A Combined Approach for Outliers Detection in Fuzzy Functional Dependency through the Typology of Fuzzy Rules
|
|
- Claire Francis
- 5 years ago
- Views:
Transcription
1 A ombined Approach for Outliers Detection in Fuzzy Functional Dependency through the Typology of Fuzzy ules Shaheera ashwan Soheir Fouad and isham Sewelam Department of omputer Science Faculty of Engineering University of Alexandria Alexandria 544 Egypt Abstract: - In this paper we are concerned with the problem of detecting the outliers ie the exceptional tuple values breaking the fuzzy dependency Using the typology of fuzzy rules we distinguish three kinds of Fuzzy Functional Dependency and propose a general framework for extending the definition of Fuzzy Functional Dependency According to a semantical view of fuzzy rules for handling exceptions this framework is based on the graded certainty/possibility of the resemblance of the consequent attributes rather than the graduality of the resemblance of these attributes Of primary interest is the exceptionality computation which is represented by the Fuzzy ertainty ule Dependency or the Fuzzy ossibility ule Dependency proposed in the paper and which will be taken in consideration by both the designer and the database user A combined approach is proposed in the paper to combine both the Fuzzy ertainty ule Dependency and the Fuzzy ossibility ule Dependency Key-ords: - Fuzzy Functional dependency Fuzzy rules Typology of Fuzzy ules Exceptionality omputation Outliers Detection Introduction In the context of regular relational databases functional dependencies have received a lot of attention since they capture some semantics about the data related to redundancy Informally a FD is a property valid on a relation stating that tuples with the same value on a set A of attributes have the same value on a set B of attributes The need to incorporate and treat information given in fuzzy terms in elational Databases has concentrated a great effort in the last years In their paper[] ubero and Vila introduced a fuzzy extension of this concept to overcome the previous anomalous behaviors and study its properties when defining a ffd if antecedent values are very similar then consequent values must be slightly similar owever the definition of Fuzzy Functional Dependencies The more resemblant the antecedent values are the more resemblant the consequent values must be [] forbids to have resemblant antecedent values and distinct consequent values which is not realistic in practice Indeed two tall persons might have different weights There may be some but very few tuple values breaking the fuzzy dependency This is called the problem of the outliers Before treating the problem we must define the fuzzy if-then rules and their different kinds[3] The representation of the fuzzy rule is used for handling exceptions Fuzzy Gradual ule Dependency Gradual rules [6]depict relations between variables X and Y according to propositions of the form The more X is A the more Y is B where A and B are Fuzzy Sets modeling certain symbolic labels ithin the context of our Fuzzy Functional Dependencies FFD framework a gradual rule reads The more resemblant the antecedent values are the more resemblant the consequent values must be ~ In terms of gradual rules if and model resemblance relations between antecedent X and consequent values Y respectively the gradual dependency relates the resemblance of X t and X t to the resemblance of Y t and Y t for two given tuples t and t in the following way: v min v u ~ u Y t Y t X t X t 5 ie the closer to ~ u and the more possible the values v and the closer to v should be In a particular case where the possibility distribution is crisp and equal to the Fuzzy Gradual ule Dependency FGD becomes ~ X t X t Y t Y t 6
2 learly the Fuzzy Functional Dependency using gradual rules in both crisp and fuzzy databases forbid to have two resemblant antecedent values which have different consequent values which is not very realistic in practice A database crisp or fuzzy may contain tuples called the outliers which are rather similar with respect to the antecedent attributes and sensibly distinct with respect to the consequent attributes Indeed two personsaving similar heights might have distinct weights onsider a relation r Zeighteight where the α β tuples satisfy the FFD eight eight as defined in []These tuples satisfy the Fuzzy Gradual ule Dependency FGD The more a person is tall the more he is heavy which can be represented as follows: w w min{ / w w y y x h } where h w h are two given tuples in w Ï r and stand for resemblance relations between the antecedent attributes and consequent attributes respectively and y y and x stand for Y t Y t and X t X t used for simplicity hen applying the FGD taking a value h such that h α it expresses that the values w such that β are fully possible v u while other values have no y / x = guaranteed possibility v u y / x owever a rather tall person can be heavy or even normal as one of just few exceptional tuples in r that are considered to break the fuzzy dependency Then the fuzzy gradual rule dependency The more a person is tall the more he is heavy is obviously not flexible in the sense that it does not allow for incorporating some tolerance toward exceptional tuples In fact the above example suggests to look for other schemes which do not only distinguish between the possibility and impossibility of consequent values but which enable us to compute the exceptionality of an input tuple and thus detect the outliers exceptional tuples For this reason we shall consider the two other kinds of fuzzy rules called: ertainty rules ossibility rules 3 Fuzzy ertainty/ossibility ule Dependency ertainty- and possibility-qualification provide a tool for distinguishing between different intended meanings that a fuzzy rule of the form "if X is A then Y is B" may convey Indeed such a rule can be interpreted in various ways according to how the ifpart qualifies the then-part[6] A first kind of fuzzy rule is the certainty rule which corresponds to statements of the form the more x is A the more certain y is B A second kind of fuzzy rule is the possibility rule which corresponds to statements of the form the more x is A the more possible Y lies in B ithin the framework of possibility theory certainty is closely related to impossibility Formally the certainty c of an event A and the possibility p of the complement A are related according to c = -p In connection with the concept of a certainty rule the fuzzy functional dependency can be understood as the more the antecedent attributes are resemblant the more certain the consequent attributes must be resemblant If ~ and model resemblance relations between antecedent X and consequent values Y respectively then the Fuzzy ertainty ule Dependency FD relates the resemblance of X t and X t to the resemblance of Y t and Y t for two given tuples t and t in the following way max ~ v u Y t Y t X t X t v u 5 Indeed when ~ u is close to the values v and which are not resemblant in the sense of should have a low degree of possibility v v u u Y t Y t X t X t Using the possibility rule the fuzzy functional dependency expresses the more the antecedent attributes are resemblant the more possible the consequent attributes must be resemblant Now If ~ and model resemblance relations between antecedent X and consequent values Y respectively the Fuzzy ossibility ule Dependency FD relates the resemblance of X t and X t to the resemblance of Y t and Y t for two given tuples t and t in the following way min v ~ u Y t Y t X t X t v u 6
