of lying between the “and”, requiring all the criteria to be satisfied, and the “or”, requiring at least one of the criteria to be satisfied. In other words, OWA operators form a family of a generalized mean operator, cf. [24.25]. In [22] it is stated that classical fuzzy connectives are not symmetric under complement. Partitioning a universe into two subsets having comparable significance suffers from the asymmetry of fuzzy connectives subsequently, a class of operations on fuzzy sets called symmetric summation is developed with intent to avoid the asymmetry drawback of classical fuzzy operators. The above attempts were based on the classical model of fuzzy sets which does not outline an aspect of negative information, cf. [14]. Several attempts have also been developed to explore negative information. An interesting extension of classical fuzzy sets introduced in [1] – intuitionistic fuzzy sets - copes with a kind of negative information that is attached to classical fuzzy set as, in general, a separate component. This additional component deals with negative information, creating a balance to positive information dispersed on the unit interval of classical fuzzy

5

Symmetrization Of Fuzzy Operators: Notes On Data Aggregation

sets. The system of intuitionistic fuzzy sets, however, does not provide a tool to combine two separated kinds of information besides a simple condition – the degree of indeterminacy. When investigating a unipolar scale, it becomes clear that the right part of the unit interval - i.e. values close to 1 - numerically expresses the states of strong positive information, e.g. (almost) full inclusion of an element into a set. It would be intuitively desired that the meaning of middle part of the unit interval – i.e. values close to 0.5 – expresses the states of weak information with a kind of neutral information or lack of information expressed by the numerical value 0.5. Likewise, the left part of the unit interval would be expected to represent the states of strong negative information, e.g. (almost) full exclusion of an element from a set. However, it is not clear whether, in the case of classical fuzzy sets, a value close to 0 expresses the state of strong negative information or rather a state of weak positive information. One may anticipate that a value close to 0 denotes a state of strong exclusion. Otherwise - assuming that numerical values monotonically increases from negative through neutral to positive information - the membership function does not provide a mechanism describing state(s) of exclusion or degrees of negative information. And then it is consistent to assume that the value 0.5 expresses the state of neutral information, lack of information, etc. Thus, symmetry of the interval − 1,1 would match the numerical representation of uncertainty better than asymmetry of the unit interval 0,1 does. And, indeed, many studies apply the interval − 1,1 as a numerical representation of information. The very early medical expert system MYCIN, cf. [3], combined positive and negative information by a somewhat ad hoc invented aggregation operator. In [6] it was shown that MYCIN aggregation operator is a particular case in a formal study on aggregation of truth and falsity values. A linear space on the space of fuzzy sets with membership grades valued on the interval −1,1 was defined in [14]. In the sequal it was extended to the interval −1,1 was done. In [11] algebraic structures of fuzzy sets with membership grades valued on the interval − 1,1 were introduced. Based on strict t-norms and t-conorms additive and multiplicative groups were defined. An immersion of fuzzy sets with membership grades valued on the interval − 1,1 into algebraic structures based on the interval − 1,1 was discussed. The immersion provides a practical approach to process fuzzy sets with membership grades in the interval − 1,1 in the framework of algebraic structures. In [15] the model of fuzzy neuron based on algebraic structures of fuzzy sets was studied. The interval − 1,1 is considered and endowed with an algebraic structure like a ring in [7]. The motivation stems from the idea of decision making, where scales which are symmetric with respect to 0 are needed, to represent a symmetry in the behavior of a decision maker. t-conorms and t-norms are utilzed and relations to uninorms are done. Symmetric pseudo-addition and symmetric pseudo-multiplications on the interval − 1,1 were defined and an attempt to build a ring is presented. It is shown that strict t-norms are of importance in algebraic structures created on the interval − 1,1 . An extensive survey of aggregation operators, as well as a discussion on various aspects of information aggregation has been provided in several studies, e.g. [5,20]. An interesting idea of re-scaling true/false concept from 0,1 values to the interval − 1,1 was developed.

[

[ ]

[

(

)

(

)

]

[

[

]

(

[

]

]

)

]

[

]

[

[

{ }

]

[

]

]

6

W. Homenda and W. Pedrycz

[

]

A number of papers considered values of membership function on the interval − 1,1 . However, no negative meaning of information is suggested, c.f. [2,10,19,28]. In [9] an extension of triangular norms to the interval − 1,1 is introduced. Yet, that extension is rather imprecisely defined and does not address an issue of inconsistency between operators’ asymmetry and the human symmetry requirement.

[

3

BALANCED EXTENSION

3.1

Motivation

]

A kind of asymmetry of the set of values of aggregation operators could be observed: if the state of (full) inclusion of the element is denoted by 1, the state of exclusion could be denoted by –1 rather than 0. This observation would be irrelevant since most of studies of fuzzy sets and all studies on crisp sets have been targeted on the unit interval − 1,1 . Yet, both scales unipolar with the unit interval

[− 1,1] are indistinguishable i (x ) = 2 x − 1 . Therefore it

[0,1]

[

]

and bipolar with the symmetric interval

in the sense of each

simple isomorphism as e.g.

would be unreasonable to convert studies between mathematically indistinguishable scales without more benefits except then forcing symmetry of the targeted scale. However, it is known that human beings handle other types of scale including the bipolar one modeled by the interval − 1,1 , cf. [8]. It has been shown that the use of bipolar scales is of particular interest in several fields since they enable the representation of symmetry phenomena in human behavior, when one is faced with positive (gain, satisfaction, etc.) or negative (loss, dissatisfaction, etc.) quantities, but also weak or neutral information with a kind of disinterest (does not matter, not interested in, etc.). These quantities can also be interpreted in context of relation between an element and a set as inclusion/exclusion/lack-of- information or – so called – positive, negative and neutral information. Furthermore, symmetry of the aggregation operator is expected by human beings. This expectation can be briefly expressed as the following: The preservation principle: aggregation operators should preserve strength of aggregated information, i.e. to keep or even to increase the strength of strong information and to keep or even increase the weakness of weak information. Considering the unipolar scale and assuming that the left end of the unit interval expresses negative information, the right end of the unit interval expresses positive information and the middle part represents weak information of positive and negative type, the following conditions illustrate preservation principle with regard to aggregation of two pieces of information: 1. if both pieces of data are strongly positive represented by numerical values close to 1 – say 0.7 and 0.9 - the aggregation result would be at least as strong as the stronger piece of data, i.e. greater or equal to 0.9 2. and vice versa, for strongly negative data pieces represented by numerical values close to 0 – say 0.3 and 0.1 - the aggregation result would be at least as strong as the stronger data piece, i.e. less or equal to 0.1,

[

]

7

Symmetrization Of Fuzzy Operators: Notes On Data Aggregation

3. for weak positive data (weak negative data, respectively) expressed by numerical values 0.55 and 0.65 the aggregation result would be as weak as the weaker data, i.e. less or equal to 0.55, though greater then 0.5, 4. having contradictory pieces of information, i.e. strong positive and strong negative – say 0.7 and 0.3 - it would be expected that result of aggregation will be neutral, equal to 0.5. The re-formulation of these conditions to a bipolar scale relies on rescaling of the unit interval 0,1 into the interval − 1,1 , c.f. Figure 1. Classical aggregation operators do not satisfy the preservation principle despite they are valued on the unit interval 0,1 , the interval − 1,1 or any other interval. To elaborate this defect in more detail, consider strong t-norm applied to non-crisp arguments 0 < a, b < 1 . The result value t a, b is, of course, less than the value of the smaller argument. It means that t-norm weakens aggregated positive data and strengthens aggregated negative data, despite the numerical values of these data. The same discussion concerns strong t-conorm - it weakens aggregated negative data and strengthens aggregated positive data. This evidence could be extended to any t-norm and any t-conorm. This phenomenon is clearly outlined when convergence of aggregation of an infinite sequence of repeated non-crisp data unit is considered:

[ ]

[

]

[ ]

[

(

]

)

(t ( (a ).... ), a),

lim t t (.... , a a ) n → 0 if a < 1 → +∞ t n times

(s (s( (a , a )....), lim s .... s a ) ,

a ) n → 1 if 0 < a → +∞ n times

Thus, it is clear that t-norms and t-conorms are inconsistent with symmetry requirements of information aggregation, cf. Figure 2.

negative -1

neutral

negative

positive

0

1

0

neutral

positive

1

1/2 t-norm

negative -1

neutral 0

positive

negative

1

0

neutral 1/2

positive

1

s-norm

Fig. 1 Preservation rule - human anticipation of aggregation symmetry

Fig. 2 Asymmetry of triangular norms

The preservation rule with respect to uninorms is satisfied only when either strong positive data units or strong negative data units are aggregated. Yet, weakness of aggregated data is not preserved. Only neutrality of data (neutral data piece is represented numerically by the identity element of uninorm) is preserved, cf. Figure 4.

8

W. Homenda and W. Pedrycz

1 between min and max

squizzed t-conorm

squizzed t-norm

between min and max

negative

0

e

0

neutral

positive

e

1

u-norm

e

Fig. 4 Uninorms – violation of the preservation rule

1

Fig. 3 Structure of uninorms

The experience with aggregation of infinite sequence of repeated non-crisp data unit gives the following result for uninorms consisting of strong t-norm and strong t-conorm:

1 if e < a ≤ 1  (u(... ),

lim u → e if a = e u ( a,  a )... a ) n → +∞   0 if 0 ≤ a < e n times  Similar behavior characterizes nullnorms. This discussion shows that classical operators become asymmetrical in information aggregation process. Thus the isomorphism i x = 2 x − 1 applied in scale symmetrization makes more confusion instead of solving the inconsistency. In fact, this isomorphism makes the scale symmetrical, but does not change asymmetry of aggregation operators. Concluding, we can formulate the following: The asymmetry observation: the idea of fuzzy sets relied on spreading positive information (information about inclusion) fitting the crisp point 1 into the interval

()

{}

(0,1] , while negative information still remain bunched into the crisp point {0}, cf. Figure 5.

The asymmetry observation interprets uncertainty handling in classical fuzzy sets. It founds an alternative model of fuzzy sets, a model that handles both kinds of information: positive and negative. It forms also a basis of a balanced extension of classical fuzzy sets. This extension relies on a distribution of crisp negative information located at the point 0 into the interval − 1, 0 . It offers symmetry of positive and negative information handling.

{ }

[

]

9

Symmetrization Of Fuzzy Operators: Notes On Data Aggregation

3.2

The extension method

The balanced extension of classical fuzzy sets and balanced extension of classical operators relies on dispersion of crisp negative information located at the point 0 into the interval

{ }

[− 1, 0] . Balanced extension does not affect classical fuzzy sets based on the unit interval [0,1] . As a consequence, classical fuzzy sets will be immersed into new space of balanced

fuzzy sets. Since both kinds of information – positive and negative – are equally important, 0 0

crisp information

crisp information

1

dispersion of the crisp positive information

1

dispersion of the crisp positive information crisp negative information

crisp negative information

0

fuzzy positive information

dispersion of the crisp negative information

1

Fig. 5 Fuzzy sets as dispersion of crisp positive information into the interval 0,1@

-1

fuzzy negative information

0

fuzzy positive information

1

Fig. 6 Dispersion of crisp negative information into the interval > 1,0

it would be reasonable to expect that such balanced extension supports the symmetry of positive/negative information with a center of symmetry placed in the middle of scale: The symmetry principle: the extension of fuzzy sets to balanced fuzzy sets relies on spreading negative information that fit the crisp point 0 of fuzzy set (information about

[

{ }

)

exclusion) into the interval − 1,0 . The extension should preserve symmetry of positive and negative information. This extension preserves properties of classical fuzzy operators handled for positive information processing. It provides the symmetry of positive/negative information handling with the center of symmetry placed in the point 0 , cf. Figures 6 and 7. A measure of inclusion of an element into a fuzzy set is expressed by a positive values, a measure of exclusion of an element from a fuzzy set is expressed by a negative value of membership function. Of course, the greater the absolute value of membership function, the more certain information about inclusion/exclusion. Classical aggregation operators, cf. f in Figure 7, have the unit square 0,1 × 0,1 as

{ }

[ ] [ ]

[ ]

domain and the unit interval 0,1 as co-domain. Balanced operators, cf. F in Figure 7, are equal to classical operators on the unit square. Due to the symmetry principle, balanced operators are defined by the formula F x, y = − f − x,− y on the square

(

)

(

)

10

[

W. Homenda and W. Pedrycz

) [

[

)

)

− 1,0 × − 1,0 . The interval − 1,0 obviously includes values produced by the balanced operator for the negative values of its arguments. This method determines the structure of the negative information space. Such a concept is consistent with the human symmetry expectation: negative information space has the same structure as its positive counterpart. So then the space of negative information is entirely constrained by its origin in the symmetrical reflection. Unlike, aggregation of positive and negative data would be defined with regard to practical applications. A fuzzy set is usually defined as a mapping from a universe X onto the unit interval 0,1 . Thus, the space of fuzzy sets over the universe X is denoted by F while space of

[ ]

balanced fuzzy sets will be denoted by

G:

F ( X ) = [0,1] = {µ | µ : X → [0,1]} X

G ( X ) = [− 1,1] = {µ | µ : X → [− 1,1]} X

4

BALANCED OPERATORS

The concept of balanced extension of aggregation operators was introduced in [12,13] as an extension and generalization of classical aggregation operators. The concept of balanced aggregation operators defined with respect to the bipolar scale is based on the asymmetry observation and symmetry principle discussed in Section 3.1. The balanced extension method was formulated in Section 3.2 and is illustrated in Figure 7.

4.1

Balanced triangular norms

In this section we introduce formal definitions of balanced fuzzy operators. The intuitive introduction of balanced operators outlined in Figure 5 and discussion on symmetry issues

-1

unconstrained area of balanced operators F(x,y) reflection of classical aggregation operators: F(x,y)=-f(-x,-y)

classical aggregation operators f:[0,1]x[0,1]->[0,1] F(x,y)=f(x,y)

unconstrained area of balanced operators F(x,y)

-1

1

f – classical operator F – balanced operator

Fig. 7 Balanced extension of fuzzy sets

11

Symmetrization Of Fuzzy Operators: Notes On Data Aggregation

in Section 3 outlined the fundamental method of fuzzy sets extension. The balanced extension is based on the assumption that classical fuzzy operators are immersed in their balanced extensions. Therefore, the classical operators are equal to their balanced

[ ] [ ]

[ ]

extensions on the unit interval 0,1 or on the unit square 0,1 × 0,1 . Precisely – identity of classical operators and their counterparts might not be satisfied in boundary points. For instance the classical negation and its balanced counterpart – the inverse operators – have different values at 0 . The immersion of classical system of fuzzy sets into its balanced extension preserves features of classical system on its domain (with isolated points excluded from this preservation, in some cases), i.e. on the unit interval 0,1 or on the unit square 0,1 × 0,1 .

{ }

[ ]

[ ] [ ]

Notice that balanced operators are mappings from the interval square

[− 1,1]

or from the

[− 1,1]× [− 1,1] into the interval [− 1,1] satisfying the symmetry principle.

Definition: Balanced t-norms T and balanced t-conorms S are mappings: P : [− 1,1]× [− 1,1] → [− 1,1] , where P stands for both balanced t-norm and balanced tconorm, satisfying the following axioms: 1. 2. 3.

associativity, commutativity & mononicity of both mappings the domain

4.

5.

[− 1,1]× [− 1,1]

T (1, a ) = a for a ∈ [0,1] S (0, a ) = a for a ∈ [0,1] P ( x, y ) = − P ( − x, − y )

and

S

on

boundary conditions symmetry

The balanced counterpart of a strong negation operator inverse operator (with respect to symmetry principle):

I : [− 1,1] → [− 1,1] ,

T

n(x ) = 1 − x is defined as an

 1− x  I ( x) =  − 1 − x  undefined 

[

] [

for

x>0   x 0   where: x* y ≤ 0 

•

ft is a continuous, strictly decreasing function undefined in 0 , ft : [− 1,1] − {0} → [− ∞,+∞] , ft (− 1) = f t (1) = 0 , f t ± → ±∞ ,

•

gt denotes the inverse of ft .

x →0

12

W. Homenda and W. Pedrycz

S is generated by an additive generator f s : S ( x, y ) = gs ( fs ( x) + fs ( y )) where: • fs is a continuous, strictly increasing function, • f s : [− 1,1] → [− ∞,+∞ ] , f s (0 ) = 0 , f s → ±∞ x → ±1

2. balanced t-conorm

• gs denotes the inverse of fs . Note: the following mappings are extensions of the additive generators used in the example in Section 2.1:

f t : [− 1,1] → [− ∞,+∞]

 (1 − x) / x f t (x ) =   undefined

f s : [− 1,1] → [− ∞,+∞]

x ≠ 0  x = 0

for

f s (x ) = x / (1 − x )

The contour plots of the above balanced t-norm and balanced t-conorm are shown in Figures 8 and 9. Note that according to the symmetry principle and the balanced extension method the right-upper quarter of the plots are equal to the plots of t-norm and t-conorm.

Y constant value 0

Y 0.9

0.6 0.3

0.6 0.3

X

X

-0.3

-0.3

constant value 0

-0.6

-0.6

-0.9

Fig. 8 The plot of balanced t-norm

Fig. 9 The plot of balanced t-conorm

The infinite sequence of information converge to the following limits:

 1  →  0 lim S (S (....S (a, a )....), a ) n →+∞  

− 1 n times   1  lim T (T (....T (a, a )....), a ) n →  0 →+∞  

− 1 n times 

if if

0 < a ≤1 a=0

if

−1 ≤ a < 0

if

a =1

if

−1< a 0 x + y < 0  x, y ∈ [− 1,1]× [− 1,1]  0 x = = y  otherwise

By analogy, the balanced t-norms min could be defined by the limit formula:

T _ min ( x, y ) = lim Tn (x, y ) n →∞

where

for

x, y ∈ [− 1,1]× [− 1,1]

Tn are balanced t-norms generated by additive generators dual to f n . Finally

min( x, y )  T _ min( x, y ) =  N (min(( N ( x), N ( y ))) 0 

for

 x ≥ 0, y ≥ 0   x ≤ 0, y ≤ 0 otherwise 

14

W. Homenda and W. Pedrycz

4.3

Balanced t-conorms vs. uninorms

Aggregation features of balanced t-conorms and uninorms are similar, i.e. both type of operators strengthen non-crisp information. On the other hand, balanced t-norm weakens non-crisp information – what is consistent with nullnorm behavior. Neutral information is neither distinguished by uninorm, nor by nullnorm, nor by balanced triangular norms. The balanced t-conorms, as defined in Section 4.1 are special cases of uninorms in the sense of the isomorphism defined at the end Section 4.1. Amazingly, balanced triangular norms as well as uninorms and nullnorms are similar products of two different viewpoints on fuzzy sets extensions. Detailed properties of balanced triangular norms and uninorms and nullnorms might differ. Despite of this, the general meaning of balanced triangular norms and of uninorms and nullnorms are the same in the sense of isomorphic mapping between them. The definition of balanced t-conorm includes the symmetry axiom in addition to other axioms that are common for uninorm and balanced t-conorm: associativity, commutativity, monotonicity and boundary conditions. The extra restriction – the symmetry axiom - means that not every uninorm is isomorphic with a balanced t-conorm while every t-conorm is isomorphic with some uninorm. Precisely, every balanced t-conorm is isomorphic with a set of uninorms that satisfy the symmetry axiom and differ in the unit elements. Of course, any two uninorms of such a set are isomorphic in the sense of an isomorphism analogous to the formula presented at the end of Section 4.1. Two sets of uninorms related to any two balanced t-conorms are disjoint assuming that respective balanced t-conorms are different. Moreover, the set of uninorms that are not isomorphic with any balanced t-conorm and the sets of uninorms related to balanced t-conorms partition the set of all uninorms, i.e., they create equivalence classes of an equivalence relation.

The same concerns observations the balanced t-norms and nullnorms. We have: u : u is a uninorm . Let us consider isomorphic mapProposition 4.3.1. Let U

^

`

pings as defined at the end of Section 4.1. Then, the pair

U , | S is an equivalence rela-

u and v , u | S v iff u and v are isomorphic with the same balanced t-conorm S or none of u and v is isomorphic with any balanced t-conorm

tion if for every two uninorms

S. 1 between min and max

squizzed t-conorm

squizzed t-norm

between min and max

unconstrained values e

-e

-1

0

between min and max

squizzed t-conorm

squizzed t-norm

between min and max

-e

e

1

unconstrained value

-1

Fig. 10 The structure of balanced uninorms

Symmetrization Of Fuzzy Operators: Notes On Data Aggregation

15

Proposition 4.3.2. The analogous proposition holds for the families of nullnorms and balanced t-norms. These propositions describe the characteristic of the set of all t-conorms (t-norms, respectively) as a family of equivalence classes of the relation | S ( |T , respectively) defined on the set of all uninorms (nullnorms, respectively).

