Appl Math Inf Sci 6 No S pp 439S-444S (0 Applied Mathematics & Infomation Sciences An Intenational Jounal @ 0 NSP Natual Sciences Publishing o Point-iseial oelation Analysis of Fuzzy Attibutes Hao-En hueh Depatment of Infomation Management, Yuanpei Univesity, 3005 Hsinhu, Taiwan oesponding autho: Email: hechueh@mailypuedutw Received June, 0; Revised Septembe, 0; Accepted 3 Septembe 0 Published online: Mach 0 Abstact: Pevious studies have pesented coelation analyses among fuzzy attibutes This study poposes a coelation analysis method among cisp attibutes and fuzzy attibutes to investigate thei linea elationships If A and ae two attibutes, A is a nominal-dichotomous attibute, and is a fuzzy attibute Thus, identifying the elationship between A and is possible by using the pointbiseial coelation analysis The fuzzy point-biseial coelation analysis is defined and deived, using the membeship gades of fuzzy attibutes Using the fuzzy point-biseial coelation analysis is feasible if A is a bimodal distibution, but not a eal nominal-dichotomous attibute, such as cime youth vesus non-cime youth, illiteate vesus non-illiteate peson, and mental deficiency vesus nomal intelligence An expeimental sample illustates that ou poposed method is simple and faste than the taditional coelation analysis Keywods: oelation Analysis, Point-iseial oelation Analysis, Nominal-Dichotomous Attibute, Fuzzy Attibutes Intoduction oelation analysis of cisp attibutes is often used to identify the elationships among attibutes in databases Vaious coelation analyses, defined on cisp attibutes, have been discussed in conventional statistics [, 5] Howeve, many ecoded attibutes in databases may be fuzzy [7,, 0] but useful, which cannot be descibed by cisp attibutes, and must be genealized to fuzzy sets, fo accuate data epesentation Exploing the attibutes is also necessay, and theefoe, methods to investigate these fuzzy attibutes ae equied oelation analyses among fuzzy attibutes ae efeed to as fuzzy coelation analyses Pevious studies [, 3, 6, 9] have pesented thee fuzzy coelation analyses (fuzzy simple coelation analysis [], fuzzy patial coelation analysis [3], and fuzzy semi-patial coelation analysis [6] on Zadeh s fuzzy sets The simple coelation coefficient between two fuzzy attibutes is called a fuzzy simple coelation coefficient [, 9], and is useful in discoveing the stength of a elationship between two vague vaiables, o fuzzy attibutes If two fuzzy attibutes ae diectly elated, a positive scoe emeges If two fuzzy attibutes ae invesely elated, a negative scoe emeges If two fuzzy attibutes ae not evidently elated, the scoe is close to zeo Fuzzy simple coelation coefficients povide a compehensive undestanding of the linea elationship between two fuzzy attibutes Howeve, in some pactical situations, fuzzy attibutes, othe than the two in question, ae also esponsible fo the obseved elationship, and may influence the elationship between the obseved fuzzy attibutes Thus, solving this poblem equies analyzing fuzzy patial coelations [3], to show the elationship between two fuzzy attibutes, when the influences of othe fuzzy attibutes ae emoved fom the obseved fuzzy attibutes In numeous fuzzy pediction models, the analysis of the fuzzy semi-patial coelation [6] is useful in choosing the pedicto fuzzy attibutes, to pedict the citeion fuzzy attibute The elationship between two fuzzy attibutes is efeed to as the fuzzy semi-patial coelation analysis,