3 Similarly when ~ u is close to v u Y t Y t X t X t = v so the values v and which are not resemblant in the sense of should have a low degree of possibility Since the certainty and the possibility rules are thought of as a constraint which holds true in general but still allows for exceptions They are more flexible than the approach based on gradual rules and seems to be particularly suitable as a formal model of FFD eturning to our example given two tuples h w h w in r the Fuzzy ertainty ule Dependency FD The more a person is tall the more certain he is heavy constrains the possibility of esemblance degrees y = w w according to certainty rule: y / x max{ x y} 7 here x = h is the membership function of the resemblance between the antecedent attributes h and h e thus obtain w w max{ w w h } y y / x h 8 The Fuzzy ossibility ule Dependency FD The more a person is tall the more possible he is heavy can be represented by: y y / x w w min{ h w w } Similarly if for instance h is close to w w the possibility bound can y y / x only be large for consequent values which are resemblant to w bounded from below To illustrate that both certainty rules and possibility rules can be used to handle exceptions while gradual rules can t be consider the following: Two tuples in r: t t satisfy the FFD α β eight eight if when h h are resemblant at level α implies that: - In case of gradual rules Figure w w are at least β resemblant ossibility where stand for resemblance relations between the antecedent attributes and consequent attributes respectively and y y and x stand for Y t Y t and X t X t used for simplicity β esemblance Figure : at least β resemblant The more resemblant the antecedent values h and h the more constrained becomes the possibility of the consequent values If for instance h is close to the possibility bound w w can only y y / x be large for consequent values which are resemblant to w bounded from above articularly if h the resulting = possibility distribution reveals complete ignorance Now according to the possibility rule: min{ y x} y / x 9 - In case of certainty rules Figure w w are at least β certainly resemblant ossibility β esemblance Figure : at least β certainly resemblant
4 3- In case of possibility rules Figure 3 w w are at least β possibly resemblant ossibility β esemblance Figure 3: at least β possibly resemblant In figure the representation of at least β resemblant when there are only two degrees of possibility complete impossibility and complete possibility coincide with the ordinary subset [ β ] which leaves completely possible the values corresponding to degrees of resemblance greater than β and hence restricts the possible values of w with respect to w In figures and 3 when there are only two degrees of resemblance non-resemblant and resemblant at least β certainly resemblant corresponds to the possibility distribution resemblant = non resemblant β ie bounded from above by β and at least β possibly resemblant corresponds to the possibility distribution non resemblant resemblant β ie bounded from below by β Attaching some uncertainty to the consequent values by expressing that the possibility degree of values outside their support is no longer strictly zero but progressively increases as the value of h moves away from the core of corresponding antecedent values allows the detection of the outliers since it does not exclude them completely In case of certainty rule ie the more a person is tall the more certain he is heavy it expresses that the values w are such that w / h if h y / x h y x is completely unspecified otherwise 3 which means that the possibility is bounded by w So the case that a person is heavy or even normal given that he is rather tall is not forbidden In case of possibility rule ie the more a person is tall the more possible he is heavy it expresses that the values w are such that w / h if h y x / h otherwise 4 y x which means that the case of a person as being heavy or even normal given that he is rather tall is not forbidden also and hence the existence of an exception tuple is not totally ignored like in the case of gradual rules 4 Exceptionality omputation learly extending the framework of Fuzzy Functional Dependency FFD using the typology of fuzzy rules allows us to compute the degree of exceptionality between a pair of tuples where the antecedent and consequent attributes are fuzzy dependent Since these tuples just very few ones were considered to break the fuzzy dependency if they do not obey to the FGD and also ignoring them to accomplish the fuzzy loss-less decomposition will cause a problem when the projection are joined again as shown in the example ence attaching an important level such as the degree of exceptionality to the framework of Fuzzy Functional Dependency FFD represented by the FD or the FD will be taken in consideration by both the designer and the user Interestingly enough a degree of exceptionality can be computed as follows ex h w = y y / x x 5 where h w is a tuple in r Substitution of the possibility distribution in 5 In case of ertainty ules: ex w max{ h } 6 where are resemblance relations of eights and eights In case of ossibility ules: ex w min{ h } 7
