A. General Type- Fzzy Clsterng There are two knds of type- fzzy sets whch are often sed n clsterng algorthms: 1) nterval and ) general. In nterval typ

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1 014 IEEE Internatonal Conference on Fzzy Systems (FUZZ-IEEE) Jly 6-11, 014, Beng, Chna A Hybrd Type- Fzzy Clsterng Technqe for Inpt Data Preprocessng of Classfcaton Algorthms Vahd Nor, Mohammad-. Akbarzadeh-T. Departments of Compter Engneerng, Islamc Azad Unversty, Mashhad Branch, Mashhad, Iran Emals: vahd.nor@mshda.ac.r, akbarzadeh@eee.org Abstract ecently, clsterng has been sed for preprocessng datasets before applyng classfcaton algorthms n order to enhance classfcaton effcency. A strong clstered dataset as npt to classfcaton algorthms can sgnfcantly mprove comptaton tme. Ths can be partclarly sefl n Bg Data where comptaton tme s eqally or more mportant than accracy. However, there s a trade-off between speed and accracy among clsterng algorthms. Specfcally, general type- fzzy c-means (GT FCM) s consdered to be a hghly accrate clsterng approach, bt t s comptatonally ntensve. To mprove ts comptaton tme we propose here a hybrd clsterng algorthm called GTFCM that combnes GT FCM wth a fast k-means algorthm for npt data preprocessng of classfcaton algorthms. The proposed algorthm shows mproved comptaton tme when compared wth GT FCM as well as FGTFCM on fve benchmarks from UCI lbrary. eywords- General tye- fzzy, k-means, clsterng, npt data preprocessng, classfcaton I. INTODUCTION Classfcaton s a common problem data mnng [] where datasets are mapped nto predefned grops called classes. Classes are defned accordng to the smlarty of characterstcs or featres of data [1]. Becase, the classes are determned before applyng the real data, ths method s known as a spervsed learnng algorthm. Classfcaton s sed n many felds and scences sch as, bo-nformatcs [15], genetcs [14], bology [16] and healthcare [17]. Several researches have shown that the comptatonal effcency of classfcaton s enhanced f the npt data s frst clstered before for classfcaton. Ths s partclarly applcable when handlng bg data, where low comptaton tme s eqally or more mportant than classfcaton accracy. The class nformaton also mproves the accracy of clsterng [1]. To have the advantages of both clsterng and classfcaton, many exstng algorthms have been developed n a seqental hybrd way [1]. For example n both [11] and [1], frst the crteron s preprocessed and optmzed by a clsterng algorthm and then n the next step the classfcaton crteron s connected wth the acheved clsterng reslts to enhance the accracy of classfcaton algorthms. Generally, there s a trade-off between accracy and comptaton tme of clsterng algorthms,.e. the hgher the accracy, the more the comptaton tme. One of the better known clsterng algorthms s k-means. -means s fast bt has low accracy [1]. On the other hand, general type- fzzy clsterng (GT Alreza owhanmanesh Department of Electrcal Engneerng, Unversty of Neyshabr Neyshabr, Iran rowhanmanesh@eee.org FCM) s a new method that has hgh accracy bt s comptatonally ntensve. In [4], a general type- fzzy clsterng algorthm s ntrodced that s based on α-planes. Ths algorthm has hgh accracy and can deal wth the ncertanty n datasets, whle k-means and FCM, whch are fast clsterng algorthms, cannot handle the ncertanty n a dataset. There are several works that focs on enhancng the speed sse of type- fzzy clsterng. A modfed verson of type- fzzy system was proposed n [6] to mprove the speed (comptatonal tme) of type- fzzy clsterng. Also, n [7]- [9] nterval type- fzzy s sed nstead of general type- fzzy for clsterng, becase nterval type- s faster than general type- fzzy. In addton Yang worked on smlarty measrements of type- fzzy clsterng algorthms on fzzy datasets [19]-[]. In these work, they redefned new smlarty measrements based on non, maxmm. These new smlarty measres affect type- fzzy clsterng performance. In ths paper, we propose a hybrd clsterng algorthm for data npt preprocessng of classfcaton algorthms to address the hgh comptaton tme of general type- fzzy clsterng algorthm. The proposed hybrd method s based on a combnaton of general type- fzzy, whch s an accrate algorthm and k-means, whch s a fast algorthm. We call the proposed approach GTFCM. GTFCM has the advantages of both general type- fzzy and k-means clsterng algorthms,.e. t has hgh accracy and low comptatonal tme. The reslts are compared wth GT FCM and FGTFCM clsterng algorthms for dfferent datasets. Unlabeled datasets are sed for clsterng algorthms; however, labeled datasets are sed for classfcaton algorthms. Snce, we se classfcaton datasets n or experments; we can measre the accracy of or clsterng algorthm. The paper s organzed as follows: secton II dscsses the proposed hybrd algorthm. The reslts and conclson are presented n sectons III and IV, respectvely. II. POPOSED METHOD Or method s based on k-means and general type- fzzy clsterng. General type- fzzy clsterng was presented n [4]. Frst, a general overvew of type- fzzy s gven, and then the proposed method s descrbed /14/$ IEEE 1131