4.4

Balanced uninorms

Neither balanced triangular norms, nor uninorms, nor nullnorms satisfy the preservation rule, cf. Figures 1 and 2. Applying the method of balanced extension to uninorms leads to an intuitively appealing definition of balanced uninorm defined as follows: Definition: Balanced uninorm is a mapping U :  1,1 u  1,1 o  1,1 , satisfying the axioms:

>

@ >

@ >

@

1.2.3.5. U is associative, commutative, monotone and symmetrical, 4. identity element: there exists an identity element e  [0,1] such that for all x  [0,1] U ( x, e) x The structure of balanced uninorm is displayed in Figure 10. As in case of balanced triangular norms, the values of balanced uninorms on the squares 0,1 u 0,1 and

> @ > @

> 1,0@u > 1,0@ are determined by the values of uninorm and symmetry principle. The values of balanced uninorm on the squares > 1,0@ u >0,1@ and >0,1@ u > 1,0@ are unconstrained and could be defined according to subjective aim of application (in fact, values in these regions are determined by monotonicity axiom). This matter, as a subject of detailed discussion, is out of the general considerations of this paper and is not investigated here. Assuming that balanced uninorm takes value 0 on the areas  1,0 u 0,1 and

>

> @ >

@ > @

@

0,1 u  1,0 , it makes it possible to get a continuous balanced uninorm on the whole domain (with separated points of non-continuity). So, continuity of balanced uninorms, important from a practical point of view, is quarantined if uninorm is continuous and takes value 0 on the border of the “between min and max” area and on the unconstrained area of balanced uninorm. Obviously, similar considerations are valid in case of nullnorms, though the values of balanced nullnorms on the unconstrained area meet different type of border conditions. Finally, balanced extension of uninorms brings an expected property: it strengthens negative -1

neutral -e

0

positive e

1

U-norm

Fig. 11. Satisfaction of the preservation rule for balanced uninorms

16

W. Homenda and W. Pedrycz

strong information (strong positive and strong negative information), it weakens weak information (weak positive and weak negative information) and retains neutrality of neutral information, cf. Figure 11. The convergence experiment gives the following results for a given balanced uninorm (assuming that original uninorm is composed of t-norm and t-conorm both being strong):

­ 1 ° 0 ° lim U U ....U a, a .... , a n o ® of  ° 1 n times °¯r e

5

if

e  a d1

if if

eae 1d a  e

if

a

re

CONCLUSIONS

An intuitively appealing aggregation rule – the preservation of strength of information - is investigated for different aggregation operators. The lack of satisfaction of the preservation rule is shown for classical aggregation operators such as triangular norms and uninorms and nullnorms. To alleviate identified difficulties, a family of balanced uninorms has been defined. It is revealed that balanced uninorms conform to the general and intuitively acceptable aggregation mechanism. Further discussion on the aggregation of mixed types of information, repetitive application of the symmetry principle and the balanced extension method to balanced operators, continuity of balanced operators is worth pursuing.

ACKNOWLEDGMENT Support from Natural Sciences and Engineering Research Council (NSERC), Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland, and Alberta Software Engineering Research Consortium (ASERC) is gratefully acknowledged.

REFERENCES 1. 2. 3. 4. 5. 6.

K. T. Atanassov, Intuitionistic Fuzzy Sets, Fuzzy Sets & Systems, 20 (1986) 87-96. J.C. Bezdek, Computing with uncertainty, IEEE Communication Magazine, Sept. 1992, pp. 24-36. B. Buchanan, and E. Shortliffe, Ruled Based Expert Systems, The MYCIN Experiment of the Stanford Heuristic Programming Project, Addison-Wesley, Reading, MA, 1984. T. Calvo at al., The functional equations of Frank and Alsina for uninorms and nullnorms, Fuzzy Sets & Systems, 120 (2001) 385-394. T. Calvo et al., Aggregation Operators: Properties, Classes and Construction Methods, in: T. Calvo, G. Mayor, R. Mesiar (Eds.), Aggregation Operators, pp. 3 – 104, Physica Verlag, Heidelberg, 2002. M. Detyniecki, Mathematical Aggregation Operators and their Application to Video Querying, PhD Thesis, l'UNIVERSITE PARIS VI, Paris, 2000.

Symmetrization Of Fuzzy Operators: Notes On Data Aggregation

7. 8.

9. 10. 11. 12. 13
14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

17

J. Fodor, R. R. Yager, A. Rybalow, Structure of uninorms, Internat. J. Uncertain Fuzziness Knowledge-Based Systems, 5 (1997) 411 –427. M. Grabisch et al., On Symmetric Pseudo-Additions and Pseudo-Multiplications: Is it possible to Built Rings on [-1,+1]?, The 9th Internat. Conf. on Inf. Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2002, 1-5 July, Annecy, France. M.M. Gupta, D.H. Rao, On the principles of Fuzzy Neural Networks, Fuzzy Sets and Systems, 61 (1994) 1-18. W. Homenda, Fuzzy Relational Equations with Cumulative Composition Operator as a Form of Fuzzy Reasoning, Proc. of the International Fuzzy Engineering Symposium'91, Yokohama, November 1991, 277-285. W. Homenda, Algebraic operators: an alternative approach to fuzzy sets, Appl. Math. and Comp. Sci., 1996, vol. 6, No. 3, 505-527. W. Homenda, Balanced Fuzzy Sets, Warsaw University of Technology, Preprint, September 2001. W. Homenda Triangular norms, uni- and nullnorms, balanced norms:the cases of the hierarchy of iterative operators, 24th Linz Seminar on Fuzzy Set Theory, Linz, February 4-8, 2003 W. Homenda, W. Pedrycz, Processing of uncertain information in linear space of fuzzy sets, Fuzzy Sets & Systems, 44 (1991) 187-198.. W. Homenda, W. Pedrycz, Fuzzy neuron modelling based on algebraic approach to fuzzy sets, Proc. of the SPIE's Intern. Symp., April 1995, Orlando, Florida. W. Homenda, W. Pedrycz, Symmetrization of fuzzy operators: notes on data aggregation, Proc. of the Internat. Conf. on Fuzzy Systems and Knowledge Discovery, Singapore, November 18-22, 2002 E.P. Klement, R. Mesiar, E. Pap, A characterization of the ordering of continuous tnorms, Fuzzy Sets and Systems 86 (1997) 189–195. E. P. Klement, R. Mesiar, and E. Pap, Triangular norms, Kluwer Academic Publishers, Dordrecht, 2000. B. Kosko, Fuzzy cognitive maps, Int. J. Man-Machine Studies, 24 (Jan. 1986). 65-75. Mesiar R. and Komorníková M., Aggregation Operators, Proceeding of the XI Conf. on Applied Mathematics PRIM' 96, Herceg D., Surla K. (Eds.), Institute of Mathematics, Novi Sad, 193-211, 1997. B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North Holland, New York, 1983. W. Silvert, Symmetric Summation: A Class of Operations on Fuzzy Sets, IEEE Trans. System, Man, Cybernetics, 9 (1979) 659-667 R. R. Yager, On Ordered Weighted Averaging Aggregation Operators, IEEE Trans. System, Man, Cybernetics, 18 (1988) 183-190. R.R. Yager, Families of OWA operators, Fuzzy Sets and Systems 59 (1993) 125–148. R.R. Yager, D. Filev, Induced ordered weighted averaging operators, IEEE Trans. Sys. Man and Cybernetics – Part B: Cybernetics 29 (1999) 141–150. R. R. Yager, A. Rybalow, Uninorm aggregation operators, Fuzzy Sets & Systems, 80 (1996) 111-120. L. A. Zadeh, Fuzzy Sets, Inform. And Control, 8 (1965) 338-353. W.R. Zhang et al., Pool2: A Generic System for Cognitive Map Development and Decision Analysis, IEEE Trans. Systems, Man Cybernetics, 19 (Jan.-Feb. 1989) 31-39.

CHAPTER 2 An Integrity Estimation Using Fuzzy Logic P. Mariño, C. A. Sigüenza, V. Pastoriza, M. Santamaría, E. Martínez, F. Machado Department of Electronic Technology, University of Vigo, 36280 Vigo, Spain Email: {pmarino, csiguenza, vpastoriza, msanta, emtnez, fmachado}@uvigo.es

Abstract: The authors have been involved in developing an automated inspection system, based on machine vision, to assess the seaming quality in metal containers (cans) for fish food. In this work we present a fuzzy model building to make the pass/fail decision for each can, and predict the closing machine adjustment state after closing each can, from the information obtained by the vision system. In addition, it is interesting to note that such models could be interpreted and supplemented by process operators. In order to achieve such aims, we use a fuzzy model due to its ability to favor the interpretability for many applications. Firstly, the can seaming process, and the current, conventional method for quality control of can seaming, are described. Then, we show the modeling methodology, that includes the generation of representative input-output data sets, and the fuzzy modeling. After that, results obtained and their discussion are presented. Finally, concluding remarks are stated.

1 Introduction Due to the demand for high-quality products, quality assurance has a major concern in all manufacturing environments, including the food industry. The growing level of production involves new challenges to seam inspection methods. Nowadays, to guarantee the desired lifespan for the target product in the food cannery sector, a manual and destructive inspection of the seam of containers is carried out. This inspection method is based on a statistical supervisory control. The worst features of this method are its inspection rate (one can each fifteen minutes) and its slowness (one check can take up to ten minutes). As an example; if an average closing machine rate is three hundred cans per minute (some closing machine can reach up to six hundred cans per minute) then only one of each 4500 cans is checked. Moreover, if one can is found defective, as the closing machine continues closing, all seamed cans after that (3000 cans) must be retired to be analyzed. Therefore, it is important to develop an automated inspection system to improve the seaming quality control process (all cans are checked, and in line). It is for this reason that we have been involved in the design and implementation of an in-line, automated machine vision system to evaluate the seaming quality of all cans. Because the quality of the can seaming depends on external and internal dimensional attributes, the current, manual inspection is also destructive when measuring the internal features (Sect. 2). However, the machine vision system, which uses CCD cameras, will be able P. Mari˜ no et al.: An Integrity Estimation Using Fuzzy Logic, Studies in Computational Intelligence (SCI) 2, 19–33 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


20

P. Mari˜ no et al.

Fig. 1. Can pieces

Fig. 2. Can shapes and their CRSC inspected points

only to measure the external dimensional attributes. The first stage will be to find a model that estimates the can seaming quality only from its external dimensions. Besides, it will be profitable to evaluate the closing machine adjustment (CMA) state after closing each can, from the same external features, with the purpose of allowing a fast maintenance of it. In addition, it will be important that such models can be interpreted and supplemented by process operators. In this work, we explore the use of fuzzy models to solve these difficulties: to make the pass/fail decision for each can and assess the CMA state after closing each can based on external dimensional attributes. Takagi-Sugeno-Kang fuzzy models are developed using a neuro-fuzzy modeling technique (Sect. 3). The remainder of this document is organized as follows: in the next section we provide an overview of the can seaming process. Then, Section 3 describes the proposed modeling methodology. The results obtained and their discussion are presented in Section 4. Finally, we state concluding remarks in Section 5.

2 Background 2.1 The can seaming process Closing of a three piece steel food can has been well supported by researchers. In 1900 the former soldering method was changed by a new double seam processing, able to be made from increasingly faster and sanitary safer closing machines. A can comprises: a lid and a body, as depicted in Figure 1. Cans can be of different shapes: round, oblong, rectangular (see Fig. 2), and have various sizes as well. In the can seaming process, firstly, a lid is mounted on the can body filled with ingredient. Then the body and lid are held between the chuck and the lifter of machine, known as a closing machine (see Fig. 3a), and after that the assembly is kept rotating while the lid is

An Integrity Estimation Using Fuzzy Logic

21

pressed against an element of the closing machine named the seaming roll. There are two types of seaming roll (double seaming mechanism): 1st roll and 2nd roll. The first roll approaches the can lid, and rolls up the lid curl and body flange sections of the can before retreating (see Fig. 3b). Next, the second roll approaches to compress the rolled-up sections st to end the seaming (see Fig. 3c). In other words, the 1 roll rolls up the can lid and can nd body, doing mainly the bending work, while the 2 roll compresses the rolled-up sections, and mainly does the seaming work.

a)

b)

c)

Fig. 3. Outline of can seaming process: a) Closing machine holding the lid over the can body, b) First operation and, c) second operation of seaming process The can seam obtained, the double seam, is depicted in Figure 4. It consists of five thicknesses of plate interlocked or folded and pressed firmly together, plus a thin layer of sealing compound. The seaming integrity is achieved by a combination of the following two elements: x The sound mechanical interlocking of the can body and can end. x The presence of sealing compound, which effectively blocks potential leakage paths.

2.2 Current can seaming quality control There are international and national entities involved in the regulation of quality control procedures for fish cannery industries. Some of them, like FDA [7] in USA, DFO [2] in Canada, SEFEL [9] in EU, and SOIVRE [8] in Spain, use a similar definition for double seam dimensional attributes, (see Fig. 4) and integrity factors. These entities have defined a quality control based on a conventional routine statistical control (CRSC) used by the canneries. The seaming control in CRSC method is destructive, and based on seaming integrity. This integrity is assessed at several points around the seam; their number and location depends on the can shape (see Fig. 2).

22

P. Mari˜ no et al.

Fig. 4. View of a double seam section and its dimensional attributes The parameters used to estimate the seaming integrity at a point of the seam are: Compactness Rating (CR), Overlap Rating (OR), and Body Hook Butting (BHB). These are named integrity factors and are computed from the double seam dimensional attributes at said point (see Fig. 4), using the following ratios: %CR = 100·(3 EF + 2 EC)/ES,

(1)

%OR = 100·a/c

(2)

%BHB = 100·b/c

(3)

The integrity in a point is acceptable if the three integrity factors in that point are inside their respective acceptation interval (Table 1). A can is only passed if the integrity is acceptable in all its inspected points.

3 Modeling methodology As mentioned in the introduction, it is important to improve the current seaming quality control process. For this reason we have been involved in the design and implementation of an in-line, automated machine vision system to evaluate the seaming quality of all cans. The can seaming quality is assessed from the integrity factors: CR, OR, and BHB (Sect. 2.2). As has been seen in Eqs. (1, 2, and 3), they depend on several external and internal Table 1. Integrity factors for round cans: acceptation range Integrity factor %CR %OR %BHB

Range [75, 95] [45, 90] [70, 95]

An Integrity Estimation Using Fuzzy Logic

23

dimensional attributes (Fig. 4). However, the machine vision system, based on CCD cameras, will be able only to measure the external dimensional attributes: ES, HS and PC. The first stage will be to verify that is possible to find a model that estimates the can seaming quality only from its external dimensions. And so it will be profitable to evaluate the closing machine adjustment (CMA) state after closing each can, from the same external features, with the purpose of allowing a fast maintenance of it. In addition, it will be important that such models can be interpreted and supplemented by process operators (linemen in this case), i.e., such models may provide enough readability of both process (can seaming and CMA) so that their interpretability can be compared with judgments made by an skilled lineman. In order to achieve such aims, we explore the use of fuzzy models due to the ability of these systems to transform human knowledge or experience in terms of linguistic rules, where each rule can easily capture the sense of a rule of thumb used by humans. For purposes aforementioned, four fuzzy inference systems were developed, one to make the pass/fail (P/F) decision for each can, and three to estimate three CMA parameters that affect the can seaming process. All these fuzzy models are limited to use of ES, HS and PC as inputs.

3.1 Data sets Representative input-output data sets of the system should be the first step to develop the different fuzzy models. We have planned a design of experiment (DOE) [1, 11] to generate these data sets. To create the DOE, we had to find out which of the main closing machine adjustments (CMA) affect the can seaming process. It was necessary to perform successive closing experiments (trial and error) using the CRSC method, finding that the main CMA that affect a can seaming process are: plate pressure (CMA1), and first and second roll maximum approach (CMA2 and CMA3). These three CMA are depicted in Figure 3. Then, the DOE assumes these three variables as experiment factors; and where each one can have one of the following categorical values: loose, normal, or tight. A full factorial design was chosen and, as the DOE considers three controllable variables with three possible values for each one, then 33=27 regions of closing machine working universe appeared. In the light of this, these CMA parameters were employed as the indicators of CMA status. We have only worked with a specific round container format named RO-85. This format was chosen because it is the most popular and this type of can imposes the most serious speed restrictions (it is the fastest seaming cadence format). RO-85 cans have been closed in a way that nine cans belonging to each region of closing machine working were gathered and then have been checked using the CRSC method to obtain an input-output data set. As RO-85 cans are round, then this method, based on destructive analysis, checks the seaming integrity at four points around the seaming perimeter (see Fig. 2). Then the final size of the input-output data set is 27 regions of closing machine working x 9 cans per region x 4 measurement points per can = 972 measurement points. Table 2 shows the list of all measured or computed parameters at each one of these points, i.e., the characteristic vector of I/O data set. See Figure 4 as well. Finally, this I/O data set was divided into two subsets, where both are representative of the closing machine working universe. One set – training data – is used to train the models, and the other – checking data – to check. Part of these are shown in Table A.1 of the Appendix.

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Table 2. Characteristic vector of I/O data set Parameter name Body wall thickness End component thickness Seam thickness Seam length Countersink depth Body hook length End hook length Overlap Internal body hook length Internal seam length Compactness rating Overlap rating Body hook butting Pass/Fail decision

Abbreviation EC EF ES HS PC GC GF a b c CR OR BHB P/F

3.2 Fuzzy Modeling Fuzzy modeling is the task of identifying the parameters of a fuzzy inference system so that a desired behavior is attained, while neuro-fuzzy modeling refers to the way of applying various learning techniques developed in the neural network literature to fuzzy inference systems [4]. The fuzzy inference system (FIS) [5, 6, 13] is a rule-based system that uses fuzzy logic to reason about data. FIS is also known as fuzzy-rule-based system, fuzzy expert system, fuzzy model, fuzzy associative memory, fuzzy logic controller, and simply fuzzy system. The basic structure of a FIS consists of four conceptual components, as illustrated in Figure 5: x A fuzzifier, which translate crisp inputs into fuzzy values. x An inference engine that applies a fuzzy reasoning mechanism to obtain a fuzzy output. x A defuzzifier, which transforms the fuzzy output into a crisp value.

Fig. 5. Fuzzy Inference System (FIS)

An Integrity Estimation Using Fuzzy Logic

25

x A knowledge base which contains a rule base containing a number of fuzzy if-then rules, and a database which defines the membership functions (MF) of the fuzzy sets used in the fuzzy rules. To solve our problem, we have chosen Takagi-Sugeno-Kang (TSK) fuzzy models [10, 12] since they are suited for modeling nonlinear systems by interpolating multiple linear models. A typical fuzzy rule in a TSK fuzzy model has the form

if x is A and y is B then z=f(x,y) where A and B are fuzzy sets in the antecedent or premise, while z = f(x,y) is a crisp in the consequent or conclusion. For the purpose of interpretability we will use only TSK models where f(x,y) is a constant or a linear combination of input variables plus a constant term. These resulting FIS are respectively named zero-order or first order TSK fuzzy models. As stated before, it is necessary to create four TSK fuzzy models, one to make the P/F decision, and three to assess each one of the three CMA parameters used to estimate the CMA status. Initially, in all models, the fuzzy rules, and the fuzzy values for inputs and outputs were defined by the system designers based on linemen’s judgment and statistical analysis of input-output data sets. In all models, the chosen method of defuzzification was weighted average, and all rules were given equal weights. After that initiation, the adaptive neuro-fuzzy inference system (ANFIS) [3, 4], a well known neuro-fuzzy modeling technique, was used to optimize the fuzzy models. ANFIS was employed using a hybrid algorithm to minimize the sum of square residuals. More specifically, in the forward pass of the hybrid learning algorithm, the consequent parameters are identified by the least-squared method; while in the backward pass, the antecedent parameters are updated by gradient descent. With the aim of providing enough readability of can seaming and CMA processes, we must take the accuracy-interpretability trade-off into account to choose the best fuzzy models. A better interpretability is usually attained at the cost of accuracy and vice versa. In the light of this criterion we pick out as best model the one that provides enough accuracy while incurring as little a loss of interpretability as possible.

4 Results and discussion This section describes the results obtained when applying the methodology exposed in the former section.

4.1 Model to make the pass/fail decision for each can The first model (P/F model) emulates the CRSC method to evaluate the seaming integrity for each can inspection point and make the P/F decision. The best P/F model found uses ES, HS, and PC as inputs, P/F decision as output, and 15 rules to define relationships between inputs and output (Table 3). This P/F model is a zero order TSK fuzzy model (singleton values for each consequent). The membership functions for each input feature are shown in Figure 6 and the singleton value for each consequent in Table 3.

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Fig. 6. Membership functions for P/F model (dashed and solid alternating lines are used to assist readability)

Table 3. Rules for P/F model Rules If ES is ES1 & HS is HS1 If ES is ES1 & HS is HS2 If ES is ES1 & HS is HS3 & PC is PC1 If ES is ES1 & HS is HS3 & PC is PC2 If ES is ES1 & HS is HS3 & PC is PC3 If ES is ES1 & HS is HS4 If ES is ES1 & HS is HS5 If ES is ES2 & HS is HS1 & PC is PC1 If ES is ES2 & HS is HS1 & PC is PC2 If ES is ES2 & HS is HS1 & PC is PC3 If ES is ES2 & HS is HS2 If ES is ES2 & HS is HS3 If ES is ES2 & HS is HS4 If ES is ES2 & HS is HS5 If ES is ES3 * S[m] = Singleton [mean].

Pass/Fail decision * S1[0], Fail S2[0], Fail S3[1], Pass S4[0], Fail S5[0], Fail S6[1], Pass S7[0], Fail S8[0], Fail S9[0], Fail S10[2], Pass S11[0], Fail S12[0], Fail S13[0], Fail S14[0], Fail S15[0], Fail

An Integrity Estimation Using Fuzzy Logic

27

4.2 Model to assess the CMA state In Section 3.1 we have established that the main CMA that affect the can seaming process were: plate pressure, and first and second roll maximum approach (see Fig. 3). Three TSK models were obtained, where each one takes charge of the estimation of a CMA after closing each can. We have assigned a numeric value between 0 and 1 to the output in each one of the models (0 is loose, 0.5 is normal, and 1 is tight). One of the fuzzy models (2R model) estimates the second roll maximum approach (CMA3). The best 2R model obtained uses only ES, and HS as inputs, CMA3 as output, and only 4 rules (Table 4). This 2R model is also a zero order TSK fuzzy model. The membership functions for each input are shown in Figure 7 and the singleton value for each consequent in Table 4.