5 The equation 5 says that the exceptionality of the tuples h w is inversely related to the possibility of observing resemblant weights for h resemblant heights as specified by the certainty rule or possibility rule The more h w and h violate these rules the more exceptional they are in the sense of 5 It is worth mentioning that 5 makes also sense in connection with the gradual rule model Applying 5 to the possibility distribution induced by a gradual rule an input tuple is either completely exceptional or not exceptional at all as follows: Assuming the possibility distribution as being crisp possibility ex w if h ie not exceptional at all ex w if h > ie completely exceptional 8 These equations reveal again the difference between the gradual rule dependency on one hand and the certainty or the possibility rule dependency on the other hand The FGD is indeed not tolerant toward exceptions in the sense that each violation of the rule is punished by classifying the involved tuples as completely exceptional ones As opposed to this exceptionality is not restricted to the crisp values in both the FD and the FD models but can be considered as a gradual property in these models In fact one possibility of regarding exceptionality as a property of an individual tuple h w is to consider the likelihood or possibility of h w to be exceptional with respect to a new tuple h w ence a degree of exceptionality can be assigned to each tuple h w representing fuzzy rules in 5 Thus one might think of generalizing 5 as follows For ertainty ules Let us apply 8 to all tuples in r we obtain the possibility distribution w/ h = min max{ h } h w 9 which follows from 8 under the application of a minimal specificity principle According to this principle each element of the domain of a possibility distribution is assigned the largest possibility in agreement with the constraints By applying ex w = w / h 3 e obtain ex w = sup ex h w 3 For ossibility ules Let us apply to all tuples in r we obtain the possibility distribution w/ h = max min{ h } h w 3 which actually represents a lower possibility bound By applying ex w = w / h 33 e obtain ex w = inf ex h w 33 Assigning a degree of exceptionality to a tuple can be interpreted as rating the reliability of the tuple Of course this degree of exceptionality depends on the formalization of the underlying rule In other words a tuple is exceptional ie an outlier tuple not by itself but only with respect to a particular rule: hanging the rule changes the degree of exceptionality of the tuple If exceptionality is equivalent to complete exceptionality as in FGD the exceptional tuples are not allowed and may be removed from the relation since the esher-gaines implication of the gradual rule is restricted to the normal cases The level of uncertainty using FD or impossibility using FD of an individual tuple resulted from the fuzzy join is increased in accordance with the degree of exceptionality of the corresponding fuzzy rule in the projection relation The generalized FFD might then be characterized as follows: The more resemblant the antecedent values are The more certain/possible the consequent values must be resemblant and the less exceptional they are 5 ombining Fuzzy ertainty ule Dependency and Fuzzy ossibility ule Dependency The use of certainty rules which belong to the class of implicative fuzzy rules leads to a constraint-based approach[7] Given an antecedent value h resemblant to h it rules out those values which are not sufficiently resemblant
6 to w Each antecedent value in an added tuple decreases the possiblity of certain consequent values On contrary possibility rules which belong to the class of conjunctive fuzzy rules leads to an exampleoriented approach[7] Given an antecedent value h resemblant to h it suggests new feasible values for w Each antecedent value in an added tuple increases the possiblity of certain consequent values onsider for instance a model which is made of the certainty rule and the possibility rule and thus combines the constraint-based and the exampleoriented approaches More precisely w / h reflects the degree of exclusion of w whereas w / h can be seen as a degree of confirmation of w Let us consider some extreme examples to illustrate this aspect a w / h = w/ h = or equivalently ex w = ex w = : This case is an expression of complete ignorance No tuples support the possibility that w = and no tuples forbid to have w = b w / h = w / h = or equivalently ex w = ex w = : This case is an expression of clear evidence against w = c w / h = w/ h = or equivalently ex w = ex w = : This case is an expression of strong support to w = These cases emphasize the advantage of the combined approach in the following manner: 6 onclusion Both the mathematical and the semantical point of view mention that assigning a degree of exceptionality to tuples is important to detect to which degree a certain tuple can be exceptional The future research will be devoted with the detection of the outliers in relation that contains more than one fuzzy dependency eferences: [] J ubero JM Medina A New Definition of Fuzzy Functional Dependency in Fuzzy elational Databases International Journal of Intelligent Systems 95 pages [] J ubero O ons MA Vila eak and Strong esemblance in Fuzzy Functional Dependencies In Third IEEE International onference on Fuzzy Systems Orlando USA pages [3] Didier Dubois enri radeat Are Fuzzy ules and ow to Use Them Fuzzy Sets and Systems Vol84 pages [4] J ubero JM Medina O ons and MA Vila Extensions of a resemblance relation Fuzzy Sets and Systems [5] J ubero JM Medina MA Vila Influence of Granularity Level in Fuzzy Functional Dependencies Symbolic and Quantitative Approaches to easoning and Uncertainty Lecture Notes in omputer Science 747 pages Springer Verlag Berlin 993 [6] D Dubois and rade Fuzzy sets in approximate reasoning Fuzzy Sets and Systems art : 4:43-; art with Lang J: 4: [7] Eyke uellermeier Dubois D rade Fuzzy ules in ase-based easoning apport IIT/99-36 Juin 999 st : The example-oriented model alone cannot distinguish between a and b It makes a great difference whether w = is not supported because no resemblant tuple values exist or there exist tuple values that forbid to have w = nd : The constraint-based model alone cannot distinguish between a and c It makes a great difference whether w = is supported because no resemblant tuple values exist or there exist resemblant tuple values that supports to have w =
On flexible database querying via extensions to fuzzy sets
On flexible database querying via extensions to fuzzy sets Guy de Tré, Rita de Caluwe Computer Science Laboratory Ghent University Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium {guy.detre,rita.decaluwe}@ugent.be
More informationUncertainty and Rules
Uncertainty and Rules We have already seen that expert systems can operate within the realm of uncertainty. There are several sources of uncertainty in rules: Uncertainty related to individual rules Uncertainty
More informationEncoding formulas with partially constrained weights in a possibilistic-like many-sorted propositional logic
Encoding formulas with partially constrained weights in a possibilistic-like many-sorted propositional logic Salem Benferhat CRIL-CNRS, Université d Artois rue Jean Souvraz 62307 Lens Cedex France benferhat@criluniv-artoisfr
More informationEfficient Approximate Reasoning with Positive and Negative Information
Efficient Approximate Reasoning with Positive and Negative Information Chris Cornelis, Martine De Cock, and Etienne Kerre Fuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematics
More informationFuzzy Systems. Possibility Theory.
