A Method for Filling up the Missed Data in Information Table

Transcription

1 A Method for Fllng up the Mssed Data Gao Xuedong, E Xu, L Teke & Zhang Qun A Method for Fllng up the Mssed Data n Informaton Table Gao Xuedong School of Management, nversty of Scence and Technology Beng, P.R. Chna, 183 Phone: , , Emal: gaoxuedong@manage.ustb.edu.cn E Xu School of Management, nversty of Scence and Technology Beng, P.R. Chna, 183 Phone: , , Emal: exu21@163.com L Teke School of Management, nversty of Scence and Technology Beng, P.R. Chna, 183 Phone: , , Emal: teke@manage.ustb.edu.cn Zhang Qun School of Management, nversty of Scence and Technology Beng, P.R. Chna, 183 Phone: , , Emal: zq@manage.ustb.edu.cn ABSTRACT Almost algorthms based on the rough sets, such as mean value method, mum frequency method, mode method are week n supportng the hdden rules n the nformaton tablel. By the breakng pont sets n the nformaton system, a new method for packng the mssed attrbute value s provded n the paper,. The method s more effcent for ndcatng the decson rules. Key words: Rough sets, Informaton table, Classng; Decson rule, Breakng pont INTRODCTION The on-gong nformaton revoluton s generatng volumes of data, from sources as dvers as bankng transactons, scentfc exploratons, telecommuncaton networks, space scence, medcal systems and so on. Indeed, there s an mperatve on the ntellgent analyss of such large volumes of data so as to derve ntrnsc knowledge (Kryskewcz, 1998, whch can mpact to optmze decson support, busness compettveness and other servcesorented portfolos. Rough sets theory (Wu et al., 23; Mchalsk, 1986; Pawlak, 1982 has been proposed by Professor Pawlak for knowledge dscovery n databases and expermental data sets. It s based on the concept of an upper and a lower approxmaton of rough sets, ndscernble matrx and so on. Rough sets theory (Kohav and Frasca, 1994 s an effcent mathematcal tool to deal wth the uncertan and ncomplete data, whch don t need some transcendental knowledge or some accessonal nformaton but ust the data tself. For now, t has been appled n many felds such as machne learnng, artfcal ntellgence and so on; especally t has been a very effcent method n data mnng, whch frequently appears n clusterng, classfcaton algorthms. In rough sets theory, data exsts n the nformaton table. But because of varous reasons such as wrong operaton, off power and so on, there are always some mssed data n the nformaton table, we must pack them so as to analyze them. For now, there have been many algorthms(lu and Setono, 1995; Pawlak, 1991; Wang, 2 for fllng up the mssed data n the nformaton table, whch can be dvded nto two categores n terms of thnkng or not thnkng about rough sets, such as mean value method that takes the whole data average on some certan attrbute as the mssed data, mum frequency method that takes the most frequent data on some certan attrbute as the mssed data, mod method that takes the mddle value on some certan attrbute as the mssed data, and so on. But those methods can not support the hdden rules n the nformaton table very well because they thnk lttle or at least don t thnk about rough sets. Communcatons of the IIMA Volume 5 Issue 4

2 A Method for Fllng up the Mssed Data Gao Xuedong, E Xu, L Teke & Zhang Qun A new approach for fllng up the mssed attrbute value s presented n the paper based on the breakng pont sets n the nformaton system, whch s presented after studyng the correlaton of condtonal and decson attrbutes. Some experments show that the method s more effcent for ndcatng the decson rules. CHIEF CONCEPTS OF ROGH SETS THEORY Rough sets theory ncludes some sgnfcant concepts (Wang, 2; Pawlak, 1998 such as an upper approxmaton, a lower approxmaton, etc. In rough set theory, annotated data s represented as an nformaton system. An nformaton system can be represented as S = (,A,V,f, { = x1, x2,..., xn Where s the unverse, a fnte set of N obects, A s a fnte set of attrbutes, whch are dvded nto dsont sets,.e. A = C D, where C s the set of condton attrbutes and D s the set of decson attrbute. V = q AVq Vq (where s a doman of the attrbute q, f : A V s the total decson functon(called nformaton f functon, such that ( x, q Vq for every q A, x. A subset of attrbutes Q A defnes an equvalence IND( Q = {( x, y : a Q, f ( x, a = f ( y, a relaton (called an ndscernble relaton on,. For a gven nformaton system S = (,A,V,f, a subset of attrbutes Q Adetermnes the approxmaton space IND( Q n S. For gven Q Aand X (a concept X, the Q lower approxmaton QX of the set X and the Q upper approxmaton QX of the set X are defned as fellows: { :[ ] A [ ] QX = x x X (1 { QX = x x I X (2 : A And meanwhle, we can defne the Q-postve regon POSQ ( D n the relaton IND( D as { POS ( D = QX : X IND( D (3 Q The postve regon POSQ ( classes defned by IND( D. So we can defne the rough set boundary as POS ( D D contans all the obects n that can be classfed wthout an error nto dstnct = B.One can form a postve regon for any two subsets of attrbutes BE, An the nformaton system S. Snce the subset of attrbutes B A defnes the ndscernble relaton IND( B, t consequently defnes the classfcaton * * B ( ( ( B IND B wth respect to the subset E. The E-postve regon of B s defned as POS ( E B = EX. (4 * X B The E-postve regon of B contans all the obects that, by usng attrbutes E, can be classfed to one of dstnct classes of the classfcaton * B. The cardnalty of the E-postve regon of B can be used to defne a measure γ E ( B of dependency of the set of attrbutes B on E: card( POSE ( B γ E ( B = (5 card( Communcatons of the IIMA Volume 5 Issue 4