Fig. 7. Membership functions for 2R model

Table 4. Rules for 2R model Rules If ES is ES1 If ES is ES2 & HS is HS1 If ES is ES2 & HS is HS2 If ES is ES3 * S[m] = Singleton [mean].

2nd roll maximum approach (CMA3) * S1[1], Tight S2[0.5], Normal S3[1], Tight S4[0], Loose

Another fuzzy model (P model) evaluates the plate pressure (CMA1). The best P model attained employs ES, HS, and PC as inputs, CMA1 as output, and a rule base of 33 rules (Table 5). This P model is a first order TSK fuzzy model (first-order polynomial in the input variables is assigned to each consequent).The membership functions for each input are shown in Figure 8, and the first-order polynomial for each consequent in Table 5.

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Fig. 8. Membership functions for: P model (solid lines) and 1R model (dashed lines)

Table 5. Rules for P model Rules for both models If ES is ES6 & HS is HS2 & PC is PC2 If ES is ES6 & HS is HS1 & PC is PC4 If ES is ES5 & HS is HS2 & PC is PC1 If ES is ES5 & HS is HS2 & PC is PC2 If ES is ES5 & HS is HS2 & PC is PC3 If ES is ES3 & HS is HS3 & PC is PC3 If ES is ES3 & HS is HS3 & PC is PC4 If ES is ES2 & HS is HS3 & PC is PC4 If ES is ES2 & HS is HS2 & PC is PC3 If ES is ES2 & HS is HS4 & PC is PC4 If ES is ES1 & HS is HS3 & PC is PC5 If ES is ES5 & HS is HS1 & PC is PC4 If ES is ES4 & HS is HS3 & PC is PC1 If ES is ES4 & HS is HS3 & PC is PC2 If ES is ES5 & HS is HS2 & PC is PC4 If ES is ES4 & HS is HS2 & PC is PC4 If ES is ES4 & HS is HS2 & PC is PC5 If ES is ES3 & HS is HS4 & PC is PC1 If ES is ES3 & HS is HS4 & PC is PC3 If ES is ES1 & HS is HS3 & PC is PC4 If ES is ES1 & HS is HS4 & PC is PC4 If ES is ES1 & HS is HS4 & PC is PC5 If ES is ES5 & HS is HS1 & PC is PC1 If ES is ES6 & HS is HS1 & PC is PC1 If ES is ES5 & HS is HS3 & PC is PC1 If ES is ES5 & HS is HS3 & PC is PC2 If ES is ES2 & HS is HS4 & PC is PC2 If ES is ES4 & HS is HS4 & PC is PC2 If ES is ES1 & HS is HS4 & PC is PC1 If ES is ES1 & HS is HS4 & PC is PC2 If ES is ES1 & HS is HS4 & PC is PC3 If ES is ES1 & HS is HS5 & PC is PC2 If ES is ES2 & HS is HS5 & PC is PC2 *

Plate pressure (CMA1) P1[-40.57 -1.836 23.37 -22.32] P2[1.55 -0.5334 -0.1888 -0.2882] P3[-17.58 -3.411 -4.116 45.72] P4[ 4.731 -0.07885 -6.375 16.44] P5[-4.716 8.918 -15.38 39.68] P6[21.45 -22.95 4.192 18.62] P7[6.962 -1.926 -16.52 57.28] P8[8.465 3.021 2.711 -26.84] P9[4.646 6.886 10.2 -58.82] P10[18.47 -11.01 -3.803 21.64] P11[-12.04 3.601 -9.694 39.74] P12[-0.02447 -0.1794 0.05579 0.7417] P13[-5.545 18.15 -16.07 17.56] P14[7.413 -3.508 2.881 -9.964] P15[20.89 1.723 3.897 -45.13] P16[0.4597 -0.1405 -0.6883 2.916] P17[-1.192 -2.774 -2.166 16.78] P18[26.61 -31.65 60.26 -156.7] P19[-21.55 5.447 13.76 -38.52] P20[29.64 -3.345 -8.234 8.814] P21[16.4 4.208 -3.928 -12.11] P22[11.54 -11.02 -6.974 43.28] P23[-3.733 -0.8367 2.96 -2.399] P24[-2.565 2.181 0.977 -3.844] P25[4.538 -2.36 1.394 -3.807] P26[-12.88 30.37 -27.89 39.34] P27[-48.08 -0.4343 -8.669 84.85] P28[-20.66 -13.69 -2.758 70.92] P29[-6.177 -1.282 -0.7175 13.21] P30[-42.45 5.764 -0.5232 30.18] P31[38.46 1.49 -9.318 -9.38] P32[15.86 -0.4132 -1.459 -8.67] P33[12.35 1.335 1.718 -22.15]

*

P[p, q, r, s] = first-order polynomial (p·ES + q·HS + r·PC + s).

An Integrity Estimation Using Fuzzy Logic

29

Finally, a third fuzzy model (1R model) assesses the first roll maximum approach (CMA2). The best 1R model found has ES, HS, and PC as inputs, CMA2 as output, and 33 rules (Table 6). This 1R model also has the same antecedents than the P model, as can be appreciated in Tables 5 and 6. This 1R model is also a first order TSK fuzzy model. The membership functions for each input are also depicted in Figure 8 and the first-order polynomial for each consequent in Table 6.

Table 6. Rules for 1R model Rules for both models 1st roll maximum approach (CMA2) * If ES is ES6 & HS is HS2 & PC is PC2 P1[1.531 -27.44 10.93 26.76] If ES is ES6 & HS is HS1 & PC is PC4 P2[-0.2884 1.603 0.9977 -5.984] If ES is ES5 & HS is HS2 & PC is PC1 P3[0.07629 -8.144 33.46 -95.05] If ES is ES5 & HS is HS2 & PC is PC2 P4[-15.44 -11.31 25.06 -40.04] If ES is ES5 & HS is HS2 & PC is PC3 P5[-0.4916 5.865 8.673 -45.09] If ES is ES3 & HS is HS3 & PC is PC3 P6[-7.731 -7.889 -5.042 48.11] If ES is ES3 & HS is HS3 & PC is PC4 P7[-33.61 14.57 40.38 -145.3] If ES is ES2 & HS is HS3 & PC is PC4 P8[-9.041 -24.38 1.943 64.64] If ES is ES2 & HS is HS2 & PC is PC3 P9[2.18 6.656 3.083 -29.41] If ES is ES2 & HS is HS4 & PC is PC4 P10[-40.17 11.66 39.39 -129.2] If ES is ES1 & HS is HS3 & PC is PC5 P11[3.8 -12.47 -3.264 40.26] If ES is ES5 & HS is HS1 & PC is PC4 P12[0.5514 1.098 0.7131 -4.875] If ES is ES4 & HS is HS3 & PC is PC1 P13[-11.41 38.62 -13.77 -34.17] If ES is ES4 & HS is HS3 & PC is PC2 P14[-2.659 -1.996 15.22 -44.72] If ES is ES5 & HS is HS2 & PC is PC4 P15[-19.45 -2.682 -36.69 164.6] If ES is ES4 & HS is HS2 & PC is PC4 P16[1.869 -0.1834 1.518 -6.282] If ES is ES4 & HS is HS2 & PC is PC5 P17[2.555 0.7514 0.916 -7.238] If ES is ES3 & HS is HS4 & PC is PC1 P18[-27.04 52.31 -66.68 127.1] If ES is ES3 & HS is HS4 & PC is PC3 P19[2.37 -0.07172 1.95 -9.561] If ES is ES1 & HS is HS3 & PC is PC4 P20[13.44 9.937 6.871 -63.13] If ES is ES1 & HS is HS4 & PC is PC4 P21[-45.1 -32.9 6.045 108.4] If ES is ES1 & HS is HS4 & PC is PC5 P22[47.38 -144.1 94.41 -30.84] If ES is ES5 & HS is HS1 & PC is PC1 P23[1.224 -0.03894 -0.9869 2.848] If ES is ES6 & HS is HS1 & PC is PC1 P24[1.41 -0.4232 -2.104 7.308] If ES is ES5 & HS is HS3 & PC is PC1 P25[4.22 -5.216 -6.709 31.2] If ES is ES5 & HS is HS3 & PC is PC2 P26[83.59 -32.27 -21.88 48.6] If ES is ES2 & HS is HS4 & PC is PC2 P27[2.148 -15.74 -0.5319 41.17] If ES is ES4 & HS is HS4 & PC is PC2 P28[-22.68 -9.386 12.58 6.64] If ES is ES1 & HS is HS4 & PC is PC1 P29[1.93 1.177 1.4 -8.861] If ES is ES1 & HS is HS4 & PC is PC2 P30[-6.858 5.18 16.21 -62.16] If ES is ES1 & HS is HS4 & PC is PC3 P31[18.76 -20.83 22.57 -46.07] If ES is ES1 & HS is HS5 & PC is PC2 P32[-26.98 -19.16 -1.659 85.72] If ES is ES2 & HS is HS5 & PC is PC2 P33[-12.89 2.659 -1.665 12.65] * P[p, q, r, s] = first-order polynomial (p·ES + q·HS + r·PC + s).

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4.3 Discussion In order to assess the prediction accuracy of the models, the root-mean-square error (RMSE) of training data and checking data sets is calculated for each model. The RMSE of each model is shown in Table 7. The results show the excellence of all fuzzy models regarding accuracy. Besides, all models had similar RMSE values for the checking data set relative to the training data set, which indicates that there is no overfitting.

Table 7. *RMSE for each model P/F model Training set 0.0291 Checking set 0.0294 * RMSE (root-mean-square error).

P model 0.0270 0.0292

1R model 0.0595 0.0768

2R model 0.0060 0.0049

Regarding the interpretability of the models, P/F and 2R models are simple models with few rules. Besides, both are zero order TSK fuzzy models (the consequent of each rule is a constant). From the rules of the P/F model (Table 3) it is deduced that the can pass decision is made when one of the three cases (rules: 3, 6, or 10) happens. From the rules of 2R model (Table 4) it is deduced that CMA3 is tight in two cases (rules 1 or 3), normal in one case (rule 2), and loose in the other case (rule 4). However, P and 1R models are more complex. The number of membership functions for each input (Fig. 8), and number of rules (Tables 5 and 6) are higher in these models. They are first-order TSK models (the consequent of each rule is computed from a first-order polynomial in the inputs). Both models have the same antecedents, and the membership functions for the inputs are very much alike. However, they have different polynomial output in the same rule. Due to the complexness of both models, the interpretation of their rules is not evident. As the performance of each model must be understood as a balance between accuracy and interpretability, then P/F and 2R models provide excellent performance, while P and 1R models provide a reduced performance, because of low interpretability.

5 Concluding remarks We have been involved in the implementation of a machine vision system to improve the quality control of the can seaming process. As mentioned in Section 2, the quality of this process depends on external and internal dimensional attributes. However, the machine vision system, which uses CCD cameras, will be able only to measure the external dimensional attributes. Thus the first stage will be to find a model that estimates the can seaming quality only from its external dimensions. Besides, it will be profitable to evaluate the CMA state after closing each can, from the same external features, with the purpose of allowing a fast maintenance of it. In addition, it will be necessary that such models can be interpreted and supplemented by process operators.

31

An Integrity Estimation Using Fuzzy Logic

In spite of the fact that the seaming process of only one can format (RO-85) has been studied, as the can seaming mechanism (double seam) is common to all formats, it is reasonable to think that fuzzy models like the P/F model can be obtained to make the P/F decision for another can format. As the three studied CMA are what affect mainly the can seaming process for any type of closing machine, then it is also reasonable to think that fuzzy models like P, 1R, and 2R models can be found to assess the CMA for any closing machine. All this leads to the conclusion that is possible to design an in-line, automated machine vision system, that only extracts the external dimensional attributes of the seamed cans, makes the P/F decision for each seamed can, and the evaluation of the CMA status after closing each can. ANFIS, the neuro-fuzzy modeling technique used to optimize the fuzzy models, provided an excellent prediction accuracy to all models. However the interpretability of P and 1R models were poor. For this reason, in the future we will study the application of other fuzzy modeling techniques that improve the interpretability without a significant loss of accuracy.

Appendix: Data sets Table A.1. Part of I/O data set gathered EC

EF

0.178 0.180 0.184 0.178 0.181 0.167 0.171 0.181

0.192 0.224 0.201 0.192 0.213 0.210 0.205 0.213

ES 1.50 1.39 1.05 1.42 1.16 1.05 1.13 1.11

HS 2.66 2.68 2.45 2.67 2.93 2.96 2.81 2.73

PC 3.57 3.57 4.13 3.54 3.52 3.63 3.56 3.58

GC 1.72 1.75 1.29 1.98 2.13 1.96 1.99 2.03

GF 1.53 1.53 1.56 1.37 1.66 1.82 1.70 1.53

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Table A.1. (cont.) a

b

c

%CR

%OR

%BHB

P/F

0.88 0.90

1.39 1.48

1.78 1.84

62 74

49 49

78 80

Fail Fail

0.78

1.04

1.56

92

50

67

Fail

0.97 1.1 1.06 1.13 1.07

1.72 1.84 1.55 1.82 1.80

1.89 2.16 2.08 2.14 2.03

66 86 92 85 90

51 51 51 53 53

91 85 75 85 89

Fail Pass Pass Pass Pass

References 1. 2. 3. 4. 5. 6. 7.

8.

Atkinson AC, Donev AN (1992) Optimum Experimental Designs. Clasendon Press, Oxford DFO (1995) Metal can defects; classification and identification manual. Inspection Services, Department of Fisheries and Oceans, Government of Canada, Ottawa Jang JSR (1993) ANFIS: Adaptive network based fuzzy inference system. IEEE, Transactions on Systems, Man, and Cybernetics, 23(3):665–685 Jang JSR, Sun CT (1995) Neuro-Fuzzy Modeling and Control. The Proceedings of the IEEE, 83(3):378–406 Klir G, Yuan B (1995) Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Upper Saddle River, NJ Kosko B (1992) Neural Networks and Fuzzy Systems: A Dynamical Systems Approach to Machine Intelligence. Prentice Hall, Englewood Cliffs, NJ Lin RC, King PH, Johnston MR (1998) Examination of Metal Containers for Integrity. In: Merker RI (ed) FDA’s Bacteriological Analytical Manual Online, 8th edn., rev. A, chap. 22. Center for Food Safety and Applied Nutrition (CFSAN), U.S. Food & Drug Administration (FDA) Paños-Callado C (1998) Cierres y defectos de envases metálicos para productos alimenticios, 1st edn. Secretaría de Estado de Comercio, Dirección General de Comercio Exterior, Subdirección General de Control, Inspección y Normalización del Comercio Exterior (CICE), Alicante

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9.

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SEFEL (1999) Recommendation SEFEL no.1, for “non easy open” steel ends. In: EUROSEAM seaming interchangeability of processed 3 piece steel food can ends (NEO) fitted on straight or necked-in bodies, 1st and 2nd Part. European Secretariat of Manufacturers of Light Metal Packaging (SEFEL), Brussels 10. Sugeno M, Kang GT (1988) Structure identification of fuzzy model. Fuzzy Sets and Systems, 28:15–33 11. Taguchi G, Wu YI (1984) Introduction to off-line Quality Control. Central Japan Quality Control Association, Nagoya 12. Takagi T, Sugeno M (1985) Fuzzy Identification of Systems and Its Applications to Modeling and Control. IEEE Transactions on Systems, Man, and Cybernetics, 15(1):116–132 13. Yager RR, Zadeh, LA (eds) (1994) Fuzzy Sets, Neural Networks, and Soft Computing. Van Nostrand Reinhold, NewYork

CHAPTER 3 Connectionist Fuzzy Relational Systems Rafał Scherer12 and Leszek Rutkowski12 1

2

Department of Computer Engineering Cze¸stochowa University of Technology Al. Armii Krajowej 36, 42-200 Cze¸stochowa, Poland, http://kik.pcz.pl Department of Artificial Intelligence Academy of Humanities and Economics in Ł´od´z, ul. Rewolucji 1905 nr 64, Ł´od´z, Poland, http://www.wshe.lodz.pl [email protected], [email protected]

Abstract: The paper presents certain neuro-fuzzy systems developed from standard neurofuzzy systems by adding an additional fuzzy relation. In this way the rules are more flexible, and data-driven learning is improved. The systems are tested on several benchmarks. Keywords: Neuro-fuzzy systems, fuzzy relations

1 Introduction There have been an enormous number of fuzzy systems developed so far. For excellent surveys the reader is referred to [1][3][5][7][9][10][18][19]. The most commonly used systems are linguistic models and functional models introduced by Takagi and Sugeno. The linguistic systems store an input-output mapping in the form of fuzzy IF-THEN rules with linguistic terms both in antecedents and consequents. The functional fuzzy systems use linguistic values in the condition part of rules, but the input-output mapping is expressed by functions of inputs in a rule consequent part. The above models are used in all fields of machine learning and computational intelligence. They all have advantages and drawbacks. The linguistic systems use intelligible and easy to express IF-THEN rules with fuzzy linguistic values. Functional systems allow modeling of input-output mapping but they suffer from lack of interpretability. Another approach, rarely studied in the literature, is based on fuzzy relational systems (see e.g. Pedrycz [9]). This relates input fuzzy linguistic values to output fuzzy linguistic values thanks to fuzzy relations. It allows the setting fuzzy linguistic values in advance and fine-tuning model mapping by changing relation elements. They were used in some areas, e.g. to classification [16] and control [2]. In this paper we propose a new neuro-fuzzy structure of the relational system (Section 3 and 5), allowing relation elements to be fine-tuned by the backpropagation algorithm. It will be also shown that relational fuzzy systems are under specific assumptions equivalent to linguistic systems with rule weights (Section 4). Moreover, another new class

R. Scherer and L. Rutkowski: Connectionist Fuzzy Relational Systems, Studies in Computational Intelligence (SCI) 2, 35–47 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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of neuro-fuzzy systems, based on a relational approach with a fuzzy certainty degree, will be suggested in Section 5. Finally, the systems are tested on problems of truck backer-upper nonlinear control and nonlinear function approximation (Section 6).

2 Fuzzy Relational Systems Fuzzy relational models can be regarded as a generalization of linguistic fuzzy systems, where each rule has more than one linguistic value defined on the same output variable, in their consequents. Fuzzy rules in SISO relational model have the following form Rk : IF x is Ak THEN y is B1 (rk1 ) , y is Bm (rkm ) , . . . , y is BM (rkM ) ,

(1)

where rkm is a weight, responsible for the strength of connection between input and output fuzzy sets. Relational fuzzy systems store associations between the input and the output linguistic values in the form of a discrete fuzzy relation R (A, B) ∈ [0, 1] .

(2)

In a general case of a multi-input multi-output system (MIMO), the relation R is a multidimensional matrix containing degree of connection for every possible combination of input and output fuzzy sets. In a multi-input single-output (MISO), there are N inputs xn , and a single output. Every input variable xn has a set An of Kn linguistic values Akn , k = 1, ..., Kn 
n . (3) An = A1n , A2n , ..., AK n Output variable y has a set of M linguistic values Bm with membership functions µBm (y), for m = 1, ..., M
 (4) B = B1 , B2 , ..., BM . For certain MIMO and MISO systems the dimensionality of R becomes quite high and it is very hard to estimate elements of R. Sometimes training data are not enough large at all. For this reason we consider a fuzzy system with multidimensional input linguistic values. Then, we have only one set A of fuzzy linguistic values 
(5) A = A1 , A2 , ..., AK , thus the relational matrix is only two-dimensional in the MISO case. Sets A and B are related to each other with a certain degree by the K × M relation matrix   r11 r11 · · · r1M  r21 r22 · · · r2M    (6) R= . . ..  .  .. .. r  . km rK1 rK2 · · · rKM

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37

3 Neuro-fuzzy Relational Systems Having given vector A¯ of K membership values µAk (¯x) for a crisp observed input value x¯ , vector B¯ of M crisp memberships µm is obtained through a fuzzy relational composition B¯ = A¯ ◦ R ,

(7)

implemented element-wise by a generalized form of sup-min composition, i.e. s-t composition K

µm = S [T (µAk (¯x) , rkm )] . k=1

The crisp output of the relational system is computed by the weighted mean m K

y¯ Sk=1 [T (µAk (¯x) , rkm )] ∑M , y¯ = m=1M x) , rkm )] ∑m=1 SK k=1 [T (µAk (¯

(8)

(9)

where y¯m is a centre of gravity (centroid) of the fuzzy set Bm . The exemplary neuro-fuzzy structure of the relational system is depicted in Fig. 1. The first layer of the system consists of K multidimensional fuzzy membership functions. The second layer is responsible for st composition of membership degrees from previous layer and KM crisp numbers from the fuzzy relation. Finally, the third layer realizes center average defuzzification. Depicting the system as a net structure allows learning or fine-tuning of system parameters through the backpropagation algorithm.

Figure 1. Neuro-fuzzy relational system

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4 Equivalence between Linguistic and Relational Fuzzy Systems In this section, we show [15] that under certain assumptions a relational fuzzy system is equivalent to a rule-based system with a number of rules being equal to the number of elements of the matrix R.