Fuzzy Systems Possibility Theory Rudolf Kruse Christian Moewes {kruse,cmoewes}@iws.cs.uni-magdeburg.de Otto-von-Guericke University of Magdeburg Faculty of Computer Science Department of Knowledge Processing
More informationFuzzy Systems. Introduction
Fuzzy Systems Introduction Prof. Dr. Rudolf Kruse Christoph Doell {kruse,doell}@iws.cs.uni-magdeburg.de Otto-von-Guericke University of Magdeburg Faculty of Computer Science Department of Knowledge Processing
More informationFrom a Possibility Theory View of Formal Concept Analysis to the Possibilistic Handling of Incomplete and Uncertain Contexts
From a Possibility Theory View of Formal Concept Analysis to the Possibilistic Handling of Incomplete and Uncertain Contexts Zina Ait-Yakoub 1, Yassine Djouadi 2, Didier Dubois 2, and Henri Prade 2 1 Department
More informationSimilarity-based Classification with Dominance-based Decision Rules
Similarity-based Classification with Dominance-based Decision Rules Marcin Szeląg, Salvatore Greco 2,3, Roman Słowiński,4 Institute of Computing Science, Poznań University of Technology, 60-965 Poznań,
More informationAccelerating Effect of Attribute Variations: Accelerated Gradual Itemsets Extraction
Accelerating Effect of Attribute Variations: Accelerated Gradual Itemsets Extraction Amal Oudni, Marie-Jeanne Lesot, Maria Rifqi To cite this version: Amal Oudni, Marie-Jeanne Lesot, Maria Rifqi. Accelerating
More informationFUZZY ASSOCIATION RULES: A TWO-SIDED APPROACH
FUZZY ASSOCIATION RULES: A TWO-SIDED APPROACH M. De Cock C. Cornelis E. E. Kerre Dept. of Applied Mathematics and Computer Science Ghent University, Krijgslaan 281 (S9), B-9000 Gent, Belgium phone: +32
More informationUncertain Logic with Multiple Predicates
Uncertain Logic with Multiple Predicates Kai Yao, Zixiong Peng Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 100084, China yaok09@mails.tsinghua.edu.cn,
More informationFuzzy Systems. Introduction
Fuzzy Systems Introduction Prof. Dr. Rudolf Kruse Christian Moewes {kruse,cmoewes}@iws.cs.uni-magdeburg.de Otto-von-Guericke University of Magdeburg Faculty of Computer Science Department of Knowledge
More informationIncompatibility Paradoxes
Chapter 22 Incompatibility Paradoxes 22.1 Simultaneous Values There is never any difficulty in supposing that a classical mechanical system possesses, at a particular instant of time, precise values of
More informationEntropy for intuitionistic fuzzy sets
Fuzzy Sets and Systems 118 (2001) 467 477 www.elsevier.com/locate/fss Entropy for intuitionistic fuzzy sets Eulalia Szmidt, Janusz Kacprzyk Systems Research Institute, Polish Academy of Sciences ul. Newelska
More informationInterpreting Low and High Order Rules: A Granular Computing Approach
Interpreting Low and High Order Rules: A Granular Computing Approach Yiyu Yao, Bing Zhou and Yaohua Chen Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail:
More informationRule-Based Fuzzy Model
In rule-based fuzzy systems, the relationships between variables are represented by means of fuzzy if then rules of the following general form: Ifantecedent proposition then consequent proposition The
More informationCorrelation Coefficient of Interval Neutrosophic Set
Applied Mechanics and Materials Online: 2013-10-31 ISSN: 1662-7482, Vol. 436, pp 511-517 doi:10.4028/www.scientific.net/amm.436.511 2013 Trans Tech Publications, Switzerland Correlation Coefficient of
More informationFrom Crisp to Fuzzy Constraint Networks
From Crisp to Fuzzy Constraint Networks Massimiliano Giacomin Università di Brescia Dipartimento di Elettronica per l Automazione Via Branze 38, I-25123 Brescia, Italy giacomin@ing.unibs.it Abstract. Several
More informationA Zadeh-Norm Fuzzy Description Logic for Handling Uncertainty: Reasoning Algorithms and the Reasoning System
1 / 31 A Zadeh-Norm Fuzzy Description Logic for Handling Uncertainty: Reasoning Algorithms and the Reasoning System Judy Zhao 1, Harold Boley 2, Weichang Du 1 1. Faculty of Computer Science, University
More informationWhy is There a Need for Uncertainty Theory?
Journal of Uncertain Systems Vol6, No1, pp3-10, 2012 Online at: wwwjusorguk Why is There a Need for Uncertainty Theory? Baoding Liu Uncertainty Theory Laboratory Department of Mathematical Sciences Tsinghua
More information3. DIFFERENT MODEL TYPES
3-1 3. DIFFERENT MODEL TYPES It is important for us to fully understand a physical problem before we can select a solution strategy for it. Models are convenient tools that enhance our understanding and
More informationPossibilistic Logic. Damien Peelman, Antoine Coulon, Amadou Sylla, Antoine Dessaigne, Loïc Cerf, Narges Hadji-Hosseini.