444 Hao-En hueh: Point-iseial oelation Analysis of Fuzzy Attibutes by the emoval of the influence of othe fuzzy attibutes fom one of the inteested fuzzy attibutes Such a coelation is moe complicated and has lage vaiety than the pevious ones we have pesented Pevious studies have pesented coelation analyses among fuzzy attibutes This study poposes a coelation analysis method among cisp attibutes and fuzzy attibutes to investigate thei linea elationships This pape genealizes the discussion of the linea elationship among fuzzy attibutes to the linea elationship among cisp attibutes and fuzzy attibutes, measued by fuzzy point-biseial coelation coefficients If A and ae attibutes, A is a nominaldichotomous attibute, and is a fuzzy attibute, which enables the utilization of the point-biseial coelation analysis to identify the elationship between A and If A is a bimodal distibution, but not a eal nominal-dichotomous attibute, such as youth cime vesus youth not involved in cime, illiteacy vesus liteacy, and mental deficiency vesus nomal intelligence, these attibutes all belong to bimodal distibution These situations also equie the possible use of the fuzzy pointbiseial coelation analysis The membeship gades of fuzzy attibutes ae defined and deived by using the fuzzy pointbiseial coelation analysis The est of this pape is oganized as follows: Section pesents seveal fuzzy coelation analyses, Section 3 illustates the developed and expeimented definition of the fuzzy point-biseial coelation analysis, and finally, Section 4 offes the conclusion Fuzzy oelation Analyses The simple coelation coefficient of fuzzy attibutes is efeed to as the fuzzy simple coelation coefficient Pevious studies have poposed numeous methods to evaluate the fuzzy simple coelation coefficient [, 9] Suppose fuzzy attibutes A and F, whee F is a fuzzy space Fuzzy attibutes A and ae defined on the domain of a cisp univesal set X, with membeship functions A and Fuzzy attibutes A and can then be expessed as: A {( x, x x X ( {( x, ( x x X (, whee A, : X [0, ] Assume that (( x, x, ( x, ( x, x, ( x,,( x n, A ( x n, ( x n is a andom sample dawn fom a cisp univesal set X The fuzzy simple coelation coefficient [] between fuzzy attibutes A and is then defined by: s (3 s s A, whee n i( xi A ( ( xi s (4 n is the covaiance of fuzzy attibutes A and s s, A A s s, A ae the vaiances of A and,, espectively The values of the fuzzy membeship functions and lie between [0, ], but the value of the A fuzzy simple coelation coefficient is constained between [-, ], which demonstates not only the elationship degee between the fuzzy attibutes, but also whethe these two attibutes ae positively o negatively elated Popeties of the fuzzy simple coelation coefficient ae stated as follows []: If, is close to, then the fuzzy attibutes A A and ae highly elated If, is close to 0, then the fuzzy attibutes A A and ae baely elated If 0, then the fuzzy attibutes A and ae positively elated If 0, then the fuzzy attibutes A and ae negatively elated If 0, then the fuzzy attibutes A and have no elationship at all In spite of the fact that values of the fuzzy membeship function ae constained between in [0,], the value of the fuzzy simple coelation coefficient lies between in [-,], which will show us not only the degee of the elationship between the fuzzy sets, but also the fact whethe these two sets ae positively o negatively elated In seveal pactical situations, fuzzy attibutes, othe than the two in question, ae also esponsible fo the obseved elationship, and these attibutes may influence the elationship between the obseved attibutes Thus, developing the analysis of the fuzzy patial coelation [3] is equied, to show the elationship between two fuzzy attibutes when the influences of othe fuzzy attibutes ae patialed out fom the obseved fuzzy attibutes Assume that simple coelation coefficients ae computed between pais of the fuzzy attibutes, and, namely,,,, and The fuzzy patial coelation coefficient between fuzzy attibutes A and, when the effects of the fuzzy
