Attribute Reduction and Information Granularity *

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1 ttribute Reuction an nformation Granularity * Li-hong Wang School of Computer Engineering an Science, Shanghai University, Shanghai, , P.R.C School of Computer Science an Technology, Yantai University, Yantai, , P.R.C an Geng-feng Wu School of Computer Engineering an Science, Shanghai University, Shanghai, , P.R.C BSTRCT n the view of granularity, this paper analyzes the influence of three attribute reucts on an information system, fining that the possible reuct an ecision reuct will make the granule view coarser, while iscernible reuct will not change the granule view. n aition, we investigate the combination of reucts from two partial information systems in parallel or in incremental ata mining an urge that the union of partial possible reucts can be regare as a possible reuct for union of partial information systems. Key wors: Rough Set, ttribute Reuct, nformation Granularity, Reucts Combination. NTRODUCTON t is necessary to approximate a colossal information system in orer to simplify the knowlege iscovery from it. n nformation System (S may be enote as S(O, T, where O is the non-empty set of objects, an T is a non-empty set of attributes. n orer to make some approximation, we collect similar objects from the set O to form a subset an name it as a granule. n a granule, one object is regare as the same as the others because the inherent ifference between two objects isappears when they are assigne to the same granule. The information system may be ivie or covere by the set consiste of these granules, which gives an approximation to the S an can be name as a granule view of it. The types of similarities range from simple equivalence relations, tolerance relations to reflective binary relations, etc. The granule view of information system epens on the similarity use to form a granule (Y.Y.Yao, 200, Skowron, Stepaniuk, 200. s an important approach to simplify an S with many attributes, the attribute reuct base on rough set theory can effectively iscar abunant information an inuct some ecision rules from ata, attracting many researchers to focus on it (Kryszkiewicz, 200, Chang, 999, Wang, 998, Miao, 997. Many efinitions on attribute reuct have been propose an some equivalence among them has been prove (Kryszkiewicz, 200. However, we are not quite sure about the influence of reuct on an information system or on a granule view of it. This paper will iscuss these problems in the view of information granularity. itionally, the restriction of memory an serial computation in single CPU make it ifficult to give an effective reuct for a gigantic information system. Generally, it is one in parallel, which means to ivie This work was supporte by Science an Technology Commission of Shanghai Municipality, No SYSTEMCS, CYBERNETCS ND NFORMTCS VOLUME - NUMBER

2 the whole S to several parts, each of which is assigne to a noe in parallel computer or to a computer in a network an then combine the results from each computer to make a reuct for the primary S (Scotney, McClean,999. Usually it is the same case in an incremental ata mining for historical an present ata, that is, how to combine the existe mining result with the new information erive from present ata to form the latest knowlege. The paper is organize as follows: Section 2 gives some basic notations in rough set theory as well as three efinitions for attribute reuct, an Section 3 shows the change of granularity of an information system after reuct, an Section 4 proves that the union of partial possible reucts can be regare as a possible reuct for union of partial information systems, but it is not true for ecision reuct or iscernible reuct. 2. TTRBUTE REDUCT DEFNTONS n this section, some basic rough set notations will be introuce, an several attribute reuct efinitions will be simply escribe (Kryszkiewicz, 200. Def. 3 Lower/Upper approximation of X Let X O, we efine { x O X} as Lower approximation of X, an X { x O X φ} as Upper approximation of X. Def. 4 Membership function membership function : O [0,], T is efine as follows: X X X X for any x O Decision Table(DT is an information system DT ( O, T { }, where, T is a istinguishe attribute calle ecision, an the elements of T are calle conitions. ecision class { x O ( x i} is a set, in which each X i element has the same ecision. Def. niscernible relation For an information system S(O,T, each subset of attributes T etermines an iniscernible relation ND( as follows: ND ( {( x, y O O a, a( x a( y} which means object x an y have the same value at any attribute in. Def. 2 Equivalence class ccoring to efinition, ND( is an equivalence relation, which can partition objects O into equivalence classes. Let (x enote the equivalence class etermine by x on. { y O ( x, y ND( } Each equivalence class can be regare as a granule. Three common attribute reucts are efine as follows: Def. 5 Possible reuct T is a possible reuct of DT for x, x O, if an only if is a set such that x ( ( T X is calle minimal possible reuct, if it is the minimal set satisfie Eq.(. With the ecrease of the carinality of, (x enlarge. However, Eq.( sets an upper boun for (x to confine the granule which contains object x. T is a possible reuct of DT if an only if is a set such that Eq. ( hols true for every x in O. There are other reucts such as approximate reuct an generalize ecision reuct, which are equivalent to possible reuct (Kryszkiewicz, 200. SYSTEMCS, CYBERNETCS ND NFORMTCS VOLUME - NUMBER 33