4.1 Relational Fuzzy System Relational fuzzy systems can be interpreted as a set of fuzzy rules with certainty degrees. Assuming we have a relational fuzzy system with K input and M output linguistic values, we can write down corresponding rules Rk : IF x is Ak THEN y is B1 (rk1 ) OR OR . . . OR y is Bm (rkm ) OR . . . OR y is BM (rkM ) ,

(10)

for k = 1 . . . K. Such a rule can be decomposed into M rules with only one linguistic value in their consequents Rk1 : IF x is Ak THEN y is B1 (rk1 ) Rk2 : IF x is Ak THEN y is B2 (rk2 ) .. . (11) Rkm : IF x is Ak THEN y is Bm (rkm ) .. . RkM : IF x is Ak THEN y is BM (rkM ) . Using a product T-norm and an arithmetic mean (the boundary case of the OWA operator) as T-conorm in s-t composition, the output of the relational system becomes m1 K

y¯ K ∑k=1 {µAk (¯x) · rkm } ∑M

, (12) y¯ = m=1 x) · rkm } ∑mj=1 K1 ∑K k=1 {µAk (¯ and reducing K, we obtain y¯ =

m K

x) · rkm } ∑M m=1 y¯ ∑k=1 {µAk (¯ , x) · rkm } ∑mj=1 ∑K k=1 {µAk (¯

(13)

K m x) · rkm } ∑M m=1 ∑k=1 {y¯ · µAk (¯ . m K {µ (¯ ∑ j=1 ∑k=1 Ak x) · rkm }

(14)

and y¯ =

4.2 Rule-based fuzzy system with rule weights A system with rule weights allows us give a certainty value to each rule. Consider a fuzzy system with the following rules Ri : IF x is Ai THEN y is Bi (wi ) ,

(15)

for i = 1 . . . L where x is the input multidimensional variable, Ai is a multidimensional premise fuzzy set, y is the output variable, Bi is a conclusion fuzzy set, and wi is a rule weight [8]. For

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39

singleton fuzzification, algebraic Cartesian product, Larsen (product) implication, and center average defuzzification the output of the system takes the form i

y¯ · µAi (¯x) · wi ∑L , (16) y¯ = i=1L ∑i=1 {µAi (¯x) · wi } where y¯i is a centre of the fuzzy set Bi . If L = K · M, every group of M rules shares the same antecedent set Ak , and every group of K rules shares the same consequent value y¯m , we can rewrite (14) as follows m x) · wkm } ∑M ∑K k=1 {y¯ · µAk (¯ , (17) y¯ = m=1 M K {µ (¯ ∑m=1 ∑k=1 Ak x) · wkm } where wkm is reindexed weight wi . It is obvious that formula (17), describing fuzzy rule-based system is equivalent to the output of a fuzzy relational system (14).

5 A Fuzzy Relational System with Linguistic Antecedent Certainty Factors The power of fuzzy systems stems from their ability to process natural language expressions. We can model nearly any term using different shapes of fuzzy sets and various modifiers, i.e. fuzzy hedges. They transform original fuzzy linguistic values (primary terms e.g. small, big) into new, more specific fuzzy sets like very fast or quite slow flow. Thanks to them, we can make expert statements more precise. The concept of fuzzy hedges is well established in the literature. In general, they can be divided into powered and shifted hedges. Powered hedges model adverbs by powering membership functions, whereas shifted hedges model adverbs by moving points of membership functions. In this section, a different view on linguistic hedges is presented [14]. Instead of modifying antecedent or consequent linguistic values, additional fuzzy sets are introduced. In this approach, a fuzzy relational system with linguistic values defined on a unitary interval is used. These values are elements of a fuzzy relation matrix R connecting antecedent or consequent linguistic values. In this case, the relation matrix contains fuzzy sets Ckm defined on a unitary interval   C11 C11 · · · C1M  C21 C22 · · · C2M    (18) R= . ..  . ..  .. .  . C km

CK1 CK2 · · · CKM Then, if we define several fuzzy linguistic values on a unitary interval (e.g. see Fig. 2), an expert can express his uncertainty concerning antecedent terms by a linguistic description. In SISO systems, or MISO systems with multidimensional antecedent fuzzy sets, the expert can define rules similar to the following exemplary ones R1 : IF x is exactly A1 THEN y is B1 R2 : IF x is more or less A1 THEN y is B2 3

1

(19)

3

R : IF x is roughly A THEN y is B . Rules (19) do not modify antecedent values. The membership degree of an antecedent fuzzy set is divided into several intervals by fuzzy sets Ckm . Instead of defining many antecedent sets

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we use a smaller number of input fuzzy sets and several sets Ckm . Every fuzzy set Ak has up to M defined linguistic values Ckm . In Fig. 2 there is also the set not at all, the meaning of which is similar to the standard hedge not. It is activated when its input fuzzy set Ak is not active. The inference in this system is similar to the sup-min composition, but min operation

Figure 2. Example of fuzzy linguistic values, expressing uncertainty in rule antecedents  is replaced by a membership degree µCkm τk , where τk is the membership degree of the k-th multivariate input fuzzy set. The vector of crisp memberships is obtained by K

µm = S [µCkm (µAk (¯x))] . k=1

(20)

Eq. (20) reflects the fuzzy hedge modifier operation. For example instead of a quadratic function for concentration operation very, we use a fuzzy set exactly (Fig. 2). Interpretation and operation of the sets in Fig. 2 is different from standard linguistic hedges. For example, the standard fuzzy hedge more or less dilates an input fuzzy set, whereas our roughly, more or less divide membership degree ranges into several intervals. The overall system output is also computed through a weighted average 

K m S [µ (µ (¯ x ))] y ¯ k ∑M C A km m=1 k=1 . (21) y¯ = K S [µ (µ (¯ x ))] k ∑M Ckm A m=1 k=1

Graphical inference (without defuzzification) in the new system is in Fig. 3. The example is based on rules (19). The neuro-fuzzy structure of the new system is presented (Fig. 4). The

Figure 3. Graphical inference of the new system on the basis of the exemplary rules (19) first layer consists of K input multivariate fuzzy membership functions. The second layer is

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41

responsible for the composition of certainty values and membership values from the previous layer. Finally, the third layer defuzzifies M output values through the center average method. All parameters of the neuro-fuzzy system can be tuned by the backpropagation algorithm. In this section a new fuzzy model with fuzzy sets linking antecedent and consequent fuzzy sets have been proposed. These linking fuzzy sets are elements of the relation. Thus, we can model descriptions like IF x is roughly fast THEN .... This approach does not involve standard fuzzy hedges, so input and output linguistic values remain intact. This improves the transparency of the model. Numerical simulations showed reliable behaviour of the new system. Its performance is comparable with the popular singleton model, but the new system allows expressing expert’s uncertainty regarding input linguistic variables.

Figure 4. The neuro-fuzzy structure of the relational system with fuzzy certainty degrees

6 Numerical Simulations This section presents application examples of the new relational systems. First we test our systems on the well-known problem of nonlinear truck control. Then we use the systems to approximate a nonlinear function. In both cases we compare our models with a singleton system. Systems were trained only by the backpropagation algorithm changing input and output fuzzy set widths and centers and relation R elements. In case of relational systems with fuzzy certainty degrees, widths of Gaussian fuzzy sets in relation R were fixed to 0.6. This value was picked by successive simulations to be the best. The systems were initialized randomly.

6.1 Truck Backer-Upper Nonlinear Control We tested proposed neuro-fuzzy systems on the well-known problem of a truck backer-upper nonlinear control [18]. The goal is to train a fuzzy model on a 282-element data set to park a truck. A fuzzy model generates the right steering angle of the truck on the basis of the

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truck position, expressed by an angle and a distance from the loading dock. We do not use distance from the truck and the loading dock, assuming that the truck has always enough space to park. The truck moves backward every time by a fixed distance. As a reference point we used the Mamdani rule-based fuzzy system with an algebraic Cartesian product and 6 rules without rule weights. To compare a new system with the conventional one, we used the relational system with 6 multidimensional input Gaussian fuzzy sets and 3 output singleton fuzzy sets. Relational models with fuzzy certainty degrees had Gaussian sets in the relation R. All parameters of each system were trained only by the backpropagation algorithm. Learning errors during training for all systems for 50,000 iterations are shown in Fig. 5. Fig. 6 shows numerical simulations of truck steering. All systems give good steering trajectories. Learning

Figure 5. Learning error for a) Mamdani system, b) relational system c) relational system with fuzzy certainty degrees

Figure 6. Truck trajectories for a) Mamdani system, b) relational system c) relational system with fuzzy certainty degrees error for the crisp relational system falls slower but in this way we can fine-tune elements of the fuzzy relation. Moreover, the truck steered by the relational system needs fewer steps to reach the loading dock. The relational system with fuzzy certainty degrees performs worst, but for this price an expert can linguistically express his uncertainty for a given rule.

6.2 Nonlinear Function Approximation Here we approximate two-input single-output nonlinear function   −1,5 2 y = 1 + x1−2 + x2 , 1
x1 , x2
5 .

(22)

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We learned and tested systems on an original 50-element data set taken from [17]. All parameters were tuned by the backpropagation algorithm. The singleton model had 6 rules and an algebraic Cartesian product. The relational system had both 6 input fuzzy sets and output fuzzy sets, related to each other by matrix. The relational system with fuzzy certainty degrees had similar structure, however crisp values were replaced by Gaussian fuzzy sets with fixed widths. Only their centres were tuned. Error during learning is shown in Fig. 7. Root mean square error (RMSE) error after 100,000 iterations for each type of model is given in Table 1.

Figure 7. Learning error for a) Mamdani system, b) relational system, c) relational system with fuzzy certainty degrees

In this problem the relational system outperforms the other systems, and the relational system

Table 1. Root mean square error for nonlinear function Singleton model

Relational model

0.084

0.044

Relational model with fuzzy certainty degrees 0.214

with fuzzy certainty degrees again performs worst.

6.3 Synthetic Normal-mixtures Data This data set consists of 2 classes with 125 objects in each class [11]. The training set consists of 250 instances and the testing set of 1000 instances (see Fig. 8). The class distribution only allows for 8% error rate. Simulation results show that relational systems outperform the standard singleton model (Tab. 2). Also the relational system with fuzzy antecedent certainty degree performs slightly better than the standard singleton model.

6.4 Synthetic Cone-torus Data This data set consists of 3 classes with 100 objects in two classes each and 200 objects from the third class [6]. The training set consists of 400 instances, as does the testing set (see Fig. 9). Simulation results (Tab. 3) show that the relational system outperform the standard singleton model. The relational system with fuzzy antecedent certainty degree performs slightly worse than the standard singleton model.

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Table 2. Learning results for normal mixture data Singleton model Number of rules

Testing error [%]

2

18.2

4

12.4

5

12.0

6

15.2

8

19.7 Relational system with min-max composition

Number of input and Testing error [%] output fuzzy sets 6

3

12.6

6

6

17.5

Relational system with multiplication-summation composition Number of input and Testing error [%] output fuzzy sets 2

2

18.9

3

2

14.0

3

3

10.8

4

2

10.1

4

3

10.6

6

2

10.8

6

3

11.3

Relational system with fuzzy antecedent certainty factor Number of input and Testing error [%] output fuzzy sets 4

3

11.5

7 Conclusion We proposed a new relational neuro-fuzzy system. The system allows learning all its parameters (relation matrix elements and membership function parameters) by the backpropagation algorithm. Obviously, we can set linguistic values in advance and fine-tune only elements of the relation. We also showed the equivalence between relational fuzzy systems and linguistic systems with rule weights. Finally, we suggested a new class of relational systems with fuzzy relation matrix elements, which can be regarded as linguistic certainty degrees. These are the subject of our future work. Simulations, carried out on the nonlinear control problem and function approximation, proved the usefulness of the relational systems trained by the backpropagation algorithm, showing that this method can be viewed as an alternative to classic IF-THEN systems. The relational system learns longer but the truck reaches the loading

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Figure 8. Synthetic normal-mixture data

Figure 9. Synthetic cone-torus data

dock faster during testing. Also, in the case of function approximation, it outperforms classic linguistic system. Further improvement can be made by better initialization of antecedent and consequent fuzzy sets, and estimation of relation R from data.

Bibliography 1. R.Babuska, Fuzzy Modeling For Control, Kluwer Academic Press, Boston, 1998.

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Table 3. Learning results for cone-torus data Singleton model Number of rules Testing error [%] 5 20.5 8 16.25 9 16.25 10 15.5 11 13.75 12 17.5 15 16.5 Relational system with multiplication-summation composition Number of input and Testing error [%] output fuzzy sets 5 3 18.9 6 4 20.0 8 4 20.75 9 4 16.5 10 3 19.5 10 4 16.0 10 6 11.5 10 7 18.75 10 8 17.25 Relational with fuzzy antecedent certainty factor Number of input and Testing error [%] output fuzzy sets 4 3 20.0 6 4 18.25 10 6 16.5

2. P.J.C. Branco, J.A. Dente, A Fuzzy Relational Identification Algorithm and its Application to Predict the Behaviour of a Motor Drive System, Fuzzy Sets and Systems, vol. 109, pp. 343-354, 2000. 3. E. Czogala, J. Leski, Fuzzy and Neuro-Fuzzy Intelligent Systems, Physica-Verlag, Heidelberg 1999. 4. H. Ischibuchi, T. Nakashima, Effect of Rule Weights in Fuzzy Rule-Based Classification Systems, IEEE Transactions on Fuzzy Systems, vol. 9, no. 4, pp.506-515, 2001. 5. R. J.-S. Jang, C.-T. Sun, E. Mizutani, Neuro-Fuzzy and Soft Computing, A Computational Approach to Learning and Machine Intelligence, Prentice Hall, Upper Saddle River, 1997. 6. Kuncheva L. I., Fuzzy Classifier Design, Physica Verlag, Heidelberg, New York, 2000. 7. D. Nauck, F. Klawon, R. Kruse, Foundations of Neuro - Fuzzy Systems, Chichester, U.K., John Wiley, 1997. 8. D. Nauck, R. Kruse, How the Learning of Rule Weights Affects the Interpretability of Fuzzy Systems, Proceedings of 1998 IEEE World Congress on Computational Intelligence, FUZZ-IEEE, Alaska pp. 1235-1240, 1998. 9. W. Pedrycz, Fuzzy Control and Fuzzy Systems, Research Studies Press, London, 1989. 10. A. Piegat, Fuzzy Modeling and Control, Physica Verlag, Heidelberg New York, 2001.

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11. B.D. Ripley, Pattern Recognition and Neural Networks, Cambridge University Press, 1996. 12. L. Rutkowski, New Soft Computing Techniques for System Modelling, Pattern Classification and Image Processing, Springer-Verlag, 2004. 13. Scherer R., Rutkowski L., Relational Equations Initializing Neuro-Fuzzy System, 10th Zittau Fuzzy Colloquium, 2002, Zittau, Germany 2002. 14. Scherer R., Rutkowski L., A Fuzzy Relational System with Linguistic Antecedent Certainty Factors, in L. Rutkowski, J. Kacprzyk (eds) Neural Networks and Soft Computing, Physica-Verlag Heidelberg, pp. 563-569, 2003. 15. Scherer R., Rutkowski L., Neuro-Fuzzy Relational Systems, 2002 International Conference on Fuzzy Systems and Knowledge Discovery, November 18-22, Singapore, 2002. 16. M. Setness, R. Babuska, Fuzzy Relational Classifier Trained by Fuzzy Clustering, IEEE Transactions on Systems, Man and Cybernetics - Part B: Cybernetics, Vol. 29, No. 5, October, pp. 619-625, 1999. 17. M. Sugeno, T. Yasukawa, A Fuzzy-Logic-Based Approach to Qualitative Modeling, IEEE Transactions on Fuzzy Systems, vol. 1, no. 1, pp. 7-31, 1993. 18. L.-X. Wang, Adaptive Fuzzy Systems And Control, PTR Prentice Hall, Englewood Cliffs, New Jersey, 1994. 19. R.R. Yager, D.P. Filev, Essentials of Fuzzy Modeling and Control, John Wiley Sons, Inc., 1994. 20. R.R. Yager, D.P. Filev, On a Flexible Structure for Fuzzy Systems Models in Fuzzy Sets, Neural Networks, and Soft Computing, R.R. Yager, L.A. Zadeh, Eds.,Van Nostrand Reinhold, New York, pp.1-28, 1994.

CHAPTER 4 Semantic Indexing and Fuzzy Relevance Model in Information Retrieval Bo-Yeong Kang1 , Dae-Won Kim2 , and Sang-Jo Lee1 1 2

Department of Computer Engineering, Kyungpook National University, 1370 Sangyuk-dong, Puk-gu, Daegu, Korea [email protected] Department of Computer Science, Korea Advanced Institute of Science and Technology, Guseong-dong, Yuseong-gu, Daejeon, Korea [email protected]

Abstract: Let us suppose that an information retrieval system comprehends the semantic content of documents and reflects the preferences of users. Then the system can more effectively search for the information better on the Internet, and may improve the retrieval performance. Therefore, in the present study, a new information retrieval system is proposed by combining a semantic based indexing and a fuzzy relevance model. In addition to the statistical approach, we propose the semantic approach in indexing based on lexical chains. The indexing extracts the semantic concepts in a given document. Furthermore, a fuzzy relevance model combined with the semantic index calculates the exact degrees of relevance between documents based on the user preference. The combination of these notions is expected to improve the information retrieval performance. Keywords: Information retrieval, Indexing, Fuzzy relevance model

1 Introduction The goal of information retrieval system is to search the documents which user wants to obtain in a fast and efficient way [1]. If there is an information retrieval system that comprehends the semantic content of documents and reflects the preferences of users, it can be very helpful to search the information on the Internet and to improve the performance of the existing systems. Because the information retrieval system must interpret the contents of information items or documents in a collection and rank them according to the degree of relevance to the user query, the representation of document contents and the user preference are important factors in the retrieval process. Therefore, in this study, we focus on the system design in the indexing and retrieval model. The proposed system indexes the documents by the semantic approach using lexical chains; it considers not only the terms in a document, but also semantic concepts in a given document. To fully exploit the performance of the semantic indexing, a fuzzy relevance model is proposed. The fuzzy relevance model ranks documents according to the exact relevance of Bo-Y. Kang et al.: Semantic Indexing and Fuzzy Relevance Model in Information Retrieval, Studies in Computational Intelligence (SCI) 2, 49–60 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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the user preference and a user query by some metrics [2]. The remainder of this chapter is organized as follows: in Section 2, the traditional indexing methods and fuzzy relevance models are discussed. In Section 3, the basic concepts and approaches for the proposed semantic indexing are presented. In Section 4, the proposed fuzzy relevance model based on user preference is addressed. The effectiveness of the proposed approach is demonstrated in Section 5. Finally we conclude our remarks in Section 6.

2 Literature Review 2.1 Indexing Schemes Indexing methods should be based not only on the occurrences of terms in a document, but also on the content of the document. Despite this obvious need, most existing indexing and the weighting algorithms analyze term occurrences and do not attempt to resolve the meaning of the text. As a result, existing indexing methods do not comprehend the topics referred to in a text, and therefore have difficulty in extracting semantically important indexes. Many weighting functions have been proposed and tested [3, 4, 5, 6]. However, most such functions developed to date depend on statistical methods or on the document’s term distribution tendency. Representative weighting functions include such factors as term frequency (TF), inverse document frequency (IDF), the product of TF and IDF(TF·IDF). A drawback of the most TF-based methods is that they have difficulties in extracting semantically exact indexes that express the topics of a document. Thus, the TF approach fails to capture the topics of the text and cannot discriminate the degree of semantic importance of each lexical item within the text. Linguistic phenomena such as the lexical chain [7], which links related lexical items in a text, have been used to enhance the indexing performance [8, 9]. Al-Halimi and Kazman developed a method for indexing transcriptions of conference meetings by topic using lexical trees, the two-dimensional version of lexical chains. However, although their method demonstrated the potential usefulness of lexical trees in text indexing and retrieval, in its present form their method is inappropriate for use in document retrieval, because the method of Al-Halimi and Kazman can extract topics as index terms from a video of a conference but it does not contain a function to estimate the weight of each extracted topic.

2.2 Fuzzy Relevance Models The boolean relevance model is a simple retrieval model based on set theory and boolean algebra. Given its inherent simplicity and neat formalism, boolean models have received great attention in past years and have been adopted by many of the early commercial systems. But the main disadvantage is that there’s no notion of a partial match to the query conditions. Thus the exact matching may lead to retrieval of too few or too many documents. To overcome this disadvantage, a fuzzy model, also called the extended boolean model, was proposed. The fuzzy model could handle the disadvantages of the classical boolean model by introducing the notion of document weight. The document weight is a measure of the degree to which the document is characterized by each index term. The document weights for all index terms lie in the range [0, 1]. However, previous fuzzy model didn’t consider the concept of user preference. In this section, we consider three fuzzy models [10, 11], i.e., MMM, PAICE, and P-NORM models.