Hao-En hueh: Point-iseial oelation Analysis of Fuzzy Attibutes 444 attibute on fuzzy sets A and ae patialed out, is defined as:, (5 ( (, Popeties of the fuzzy patial coelation coefficient ae stated as follows [3]: The fuzzy patial coelation coefficient is equal to the fuzzy patial coelation coefficient, A, A The fuzzy patial coelation coefficient o is not defined, if thee exists a linea elationship between fuzzy sets A and, that is, Similaly, o is not defined, if, A, If 0, then, Similaly, if A 0,, then, A If 0 and 0, then,, A, A If thee is no elationship between A and, that is, if 0, this does not necessaily mean that thee is no patial elationship between fuzzy sets A and when the influence of fuzzy set is emoved Next, we discuss the fuzzy semi-patial coelation, the coelation between two fuzzy attibutes when the influences of othe fuzzy attibutes ae emoved fom only one of the two inteested fuzzy attibutes [6] Assume that the simple coelation coefficients,,, and of fuzzy attibutes and have been calculated accoding to fomula (3 Then the semi-patial coelation coefficient between the fuzzy attibutes A and, with the influence of fuzzy attibute emoved fom the fuzzy attibute, can be calculated as:, (6 ( (, Similaly, the semi-patial coelation,( A can be calculated as:, (7,( A (, ( A illustates the elationship between the fuzzy attibutes A and, with the influence of the fuzzy attibute emoved fom fuzzy attibute A Popeties of the fuzzy semi-patial coelation coefficient ae stated as follows [6]: Thee ae six diffeent fuzzy semi-patial coelation coefficients among thee fuzzy attibutes and, say,,, (, ( A, ( A, and, although thee ae only, ( A (, ( A thee diffeent fuzzy patial coelation coefficients among the same attibutes, say,, and The fuzzy semi-patial coelation coefficient,, is diffeent fom because of thei, ( A ( diffeent meanings means the elationship ( between the fuzzy attibutes A and with the influence of fuzzy attibute emoved fom the fuzzy attibute, but compaatively, means, ( A the elationship between the fuzzy attibutes A and with the influence of fuzzy attibute emoved fom the othe fuzzy attibute, A When =, the semi-patial coelation, coefficient between the fuzzy attibutes A and with the influence of fuzzy attibute emoved fom the fuzzy attibute, (, is not defined Similaly, when =, is also not defined, ( A If thee is no linea elationship between A and, then the elationship between the fuzzy attibutes A and with the influence of fuzzy attibute emoved fom the fuzzy attibute is the same as the elationship between the fuzzy attibutes A and with the influence of fuzzy attibute emoved fom both of the two fuzzy attibutes A and That means, when, =0, A,( Similaly, when, =0, then,( A is the same as, A When =0 and, =0, (, ( A That is, the two semi-patial coelation coefficients will be the same as the simple coelation coefficient in this paticula situation If =0, then whethe the influence of fuzzy attibute is emoved fom fuzzy attibute A o, it is not necessaily to discuss the semi-patial elationship between fuzzy attibutes A and The value of a fuzzy semi-patial coelation coefficient is usually smalle than value of a fuzzy patial coelation coefficient (, the value of fuzzy ( simple coelation coefficient, and thus the value of A, lies in, ( lies in, A,, 0, so the value of fuzzy semi-patial coelation coefficient,, is smalle than the value of fuzzy patial ( coelation coefficient, A,