Possibilistic Logic Damien Peelman, Antoine Coulon, Amadou Sylla, Antoine Dessaigne, Loïc Cerf, Narges Hadji-Hosseini November 21, 2005 1 Introduction In real life there are some situations where only
More informationFeature Selection with Fuzzy Decision Reducts
Feature Selection with Fuzzy Decision Reducts Chris Cornelis 1, Germán Hurtado Martín 1,2, Richard Jensen 3, and Dominik Ślȩzak4 1 Dept. of Mathematics and Computer Science, Ghent University, Gent, Belgium
More information2) There should be uncertainty as to which outcome will occur before the procedure takes place.
robability Numbers For many statisticians the concept of the probability that an event occurs is ultimately rooted in the interpretation of an event as an outcome of an experiment, others would interpret
More informationCorresponding Regions in Euler Diagrams
orresponding Regions in Euler Diagrams John Howse, Gemma Stapleton, Jean Flower, and John Taylor School of omputing & Mathematical Sciences niversity of righton, righton, K {John.Howse,G.E.Stapleton,J..Flower,John.Taylor}@bton.ac.uk
More informationFrom fuzzy dependences to fuzzy formulas and vice versa, for Kleene-Dienes fuzzy implication operator
From fuzzy dependences to fuzzy formulas and vice versa, for Kleene-Dienes fuzzy implication operator NEDŽAD DUKIĆ, DŽENAN GUŠIĆ, AMELA MURATOVIĆ-RIBIĆ, ADIS ALIHODŽIĆ, EDIN TABAK, HARIS DUKIĆ University
More informationLaurent UGHETTO Λ, Didier DUBOIS ΛΛ and Henri PRADE ΛΛ
From: AAAI-99 Proceedings. Copyright 999, AAAI (www.aaai.org). All rights reserved. Implicative and conjunctive fuzzy rules A tool for reasoning from knowledge and examples Laurent UGHETTO Λ, Didier DUBOIS
More informationLecture 6. Probability events. Definition 1. The sample space, S, of a. probability experiment is the collection of all
Lecture 6 1 Lecture 6 Probability events Definition 1. The sample space, S, of a probability experiment is the collection of all possible outcomes of an experiment. One such outcome is called a simple
More informationInvestigating Measures of Association by Graphs and Tables of Critical Frequencies
Investigating Measures of Association by Graphs Investigating and Tables Measures of Critical of Association Frequencies by Graphs and Tables of Critical Frequencies Martin Ralbovský, Jan Rauch University
More informationOutline. Introduction, or what is fuzzy thinking? Fuzzy sets Linguistic variables and hedges Operations of fuzzy sets Fuzzy rules Summary.
Fuzzy Logic Part ndrew Kusiak Intelligent Systems Laboratory 239 Seamans Center The University of Iowa Iowa City, Iowa 52242-527 andrew-kusiak@uiowa.edu http://www.icaen.uiowa.edu/~ankusiak Tel: 39-335
More informationIt rains now. (true) The followings are not propositions.
Chapter 8 Fuzzy Logic Formal language is a language in which the syntax is precisely given and thus is different from informal language like English and French. The study of the formal languages is the
More informationConcept Lattices in Rough Set Theory
Concept Lattices in Rough Set Theory Y.Y. Yao Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: yyao@cs.uregina.ca URL: http://www.cs.uregina/ yyao Abstract
More informationBasic Structure of Some Classes of Neutrosophic Crisp Nearly Open Sets & Possible Application to GIS Topology
eutrosophic Sets and Systems, Vol 7, 2015 18 Basic Structure of Some lasses of eutrosophic risp early Open Sets & Possible Application to GIS A A Salama Department of Mathematics and omputer Science, Faculty
More informationCompenzational Vagueness
Compenzational Vagueness Milan Mareš Institute of information Theory and Automation Academy of Sciences of the Czech Republic P. O. Box 18, 182 08 Praha 8, Czech Republic mares@utia.cas.cz Abstract Some
More informationTopic 1: Propositional logic
Topic 1: Propositional logic Guy McCusker 1 1 University of Bath Logic! This lecture is about the simplest kind of mathematical logic: propositional calculus. We discuss propositions, which are statements
More informationNested Epistemic Logic Programs
Nested Epistemic Logic Programs Kewen Wang 1 and Yan Zhang 2 1 Griffith University, Australia k.wang@griffith.edu.au 2 University of Western Sydney yan@cit.uws.edu.au Abstract. Nested logic programs and
More informationBackground on Coherent Systems
2 Background on Coherent Systems 2.1 Basic Ideas We will use the term system quite freely and regularly, even though it will remain an undefined term throughout this monograph. As we all have some experience
More informationA Systematic Approach to the Assessment of Fuzzy Association Rules
A Systematic Approach to the Assessment of Fuzzy Association Rules Didier Dubois IRIT-UPS, Toulouse, France Eyke Hüllermeier Faculty of Computer Science University of Magdeburg, Germany Henri Prade IRIT-UPS,
More informationUncertain Satisfiability and Uncertain Entailment
Uncertain Satisfiability and Uncertain Entailment Zhuo Wang, Xiang Li Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China zwang0518@sohu.com, xiang-li04@mail.tsinghua.edu.cn
More informationOn Tuning OWA Operators in a Flexible Querying Interface
On Tuning OWA Operators in a Flexible Querying Interface Sławomir Zadrożny 1 and Janusz Kacprzyk 2 1 Warsaw School of Information Technology, ul. Newelska 6, 01-447 Warsaw, Poland 2 Systems Research Institute
More informationSets with Partial Memberships A Rough Set View of Fuzzy Sets
