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
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
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
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
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
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 44-448994 [] 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 6-66994 [3] Didier Dubois enri radeat Are Fuzzy ules and ow to Use Them Fuzzy Sets and Systems Vol84 pages69-85 996 [4] J ubero JM Medina O ons and MA Vila Extensions of a resemblance relation Fuzzy Sets and Systems 86 997 [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 73-78 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: 4:3-4499 [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 =