3 Def. 6 ecision reuct T is a ecision reuct of DT for x, x O, if an only if is a set such that T (2 The variation of granules after possible reuct First, we efine granules as equivalence classes of objects, that is, for x, x O, is a granule embracing x before reuct while ( T { } x { } ( x is the where ( ( x,...,, (x enotes n j granule after reuct. For any y ( T { } x, its an Xj is the jth ecision class. Eq.(2 Xj requires the membership of x to all ecision classes preserve. is calle minimal ecision reuct if it is the minimal set satisfie Eq. (2. T is a ecision reuct of DT if an only if is a set such that Eq. (2 hols true for every x in O. There is reuct which is equivalent to ecision reuct (Kryszkiewicz, 200. Def. 7 Discernible reuct For an information system S(O,T, T is the iscernible reuct for x O if an only if (x T (x. T is the iscernible reuct of DT if an only if (x T (x hols true for any object x. Discernible reuct preserves the istinction between two ifferent objects in S. 3. NFORMTON GRNULRTY DFFERENCE FTER TTRBUTE REDUCT ttribute reuct simplifies an information system by iscaring some reunant attributes. n the view of approximation, the information system is reuce to a granule view of it. However, it is necessary to analyze the approximation by these reucts. We prove that possible reuct an ecision reuct lea to a coarser view of original system, while the iscernible reuct oes not change the granularity of an information system. conitions an ecision are same with those of x. By efinition 5, possible reuct shoul satisfy that (x T X (x, while T X { y O T ( y φ} ( T { } x X (x requires the element y in (x either has the same ecision with x or there exist one element, which have the same conitions with those of y an whose ecision is (x. The loose requirement makes larger than {} (x in carinality an coarser in granularity. For example, table is an information system S, we can get a coarser approximation for it by possible reuct. Table nformation System S No. a a2 a3 a4 a By efinition 5, {a,a2} is a possible reuct of T{a,...,a5}. Let s regar granule embracing x before reuct an as the the granule after reuct. From this example we can fin that the granule view of the system has been change, the later view is coarser. The granule view before reuct: ( T { } x ( T { } {}, (2 T { } {2}, { } ( x as 34 SYSTEMCS, CYBERNETCS ND NFORMTCS VOLUME - NUMBER

4 (3 T { } {3}, (4 T { } {4}. The granule view after reuct: (3 {,3}, { } ( { } (2 { } {2}, (4 { } {4}. The variation of granules after ecision reuct The granule is efine as above, accoring to efinition 6, reuct shoul satisfy that that is, T T T ( x,..., n (,..., n ( an further X for i..n i T X Table 2 is an example for T i ecision reuct. Through efinition 6,we know that {a,a2} is a ecision reuct for T{a,...,a5}. The variation of granule view shows the approximation influence of ecision reuct. Table 2 nformation System S 2 No. a a2 a3 a4 a The granule view before reuct: ( T { } {}, (2 T { } {2}, (3 T { } {3}, (4 T { } {4}. The granule view after reuct: { } ( { } (3 {,3} { } ( 2 { } (4 {2,4} These examples show that possible reuct an ecision reuct will make the granule view of an information system coarser. Discernible reuct preserves the granule view By efinition 7, the unchange equivalence classes of an information system after iscernible reuct prevent the granule view from varying. 4. REDUCT COMBNTON n parallel or in incremental ata mining, partial mining results shoul be integrate to get present efficient knowlege (Scotney, McClean,999. We recommen the union of partial reucts as a reuct of whole information system, but this conclusion hols true only for possible reuct. Theorem Let S ( O, T { } an 2 S ( O 2, T { } be two homogenous information system, an attributes set reuct for S / S 2, then / 2 2 is the possible is a possible reuct for S ( O O2, T { }, but this is not true for Proof: ecision reuct or iscernible reuct. Let be a possible reuct for S, then x hols true in S, which can be ( T X enote as ( S, an we know that ( 2 x T X ( x hols true for any subset 2, an S T X ( S S hols true, then ( x ( 2 T X SYSTEMCS, CYBERNETCS ND NFORMTCS VOLUME - NUMBER 35