Sets with Partial Memberships A Rough Set View of Fuzzy Sets T. Y. Lin Department of Mathematics and Computer Science San Jose State University San Jose, California 95192-0103 E-mail: tylin@cs.sjsu.edu
More informationA new Approach to Drawing Conclusions from Data A Rough Set Perspective
Motto: Let the data speak for themselves R.A. Fisher A new Approach to Drawing Conclusions from Data A Rough et Perspective Zdzisław Pawlak Institute for Theoretical and Applied Informatics Polish Academy
More informationA Little Deductive Logic
A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that
More informationSolving Fuzzy PERT Using Gradual Real Numbers
Solving Fuzzy PERT Using Gradual Real Numbers Jérôme FORTIN a, Didier DUBOIS a, a IRIT/UPS 8 route de Narbonne, 3062, Toulouse, cedex 4, France, e-mail: {fortin, dubois}@irit.fr Abstract. From a set of
More informationOn Proofs and Rule of Multiplication in Fuzzy Attribute Logic
On Proofs and Rule of Multiplication in Fuzzy Attribute Logic Radim Belohlavek 1,2 and Vilem Vychodil 2 1 Dept. Systems Science and Industrial Engineering, Binghamton University SUNY Binghamton, NY 13902,
More information2 Interval-valued Probability Measures
Interval-Valued Probability Measures K. David Jamison 1, Weldon A. Lodwick 2 1. Watson Wyatt & Company, 950 17th Street,Suite 1400, Denver, CO 80202, U.S.A 2. Department of Mathematics, Campus Box 170,
More informationENTROPIES OF FUZZY INDISCERNIBILITY RELATION AND ITS OPERATIONS
International Journal of Uncertainty Fuzziness and Knowledge-Based Systems World Scientific ublishing Company ENTOIES OF FUZZY INDISCENIBILITY ELATION AND ITS OEATIONS QINGUA U and DAEN YU arbin Institute
More informationWhere are we? Knowledge Engineering Semester 2, Reasoning under Uncertainty. Probabilistic Reasoning
Knowledge Engineering Semester 2, 2004-05 Michael Rovatsos mrovatso@inf.ed.ac.uk Lecture 8 Dealing with Uncertainty 8th ebruary 2005 Where are we? Last time... Model-based reasoning oday... pproaches to
More informationHybrid Logic and Uncertain Logic
Journal of Uncertain Systems Vol.3, No.2, pp.83-94, 2009 Online at: www.jus.org.uk Hybrid Logic and Uncertain Logic Xiang Li, Baoding Liu Department of Mathematical Sciences, Tsinghua University, Beijing,
More informationApplication of Fuzzy Relation Equations to Student Assessment
American Journal of Applied Mathematics and Statistics, 018, Vol. 6, No., 67-71 Available online at http://pubs.sciepub.com/ajams/6//5 Science and Education Publishing DOI:10.1691/ajams-6--5 Application
More informationImplications from data with fuzzy attributes vs. scaled binary attributes
Implications from data with fuzzy attributes vs. scaled binary attributes Radim Bělohlávek, Vilém Vychodil Dept. Computer Science, Palacký University, Tomkova 40, CZ-779 00, Olomouc, Czech Republic Email:
More informationarxiv: v1 [cs.lo] 16 Jul 2017
SOME IMPROVEMENTS IN FUZZY TURING MACHINES HADI FARAHANI arxiv:1707.05311v1 [cs.lo] 16 Jul 2017 Department of Computer Science, Shahid Beheshti University, G.C, Tehran, Iran h farahani@sbu.ac.ir Abstract.
More informationA Little Deductive Logic
A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that
More informationMotivation. From Propositions To Fuzzy Logic and Rules. Propositional Logic What is a proposition anyway? Outline
Harvard-MIT Division of Health Sciences and Technology HST.951J: Medical Decision Support, Fall 2005 Instructors: Professor Lucila Ohno-Machado and Professor Staal Vinterbo Motivation From Propositions
More informationFuzzy Limits of Functions
Fuzzy Limits of Functions Mark Burgin Department of Mathematics University of California, Los Angeles 405 Hilgard Ave. Los Angeles, CA 90095 Abstract The goal of this work is to introduce and study fuzzy
More informationLecture 1: Introduction & Fuzzy Control I
Lecture 1: Introduction & Fuzzy Control I Jens Kober Robert Babuška Knowledge-Based Control Systems (SC42050) Cognitive Robotics 3mE, Delft University of Technology, The Netherlands 12-02-2018 Lecture
More informationA Comparative Study of Different Order Relations of Intervals
A Comparative Study of Different Order Relations of Intervals Samiran Karmakar Department of Business Mathematics and Statistics, St. Xavier s College, Kolkata, India skmath.rnc@gmail.com A. K. Bhunia
More informationInclusion of the Intuitionistic Fuzzy Sets Based on Some Weak Intuitionistic Fuzzy Implication
LGRIN DEMY OF SIENES YERNETIS ND INFORMTION TEHNOLOGIES Volume No 3 Sofia 0 Inclusion of the Intuitionistic Fuzzy Sets ased on Some Weak Intuitionistic Fuzzy Implication Piotr Dworniczak Department of
More informationOPTIMIZATION TECHNIQUES FOR EDIT VALIDATION AND DATA IMPUTATION
Proceedings of Statistics Canada Symposium 2001 Achieving Data Quality in a Statistical Agency: a Methodological Perspective OPTIMIZATION TECHNIQUES FOR EDIT VALIDATION AND DATA IMPUTATION Renato Bruni
More informationInformation Systems for Engineers. Exercise 8. ETH Zurich, Fall Semester Hand-out Due
Information Systems for Engineers Exercise 8 ETH Zurich, Fall Semester 2017 Hand-out 24.11.2017 Due 01.12.2017 1. (Exercise 3.3.1 in [1]) For each of the following relation schemas and sets of FD s, i)
More informationFuzzy and Rough Sets Part I
Fuzzy and Rough Sets Part I Decision Systems Group Brigham and Women s Hospital, Harvard Medical School Harvard-MIT Division of Health Sciences and Technology Aim Present aspects of fuzzy and rough sets.
More informationSo, we can say that fuzzy proposition is a statement p which acquires a fuzzy truth value T(p) ranges from(0 to1).