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The MMM Model In the Mixed Min and Max (MMM) model, each index term has a fuzzy set associated with it. The document weight of a document with respect to an index term A is considered to be the degree of membership of the document in the fuzzy set associated with A. Thus given a document D with index-term weight (dA1 , dA2 , ..., dAn ) for terms A1 , A2 , ..., An , and the queries Qand = (A1 and A2 and ... and An ) Qor = (A1 or A2 or ... or An )

(1) (2)

The query-document similarity in the MMM model is computed in the following manner. SIM(Qand , D) = Cand1 × γ(A) +Cand2 × δ(A)

(3)

SIM(Qor , D) = Cor1 × δ(A) +Cor2 × γ(A)

(4)

γ(A) = min(dA1 , dA2 , ..., dAn )

(5)

δ(A) = max(dA1 , dA2 , ..., dAn )

(6)

where Cor1 ,Cor2 are softness coefficients for the or operator, and Cand1 ,Cand2 are softness coefficients for the and operator.

The PAICE Model This model is similar to the MMM model in that it assumes that there is a fuzzy set associated with each index term and document weight of a document. However, while the MMM model considers only the maximum and minimum document weights for the index terms while calculating the similarity, the PAICE model takes into account all of the document weights. SIM(Q, D) =

n

n

i=1

i=1

∑ ri−1 di / ∑ ri−1

(7)

where Q, ri mean the query and the constant coefficient, respectively. di means index term weights that is considered in ascending order for and operation and descending order for or operation.

The P-NORM Model The previous two fuzzy relevance models, MMM and PAICE models, do not provide a way of evaluating query weights. They only consider the document weights. The P-NORM model explicitly reflects the query weight in its model. Given a document D with index-term weights (dA1 , dA2 , ..., dAn ) for terms A1 , A2 , ..., An , and the queries Q with weights (qA1 , qA2 , ..., qAn ), the query-document relevance is calculated as  1/p ∑ni=1 (1 − dAi ) p (qAi ) p (8) SIM(Qand , D) = 1 − ∑ni=1 (qAi ) p  1/p ∑ni=1 (dAi ) p (qAi ) p (9) SIM(Qor , D) = ∑ni=1 (qAi ) p where p is a control coefficient ranged from 1 to ∞. In general, the P-NORM model has shown its superior effectiveness to other fuzzy relevance models.

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3 Semantic Approach to Indexing Scheme Words and phrases in a document are often too specific in representing text contents preventing generic searches of information in texts or vice versa. However, before selecting indexes, the present work extracts semantically important concepts, and then identifies which indexes are really significant.

3.1 Representing Semantic Concepts Most of the traditional indexing schemes based on statistical methods suffer from limitations that diminish the precision of the extracted indexes [12]. TF is useful when indexing long documents, but not short ones. However, TF algorithms do not generally represent the exact TF because they do not take into account semantic characteristics such as anaphoras, synonyms, and so on. Even though the normalized TF × IDF method was proposed to account for the fact that TF factors are numerous for long documents but negligible for short ones which obscures the real importance of terms, this approach still uses the TF function that suffers from the same shortcomings as the TF method does. Documents generally contain various concepts, and we must determine those concepts if we are to comprehend the aboutness of a document. In accordance with the accepted view in the linguistics literature that lexical chains provide a good representation of discourse structures and topicality of segments [7], here we take each lexical chain to represent a concept that expresses one aspect of the meaning of a document. In practice, we define each lexical chain derived from a document as a concept cluster that captures one of the concepts of the document. The proposed method first extracts concept clusters that represent the semantic content of the text and assigns scores to the extracted concept clusters and lexical items in the clusters for representing semantic importance degree. Then, each concept cluster is evaluated to identify the representative concepts, which are used to guide the semantic indexes and their weights.

3.2 Extracting Representative Concepts Concept clusters are lexical chains that represent the concept or topic of a document. It is generally agreed that lexical chains represent the discourse structure of a document. For the semantic importance estimation of terms within a document, the representative concepts should be identified for a given document. To achieve this, we define two scoring functions for each concept cluster and the terms in the cluster. To compose the concept cluster by related lexical items, we use five relations – identity, synonymy, hypernymy, hyponymy, meronymy – where the relation weight is in the order listed (i.e., identity highest and meronymy lowest). The scoring function of each concept cluster (cx ) and the nouns (wi ) in the cluster, denoted by λ(cx ) and φ(wi ) respectively, are defined as follows: φ(wi ) =

∑ τ(k) × υ(k)

k∈R

λ(cx ) =

∑

wi ∈cx

φ(wi )

where τ(k) and υ(k) mean the number of weights of relation k respectively. From the weighted concept clusters, we discriminate a representative concept, since we cannot deal with all the

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concepts of a document. A concept cr will be considered a representative concept if it satisfies the following criterion. ∑ λ(cx ) (10) λ(cr ) ≥ α · x |cx | where |cx | is the number of concepts in a given document. After extracting the representative concept clusters of a document, we extract the terms in each representative concept cluster as index terms that capture the aboutness of a document, and regard the assigned term scores to those terms as the index weights that represent the semantic importance within a document.

4 Preference-Based Fuzzy Relevance Model In the previous section, we presented a new indexing scheme using lexical concepts within a document. The semantically important terms obtained through the representative concepts are used as the indexes, which are referred to as the documents weights in a fuzzy information retrieval system. In this section, a new fuzzy relevance model is described. Based on fuzzy set theory, the proposed model defines the similarity between a query and a document.

4.1 Overview As already mentioned before, the previous fuzzy models were introduced to overcome the disadvantages of the classical boolean models. However, they ignored the fact that each index term has vagueness and that the user preference is a very important factor in an information retrieval system. Besides the weight of each index term, the preference on the distribution of index term weight is also essential in designing similarity model. Let us show an example. Query (and) is given in the form of composite of (index term, index term weight). Q = ( f uzzy, 0.8), (system, 0.7), (korea, 0.3), (author, 0.2)

(11)

The above query can be represented by a fuzzy set with element and membership value like this. Q = {( f uzzy, 0.8), (system, 0.7), (korea, 0.3), (author, 0.2)} (12) Then suppose that four documents (D1 , D2 , D3 , D4 ) are stored in the document collection. Each document is represented as a set of index term and its weight. D1 = {( f uzzy, 0.8), (system, 0.7)} D2 = {( f uzzy, 0.2), (system, 0.2), (korea, 0.3), (author, 0.2)} D3 = {(korea, 0.7), (author, 0.8)} D4 = {( f uzzy, 0.8), (system, 0.7), (korea, 0.3), (author, 0.2)} Given a query Q and the document collection, what is the rank result of relevance degree in document retrieval? Intuitively, D4 is the most relevant document and D3 is the least relevant one. Then how about the rank of D1 and D2 ? The classical models and previous fuzzy models can’t decide the rank of two documents in a deterministic way. The difference of similarity value between D1 and D2 is very small because the similarity value is calculated by adopting a monotonic increasing function. Thus the sum of two index terms with large weight in D1 is nearly equal to the sum of four index terms with small weight in D2 .

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Hence, we take an approach to resolve the above problem by adopting the concept of user preference. User preference can provide a clearer ranking result. In the above example, a user might want to obtain a document with a high degree of membership values. In other words, by weighing a preference for high membership area, D1 has higher similarity value than D2 even though the number of matched words in D1 is smaller than D2 . In this way, our proposed model can clarify the vagueness which can occur in the retrieval process.

4.2 Relevance Model using Preference To retrieve relevant documents, the query-document similarity for documents in the collection must be calculated. The query-document similarity is an attempt to predict the relevance of a document to the query. Given N fuzzy sets, which represent documents in the collection, the goal is to find the closest one in the collection for the new fuzzy set which stands for a query. So the relevance calculation model can be transformed to the similarity computation process in fuzzy set theory. Most previous work related to the similarity calculation is based on Euclidean or Hamming measures. They simply calculate the vertical distance on each data in fuzzy set. But these kind of methods are not complete and have several error factors. And, as already mentioned, fuzzy sets representing query and documents have a vague and ambiguous property, so we must consider the possibility distribution of truth in order to obtain the similarity value. In addition to this fact, the result of a similarity comparison might depend on user preference. The change of preference viewpoints imply the change the similarity result. Several recent studies reflect these factors, but many of them lack a consistency [13, 14, 15, 16, 17, 18, 19, 20, 21] Thus, we developed a new similarity measure between fuzzy sets, which can generate a better relevance degree and reflect user preference and weight. To accomplish this, we proposed two new concepts, domain preference on the domain axis and membership degree preference on the membership axis. By introducing these concepts, a user can give a weight to the specific part which they think more important. The final similarity value is a composition of domain preference and membership preference using integration function. The approach taken in this paper has a contribution in making a generalization of similarity comparison and providing the preference degree in an information retrieval system and in fuzzy set theory.

4.3 Similarity Measure To generally extend the comparison of similarity between fuzzy sets, we must consider the user preference or intention. We define two preference values in order to reflect the user’s will.

Domain Preference and Membership-value Preference This preference is a weight on the domain axis of comparing fuzzy sets. The definition of this value is shown in Eq. 13. dDomain = f (x)dx (13) Figure 1 shows sample fuzzy sets. Figure 1(a) represents the two comparing fuzzy sets. Figure 1(b) means the applied domain preference in computing a similarity between fuzzy set A and B, in this case the user puts more weight on the right side of domain.

Semantic Indexing and Fuzzy Relevance Model in Information Retrieval

A

1.0

55

B

0.5

membership preference

(a)

(c) (b)

domain preference

Fig. 1. (a) Two fuzzy sets A, B (b) Domain preference function (c) Membership preference function The second preference value is put on the membership degree of fuzzy sets. This one changes the weight of membership value axis to reflect the user’s will. Membership preference function is defined as Eq. 14. (14) dMV = fMV (y)dy Figure 1(c) shows a simple shape of membership-value preference function.

Similarity Computation using Preference The algorithm requires two steps. First, the preliminary similarity value is calculated using the domain preference function. After that, we compute the whole similarity value by applying the membership preference function. For a point x on the domain, the similarity value ψA,B (x, y) which corresponds to specific membership value y is calculated by Eq. 15.

1 if y ≤ MIN(µA (x), µB (x)) ψA,B (x, y) = (15) 0 otherwise According to the above equation, if fuzzy set A and B is similar at specific level y, it means both fuzzy sets are greater than membership value y. By Eq. 13, domain preference can be applied, the integral form is given in Eq. 16. ξ(y) =

 

Domain

= Domain

ψA,B (x, y)dDomain

(16)

ψA,B (x, y) fDomain (x)dx

(17)

Figure 2 shows this situation. Given a domain preference fDomain (x) and a domain area r, the similarity calculation process is carried out at specific membership value y. For all x ∈ r, ψA,B (x, y) is set to 1 by Eq. 15. The integration value of domain function in Eq. 16 becomes the similarity value which reflects the preference at specific level y. The similarity value obtained by domain preference is based on the specific membership level y. So it’s necessary to add the membership preference concept in the second step of algorithm. This computation is represented in Eq. 18.

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membership value 1.0

A B

y 0.0

r

domain

fDomain(x) : domain preference

Fig. 2. Apply the domain preference membership value fMV(y)

1.0

A B

y domain

0.0

membership preference domain preference

Fig. 3. Apply the membership value preference δA,B =

 MV  1

= y=0

ξ(y)dMV

(18)

ξ(y) fMV (y)dy

(19)

δA,B is a final similarity value, which is obtained by integrating ξ(y) on membership area (MV , membership-value). This can be rewritten in Eq. 19 using membership function fMV (y), as shown in Fig. 3. In other words, a similarity value is calculated by integrating a membership preference function ( fMV (y)) and a similar value at specific level y (ξ(y)). The final similarity value (δA,B ) between fuzzy set A and B is determined by calculating for a whole membership area. Now, we’ll go back to the problem proposed in the document retrieval. Given N fuzzy sets in a document collection, the problem is find the closest fuzzy set, as represented in Eq. 20. In this equation, we denote that χ is the closest fuzzy set which corresponds to the most relevant document, A to Z are fuzzy sets which represent documents in the collection, and I is the fuzzy set which stands for a query.

Semantic Indexing and Fuzzy Relevance Model in Information Retrieval χ = MAX(δA,I , δB,I , ..., δZ,I )

57

(20)

5 Experimental Results In this section we discuss document retrieval experiments in which the proposed fuzzy information retrieval models were applied to the TREC-2 collection of 1990 Wall Street Journal (WSJ) documents, which comprises 21,702 documents. The preference-based fuzzy relevance model was compared with the well-known PAICE and P-NORM models, and moreover, the semantic-based indexing was compared with TF-based indexing by applying each indexing scheme to the three relevance models. The parameters of PAICE and P-NORM models were experimentally set to be r = 1.0 and p = 2.0 respectively. The domain preference function was set to be constant, and the membership preference of the proposed model was given in a step function; 1.0 if y ≥ 0.6 and 0.1 if y ≤ 0.6. Because the TF and semantic weight only represent the importance degree of a term within a document, and cannot express the importance degree of the term within a document collection, we used IDF in both weighting schemes for representing the importance degree of the term within a document collection. For the relevance judgement of the TREC-2 collection, 50 queries from query number 101 to 150 are supported. However, we only used 40 queries because there are no 1990 WSJ documents for the other 10 queries and we have no answer set to compare the relevance for those 10 queries. When a document is searched, the retrieval system will already have loaded the index file to memory. When a user inputs a query to the system, it constructs the inverted file appropriate to the query using the index file. As the size of the index file decreases, the time required for loading and constructing the inverted file becomes less, and the overall search time is reduced accordingly. The critical determinant of the index file size is the index term dimension; thus, a reduction in the number of index terms will decrease the size of the indexing file. When a document is indexed based on the TF, all the terms in the document are used as indexes, and hence the index term dimension simply equals the number of words in the document. However, when we index a document using the proposed indexing scheme, we first extract representative concepts from the document and then extract index terms from those concepts. Hence, the index term dimension of the document will be less than that obtained using the TF approach. This is clearly demonstrated in the present experiments on the 1990 WSJ documents, for which the average index term dimension was 89.55 using the TF method but 31.90 using the proposed method. Thus, on average the TF method represents the 1990 WSJ documents using about 90 words as index terms whereas the proposed method requires only about 32 words, indicating that the proposed scheme reduces the index term dimension by about 64.40% compared to the TF method. In the query, there are various components for retrieval clues, for example narrative, concept, and summary. We used nouns of the narrative part as the query, and processed the traditional query processing. The weighting scheme for the query was TF. After performing query processing, we searched the WSJ documents and compared the search results obtained using TF weighting with those obtained using the proposed semantic weight scheme. Table 1 lists the overall search results of the three fuzzy models using TF-based indexes, showing average precision of each method for Top 1 to Top 5 documents. The search results show that the proposed preference-based model outperforms PAICE and P-NORM models. The average precision of PAICE and P-NORM models are 8.90% and 9.07%, respectively. We see that the

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12.50 10.00 7.50 7.50 7.00

7.50 8.75 8.33 11.25 9.50

10.00 10.00 13.33 11.88 11.50

Avg.

8.90

9.07

11.34

Table 2. Average precisions of three relevance models using the semantic index Top N PAICE P-NORM Proposed Top 1 Top 2 Top 3 Top 4 Top 5

20.00 12.50 12.50 11.25 11.50

22.50 21.25 20.00 20.63 19.50

20.00 22.50 23.33 21.25 20.00

Avg.

13.55

20.78

21.42

precision of the proposed system for the top-ranked document (11.34%) is higher than the other two models. Table 2 lists the average precision of the three fuzzy models using the proposed semantic indexes. We find that the proposed preference-based model also outperforms PAICE and P-NORM models. The average precision of PAICE and P-NORM models are 13.55% and 20.78%, respectively. The precision of the proposed system for the top-ranked document (21.42%) is higher than the other two models. Moreover, we see that the semantic approach to indexing shows its superior performance to TF-based indexing. The average precisions of the three fuzzy systems were markedly improved by the semantic index, especially 88.89% improved in the proposed fuzzy model.

6 Concluding Remarks In the present study a new information retrieval system has been proposed by combining the semantic based indexing and the fuzzy relevance model. So far there are few index term extraction systems that use semantic relations between words or sentences for index term detection. However, this is an important factor to understand the content of a document, thus we should not discard this semantic aspect in detecting index terms that can fully represent the content of a document. To find out index terms in documents, lexical chains were employed. Using lexical chains we detected index terms and their semantic weights. Furthermore, we

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proposed an extended fuzzy information retrieval system that can handle the shortcomings of the previous models. In the proposed model, a new fuzzy similarity measure was introduced in order to calculate the relevance degree among documents that exploit user preference. The improvements of these problems will enable us to implement a robust system, by which we are able to guess the content of a document better and retrieve the exact documents.

References 1. Baeza-Yates R, Ribeiro-Neto B (1999) Modern information retrieval, Addison-Wesley 2. Kim DW, Lee KH (2001) Proceedings of the 10th IEEE International Conference on Fuzzy Systems 3. Salton G, McGill MJ (1983) Introduction to modern information retrieval, New York 4. Salton G, Buckley C (1988) Term-weighting approaches in automatic text retrieval, Information Processing and Management 24:513–523 5. Fuhr N, Buckley C (1991) A probabilistic learning approach for document indexing, ACM Tracsactions on Information Systems 9:223–248 6. Lee JH (1995) Combining multiple evidence from different properties of weighting schemes, Proceedings of the 18th SIGIR Conference 180–188 7. Morris J, Hirst G (1991) Lexical cohesion computed by thesaural relations as an indicator of the structure of text, Computational Linguistics 17(1):21–43 8. Kominek J, Kazmand R (1997) Accessing multimedia through concept clustering, Proceedings of CHI’97 19–26 9. Kazmand R et al (1996) Four paradigms for indexing video conferences, IEEE Multimedia 3(1):63–73 10. Salton G, Fox EA, Wu H, Extended boolean information retrieval, Communications of the ACM 26(12): 1022–1036 11. Lee JH (1994) On the evaluation of Boolean operators in the extended Boolean retrieval framework, Proceedings of the 17th ACM-SIGIR conference on Research and development in information retrieval 182–190. 12. Moens MF (2000) Automatic Indexing and Abstracting of Document Texts, Kluwer Academic Publishers 13. Lee JH (1999) Comparison, ranking and determination of the ranks of fuzzy numbers based on a satisfaction function, Ph.D. Dissertation, Department of Computer Science, KAIST. 14. Wang WJ (1997) New similarity measures on fuzzy sets and on elements, Fuzzy Sets and Systems 85:305–309 15. Lee KH (1994) Similarity measure between fuzzy Sets and between elements, Fuzzy Sets and Systems 62:291–293 16. Hong DH, Hwang SY (1994) A note on the value similarity of fuzzy systems variables, Fuzzy Sets and Systems 66:383–386 17. Fan J, Xie W (1999) Some notes on similarity measure and proximity measure, Fuzzy Sets and Systems 101:403–412 18. Koczy LT, Hirota K (1993) Ordering, distance and closeness of fuzzy sets, Fuzzy Sets and Systems 59:281–293 19. Pappis CP, Karacapilidis NI (1993) A comparative assessment of measures of similarity of fuzzy values, Fuzzy Sets and Systems 56:171–174 20. Xuecheng L (1992) Entropy, Distance measure and similarity measure of fuzzy sets and their relations, Fuzzy Sets and Systems 52:305–318

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21. Chen SM et al (1995) A comparison of similarity measures of fuzzy values, Fuzzy Sets and Systems 72:79–89

CHAPTER 5 Backward Reasoning on Rule-Based Systems Modeled by Fuzzy Petri Nets Through Backward Tree Rong Yang1 , Pheng-Ann Heng2 , and Kwong-Sak Leung3 1 2 3

Dept. Computer Science and Engineering, The Chinese University of Hong Kong [email protected] Dept. Computer Science and Engineering, The Chinese University of Hong Kong [email protected] Dept. Computer Science and Engineering, The Chinese University of Hong Kong [email protected]

Abstract: The crux in rule-based systems modeled by Fuzzy Petri Nets (FPN) is to decide the sequence of transitions firing. In this problem, backward reasoning shows advantages over forward reasoning. In this paper, given goal place(s), an FPN mapped from a rule-based system is mapped further into a backward tree, which has distinct layers from the bottom to the top. The hierarchical structure of the backward tree provides the order of transitions firing. The nearer the top the transition, the earlier it fires. An innovative and efficient algorithm on backward reasoning through the backward tree with detailed descriptions on data structure is proposed in this paper.

1 Introduction Petri Net (PN) is a sucessful tool for describing and studying information systems. Incorporating the fuzzy logic introduced in [9], Fuzzy Petri Net (FPN) has been widely used to deal with fuzzy knowledge representation and reasoning. It has also proved to be a powerful representation method for the reasoning of a rule-based system. However, most reasoning algorithms on rule-based systems modeled by FPNs are focus on forward reasoning [5, 2, 3, 6]. In fact, for a rule-based system modeled by FPNs, forward reasoning has an inherent limitation on determining the sequence of transitions firing. When an FPN has a complex structure, the traditional forward reasoning becomes difficult and time-consuming, because all information, not matter if it is relative to the goal or not, are considered during the whole reasoning process. Due to these reasons, backward reasoning is considered to be a better alternative for many cases. The advantage of backward reasoning is that only information relative to the goal will be considered. This feature makes the reasoning process more flexible and more R. Yang et al.: Backward Reasoning on Rule-Based Systems Modeled by Fuzzy Petri Nets Through Backward Tree, Studies in Computational Intelligence (SCI) 2, 61–71 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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intelligent. Although the algorithm presented in [1] could perform fuzzy backward reasoning, it was not efficient enough in real-world applications due to the fact that it was based on interval-valued fuzzy sets for performing fuzzy backward reasoning and took a large amount of time to perform the task. In [3], Chen provided a new fuzzy backward reasoning algorithm, where an FPN was transformed into an AND-OR graph. However, it is difficult to realize this algorithm into practice for the following reasons: • The data structure of the fuzzy AND-OR graph and how to sprout it in computer language have not been explicitly provided; • Since the sequence of transitions firing has not been provided, when the algorithm is applied to an FPN with a complex structure, it meets with trouble, because the IBIS of a current nonterminal node may have the degree of truth not available when it is considered; • The lack of description on data structure of the fuzzy AND-OR graph makes it difficult to separate the four cases in programming. We now proposed an improved method from an innovative viewpoint. Here, given the goal place(s), an FPN is mapped into a tree structure, which we call the Backward Tree (BT). The BT has an explicit data structure. It comprises two parts, the layers and the terminal places. Each layer consists of several transitions and their output places. After an FPN has been mapped into a backward tree, the reasoning process can be performed easily by firing the transitions from the top layer to the bottom layer in turn. This chapter is organized as follows. Section 2 gives the preliminary knowledge on the FPNs and the fuzzy rule-based reasoning represented by FPNs. In Section 3, structure of the BT is introduced. The mapping from an FPN into a BT is also presented in this section. In Section 4, a new efficient, complete, and practical algorithm is proposed. Some examples are provided in Section 5 to illustrate the reasoning process through our algorithm. The final section gives the conclusion and the future work.