3 A Method for Fllng up the Mssed Data Gao Xuedong, E Xu, L Teke & Zhang Qun Rough sets defne a measure of sgnfcance (coeffcent of sgnfcance of the attrbute a A from the set E wth respect to the subset B: sg( a = γ ( B γ ( B. (6 \{ A A a PMMDVBBP ALGORITHM PMMDVBBP (Packng Method for Mssed Data Value Based on Breakng Ponts algorthm s based on the breakng ponts n the nformaton table wth respect to the correlaton of condton and decson attrbutes, so t can padded the mssed data better. Relevant Defnton and Theorem Gven an nformaton system S=(,A,V,f, = { = 1,..., attrbutes set follows: MAS, ndscernble set A a m s the attrbute set, assumed X,then t s mssed NS and the mssed obect set of the nformaton system MOS are defned as = { ( = *, = 1,..., = {, = 1,..., { (,,, 1,..., MAS a a x k m (7 k k MOS MAS n (8 NS = M = = n (9 From the nformaton system table, we can proof a theorem as follows: Theorem: Gven an nformaton system S =, AV,, F, where =, complete attrbute values, s the sample set that we only know the partal values. sgnfcant attrbute set, C s the redundant attrbute set, D s the decson attrbute set. If ca ( = cb (, then conclude that the nformaton system s certanty s stable. Proof: We can assume any classfcaton of the system, ( ( = classfcaton dvded by the condton attrbute set C, { 1, 2,..., n = ( classfcaton E IND( C, t s certanty to the decson attrbute class s as follows: µ ( E = E I X / E : X IND( D ({ s the total sample set wth A = C C D, Communcatons of the IIMA Volume 5 Issue 4 C s the a, b, c C, E IND C, 1, 2,..., m, m s the number of the X X X IND D, then for some certan (1 We can nduce the nformaton system certanty as follows: m E µ ( S = µ ( E (11 = 1 Based on the formula (11, we can dscuss the above theorem from two angles: If C has only one element c, then we can regard the formula E IND( C ( c, ( = 1,2,..., m as the classes determned by the condton attrbute set C = C { c. Snce c s the redundant attrbute, we consequently nduce the equaton IND( C IND( C { c =, namely addng redundant attrbute can t affect the classes n the nformaton system S,.e. E = E, therefore,we can nduce EE, INDC, then there must be E IND C { c, we can conclude the result that f ( formula ( E µ ( E µ = ; so obtan the concluson that ( S changed. In the same way, f { C1, C2,..., C m nformaton table wll not changed PMMDVBBP Algorthm Descrpton ( µ n the nformaton table s not µ n the C = s the redundant attrbute set, then the ( S Based on the chef concepts of rough sets theory, especally the above formula (8, formula (9, formula (1 and formula (11, the paper puts forward an algorthm for fllng up the mssed data n the nformaton system S,

4 A Method for Fllng up the Mssed Data Gao Xuedong, E Xu, L Teke & Zhang Qun PMMDVBBP algorthm. The man steps of PMMDVBBP algorthm can be descrbed as follows, gven the ntal nformaton system table S =, A, V, f. Step 1: Calculate the relevant coeffcent among varous attrbutes, construct the matrx n \ MOS. Step 2: Cluster the condton attrbutes n the relevant coeffcent matrx, then select one representatve attrbute from every cluster to form a new condton attrbute set. Step 3: Calculate the condton attrbute mportance n \ MOS, sort them by the mportance, and then place MOS at the end of the nformaton table to form a new nformaton table S =, A, V, f. Step4: Assumed that P s the breakng pont set of anyone attrbute A, A \ MOS, the breakng pont s P,=1,2,,n-1ts nearest neghbors are x x + 1, furthermore, x < P < x + 1,we can deal wth A as follows: FOR (=1; n; ++ {f α = x ; x = x+ 1,the nformaton table has no conflcts, then P = P \ { P ; else x ; = α x+ 1 = x+ 1.. Step 5: Classfy the samples n the nformaton table, sort the classes by ther cardnalty, and then operate them as Step 6. Step 6: We can assume any condton attrbutes set B B A, x MOS, and operate them as follows n one class: ⑴ If B r F = r F,then let a ( x s equal to any number. ak and ( ( ⑵ Else operate as follows: B B\ B 1 If x MOS calculate ( 2 If NS( x = y then let a ( x = a ( y 3If NS( x { y, y,..., y a ( x = a ( y k k NS x n the attrbute set \{ k k = 1 2 m then let k B B y = y the cardnalty of y s n the NS( x and ⑶ Operate the nformaton table as above untl x MOS s the fnal attrbute. Step 7: Select the second class n the decson table and operate t as Step 6. Step 8: Operate the decson table as Step 4 untl the fnal class, then we can export a new nformaton table S =, A, V, f. The above steps are the contents of PMMDVBBP algorthm proposed by the paper. DEMONSTRATION In order to explan the thnkng and contents of PMMDVBBP algorthm, we gve a demonstraton. Example: Gven an ntal nformaton system S =, A, V, f, where s the unverse, a fnte set of N obects { = x1, x2,..., xn, A s a fnte set of attrbutes, whch are dvded nto dsont sets,.e. A = C D, where C s the set of condton attrbutes and D s the set of decson attrbute. The contents of t n detals are showed as Table 1. a b c d * * 1 Communcatons of the IIMA Volume 5 Issue 4