Chapter 4 Fuzzy Proposition Main difference between classical proposition and fuzzy proposition is in the range of their truth values. The proposition value for classical proposition is either true or
More informationMutuality Measures Corresponding to Subjective Judgment of Similarity and Matching
Mutuality Measures Corresponding to Subjective Judgment of Similarity and Matching Ayumi Yoshikawa Faculty of Education, Okayama University 1-1, Naka 3-chome, Tsushima, Okayama 700-8530, Japan ayumi@sip.ed.okayama-u.ac.jp
More informationConvex Hull-Based Metric Refinements for Topological Spatial Relations
ABSTRACT Convex Hull-Based Metric Refinements for Topological Spatial Relations Fuyu (Frank) Xu School of Computing and Information Science University of Maine Orono, ME 04469-5711, USA fuyu.xu@maine.edu
More informationABDUCTIVE reasoning is an explanatory process in
Selecting Implications in Fuzzy bductive Problems drien Revault d llonnes Herman kdag ernadette ouchon-meunier bstract bductive reasoning is an explanatory process in which potential causes of an observation
More informationA PRIMER ON ROUGH SETS:
A PRIMER ON ROUGH SETS: A NEW APPROACH TO DRAWING CONCLUSIONS FROM DATA Zdzisław Pawlak ABSTRACT Rough set theory is a new mathematical approach to vague and uncertain data analysis. This Article explains
More informationA Preference Logic With Four Kinds of Preferences
A Preference Logic With Four Kinds of Preferences Zhang Zhizheng and Xing Hancheng School of Computer Science and Engineering, Southeast University No.2 Sipailou, Nanjing, China {seu_zzz; xhc}@seu.edu.cn
More informationOn (Weighted) k-order Fuzzy Connectives
Author manuscript, published in "IEEE Int. Conf. on Fuzzy Systems, Spain 2010" On Weighted -Order Fuzzy Connectives Hoel Le Capitaine and Carl Frélicot Mathematics, Image and Applications MIA Université
More informationDrawing Conclusions from Data The Rough Set Way
Drawing Conclusions from Data The Rough et Way Zdzisław Pawlak Institute of Theoretical and Applied Informatics, Polish Academy of ciences, ul Bałtycka 5, 44 000 Gliwice, Poland In the rough set theory
More informationNeale and the slingshot Fabrice Correia
a Neale and the slingshot Fabrice Correia 'Slingshot arguments' is a label for a class of arguments which includes Church's argument to the effect that if sentences designate propositions, then there are
More informationGroup Decision-Making with Incomplete Fuzzy Linguistic Preference Relations
Group Decision-Making with Incomplete Fuzzy Linguistic Preference Relations S. Alonso Department of Software Engineering University of Granada, 18071, Granada, Spain; salonso@decsai.ugr.es, F.J. Cabrerizo
More informationFuzzy Expert Systems Lecture 3 (Fuzzy Logic)
http://expertsys.4t.com Fuzzy Expert Systems Lecture 3 (Fuzzy Logic) As far as the laws of mathematics refer to reality, they are not certain, and so far as they are certain, they do not refer to reality.
More informationMulti-Criteria Optimization - an Important Foundation of Fuzzy System Design
University of Texas at El Paso DigitalCommons@UTEP Departmental Technical Reports (CS) Department of Computer Science 1-1-1997 Multi-Criteria Optimization - an Important Foundation of Fuzzy System Design
More informationFrom Fuzzy- to Bipolar- Datalog
From Fuzzy- to Bipolar- Datalog Ágnes Achs Department of Computer Science, Faculty of Engineering, University of Pécs, Pécs, Boszorkány u. 2, Hungary achs@witch.pmmf.hu Abstract Abstract. In this work
More informationAxiomatic set theory. Chapter Why axiomatic set theory?
Chapter 1 Axiomatic set theory 1.1 Why axiomatic set theory? Essentially all mathematical theories deal with sets in one way or another. In most cases, however, the use of set theory is limited to its
More informationIntroduction to Metalogic
Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)
More informationA Class of Partially Ordered Sets: III
A Class of Partially Ordered Sets: III by Geoffrey Hemion 1 Definitions To begin with, let us fix a certain notation. Let the pair (X, ) be a given poset. For a X, let a = {x X : x < a} and a = {x X :
More informationComparison of two versions of the Ferrers property of fuzzy interval orders
Comparison of two versions of the Ferrers property of fuzzy interval orders Susana Díaz 1 Bernard De Baets 2 Susana Montes 1 1.Dept. Statistics and O. R., University of Oviedo 2.Dept. Appl. Math., Biometrics
More informationRegular finite Markov chains with interval probabilities
5th International Symposium on Imprecise Probability: Theories and Applications, Prague, Czech Republic, 2007 Regular finite Markov chains with interval probabilities Damjan Škulj Faculty of Social Sciences
More informationOn the Amount of Information Resulting from Empirical and Theoretical Knowledge
On the Amount of Information Resulting from Empirical and Theoretical Knowledge Igor VAJDA, Arnošt VESELÝ, and Jana ZVÁROVÁ EuroMISE Center Institute of Computer Science Academy of Sciences of the Czech
More informationFunctional Dependencies and Normalization
Functional Dependencies and Normalization There are many forms of constraints on relational database schemata other than key dependencies. Undoubtedly most important is the functional dependency. A functional
More informationInterval based Uncertain Reasoning using Fuzzy and Rough Sets