2 Fuzzy Reasoning via Petri Nets A Fuzzy Petri Net (FPN) is a directed, weighted, bipartite graph consisting of two kinds of nodes called places and transitions, where arcs are either from a place to a transition or from a transition to a place. In graphical representation, places are drawn as circles, transitions as bars. Figure 1 shows a graph of the simplest FPN, where transition t has one input place p1 and one output place p2 . According to the definition in [2], an FPN can be defined as an 8-tuples: FPN = (P, T, D, I, O, f , ω, β) where,

(1)

Backward Reasoning on Rule-Based Systems Modeled

  p1

t

63

  p2

Fig. 1. The simplest FPN

P = {p1 , p2 , · · · , pn } is a finite set of places; T = {t1 ,t2 , · · · ,tn } is a finite set of transitions; D = {d1 , d2 , · · · , dn } is a finite set of propositions; I : T → P is an input function, mapping transitions to their input places; O : T → P is an output function, mapping transitions to their output places; f : T → [0, 1] is a function, mapping transitions to real values between zero and one to express the Certainty Factors (CF) of the corresponding rules; ω : P → [0, 1] is a function, mapping places to real values between zero and one to denote the degrees of truth of the corresponding propositions; β : P → D is an association function, mapping places to propositions. A transition t may fire if ω(pi ) ≥ λi , ∀pi ∈ I(t), where λi is a set of threshold values between zero and one. When transition t fires, the degrees of truth of its input places are used to calculate the degree of truth of its output places according to certain firing schemes. A typical fuzzy production rule is represented in the form of “IF antecedent proposition THEN consequence proposition (CF=µ)”. Such a rule can be mapped to an FPN shown as Fig. 1, where the antecedent and consequence propositions are represented as the input and output places, respectively, and the causality between them is represented as the transition. In real applications of fuzzy rules reasoning, there exist some cases where the antecedent part or consequence part of a fuzzy rule contain “AND” or “OR” connectors. According to [2], these composite fuzzy production rules can be distinguished into the following types and their corresponding FPN structures are depicted along with their firing schemes: Type 1: IF d1 AND d2 AND · · · AND dm , THEN dz (CF = µ). Figure 2 shows the FPN structure and the firing scheme of this case, where pi = β(di ), i = 1, 2, . . . , m, z. Type 2: IF d1 THEN da AND db AND · · · AND dz (CF = µ). Figure 3 shows the FPN structure and the firing scheme of this case, where pi = β(di ), i = 1, a, b, . . . , z. Type 3: IF d1 OR d2 OR · · · OR dm , THEN dz (CF = µ). Figure 4 shows the FPN structure and the firing scheme of this case, where pi = β(di ), i = 1, 2, . . . , m, z. Let pi be a place, and t j be a transition in an FPN. If pi ∈ I(t j ), then pi is called the Nearest Backward Place of t j . All the nearest backward places of t j constitute a Set of the Nearest Backward Place of t j , denoted by SNBP(t j ). If pi ∈ O(t j ), then pi is called the Nearest Forward Place of t j . All the nearest forward places of t j

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 p1

ω(p )

1 Q  Q  Q p2 ω(p2 ) XX Q f (t) = µ X s Q  XX z 3  ..  t .    pm ω(pm )  

 - ω(pz ) pz 

ω(pz ) = min ω(pi ) · f (t) 1≤i≤m

Fig. 2. Fuzzy reasoning process of Type 1

 ω(pb )

 p1

ω(p1 )



pb

 3   f (t) = µ   pa ω(pa ) : -  Q .. Q t . Q Q  s ω(p ) p Q z z 

ω(pa ) = ω(pb ) = · · · = ω(pz ) = ω(p1 ) · f (t)

Fig. 3. Fuzzy reasoning process of Type 2

 p1

ω(p1 )

 

-

µ1 = f (t1 )

Q t1

Q Q - XX Q  XXQ s t2  z ω(pz ) pz X 3  ..  ω(pz ) = max ω(pi ) · f (ti ) . 1≤i≤m    µm = f (tm )  pm ω(pm ) tm  p2

ω(p2 )

µ2 = f (t2 )

Fig. 4. Fuzzy reasoning process of Type 3

constitute a Set of the Nearest Forward Place of t j , denoted by SNFP(t j ). In an FPN, we define pi as a Terminal Place if there does not exist a transition t j such that pi ∈ O(t j ). The reasoning always starts from the terminal places, which are also called Starting Place(s). The degrees of truth of the starting places are either already known before the reasoning starts or provided by the user during the reasoning process. Another kind of place with degrees of truth that we are interested in is called the Goal Place(s).

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3 Backward Tree The difficulty in fuzzy reasoning through an FPN is to determine the sequence of transition firing. A transition can fire under the condition that the degrees of truth of all its input places are not nulls and greater than certain threshold values. If any of its input places has no information about its degree of truth, we need to go backward to find which transition infers this place. Such a backward process proceeds continuously until all input places of the current transition are available. Actually, this backward process can be expressed explicitly as a tree structure, with root(s) as the goal place(s) and leaves as the starting places. We call such a tree structure mapped from an FPN the Backward Tree (BT). The transformation from FPN to BT considers the transitions as the key element. It is different from the existing approaches [2, 3, 5, 6], where only the relationship between places are considered. This innovative idea proposed below makes it easy to perform the transformation, and furthermore, straightforward to implement programs in a computer language. As shown in Section 2, the composite fuzzy production rules can be distinguishably mapped into three types of FPNs. Here, these three types of FPNs can also be transformed into three cases of BTs, respectively: Case 1: Transition t has more than one input places p1 , p2 , . . . , pm and only one output pz (corresponding to Type 1 in Section 2). Figure 5 shows the mapping. Case 2: Transition t has only one input place p1 , and more than one output places pa , pb , . . . , pz (corresponding to Type 2 in Section 2). Figure 6 shows the mapping. Case 3: Transition t has an input place pi and an output place p j (corresponding to Type 3 in Section 2). Figure 7 shows the mapping.

p1

p2

pm

m m... m Q A  Q  s Q UA +t =⇒ ? mpz

p p

p

s 1 s .2. . s m QQ  A t s pz

Fig. 5. Mapping from FPN to BT: Case 1

The transformation from FPNs to BTs starts at the goal place(s). For each goal place in an FPN, find all transitions with their sets of output places connecting to this goal place. These transitions and the goal places form Layer 1 of the BT. For Layer 2, all input places of the transitions of Layer 1 are selected. All transitions with their sets of output places connecting to these input places are chosen. These transitions

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m

p

?t =⇒  Q
 Q +
  s Q m m... m

s1 t  s
s .Q .Q . s

pa pb

pz

pa pb pz Fig. 6. Mapping from FPN to BT: Case 2 pi

m ?t ? m

p

si =⇒

t

s

pj

pj Fig. 7. Mapping from FPN to BT: Case 3

and their output places form Layer 2. Such a process is repeated until all input places of transitions in current layer are the terminal places. The BT has a hierarchical structure constituted by distinct layers from the top to the bottom. Each layer consists of several transitions and their output places. Two adjacent layers are connected so that all the input places of the transitions in the lower layer are included in the set of the output places of the next layer nearer to the top. Obviously, a BT of an FPN comprises two kinds of elements, the layers and the terminal (starting) places. Figure 8 shows an example, where the FPN shown on the left hand side is mapped into its BT shown on the right hand side.

4 Reasoning Algorithm Through the Backward Tree The hierarchical structure in the BT gives the order of transitions firing. The transitions in upper layers should fire before those in lower layers so as to guarantee that every transition has its input places available before firing. Here, the term “input places available” means that the degrees of truth of all input places of the considered transition are already derived. They have been either provided by user or inferred by other transitions and they are represented by values between zero and one. Before illustrating our algorithm, some definitions are given as follows: Layer: An array of the layer structure, which is defined as a Transitions List (TL) and a Places List (PL). Places in the PL are the output places of transitions in the TL.

Backward Reasoning on Rule-Based Systems Modeled p1

p2

m

m

@ R t1 @ p3

p4 ?

?t2

?t3

p5 ? m

p7 ? m

s p1 s p2 QQ  t1

m

m

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=⇒

@ R t4 @ p7 ? m

?t5

s p3

s p4

t2

t3

s p5 s p6 Q  Q t4 s p7 t5

s p8

} } } }

Layer 4

Layer 3 Layer 2

Layer 1

p8 ? m Fig. 8. Mapping from FPN to BT: An example

Known Place Set (KPS): A list of places with their degrees of truth already known either by user input or by inference during the reasoning process. Current Layer Place Set (CLPS): A copy of the PL of current layer. Let ti be a transition and p j be a place in an FPN. Let f (ti ) represent the degree of certainty of ti , ω(p j ) denote the degree of truth of p j . A complete and practical algorithm is represented as follows: Step1: Input – FPN structure (every transition, with its input and output places), and the name of the goal place(s). Step2: Finding the terminal (strating) places, and put them into KPS – For each ti ∈ FPN, put their output places to a list called Output-Place-List; / Output-Place-List, then put p j into KPS (if overlap, For each p j ∈ FPN, if p j ∈ remain one); Put the goal place(s) into PL of Layer[1]; Set num = 1, where num is the sequence number of layer. Step3: Construct the backward tree – copy PL of Layer[num] to CLPS Start do-while loop if CLPS = Ø { for each ti ∈ FPN { for each p j ∈ O(ti ) if p j ∈ CLPS, then put ti into T L of Layer[num]

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(if overlap, remain one); } Reset CLPS; for each ti ∈ T L of Layer[num] { for each p j ∈ I(ti ) { if p j is not a terminal place, then put p j into PL of Layer[num + 1] (if overlap, remain one); copy PL of Layer[num + 1 to CLPS } } num + + } Set num = num − 1. Now num is the sequence number of the highest layer. Step4: Start reasoning layer by layer from the top to the bottom – Start do-while loop if num ≥ 1 { for each ti ∈ T L of Layer[num] { ti fires; for each p j ∈ O(ti ) { ω(p j ) = min(ω(pk ) f˙(ti )), where pk ∈ I(ti ) insert p j into KPS  (if p j already exists in KPS, denoted as p j ,   then ω(p j ) = ω(p j ) if ω(p j ) < ω(p j ) ) } } num − − } Step 5: Output – Degrees of truth of the goal place(s), and degrees of truth of the places related to the goal place(s). The algorithm has two phases. One is the construction of the BT, and the other is the forward reasoning layer by layer. The layer structure provides the sequence of transitions firing, and guarantees that all input places of transitions are available before they fire. Meanwhile, during the construction of the BT, all transitions and places which have no relationship with the goal places are excluded. This makes the algorithm more efficient. After the BT is constructed, the reasoning process is just firing the transitions in turn from the most top layer to the most bottom layer.

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5 Illustration Example Let d0 , d1 , d2 , d3 , d4 , d5 , and d6 be seven propositions. Assume that the knowledge base of a rule-based system contains the following fuzzy production rules: R1 : R2 : R3 : R4 : R5 : R6 : R7 :

IF d0 IF d0 IF d2 IF d1 IF d3 IF d3 IF d0

THEN d1 (CF = 0.90) THEN d2 (CF = 0.85) THEN d3 (CF = 0.95) AND d4 THEN d5 (CF = 0.85) THEN d4 (CF = 0.90) AND d5 THEN d6 (CF = 0.75) THEN d6 (CF = 0.90)

Those rules can be modeled by an FPN shown in Fig. 9. p0

m PP  q t2 P )  t1 p1 ? m

?t7

p2 ? m

?t3 p4

m  
?
 t4

p3 ? m

t5

p5 ? m

j?t6 p6 ? - m Fig. 9. The FPN structure of the illustration example

Assume that we want to know the degree of truth of proposition d6 (in the FPN, it corresponds to p6 ), then the proposition d6 is called the goal proposition, and its corresponding place p6 is called the goal place. This FPN has one starting place p0 and its degree of truth is given as 0.90. The FPN structure being considered can be easily transformed into a BT, as shown in Fig. 10. The corresponding BT has five layers: Layer[1].T L = {t6 ,t7 }, Layer[1].PL = {p6 }; Layer[2].T L = {t3 ,t4 }, Layer[2].PL = {p3 , p5 };

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s p0 t2

s p2 t3

s p0

s p0

s p3

t2

t1

t5

s p2

s p1 s p4 Q  Q t

t3

4

s p0 s p3 s p5 PP HH   P 7 t6 HtH HHs p

6

} } } } }

Layer 5

Layer 4

Layer 3

Layer 2

Layer 1

Fig. 10. Backward tree mapped from the FPN shown in Fig. 9

Layer[3].T L = {t2 ,t1 ,t5 }, Layer[3].PL = {p2 , p1 , p4 }; Layer[4].T L = {t3 }, Layer[4].PL = {p3 }; Layer[5].T L = {t2 }, Layer[5].PL = {p2 }; According to our proposed algorithm, after constructing the BT, transitions can fire in turn from the top layer (Layer[5]) to the bottom layer (Layer[1]). The reasoning process is shown as follows: Layer[5]: t2 Layer[4]: t3 Layer[3]: t2 t1 t5 Layer[2]: t3 t4 Layer[1]: t6 t7

fires, then KPS = {{p0 , 0.90}, {p2 , 0.77}} ; fires, then KPS = {{p0 , 0.90}, {p2 , 0.77}, {p3 , 0.73}} ; fires, then KPS = {{p0 , 0.90}, {p2 , 0.77}, {p3 , 0.73}} ; fires, then KPS = {{p0 , 0.90}, {p2 , 0.77}, {p3 , 0.73}, {p1 , 0.81}} ; fires, then KPS = {{p0 , 0.90}, {p2 , 0.77}, {p3 , 0.73}, {p1 , 0.81}, {p4 , 0.73}} ; fires, then KPS = {{p0 , 0.90}, {p2 , 0.77}, {p3 , 0.73}, {p1 , 0.81}, {p4 , 0.73}} ; fires, then KPS = {{p0 , 0.90}, {p2 , 0.77}, {p3 , 0.73}, {p1 , 0.81}, {p4 , 0.73}, {p5 , 0.58}} ; fires, then KPS = {{p0 , 0.90}, {p2 , 0.77}, {p3 , 0.73}{p1 , 0.81}, {p4 , 0.73}, {p5 , 0.58}, {p6 , 0.44}} ; fires, then KPS = {{p0 , 0.90}, {p2 , 0.77}, {p3 , 0.73}, {p1 , 0.81}, {p4 , 0.73}, {p5 , 0.58}, {p6 , 0.81}} ;

Now, the degree of truth of the goal place p6 has been derived. It is valued as 0.81. Other places, which are relative to place p6 , also have their degrees of truth be achieved.

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6 Conclusion We have presented a new backward reasoning technique for deriving possible conclusions in fuzzy rule-based systems modeled by FPNs. The crux in such systems is to determine the sequence of transitions firing. Our technique solves this problem successfully over existing forward and backward reasoning methods. In our approach, a fuzzy rule-based system is transformed into an FPN first. Given the goal place(s), an FPN is mapped further into a BT, which has distinct layers from the top to the bottom. Each layer consists of several transitions and their output places. The hierarchical structure of the BT provides the order of transitions firing. The nearer to the top the transition, the earlier it fires. The transition nearer to the top layer fires before those nearer to the bottom layer, so as to guarantee that every transition has its input places available before firing. To fire the transition, we follow the most common model, that is, the degrees of truth of output places are equal to the minimum of degrees of truth of input places multiplying the degree of certainty of the transition. This model holds a basic supposition that there is no interaction among the input places of a transition. However, in many real applications, such inherent interactions cannot be ignored. In our future work, we will consider the new nonlinear multiregression model proposed in [7, 8], and merge this model with the backward reasoning technique presented here.

Acknowledgment The work described in this paper was fully supported by the Research Grants Council of the Hong Kong Special Administrative Region (Project No. CUHK 4185/00E).

References 1. Arnould T, Tano S (1995) Interval-valued fuzzy backward reasoning. In: IEEE Trans Fuzzy Systems 3:425-437 2. Chen SM (1990) Knowledge representation using fuzzy Petri nets. In: IEEE Trans Knowledge and Data Engineering 2:311–319 3. Chen SM (2000) Fuzzy backward reasoning using fuzzy Petri nets. In: IEEE Trans SMCPart B: Cybernetics 30:846–855 4. Li X, Lara-Rosano F (2000) Adaptive fuzzy Petri nets for dynamics knowledge representation and inference. In: Expert System Applications 19:235-241 5. Looney CG (1988) Petri fuzzy nets for rule-based decision making. In: IEEE Trans SMC 18:178–183 6. Pedrycz W, Gomide F (1994) A generalized fuzzy Petri net model. In: IEEE Trans Fuzzy Systems 2:295–301 7. Wang Z, Leung KS, Wang J (1999) A genetic algorithm for determining nonadditive set functions in information fusion. In: Fuzzy Sets and Systems 102:463-469 8. Wang Z, Leung KS, Wang J, Xu K (2000) Nonlinear nonnegative multiregression based on Choquet integrals. In: International Journal of Approximate Reasoning 25:71-87 9. Zadeh LA (1965) Fuzzy sets. In: Information and Control 8:338–353

CHAPTER 6 On The Generalization of Fuzzy Rough Approximation Based on Asymmetric Relation Rolly Intan1 and Masao Mukaidono2 1 2

Petra Christian University, Jl. Siwalankerto 121-131, Surabaya 60236, Indonesia. [email protected] Meiji University, Higashi-mita 1-1-1, Kawasaki-shi, Kanagawa-ken, Japan, [email protected]

Abstract: An asymmetric relation, called a weak similarity relation, is introduced as a more realistic relation in representing the relationship between two elements of data in a real-world application. A conditional probability relation is considered as a concrete example of the weak similarity relation by which a covering of the universe is provided as a generalization of a disjoint partition. A generalized concept of rough approximations regarded as a kind of fuzzy rough set is proposed and defined based on the covering of the universe. Additionally, a more generalized fuzzy rough approximation of a given fuzzy set is proposed and discussed as an alternative to provide interval-valued fuzzy sets. Their properties are examined. Keywords: Rough Sets, Fuzzy Sets, Fuzzy Rough Sets, Conditional Probability Relations

1 Introduction Rough set theory generalizes classical set theory by allowing an alternative to formulate sets with imprecise boundaries. A rough set is basically an approximate representation of a given crisp set in terms of two subsets, called lower and upper approximations, derived from a crisp partition defined on the universal set involved [9]. In the partition, every element belongs to one equivalence class and two distinct equivalence classes are disjoint. Formally, let U denote a non-empty universal set, and E be an equivalence relation on U. A partition of the universe is referred to as a quotient set expressed by U/E, where [x]R denotes an equivalence class in U/E that contains x ∈ U. A rough set of A ⊆ U is represented by a pair of lower and upper approximations. The lower approximation, apr(A) = {x ∈ U|[x]E ⊆ A}, =



{[x]E ∈ U/E|[x]E ⊆ A},

is the union of all equivalence classes in U/E that are contained in A. The upper approximation, R. Intan and M. Mukaidono: On The Generalization of Fuzzy Rough Approximation Based on Asymmetric Relation, Studies in Computational Intelligence (SCI) 2, 73–88 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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/ apr(A) = {x ∈ U|[x]E ∩ A = 0}, =



/ {[x]E ∈ U/E|[x]E ∩ A = 0},

is the union of all equivalence classes in U/E that overlap with A. However, as pointed out in [14, 11], even if it is easy to analyze, the rough set theory built on a partition induced by equivalence relations may not provide a realistic view of relationships between elements in a real-world application. Instead a covering of the universe [14] might be considered as an alternative to provide a more realistic model of rough sets. A covering of the universe, C = {C1 , ...,Cn }, is a family of subsets of a non-empty universe U such  that U = {Ci |i = 1, ..., n}. Two distinct sets in C may have a non-empty overlap. An arbitrary element x of U may belong to more than one set in C. The sets in C may describe different types or various degrees of similarity between elements of U. This paper, as an extension of [5], introduces a weak similarity relation with intent to provide a more realistic model of relation in representing relationships between two elements of data. Naturally, a relationship between two elements in a real-world application is not necessarily symmetric and transitive as characterized by the weak similarity relation. Fuzzy conditional probability relations proposed in [2] is regarded as a concrete example of the weak similarity relation. Moreover, weak similarity relation as well as conditional probability relations can be considered as generalizations of the similarity relation proposed in [17]. A covering, called α-coverings of the universe, is then constructed dealing with the conditional probability relations. A generalized concept of rough approximations is introduced and defined based on α-coverings of the universe into two interpretations, element-oriented generalization and similarity-class-oriented generalization. The generalized concept of rough approximations is regarded as a kind of fuzzy rough set. Some symmetric relations can be performed dealing with the (fuzzy) conditional probability relation. Through the symmetric relations, we introduce various formulations of generalized rough approximations, and examine their properties in the relation to the generalized rough approximations induced by the conditional probability relation. Additionally, a more generalized fuzzy rough approximation of a given fuzzy set is proposed and discussed as an alternative to provide interval-valued fuzzy sets from an information system. Their properties are examined.