444 Hao-En hueh: Point-iseial oelation Analysis of Fuzzy Attibutes Also, we can see that sign of a fuzzy semipatial coelation coefficient is the same as sign of a fuzzy patial coelation coefficient 3 Fuzzy Point-iseial oelation Analysis A coelation analysis method among cisp sets and fuzzy sets is poposed Assume that two attibutes, A and, ae pesent A is a nominaldichotomous attibute, whee one value is a, and anothe value is a F is a fuzzy attibute, whee F is a fuzzy space The fuzzy attibute is defined on the domain of a cisp univesal set X, with the membeship function The fuzzy attibute can then also be expessed with fomula ( Assume that (( x, x, ( x, ( x, x, ( x,, ( xn, xn, ( xn is a andom sample dawn fom a cisp univesal set X The fuzzy pointbiseial coelation coefficient [, 5] between the cisp attibute A and the fuzzy attibute is then defined by: SD p p (3, whee and ae the means of the fuzzy attibute when x i a and x a i, espectively p is the pecentage of x a i, and p p is the pecentage of x i a SD is the standad deviation of the fuzzy attibute Table Degee of intelligence and gende of 5 students Student Gende Degee of Intelligence x M 065 x F 070 x3 M 03 x4 F 049 x5 F 00 x6 M 050 x7 F 035 x M 00 x9 M 0 x0 F 069 x F 07 x M 055 x3 F 077 x4 F 090 x M 04 5 Fo example, let X ={all the students in a univesity} when a eseache wishes to lean the coelation between the degee of intelligence and student gende of the univesity in question All the students on campus cannot be measued, and theefoe, a sample of 5 students is taken at andom fom the campus, x, x,, x [] Assume 5 that the cisp attibute A is gende whee a is M, a is F, and the fuzzy attibute is the degee of intelligence Table shows the degee of intelligence, and the gende of 5 andomly selected students p 7 04667, (3 5 p 05334, (33 5 334 0477, (34 7 54 0650, (35 ( 5936 SD 5 077, (36 5 0477 0650 (04667 (05334 047764 077 (37 If we use to epesent M in gende, and use 0 to epesent F in gende, and then use the poductmoment coelation fomula, to compute the coelation coefficient between the degee of intelligence and the sex, we can then obtain the same value To test whethe the fuzzy point-biseial coelation coefficient is significantly diffeent fom 0, the following fomula is employed [, 5, 6]: 047764 0 t 960 (3 ( 047764 N 5 Let the degee of confidence be 95 %, that is, 005, and theefoe, accoding to the citical value fom the statistics table, is calculated as follows: t t 60 (005(3 (39 ( N The value of does not fall into the egion of ejection, and theefoe, should accept the null hypothesis The test esult above shows that no coelation exists between the degee of intelligence and the gende of students in this univesity Anothe expeimental dataset used in this study esulted fom the andomly sampled custome etention activities, and the esponses of customes,
Hao-En hueh: Point-iseial oelation Analysis of Fuzzy Attibutes 444 of a telecom company in Taiwan, whose contacts wee due to expie between June and July 00 [4] Table The esults of custome etention activities S EM P MM E June 0 ~ 300 TM 0 June 0 ~ 300 DM 0 3 June 30 ~ 00 TM 04 4 June 30 ~ 00 DM 004 5 July 0 ~ 300 TM 07 6 July 0 ~ 300 DM 05 7 July 30 ~ 00 TM 04 July 30 ~ 00 DM 004 S: ustome subgoup EM: Expied Month of contact P: ill Payment (NT$ MM: Maketing Method E: Degee of willing to extend contact TM: Telemaketing DM: Diect mail Among the customes whose contacts wee due to expie in June and July 00, 400 customes wee andomly selected fom each of the following goups: customes with monthly bills of NT$0 ~ NT$300 and customes with monthly bills of NT$30 ~ NT$00 Each goup of 400 customes was then divided futhe into two subgoups of 00 customes each ustome etention maketing pogams wee implemented using diect mail (indiect maketing and telemaketing (diect maketing This telecom company developed maketing pogams to etain thei customes Duing this etention maketing pocess, customes could choose the maketing pogams they wanted Table displays the esults of the entie etention maketing pocess The telecom company needs to know the coelation between the degee of willing to extend contact and othe attibutes in Table Assume that the cisp attibute A is the expied month, whee a is June, a is July, and the fuzzy attibute is the degee of willing to extend contact p 