Interval based Uncertain Reasoning using Fuzzy and Rough Sets Y.Y. Yao Jian Wang Department of Computer Science Lakehead University Thunder Bay, Ontario Canada P7B 5E1 Abstract This paper examines two
More informationCombining Interval, Probabilistic, and Fuzzy Uncertainty: Foundations, Algorithms, Challenges An Overview
Combining Interval, Probabilistic, and Fuzzy Uncertainty: Foundations, Algorithms, Challenges An Overview Vladik Kreinovich 1, Daniel J. Berleant 2, Scott Ferson 3, and Weldon A. Lodwick 4 1 University
More informationFormal Logic. Critical Thinking
ormal Logic Critical hinking Recap: ormal Logic If I win the lottery, then I am poor. I win the lottery. Hence, I am poor. his argument has the following abstract structure or form: If P then Q. P. Hence,
More informationtype-2 fuzzy sets, α-plane, intersection of type-2 fuzzy sets, union of type-2 fuzzy sets, fuzzy sets
K Y B E R N E T I K A V O L U M E 4 9 ( 2 3 ), N U M B E R, P A G E S 4 9 6 3 ON SOME PROPERTIES OF -PLANES OF TYPE-2 FUZZY SETS Zdenko Takáč Some basic properties of -planes of type-2 fuzzy sets are investigated
More informationFuzzy Expert Systems Lecture 3 (Fuzzy Logic)
Fuzzy Expert Systems Lecture 3 (Fuzzy Logic) As far as the laws of mathematics refer to reality, they are not certain, and so far as they are certain, they do not refer to reality. Albert Einstein With
More informationTowards Formal Theory of Measure on Clans of Fuzzy Sets
Towards Formal Theory of Measure on Clans of Fuzzy Sets Tomáš Kroupa Institute of Information Theory and Automation Academy of Sciences of the Czech Republic Pod vodárenskou věží 4 182 08 Prague 8 Czech
More informationOn the Relation of Probability, Fuzziness, Rough and Evidence Theory
On the Relation of Probability, Fuzziness, Rough and Evidence Theory Rolly Intan Petra Christian University Department of Informatics Engineering Surabaya, Indonesia rintan@petra.ac.id Abstract. Since
More informationLearning Multivariate Regression Chain Graphs under Faithfulness
Sixth European Workshop on Probabilistic Graphical Models, Granada, Spain, 2012 Learning Multivariate Regression Chain Graphs under Faithfulness Dag Sonntag ADIT, IDA, Linköping University, Sweden dag.sonntag@liu.se
More informationA Generalized Decision Logic in Interval-set-valued Information Tables
A Generalized Decision Logic in Interval-set-valued Information Tables Y.Y. Yao 1 and Qing Liu 2 1 Department of Computer Science, University of Regina Regina, Saskatchewan, Canada S4S 0A2 E-mail: yyao@cs.uregina.ca
More informationFrom imprecise to granular probabilities
Fuzzy Sets and Systems 154 (2005) 370 374 www.elsevier.com/locate/fss Discussion From imprecise to granular probabilities Lotfi A. Zadeh,1 Berkeley initiative in Soft Computing (BISC), Computer Science
More informationOn Urquhart s C Logic
On Urquhart s C Logic Agata Ciabattoni Dipartimento di Informatica Via Comelico, 39 20135 Milano, Italy ciabatto@dsiunimiit Abstract In this paper we investigate the basic many-valued logics introduced
More informationMining Positive and Negative Fuzzy Association Rules
Mining Positive and Negative Fuzzy Association Rules Peng Yan 1, Guoqing Chen 1, Chris Cornelis 2, Martine De Cock 2, and Etienne Kerre 2 1 School of Economics and Management, Tsinghua University, Beijing
More informationOn Objectivity and Models for Measuring. G. Rasch. Lecture notes edited by Jon Stene.
On Objectivity and Models for Measuring By G. Rasch Lecture notes edited by Jon Stene. On Objectivity and Models for Measuring By G. Rasch Lectures notes edited by Jon Stene. 1. The Basic Problem. Among
More informationComparison of 3-valued Logic using Fuzzy Rules
International Journal of Scientific and Research Publications, Volume 3, Issue 8, August 2013 1 Comparison of 3-valued Logic using Fuzzy Rules Dr. G.Nirmala* and G.Suvitha** *Associate Professor, P.G &
More informationA Crisp Representation for Fuzzy SHOIN with Fuzzy Nominals and General Concept Inclusions
A Crisp Representation for Fuzzy SHOIN with Fuzzy Nominals and General Concept Inclusions Fernando Bobillo Miguel Delgado Juan Gómez-Romero Department of Computer Science and Artificial Intelligence University
More informationPairing Transitive Closure and Reduction to Efficiently Reason about Partially Ordered Events
Pairing Transitive Closure and Reduction to Efficiently Reason about Partially Ordered Events Massimo Franceschet Angelo Montanari Dipartimento di Matematica e Informatica, Università di Udine Via delle
More informationAn Approach to Classification Based on Fuzzy Association Rules
An Approach to Classification Based on Fuzzy Association Rules Zuoliang Chen, Guoqing Chen School of Economics and Management, Tsinghua University, Beijing 100084, P. R. China Abstract Classification based
More informationInternational Journal of Uncertainty, Fuzziness and Knowledge-Based Systems c World Scientific Publishing Company
International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems c World Scientific Publishing Company ON AN INTERPRETATION OF KEYWORDS WEIGHTS IN INFORMATION RETRIEVAL: SOME FUZZY LOGIC BASED
More informationFuzzy Rules and Fuzzy Reasoning (chapter 3)
Fuzzy ules and Fuzzy easoning (chapter 3) Kai Goebel, Bill Cheetham GE Corporate esearch & Development goebel@cs.rpi.edu cheetham@cs.rpi.edu (adapted from slides by. Jang) Fuzzy easoning: The Big Picture
More information