2 Conditional Probability Relations The similarity relation proposed by Zadeh [17] generalizes equivalence relations for dealing with fuzzy data. A similarity relation on a given universe, U, maps each pair of elements in the universe to an element in the closed interval [0,1]. Definition 1. [17] A similarity relation is a mapping, s : U ×U → [0, 1], such that for x, y, z ∈ U, (a) Reflexivity : s(x, x) = 1, (b) Symmetry : s(x, y) = s(y, x), (c) Max−min transitivity : s(x, z) ≥ maxy∈U min[s(x, y), s(y, z)]. There are some considerable criticisms [12, 11] about the use of similarity relation, especially for the property of max-min transitivity. Therefore, a proximity relation [1] (also called a resemblance relation) is proposed by (just) considering reflexive and symmetric properties. Moreover, it should be mentioned that even symmetry as required in similarity and proximity relations is too strong property to represent relationships between two elements or objects in a real-world application (see [2]). Although it is true to say that if “x is similar to y” then “y is

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similar to x”, but these two statements might have different degrees of similarity. Hence, we consider a conditional probability relation as a more realistic relation in representing relationships between two elements. Definition 2. A conditional probability relation is a mapping, R : U × U → [0, 1], such that for x, y ∈ U, |x ∩ y| , (1) R(x, y) = P(x | y) = P(y → x) = |y| where R(x, y) means the degree y supports x or the degree y is similar to x. In the definition of conditional probability relations, the probability values may be estimated based on the semantic relationships between two elements by using the epistemological or subjective view of probability theory. The conditional probability relation could be used for calculating the degree of similarity relationships between elements (objects) in the universe U in terms of an information table. When every object in U is represented by a subset of attributes as in the case of a binary information table, we have a simple procedure for estimating the conditional probability relation as shown in Definition 2, where | · | denotes the cardinality of a set. Consider a binary information table given in Table 1, where the set of objects, U = {O1 , O2 , O3 }, is characterized by a set of attributes, At = {a1 , a2 , . . . , a8 }.

Table 1. Binary Information Table Obj O1 O2 O3

a1 0 1 0

a2 0 1 0

a3 1 0 1

a4 0 1 1

a5 1 0 0

a6 0 0 0

a7 0 1 1

a8 0 0 1

As shown in Table 1, O1 = {a3 , a5 }, O2 = {a1 , a2 , a4 , a7 }, and O3 = {a3 , a4 , a7 , a8 }. Therefore, we have: R(O1 , O2 ) = 0, R(O1 , O3 ) = 1/4, R(O2 , O3 ) = 2/4, R(O2 , O1 ) = 0, R(O3 , O1 ) = 1/2, R(O3 , O2 ) = 2/4. The notion of binary information table can be easily generalized to a fuzzy information table by allowing a number in the unit interval [0, 1] for each cell of the table. The number is the degree to which an object has a particular attribute. Each object might be represented as a fuzzy set on the set of attributes. The degree of similarity between two objects represented by two fuzzy sets on the set of attributes can be calculated by a fuzzy conditional probability relation [2, 3]. In this case, |x| = ∑a∈At µx (a), where µx is a membership function of x over a set of attribute At, and intersection is defined by minimum in order to obtain a property of reflexivity, even though some operations of t-norm could be used.

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Definition 3. Let µx and µy be two fuzzy sets over a set of attribute At for two elements x and y of a universe of elements U. A fuzzy conditional probability relation is defined by: min{µx (a), µy (a)} ∑ R(x, y) = a∈At . (2) ∑a∈At µy (a) It can be easily verified that (fuzzy) conditional probability relation R satisfies properties as shown in Proposition 1. Proposition 1. [6] Let R(x, y) be (fuzzy) conditional probability relation of x given y and R(y, x) be (fuzzy) conditional probability relation of y given x, such that for x, y, z ∈ U, (r1) R(x, y) = R(y, x) = 1 ⇐⇒ x = y, (r2) [R(y, x) = 1, R(x, y) < 1] ⇐⇒ x ⊂ y, (r3) R(x, y) = R(y, x) > 0 =⇒ |x| = |y|, (r4) R(x, y) < R(y, x) =⇒ |x| < |y|, (r5) R(x, y) > 0 ⇐⇒ R(y, x) > 0, (r6) [R(x, y) ≥ R(y, x) > 0, R(y, z) ≥ R(z, y) > 0] =⇒ R(x, z) ≥ R(z, x), (r7) R(x, z) ≥ max(R(x, y) + R(z, y) − 1, 0) ∗ R(y, z)/R(z, y). Proof. (r1) [R(x, y) = |x∩y| |y| = 1 ⇒ x ∩ y = y, R(y, x) = 1 ⇒ x ∩ y = x] =⇒ x = y, (r2) [R(y, x) = 1 ⇒ x ∩ y = x, R(x, y) < 1 ⇒ x ∩ y ⊂ y] =⇒ x ⊂ y, |x∩y| (r3) |x∩y| |y| = |x| > 0 =⇒ |y| = |x|, (r4)

|x∩y| |y|

|x∩y| |x| =⇒ |x| < |y|, [R(x, y) = |x∩y| |y| > 0 ⇒ |x ∩ y| > 0


0, (r6) [R(x, y) ≥ c j (y, x) > 0 ⇒ |x| ≥ |y|, R(y, z) ≥ R(z, y) > 0 ⇒ |y| ≥ |z|] =⇒ [|x| ≥ |z| ⇒ R(x, z) ≥ R(z, x)], (r7) R(x, z) ≥ max(R(x, y) + R(z, y) − 1, 0) ∗ R(y, z)/R(z, y) ⇒ x ∩ z ≥ max(x ∩ y + z ∩ y − y, 0), where max(x ∩ y + z ∩ y − y, 0) could be verified as the least possible solution for x and z have intersection through y. From Proposition 1, (r7) follows the formulation of the Lukasiewicz intersection (bounded difference) to perform relationships between x and z through y, and in the relation to the properties (r1), (r5) and (r6), we can define an interesting mathematical relation, called the weak similarity relation, based on their constraints represented by axioms in representing the similarity level of a fuzzy relation as follows. Definition 4. A weak similarity relation is a mapping, s : U × U → [0, 1], such that for x, y, z ∈ U, (a) Reflexivity : s(x, x) = 1, (b) Conditional symmetry : if s(x, y) > 0 then s(y, x) > 0, (c) Conditional transitivity : if s(x, y) ≥ s(y, x) > 0 and s(y, z) ≥ s(z, y) > 0 then s(x, z) ≥ s(z, x).

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By definition, the similarity relation is regarded as a special case (type) of the weak similarity relation, and (fuzzy) conditional probability relation is a concrete example of the weak similarity relation.

3 Generalized Rough Approximations In this section, we generalize the classical concept of rough approximations based on a covering of the universe induced by the conditional probability relation. Two different interpretations of lower and upper approximations are introduced in the presence of an α-covering of the universe. 3.1 Generalizations Based on The Asymmetric Property First, based on the asymmetric property of conditional probability relations, two asymmetric similarity classes of a particular element x ∈ U as a basis of constructing a covering is given by the following definition. Definition 5. Let U be a non-empty universe, and R be a conditional probability relation on U. For any element x ∈ U, Rαs (x) and Rαp (x) are defined as a set that supports x and a set supported by x, respectively, such that Rαs (x) = {y ∈ U|R(x, y) ≥ α}, Rαp (x) = {y ∈ U|R(y, x) ≥ α},

(3) (4)

where α ∈ (0, 1]. Rαs (x) can also be interpreted as a set that is similar to x. On the other hand, Rαp (x) is a set to which x is similar. The relationship between Rαs (x) and Rαp (x) satisfies the following theorems. Theorem 1. y ∈ Rαp (x), |x| ≥ |y| ⇒ y ∈ Rαs (x), y ∈ Rαs (x),

(5)

|x| ≤ |y| ⇒ y ∈ Rαp (x).

(6)

Proof. Through Bayes’ theorem: P(x|y) = |x|×P(y|x) , (1) in Definition 2 can be ex|y| pressed by: |x| × R(y, x). R(x, y) = |y| Proof of (5): y ∈ Rαp (x) ⇔ R(x, y) ≥ α ⇒ |y| |x|

|x| |y|

× R(y, x) ≥ α ⇒ R(y, x) ≥

|x| ≥ |y| ⇒ × α ≤ α ⇒ R(y, x) ≥ α ⇔ Likewise (6) can be proved.

y ∈ Rαs (x).

|y| |x|

× α,

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Theorem 2. Let x ∈ U, S1 = Rαp (x) − (Rαp (x) ∩ Rαs (x)) and S2 = Rαs (x) − (Rαs (x) ∩ Rαp (x)). Then, (7) ∀u ∈ S1 , ∀v ∈ S2 , |v| < |u|. Proof. Every u ∈ S1 satisfies R(u, x) > R(x, u), for u ∈ Rαp (x) and u ∈ Rαs (x), such that u ∈ S1 ⇒ R(u, x) > R(x, u) ⇒ |x∩v| |x|

|x∩v| |v|

|x∩u| |x|

>

|x∩u| |u|

⇒ |x| < |u|. Similarly, v ∈ S2 ⇒ R(v, x)
0}, for short. µapr (A) (x) and µapres (A) (x) are grades of membership of x in apres (A) and apres (A), es respectively. Similarly, µapr∗ (A) (y) and µapr∗cs (A) (y) are grades of membership of y in cs apr∗cs (A) and apr∗cs (A), respectively (Note: ∗ ∈ {m, M}). Using p-similarity classes, similarly Definition 8 will provide µapr µ

aprm (A) cp

(x), µ

aprm cp (A)

ep

(A) (x), µaprep (A) (x),

(x), µaprM (A) (x) and µaprMcp (A) (x). cp

Since µRs (x) (y) = µR p (y) (x) as shown in Definition 7, we may represent Definition 8 by using R p as follows: (i) Element-oriented generalization:
 µapr (A) (x) = inf min(µR p (y) (x), µA (y)) , (40) es y∈U
 (41) µapres (A) (x) = sup min(µR p (y) (x), µA (y)) . y∈U

(ii) Similarity-class-oriented generalization, for y ∈ U:

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 µ (y) = inf inf min(µR p (z) (x), µA (z)) , ψ z∈U 


 , µaprM (A) (y) = sup inf min(µR p (z) (x), µA (z)) aprm (A) cs

ψ

cs

z∈U


 , µaprmcs (A) (y) = inf sup min(µR p (z) (x), µA (z)) ψ z∈U 


 µaprMcs (A) (y) = sup sup min(µR p (z) (x), µA (z)) , ψ

(42) (43) (44) (45)

z∈U

where ψ = {x ∈ U|µR p (y) (x) > 0}, for short. Obviously, apres (A) and apres (A) as well as apr∗cs (A) and apr∗cs (A) are fuzzy sets, where we have, ∀y ∈ U, µapr

es

(A) (y) ≤ µA (y) ≤ µapres (A) (y),

µaprm (A) (y) ≤ µaprM (A) (y) ≤ µA (y) ≤ µaprmcs (A) (y) ≤ µaprMcs (A) (y). cs

cs

Moreover, the relationship between element-oriented generalization and similarityclass-oriented generalization is represented by, µapr

es

(A) (y) ≤ µaprM (A) (y), cs

µapres (A) (y) ≤ µaprMcs (A) (y), where relationship between µapr (A) (y) and µaprm (A) (y) as well as relationship bees cs tween µapres (A) (y) and µaprmcs (A) (y) cannot be queried. Also, the pairs of µapr (A) (y), µapres (A) (y), µaprm (A) (y), µaprmcs (A) (y) and es cs µaprM (A) (y), µaprMcs (A) (y) are regarded as lower and upper membership functions of cs y in A given on s-fuzzy similarity class. Lower and upper membership functions are the bounds of an interval value characterized by an interval-valued fuzzy set, and all pairs of lower and upper membership functions can be combined to represent a fuzzy set of type 2. In this case, Definition 8 shows an alternative to obtain an interval-valued fuzzy set from an information system via a generalized fuzzy rough approximation of a fuzzy set. Let A be defined as an interval-valued fuzzy set given an ordinary fuzzy set A. For y ∈ U, µes A (y) = [µapr

es

(A) (y), µapres (A) (y)],

m M (y)], µcs (y)]. µcs A (y) = [µaprm (A) (y), µaprm A (y) = [µaprM (A) (y), µaprM cs (A) cs (A) cs

cs

A fuzzy set of type 2 represented by a membership function, ϒ : U → F ([0, 1]), where F ([0, 1]) is a fuzzy power set of [0, 1], could be defined based on element-oriented generalization and similarity-class-oriented generalization by: ϒAes (y) = {

1 β , }, (A) (y) µA (y) µapres (A) (y)

β µapr

es

,

On The Generalization of Fuzzy Rough Approximation

ϒAcs (y) = {

87

β2 1 β2 β1 β1 , , , , }, µaprm (A) (y) µaprM (A) (y) µA (y) µaprmcs (A) (y) µaprMcs (A) (y) cs

cs

where β, β1 , β2 ∈ (0, 1], β1 ≤ β2 . Both ϒAes (y)(µA (y)) and ϒAcs (y)(µA (y)) should be equal to 1 representing the most accurate membership function of y in A, and its membership function will decrease when difference between the value and µA (y) is greater (represented by β1 ≤ β2 ). As discussed in the last section, symmetric relations defined in (19)-(21) can be used to provide various formulations of the generalized fuzzy rough approximations as proposed in this section. Relationship between the generalized fuzzy rough approximations based on the conditional probability relation (regarded as a asymmetric relation) and the generalized fuzzy rough approximations based on the symmetric relations would give interesting properties and contributions in the relation to fuzzy granularity. We will discuss this issue in future work.

5 CONCLUSIONS A notion of asymmetric relation, called the weak similarity relation was introduced. Two examples of such relations, conditional probability and fuzzy conditional probability, were suggested for construction and interpreting coverings of the universe. Based on such covering, rough approximations were generalized based on elementoriented generalization and similarity-class-oriented generalization. Such generalization was considered as a type of fuzzy rough set. Some symmetric relations were proposed dealing with the (fuzzy) conditional probability relation. Through symmetric relations, we introduced various formulations of generalized rough approximations, and examined their properties in relation to the generalized rough approximations induced by the conditional probability relation. Additionally, a notion of a generalized fuzzy rough set given a fuzzy set was proposed and discussed as an alternative to provide interval-valued fuzzy sets as well as a fuzzy set of type 2 from an information system.

References 1. Dubois D, Prade H (1990) Rough Fuzzy Sets and Fuzzy Rough Sets. Intern. J. of General Systems 17(2-3): 191-209 2. Intan R, Mukaidono M (2000) Conditional Probability Relations in Fuzzy Relational Database. Proceedings of RSCTC’00, LNAI 2005, Springer-Verlag, 251-260 3. Intan R, Mukaidono M, Yao Y.Y (2001) Generalization of Rough Sets with α-coverings of the Universe Induced by Conditional Probability Relations. Proceedings of International Workshop on Rough Sets and Granular Computing, 173-176 4. Intan R, Mukaidono M (2002) Generalization of Rough Membership Function based on α-Coverings of the Universe. Proceedings of AFSS’02, LNAI 2275, Springer-Verlag, 129-135

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5. Intan R, Mukaidono M (2002) Generalization of Fuzzy Rough Sets By Fuzzy Covering Based On Weak Fuzzy Similarity Relation. Proceeding of Fuzzy Sets and Knowledge Discovery 2002, 344-348 6. Intan R, Mukaidono M (2002) Generalized Fuzzy Rough Sets By Conditional Probability Relations. International Journal of Pattern Recognition and Artificial Intelligence 16(7), World Scientific, 865-881 7. Inuiguchi M, Tanino T (2001) On Rough Sets under Generalized Equivalence Relations. Proceedings of International Workshop on Rough Sets and Granular Computing, 167-171 8. Komorowski J, Pawlak Z, Polkowski L, Skowron A (1999) ‘Rough Sets: A Tutorial. In: Pal S.K, Skowron A (eds) Rough Fuzzy Hybridization. Springer, 3-98 9. Klir G.J, Yuan B (1995) Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, New Jersey 10. Pawlak Z (1982) Rough Sets. International Journal Computation & Information Science 11: 341-356 11. Slowinski R, Vanderpooten D (2000) A Generalized Definition of Rough Approximations Based on Similarity. IEEE Transactions on Knowledge and Data Engineering 12(2): 331-336 12. Tversky A (1977) Features of Similarity. Psychological Rev. 84(4): 327-353 13. Yamaguchi Y., Intan R., Emoto M., Mukaidono M. (2003) Generalization of Rough Sets Using Active and Passive Relations. Proceeding of Intech 2003, 539-544 14. Yao Y.Y, Zhang J.P (2000) Interpreting Fuzzy Membership Functions in the Theory of Rough Sets. Proceedings of RSCTC’00, LNAI 2005, Springer-Verlag, 82-89 15. Yao Y.Y (1996) Two View of the Theory of Rough Sets in Finite Universe. International Journal of Approximate Reasoning 15: 291-317 16. Yao Y.Y (1998) A comparative study of fuzzy sets and rough sets. International Journal of Information Science 109: 227-242 17. Zadeh L.A (1970) Similarity Relations and Fuzzy Orderings. Inform. Sci.3(2): 177-200

CHAPTER 7 A new approach for the fuzzy shortest path problem Tzung-Nan Chuang1 and Jung-Yuan Kung2 1

Department of Merchant Marine, National Taiwan Ocean University Department of Information Management, Chinese Naval Academy

2

Abstract: Many researchers have paid much attention to the fuzzy shortest path problem since it is central to many applications. In this problem, the fuzzy shortest length and the corresponding shortest path are useful information for decision makers. In this paper, we propose a new approach that can obtain the important information. First, we propose a heuristic procedure to find the fuzzy shortest length among all possible paths in a network. It is based on the idea that a crisp number is a minimum number if and only if any other number is larger than or equal to it. It owns a firm theoretic base in fuzzy sets theory and can be implemented effectively. Secondly, we propose a way to measure the similarity degree between the fuzzy shortest length and each fuzzy path length. The path with the highest similarity degree is the shortest path. An illustrative example is given to demonstrate our proposed approach.

1. INTRODUCTION In a network, the arc length may represent time, or cost. The shortest path problem has received lots of attention from researchers in past decades because it is important to many areas such as communication, transportation, scheduling, and routing. In practical situations it is reasonable to assume that each arc length is a fuzzy set. There were several methods reported to solve the shortest-path problem in the open literature [1-10]. Dubois and Prade [1] first treated the fuzzy shortest problem. In their method, the shortest length can be found, but there may not exist a corresponding actual path in the network. To overcome this shortcoming, Kelin [6] proposed a dynamical programming recursion based fuzzy algorithm that specified each arc length with an integer value between 1 and a fixed number. However, such a constraint seems impractical in real applications. On the other hand, Yager [4] introduced the idea of possibilistic production system, which was composed of states and was possibilistic production rules necessary to traverse between states, to investigate a path problem. The concept of possibility was utilized for traversing between states and similar to that of uncertainty introduced in [15]. He developed a general algorithm to find a path with maximal possibility from an initial state (or node) to a goal state (or node) and called it a path of least resistance. In [10], Okada and Soper proposed a fuzzy algorithm, which was based on multiple labeling method [2,3] for solving a multi-criteria shortest path problem, to offer non-dominated paths to a decision maker. In order to reduce the number of nondominated paths, they [10] also used the concept of the possibility level h 䯴 the degree of optimism of a decision maker䯵 for order relation which was introduced by [13] and easily T.-N. Chuang and J.-Y. Kung: A new approach for the fuzzy shortest path problem, Studies in Computational Intelligence (SCI) 2, 89–100 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com


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implemented for computer computation by [14]. Nevertheless, the fuzzy path length that corresponds to the obtained shortest path via such an algorithm is not necessarily the actual shortest length. In this paper, to avoid the problems stated above, we propose an approach that can obtain the fuzzy shortest length and the corresponding shortest path. This approach is composed of two parts. First, we propose a heuristic procedure to find the fuzzy shortest length among all possible paths in a network. This procedure can yield a permissible solution and is meaningful, rational, computationally efficient, and general in nature. It is based on the idea that a crisp number is a minimum number if and only if any other number is larger than or equal to it. It owns a firm theoretic base in fuzzy sets theory and can be implemented effectively. Secondly, we propose a way to measure the similarity degrees between the fuzzy shortest length and each fuzzy path lengths. The path with the highest similarity degree is the shortest path.