4 05, (30 p 4 05, (3 09 03, 4 (3 094 035, 4 (33 (6 0636 SD 05959, (34 03 035 05956 (05 (05 00567 (35 Similaly, we can obtain the fuzzy point-biseial coelation coefficient between the degee of willing to extend contact and the bill payment Assume that the cisp attibute is bill payment, whee c is NT$0 ~ NT$300, c is NT$30 ~ NT$00, and the fuzzy attibute is also the degee of willing to extend contact p 4 05, (36 p 4 05, (37 0 005, 4 (3 04 06, 4 (39 (6 0636 SD 05959, (30 005 06 05956 (05 (05 073 (3 Assume that the cisp attibute D is maketing method, whee d is telemaketing, d is diect mail, and the fuzzy attibute is still the degee of willing to extend contact p 4 05, (3 p 4 05, (33 3 0375, 4 (34 055 0375, 4 (35 (6 0636 SD 05959, (36 0375 0375 (05 (05 059579 05956 (37 The fuzzy point-biseial coelation coefficient between the degee of willing to extend contact and expied month is equal to -00576; the fuzzy point-biseial coelation coefficient between the degee of willing to extend contact and the bill payment is equal to -073; the fuzzy pointbiseial coelation coefficient between the degee of willing to extend contact and maketing method is equal to 059579 Accoding to the above analyses, the most impotant attibute of the etention maketing pocess is maketing method
444 Hao-En hueh: Point-iseial oelation Analysis of Fuzzy Attibutes Pevious eseach [4] used the data mining technique to constuct a decision tee based chun etention model and educe chun ates based on custome esponses to etention activities pefomed by custome sevice centes The constucted chun etention model confims the key factos detemined by fuzzy point-biseial coelation analysis 4 onclusions Infomation egading the coelations between attibutes is a necessity in numeous data analysis tasks oelation analyses ae commonly used when an analysis of the elationships between database attibutes is equied Vaious coelation analyses among cisp attibutes have been widely discussed in conventional statistics Pevious studies have also pesented numeous coelation analyses among fuzzy attibutes This study is pimaily concened with coelations among cisp attibutes and fuzzy attibutes Most eseaches may select the poduct-moment coelation fomula to compute the coelation among cisp sets and fuzzy sets This pape pesents a new method, called fuzzy point-biseial coelation analysis to solve this poblem, and is a simple and faste method than taditional coelation analyses Refeences [] S F Anold, Mathematical Statistics, Pentice- Hall, New Jesey (990 [] D A hiang and N P Lin, oelation of Fuzzy Sets, Fuzzy Sets and Systems, Vol 0 (999, -6 [3] D A hiang and N P Lin, Patial oelation of Fuzzy Sets, Fuzzy Sets and Systems, Vol 0 (000, 09-5 [4] H E hueh, S Lin and N Y Jan, Mining the Telecom Maketing Infomation to Optimizing the ustome Retention Stategies, II Expess Lette, Pat : Application, Vol, No (0, -6 [5] S Dowdy and S Weaden, Statistics fo Reseach, John Wiley & Sons (93 [6] N P Lin and H E hueh, Fuzzy Semi-Patial oelation Analysis, WSEAS Tansactions on omputes, Vol 5, No (006, 970-976 [7] G J Kli and T A Folge, Fuzzy Sets, Uncetainty, and Infomation, Pentice-Hall Intenational (9 [] G J Kli and Yuan, Fuzzy Sets and Fuzzy Logic: Theoy and Applications, Pentice-Hall Intenational (995 [9] Yu, oelation of Fuzzy Numbes, Fuzzy Sets and Systems, Vol 55 (993, 303-307 [0] L A Zadeh, Fuzzy sets, Infomation and ontol, Vol (965, 33-353 Hao-En hueh eceived the PhD in ompute Science and Infomation Engineeing fom Tamkang Univesity, Taiwan He is cuently an Assistant Pofesso with the Depatment of Infomation Management at Yuanpei Univesity, Taiwan His eseach inteests ae in the aeas of fuzzy set theoy, data dining, database system and its applications He is now the Edito In hief of Intenational jounal of Web & Semantic Technology (IJWesT