5 A Method for Fllng up the Mssed Data Gao Xuedong, E Xu, L Teke & Zhang Qun * Table 1: The ntal nformaton table. Accordng to some mportant concepts of rough set and the chef steps of PMMDVBBP algorthm, we can do as follows: INDabc (,, = 1, 2,..., 8 {{{ { ( {{{ 1, 2 {, 3 {, 4,5,{ 7,{ 8 ( {{ 1,5,{ 2,{ 3,7,8,{ 4,6 ( {{ 1,2,4,5,6,{ 3,7,8 ( ( C ( / ( IND a = IND b = IND c = r D = card POS D card =8/8=1 C ( ( ( ( ( ( r D = card POS D card =3/8 {( {( / C\ a C\ a r D = card POS D card =5/8 {( {( / C\ b C\ b r {( D \ card POS \{( D / card C c C c So, the mportance of attrbute a s equal to rc ( D rc\ {( D a the mportance of attrbute b s equal to C ( C\ {( b the mportance of attrbute c s equal to ( {( = =8/8 =1-.375=.625, r D r D =1-.625=.375, rc D r D =1-1=obvously, the attrbute c s redundant. Followng C\ c the other steps, we can obtan the Table 2.and Table 3 from Table 1. a b c d * * * Table 2: The mddle table. a b c d Table 3: The fnal table that has been flled up. Communcatons of the IIMA Volume 5 Issue 4

6 A Method for Fllng up the Mssed Data Gao Xuedong, E Xu, L Teke & Zhang Qun For test the performance of the algorthm above, an experment has been done about the famous Irs classfcaton problem n the CI machne learnng database. There are only nstances wth contnuous attrbutes n Irs database. Every nstance has four contnuous attrbutes, whch are petal-length, petal-wdth, sepal-length, sepal-wdth. And the sum number of the nstances, whch belong to three classes, s 15. The experment result s as Table 4. PMMDVBBP algorthm s proved to be relable and effcacous from Table 4. Number( mssed data (mssed data ( Number( d d padded correctly (padded correctly Table 4: Expermental result. CONCLSION Ths algorthm s relable, effcent and smple. It s proposed on the correlaton of condton attrbutes and decson attrbutes, especally on the mportance of the breakng ponts n the nformaton system. So t avods conflcts n the nformaton system that general algorthms, such as the mean method, the mum method, the mod method and so on, made n the nformaton system and the data that t flled up can ndcate the hdden knowledge better. REFERENCES Kohav R, Frasca B. (1994. seful Feature Subsets and Rough Set Reducts. The 3th Internatonal Workshop on Rough Sets and Soft Computng. Kryskewcz M. (1998. Rough Set Approach to Incomplete Informaton System. Informaton Scences, 112, 39~49. Lu H., Setono R. (1995. Feature Selecton and Dscretzaton of Numerc Attrbutes. Proceedngs of the 7 th Internatonal Conference on Tools wth Artfcal Intellgence, Washngton D.C. Mchalsk R S. (1986. Mult-Purpose Incremental Learnng System AQ15 and Its Testng Applcaton to Three Medcal Domans. Proceedngs of the 5 th AAAI, CA. Pawlak Z. (1982. Rough Set. Internatonal Journal of Computer and Informaton Scence. 1, Pawlak Z. (1991. Rough Sets-Theoretcal Aspects of Reasonng about Data. Kluwer: Kluwer Academc Publsher. Pawlak Z. (1998. Rough Set Theory and Its Applcaton to Data Analyss. Cybernetcs and Systems, 29 (9, Wang Guo-yn. (2 Rough Set Theory and Knowledge Acquston (n Chna. X an, Chna: X an Communcaton nversty Publsher. Wu Sen, Gao Xue-dong, M. Bastan. (23. Data House and Data Mnng. Beng, Chna, Metallurgy Industral Publsher. Communcatons of the IIMA Volume 5 Issue 4