2. FUZZY SHORTEST LENGTH HEURISTIC PROCEDURE In this section, we design a procedure to determine the fuzzy shortest path length. If the path length is crisp, lmin(prv) is defined as the minimum path length among all given path lengths l(prv). That is, a number (l(prv)) is the minimum number (lmin(prv)) if and only if any other number is greater than or equal to it. This idea is generalized to the fuzzy shortest path length Lmin(prv). Assume there are m paths pirv from node r to v in a network, i = 1, 2, …, m. L(pirv) is ith fuzzy path length. We want to determine Lmin(prv). For the following derivation, L(pirv) and Lmin(prv) is replaced by Li and Lmin, respectively, for case of description. Let µL1(x) denote the membership function for the possible time x with respect to L1. As noted for crisp value, x is the minimum if any other number is greater than or equal to x. In other words, if x with respect to L1 is the shortest time, x does exist and the other time with respect to Lk (k z 1) smaller than x does not exist. This idea is extended to determine Lmin. We define SL1 as the fuzzy set that the possible time x with respect to L1 is the shortest one. Let µSL1(x) be the membership function for x with respect to SL1. Therefore, µSL1(x)= µL1(x) š µL’2(x) š µL’3(x) š…š µL’m(x) = min[µL1(x), µL’2(x), µL’3(x),…, µL’m(x)] = min[µL1(x),

min [µL’k(x)]]

(1)

k z1

Formula (1) can also be expressed as follows䰆 SL1 = L1 ŀ L’2 ŀ…ŀ L’m = L1 ŀ (

 L’k)

(2)

k z1

where L’k denotes the fuzzy set that the other time y with respect to Lk (k z 1) smaller than x does not exist, µL’k(x) is the membership function for x with respect to L’k and the minimum operator is used for the intersection of two fuzzy sets. Further, µL’k(x) can be detremined as y x

µL’k(x) = 1 - Max µLk(y)(3) yLK

Assume the triangular membership function of Lk is represented as (ak, bk, ck). The following two cases may exist for calculatingµL’k(x):

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y x

Case (a): x is at the right of bk. For this case, µL’k(x) = 1 - max µLk(y) = 1µLk(bk) = yLK

y x

11.0 = 0.0. Case (b): x is at the left of bk. For this case, µL’k(x) = 1 - max µLk(y) = yLK

1µLk(x). Therefore, for a triangular membership function of Lk = (ak, bk, ck), called a half-inverse fuzzy set and can be found as follows:

, x ! bk ­0 ½ ° ° x a x b 1  P ( ) , d d ® Lk k k¾ °1 ° , x  ak ¯ ¿

P L' ( x ) k

µL’k(x) is

(4)

Substitution of (4) into (1) can yieldµSL1(x). Similarly, we can get the membership functionµSLk(x).Let µLmin(x) be the membership function for x with respect to Lmin. It can be determined as follows. µLmin(x) = µSL1(x) › µSL2(x) › µSL3(x) ›…› µSLm(x) m

=

max µSLt(x) t 1 m

= max [min[µLt(x), t 1

min µL’k(x)]]. k zt

(5)

According to the above derivation, the fuzzy shortest path length procedure can be performed as follows.

Fuzzy shortest path length procedure Input: m triangular fuzzy sets, namely, L1, L2, …, Lm. Output: The fuzzy shortest path length Lmin. Step 1: For each fuzzy set Lk, k = 1 to m, find its half-inverse membership function L’k. Step 2: For each fuzzy set Lt, t = 1 to m, find: µSLt(x) = min[µLt(x), min µL’k(x)]. k zt

m

Step 3: SetµLmin(x) =

max µSLt(x). t 1

Step 4: Normalize Lmin where Lmin is then output as the fuzzy shortest path length. Although the fuzzy shortest path length procedure stated above can be used to find the fuzzy shortest path length, the computation associated with it is somewhat complicated. We hope to reduce the computational complexity and membership-function complexity. First, we discuss the fuzzy shortest path length between two fuzzy path lengths. We simply want to obtain three points to describe the new intermediate results, with the b point representing the highest membership function. Take two fuzzy path lengths L1 and L2 for an example.

There are 24 possible combinations of shapes for L1 and L2. According to the fuzzy shortest path length procedure, we can find the fuzzy shortest path length for each case.

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The fuzzy shortest path length for the other cases can be similarly derived. Summarizing the 24 cases, we can derive a formula for finding fuzzy shortest path length as follows. The fuzzy shortest path length formula on two fuzzy path lengths

Let two fuzzy path length L1 = (a1, b1, c1) and L2 = (a2, b2, c2). Then the fuzzy shortest path length Lmin = (a, b, c) can be determined by if min(b1 , b2 ) d max(a1 , a 2 ) ­ min(b1 , b2 ), ° (b u b )  ( a u a ) 1 2 ® 1 2 , if min(b1 , b2 ) ! max(a1 , a 2 ) °¯ (b1  b2 )  ( a1  a 2 ) a = min (a1, a2) c = min [min (c1, c2), max (b1, b2)] b

If there are more than two fuzzy path lengths, we can design the procedure based on the above formula. First, we form the set Q by sorting Li in ascending order of bi. Then, we find the fuzzy shortest path length from the first and second fuzzy path length in Q using the above formula. Next, we find the fuzzy shortest path length from the previous obtained fuzzy shortest path length and the third fuzzy path length in Q using the same formula. By repeating the step, the final fuzzy shortest path length can be found. According to the above derivation, the fuzzy shortest path length heuristic procedure can be performed as follows: Fuzzy shortest length heuristic procedure

Input: Li = (ai, bi, ci), i = 1, 2, 3 ….., m, where Li denotes the triangular fuzzy length. Output: Lmin = (a, b, c) where Lmin denotes the fuzzy shortest length. Step 1: Form the set Q by sorting Li in ascending order of bi; Q = {Q1, Q2, Q3, …, Qm} where Qi = (a’i, b’i, c’i), i = 1, 2, ….., m. Step 2: Set Lmin = (a, b, c) = Q1 = (a’1, b’1, c’1). Step 3: Let i = 2. Step 4: Calculate

b

­ b; if b d a i' ° ' ' ® (b u bi )  ( a u a i ) ' ° (b  b ' )  ( a  a ' ) ; if b ! a i i i ¯

a = min (a, a’i); c = min (c, b’i).; Step 5: Set Lmin = (a, b, c). Step 6: Set i = i+1. Step 7: Repeat Step 4 to Step 6 until i = m +1.

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Remark: In Step 1, if there are more than two Lis with the same bi, then the corresponding sorting order for Li can be made arbitrarily and it will not affect the result as we use this procedure. It can be easily seen that the heuristic procedure is very simple and needs only a light computational load. Example 1: It is supposed that there are five fuzzy arc lengths which are L1 = (201,

248, 281), L2 = (199, 253, 292), L3 = (148, 195, 227), L4 = (186, 234, 259), L5 = (174, 222, 245), respectively. This procedure can be executed as follows: Step 1: Q1=(177, 195, 256 ) = L3, Q2 = (160, 222, 235 ) = L5, Q3 = (159,234,249 ) = L4, Q4 = (201, 248, 271 ) = L1, Q5 = (196, 253, 282 ) = L2, Step 2: Set Lmin = (a, b, c) = Q1 = (a’1, b’1, c’1) =(177, 195, 256). Step 3: Let i =2. Step 4: Calculate (a, b, c).  b = 195 > a’2 = 160, ?b

(195 u 222 )  (177 u 160 ) (195  222 )  (177  160 )

187 .125 ;

a = min(177,160) = 160 ; c = min(256,222) = 222. Step 5: Set Lmin = (160, 187.125 ,222). Step 6: Set i = i+1=2+1=3 . Step 7: Repeat Step 4 to Step 6 until i = 6. After the above procedure is executed, the shortest length is (159, 179.65, 222).

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3. A NEW APPROACH FOR THE FUZZY SHORTEST PATH PROBLEM Several approaches had been presented to treat the fuzzy shortest path problem and appropriately provide decision makers with the shortest path [1-10]. In this section, we intend to propose a new approach, which consists of a new fuzzy shortest length heuristic procedure and a way for determining the similarity degree between two fuzzy lengths, to find the fuzzy shortest length and offer a corresponding shortest path for decision makers. The fuzzy shortest length heuristic procedure has already been presented in Section 2 and an illustration has also been given in Example 1. Now, we want to show a way to measure the similarity degree between two fuzzy lengths. Based on the idea that the larger the intersection area of two triangles is, the higher the similarity degree between them is, we use the intersection area of two triangular fuzzy sets to measure the similarity degree between them. Let the i-th fuzzy path length Li = (ai, bi, ci) and the fuzzy shortest length Lmin = (a, b, c). Then the similarity degree si between Li and Lmin can be calculated as

si

areaLi  Lmin ­0; if Li  Lmin ) ° (1) c  ai 2 ® ° 2>c  b  b  a @ ; if Li  Lmin z ) i i ¯

It should be noted that a, b, and c is always smaller than or equal to ai, bi, and ci, respectively. Integrating the new fuzzy shortest length heuristic procedure and the way for finding the similarity degree between two triangular fuzzy sets, we propose the following fuzzy shortest path approach:

An approach for fuzzy shortest path problem Input: A directed network of n nodes with fuzzy arc lengths necessary to traverse from one node to another node. Output: The fuzzy shortest length and the corresponding shortest path. Step 1: Form the possible paths from source node to destination node and compute the corresponding path lengths Li = (ai, bi, ci), i = 1, 2, ..., m, for possible m paths. Step 2: Find the Lmin = (a, b, c) by using The fuzzy shortest length heuristic procedure. Step 3: Find the similarity degree si between Lmin and Li, i = 1, 2,..., m. Step 4: Determine the actual shortest path with the highest similarity degree si. Some remarks concerning the above approach are worth mentioning as below: (R1): For an easier demonstration of our proposed approach, in step 1 we form the

possible paths directly instead of utilizing multiple labeling method [2,3,12] which can also be applied to our approach and is capable of reducing the computational load.

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(R2): In step 3, one can also use similarity measures [13-14] to measure the simi-

larity degree between fuzzy sets if one does not choose our proposed way as a measuring one. Next, one example is provided to illustrate our proposed approach and shown as below. Example 2: A classical network with fuzzy arc lengths is shown in Figure 1.

(56, 58, 72) (33, 45, 50) 1

2

(36, 38, 47)

4

(51, 79, 85)

(88, 92, 134)

(32, 40, 46)

6

(42, 57, 61) 3

5

(43, 55, 60)

(75, 110, 114)

Fig. 1. Classical Network. Let rn denote the arc length from node r to n. In Fig.11, suppose that 12 = (33, 45, 50), 13 = (42, 57, 61), 23 = (36, 38, 47), 24 = (56, 58, 72), 25 = (51, 79, 85), 35 = (43, 55, 60), 45 = (32, 40, 46), 46 = (88, 92, 134), and 56 = (75, 110, 114), then the possible paths are 12-3-5-6 with length L1 = (187, 248, 271), 1-2-4-5-6 with L2 = (196, 253, 282), 1-2-4-6 with length L3 = (177, 195, 256), 1-2-5-6 with L4 = (159, 234, 249), and 1-3-5-6 with L5 = (160, 222, 235), respectively. In Step 2, we can obtain the shortest length Lmin = (159, 179.65, 222) through the proposed fuzzy shortest length heuristic procedure, which is illustrated in Example 1. Next, substituting Lmin and Li into (1), one can get the similarity degree s1 = 16.78, s2 = 18.42, s3 = 16.91, s4 = 2.47, and s5 = 3.40. Finally, in step 4, we choose 1-3-5-6 as the shortest path since the corresponding length L5 (= Q2 ) has the highest similarity degree (=18.42) to the fuzzy shortest length Lmin.

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4. CONCLUSIONS The fuzzy shortest length and the corresponding shortest path are useful information for decision makers in a fuzzy shortest path problem. In this paper, we proposed the fuzzy shortest length heuristic procedure that can find the fuzzy shortest length among all possible paths in a network. It is based on the idea that a crisp number is a minimum number if and only if any other number is larger than or equal to it. In addition, we proposed a way to decide the shortest path that corresponds to the fuzzy shortest length. It is based on the calculation of the similarity degree between the fuzzy shortest length and each fuzzy path lengths. The approach that combines the above methods can provide the fuzzy shortest length and the corresponding shortest path for the decision makers. Some illustrative examples are included to demonstrate our proposed methods.

5. REFERENCES 1. Dubois D, Prade H (1980) Theory and Applications. Fuzzy Sets and Systems, Academic Press, New York 2. Hansen P (1980) Bicriterion path problems. In:. Beckmann M, Kunzi H.P (Eds.), Multiple Criteria Decison Making: Theory and Applications, Lecture Note in Economics and in Mathematical Systems, vol. 177, Springer, Berlin, pp. 109-127 3. Martins EQV (1984) On a multicriteria shortest path problem. Eur. J. Oper. Res. 16: 236-245 4. Yager RR (1986) Paths of least resistance on possibilistic production systems. Fuzzy Sets and Systems 19: 121-132 5. Broumbaugh-Smith J, Shier D (1989) An empirical investigation of some bicriterion shortest path algorithm. Eur J Oper Res 43: 216-224. 6. Klein CM (1991) Fuzzy shortest paths. Fuzzy Sets and Systems 39: 27-41 7. Lin K, Chen M (1994) The fuzzy shortest path problem and its most vital arcs. Fuzzy Sets and Systems 58: 343-353 8. Okada S, Gen M (1994) Fuzzy shortest path problem. Comput. Indust. Eng. 27: 465-468 9. Henig MI (1994) Efficient interactive methods for a class of multiattribute shortest path problems. Management Science 40 (7): 891-897 10.Okada S, Soper T (2000) A shortest path problem on a network with fuzzy lengths. Fuzzy Sets and Systems 109: 129-140 11.Hyung LK, Song YS and Lee KM (1994) Similarity measure between fuzzy sets and between elements. Fuzzy Sets and Systems 62: 291-293 12.Wang WJ (1997) New similarity measures on fuzzy sets and on elements. Fuzzy Sets and Systems 85:305-309 13.Tanaka H, Ichihashi H, Asai K (1984) A formulation of fuzzy linear programming problem based on comparison of fuzzy numbers. Control and Cybernetics 13: 185-194 14.Ramik J, Rimanek J (1985) Inequality relation between fuzzy numbers and its use in fuzzy optimization. Fuzzy Sets and Systems 16: 123-138 15.Zadeh LA (1978) Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets ans Systems 1:3-28.

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Appendix: 24 possible combinations of shapes for L1 and L2 L1

L2

(1)

a2

b2

c2

a1

L2

(2)

b1

a2

c1

L

L

2

a1

c2

b1

c1

Lmin = (a, b, c) = (a2, b2, c2)

Lmin = (a, b, c) = (a2, b2, c2) (3)

b2

L1

(4)

L2

L1

b2

c 2 b1

1

bL1

cL1

a2

a1

f b2

c2

b1

c1

a1

Lmin = (a, b, c) = (a2, f, c2) L2

(5)

a2

b2

L1

a1

b1

Lmin = (a, b, c) = (a2, b2, b1)

c1

Lmin = (a, b, c) = (a1, f, c2) (6)

c2

a2 f

c1

L2

a2

b2

L1

a1

b1

c1

Lmin = (a, b, c) = (a2, b2, b1)

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(7)

a2

L2

L1

a1 f b2

b1

(8)

a1

c2 c1

a2

L2

L1

f b2

b1

c2 c1

Lmin = (a, b, c) = (a1, f, b1)

Lmin = (a, b, c) = (a2, f, b1) L1

(9)

L2

L2

(10)

L1

aL2

a2 a1 f

b2

b1

c1 c2

a1 a2 f

Lmin = (a, b, c) = (a2, f, b1) L2

(11)

a1 f

b2 b1

b1

c1

Lmin = (a, b, c) = (a1, f, b1) L1

(12)

L1

a2

b2

L2

c2

c1

Lmin = (a, b, c) = (a2, f, b1)

a1

a2 f b2 b1

c2

Lmin = (a, b, c) = (a1, f, b1)

c1

c2

A new approach for the fuzzy shortest path problem

L2

(13)

a2

a1

f

b1b2

L1

(14)

L1

c1

L2

c2

a1

Lmin = (a, b, c) = (a2, f, b2)

(15)

a2

a1

L1

L2

f b1

b2

(16)

c2

a1

c1

b1 b2

c1

c2

a2 f

L1

L2

b1

b2

c2

c1

Lmin = (a, b, c) = (a1, f, b2)

L2

(17)

a2

a2 f

Lmin = (a, b, c) = (a1, f, b2)

Lmin = (a, b, c) = (a2, f, b2) L1

99

L1

L2

b1

b2

(18)

a1 f

b1

b2

Lmin = (a, b, c) = (a2, f, b2)

c1 c2

a1

a2 f

Lmin = (a, b, c) = (a1, f, b2)

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L1

(19)

L2

L1

(20)

L2

aL2

a1

b1

a2

b2

c2

c1

a1

b1 a2

Lmin = (a, b, c) = (a1, b1, b2)

a2

a1

f b1

c1 b2

c2

a1

b1 a2

c1

b2

c2

L2

a1 a2

f b1

c1 b2

c2

Lmin = (a, b, c) = (a1, f, c1)

L2

L1

L1

(22)

Lmin = (a, b, c) = (a2, f, c1)

(23)

c1

Lmin = (a, b, c) = (a1, b1, b2)

L2

L1

(21)

b2

(24)

c2

Lmin = (a, b, c) = (a1, b1, c1)

a1

L1

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c1 a2

b2

Lmin = (a, b, c) = (a1, b1, c1)

c2

CHAPTER 8 Distances Between Intuitionistic Fuzzy Sets and their Applications in Reasoning Eulalia Szmidt and Janusz Kacprzyk Systems Research Institute, Polish Academy of Sciences ul. Newelska 6, 01–447 Warsaw, Poland E-mail: {szmidt, kacprzyk}@ibspan.waw.pl Abstract: In this article we propose the use of intuitionistic fuzzy sets (Atanassov [2]) as a tool for reasoning under imperfect facts and imprecise knowledge, particularly via distances between intuitionistic fuzzy sets. We consider two issues: (1) a method to evaluate a degree (extent) of agreement (meant here as a distance from consensus) in a group of experts (individuals), and (2) a new approach to supporting medical diagnosis based on a reasoning scheme using a distance between intuitionistic fuzzy sets. Keywords: intuitionistic fuzzy sets, distances between intuitionistic fuzzy sets, medical databases, medical diagnosis

1 Introduction Intuitionistic fuzzy sets (Atanassov [1], [2]), because of an additional degree of freedom in comparison with fuzzy sets (Zadeh [22]), can be viewed as their generalization. The additional degree of freedom makes it possible to better model imperfect information which is omnipresent in reality, notably in most reasoning schemes. First, we propose a method to evaluate a degree (extent) of agreement (meant as a degree of consensus) in a group of experts (individuals) when individual testimonies are intuitionistic fuzzy preference relations, as opposed to traditional fuzzy preference relations commonly employed. For a comprehensive review of group decision making and (soft) measures of consensus under fuzzy preferences and fuzzy majority, see Kacprzyk and Nurmi [5]. Basically, in those works, the point of departure is a set of individual fuzzy preference relations which associate with each pair of options a number from [0, 1]. So, we have a set of n options, S = {s1 , . . . , sn }, and a set of m individuals, I = {1, . . . , m}. Each individual k provides his or her individual fuzzy preference relation µRk : S × S → [0, 1], conveniently represented by a matrix [rikj ] such that rikj = µRk (si , s j ); i, j = 1, . . . , n; k = 1, . . . , m; [rikj ] + [rkji ] = 1. The elements

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of the matrix 0 < rikj < 1 are such that the highe r the preference o f individual k of si over s j the higher rikj : from rikj = 0 indicating a definite preference s j over si , through rikj = 0.5 indicating indifference between si and s j , to rikj = 1 indicating a definite preference si over s j . In Szmidt [8], Szmidt and Kacprzyk [11, 13, 14, 17], Kacprzyk and Szmidt [16] the use of intuitionistic fuzzy sets for formulating and solving group decision problems, and for the determination of soft measures of consensus was considered. The starting point of this approach is the use of intuitionistic fuzzy preferences instead of fuzzy preferences. In effect, each individual k provides his or her (individual) intuitionistic fuzzy preference relation, giving not only Rk , (µRk ), but also Πk – so-called hesitation margins, πk : S × S → [0, 1], conveniently represented by a matrix [πkij ]; i, j = 1, . . . , n; k = 1, . . . , m. Such a representation of individual preferences (taking into account another degree of freedom, i.e. hesitation margins) makes it possible to express the hesitation of individuals, and leads to finding out a degree of soft consensus – in this article using distances between preferences of individuals. Second, a medical database is considered. Employing intuitionistic fuzzy sets, we can simply and adequately express a hesitation concerning the objects considered – both patients and illnesses, and render two important facts. First, values of symptoms change for each patient as, e.g., temperature goes up and down, pain increases and decreases, etc. Second, for different patients suffering from the same illness, values of the same symptom can often be different. Our task, i.e. to find an illness for each patient, is fulfilled by looking for the smallest distance [cf. Szmidt and Kacprzyk [12, 15]] between symptoms that are characteristic for a patient and symptoms describing the illnesses considered.

2 Brief introduction to intuitionistic fuzzy sets As opposed to a fuzzy set in X(Zadeh [22]), given by

A = {< x, µA (x) > |x ∈ X}

(1)

where µA (x) ∈ [0, 1] is the membership function of the fuzzy set A , an intuitionistic fuzzy set (Atanassov [1], [2]) A is given by A = {< x, µA (x), νA (x) > |x ∈ X}

(2)

where: µA : X → [0, 1] and νA : X → [0, 1] such that 0