5 we can get ( S S T X, which means is a possible reuct of information system S ( O O, 2 T { }. However, this conclusion is not true for ecision reuct or iscernible reuct. Here is an example for ecision reuct. Table 3 nformation System S 3 an S 4 No. a a2 a3 a4 a n table 3, {a5} can be regare as a ecision reuct in S 3 ({,2,3},{a,...a5,}, an {a2,a3} as a ecision reuct in S 4 ({4,5,6,7}, {a,...a5,}, while { a2, a3, a5} { a5} { a2, a3} is not a ecision reuct for S 3 S 4 T ( (0.5,0,0.5 ( (,0,0 ( T ( S5 S 6, because for object No. which implies the istribution in ecision classes has been change by the reuct. Similarly, table 4 shows an example for iscernible reuct. By efinition 7, {a3,a4} is a iscernible reuct for S 5 ({,2,3,4},{a,...a5,}, an {a5,} is a iscernible reuct for S 6 ({5,6,7,8},{a,...a5,}. However, { a3, a4, a5, } { a3, a4} { a5, } is not a iscernible reuct for an No.5 become iniscernible after reuct. because object No. Table 4 nformation System S 5 an S 6 No. a a2 a3 a4 a This is the en of proof. 5. CONCLUSON REMRKS n the view of information granularity, the paper iscusse the influence of attribute reucts on the granule view of an information system, proposing that the possible reuct an ecision reuct will make the granule view coarser, while iscernible reuct will not change the granule view. n aition, we gave a simple algorithm for the integration of reucts of two partial information systems in parallel or incremental ata mining, an prove that the union of possible reucts is a possible reuct for union of partial information systems. REFERENCES []nrzej Skowron, J.Stepaniuk nformation Granules: Towars Founations of Granular Computing, nternational Journal of ntelligent Systems, Vol.6, p57-85 [2]B.Scotney, S.McClean Efficient Knowlege Discovery Through the ntegration of Heterogeneous Data, nformation an Software Technology 4 p [3]Chang Li-yun, Wang Guo-yin, Wu Yu.999. n pproach for ttribute Reuction an Rule Generation Base on Rough Set Theory, Journal of software, Vol.0 No., p206-2(in Chinese [4]Miao Duo-qian.997. Rough Set an ts pplication in Machine Learning, [Ph.D. Thesis]. nstitute of utomation, The Chinese caemy of Sciences, (in Chinese [5]M.Kryszkiewicz Comparative Stuy of 36 SYSTEMCS, CYBERNETCS ND NFORMTCS VOLUME - NUMBER

6 lternative Types of Knowlege Reuction in nconsistent Systems. nternational Journal of ntelligent Systems, Vol.6, p05-20 [6]Wang jue, Wang ren, Miao Duo-qian Data Enriching Base on Rough Set Theory, Chinese Journal of Computers, 2(5, p (in Chinese [7]Y.Y.Yao.200.nformation Granulation an Rough Set pproximation, nternational Journal of ntelligent Systems, Vol.6, p87-04 Li-hong Wang, an associate professor of Yantai university, is now pursuing her Ph.D in school of computer engineering an science in Shanghai university, P.R.C. Her research interests inclue knowlege iscovery, intelligent information processing an the architecture of computer network. Geng-feng Wu is presently a professor of Shanghai university, an associate member of the thir worl acaemy of sciences an a councilor of Chinese computer feeration. His main research interests inclue ata mining an knowlege iscovery in atabase, intelligent control, intelligent information processing an CSCW (computer supporte cooperative work. SYSTEMCS, CYBERNETCS ND NFORMTCS VOLUME - NUMBER 37

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