2 A. General Type- Fzzy Clsterng There are two knds of type- fzzy sets whch are often sed n clsterng algorthms: 1) nterval and ) general. In nterval type- fzzy, the secondary membershp fncton s always one, whle n general type- fzzy t s a vale n range of [0,1]. General type- fzzy clsterng s based on FCM (Fzzy C-Means) algorthm. So, the same as FCM, t ntalzes the centers randomly. The FCM algorthm ses lngstc terms sch as Small, Medm or Hgh, modeled by type-1 fzzy sets for the fzzfer parameter M (Fg.1). The FCM algorthm s sed by the GT FCM clster membershp fnctons. The general type- fzzy clsterng proposed n [4] ses α-planes. α-planes manage the ncertanty of general type- fzzy sets. The GT FCM algorthm explots the lngstc fzzfer M for ts secondary membershp fnctons of the general type- fzzy partton matrx as shown n (1). Eqaton (), that s a membershp grade ) expressed as type-1 fzzy sets, s sed to descrbe the membershp degree of pattern x to clster v. ) (1) x X (4): ) [0,1] [0,1] [ s S ), s s ) )] Where s ) and ) are calclated by (3) and s max s ) c l 1 ) d d l 1 sm ( ) 1, c l 1 d d l 1 () sm ( ) (4) mn, c c d sm ( ) 1 s d M ( ) 1 l 1 dl l 1 dl Accordng to [4], centrod C can be calclated as a weghted composton of the nterval centrods of ndvdal α- planes sng (5). The npt of (5) s. Here, d s the dstance of th data from th centrod. Intal centrods are sed for the frst teraton. s and s are obtaned as shown n M M Fg. for each α-planes and c s the nmber of clsters. (3) Fg. 1. ngstc varables for ntalzng the membershp fnctons [4] To compte the precse clster poston, (6) s sed to defzzfy the clster centrod C. [0,1] [ c ( ), c C (5) v 1 1 y C C (6) ( )] In (6), s the nmber of steps that the doman of the centrod has been dscretzed nto and y s the poston vector of th dscretzed step. In ths algorthm the hard-parttonng s done based on the defzzfed vale of the type-1 fzzy membershp grade. So, the followng rle s sed for hardparttonng: If ( ) k )), k 1,..., c, k (7) Then x Clster Bt n [4] formla (8) s sed for hard-parttonng nstead of (7). In (7), snce the Ecldan dstance norm s sed to calclate the membershp of pattern x to clster n the mltdmensonal space, t seems redndant to separately aggregate dentcal membershp vales for each dmenson. So n [4] the athors se (8) for hard-parttonng: ( ) c ( ) ), k 1,..., c, k If ( c x k x Then x Clster (8) The centrod of the type-1 fzzy membershp grade c ) can be calclated sng (9): c ) 1 1 y In ths eqaton, and y have the same defntons as n c ) s the centrod of the th clster. (6), where (9) 113

3 Schematc vew of GT FCM s depcted n Fg.. andom selecton of ntal centrods cases the algorthm to have more teratons, hence, more comptaton tme (lower speed). So, f a clsterng algorthm, sch as k-means, fnds the centers one step before GT FCM and passes them to GT FCM, the comptaton tme of GT FCM wll be redced. α- Plane epresentaton α=0 S M (0), SM (0) (0) α= 1 S M ( 1 ), SM ( 1) ( 1 ) 1 M ( S ) Fzzfer M M ( S ) B. -means -means s one of the most common algorthms n clsterng. In ths method, k denotes the nmber of clsters. -means algorthm has three steps ncldng: Step 1) k clster centers are specfed, randomly.e. one center for each clster, step ) for each npt, dstance from each clster center s calclated. The data belongs to the clster whch has the closest dstance from the center. Ths step s repeated for all npt data, and step 3) the barycenters of clsters (whch are generated n step ) are calclated and consdered as new clster centers and then the algorthm goes to step [1]. These steps are repeated ntl centers do not change for the two consectve teratons. The goal of ths algorthm s to mnmze ts cost fncton denoted as (10) [1],[3]. n J x c (10) 1 1 Here, n s the nmber of samples, s the nmber of clsters, c shows the th clster and x shows th sample of pattern. In ths paper we se Ecldan dstance whch s a tradtonal metrc for dstance measrement of k-means. Followng eqaton presents the Ecldan dstance [10]: n d, y) y ) (11) 1 In ths eqaton d, y) s the Ecldean dstance between sample x wth clster y. C. Prevos work To mprove the comptaton tme of GT FCM, the algorthm at [10] ses the otpt of k-means algorthm as the npt for FCM. Then, the otpts of FCM are sed as the ntal clster centrods for GT FCM. In the followng the detals are dscssed. In the conventonal FCM, the ntal vales of membershp fnctons are random nmbers n the range of [0, 1]. However, f the ntal vales of membershp fnctons are selected more wsely, t s expected to need less nmber of teratons by FCM, hence, the comptaton tme s mproved. To do so, the athors of [10] se k-means to determne the centrods of npt data and then calclate the dstances of each data from all centrods. The normalzed dstances are assmed as ntal vales of membershp fnctons of npt data of FCM. By dong so, FCM wold have a better startng pont and t helps to redce the execton tme and teratons of FCM. Ths algorthm s called FGTFCM. However, sng both k-means and FCM clsterng algorthms for ntalzng GT FCM cases almost a large overhead. X α= S 1 M ( 1), SM ( 1) Type- edcton ( 1) C [ c ( ), c ( )] [01,] α=1 S (1), (1) M S M (1) Fg.. Schematc vew of GT FCM [4] Center Update D. Proposed Method To make the algorthm even faster, we can omt one of the k-means or FCM algorthms whch generate ntal centrods for GT FCM. We omt FCM algorthm from the flow becase k-means s faster than FCM. Also, clsters centrods detected by FCM and k-means clsterng algorthms are close to each other. However, the complexty tme of k-means and FCM are O(ncd) and O(ndc),respectvely [3], where, n denotes nmber of dmensons, s the nmber of teratons, n shows the nmber of sample of dataset and c s the nmber of clsters. Accordng to [3], for n=100, d=3, =0 and for a constant nmber of dataset s samples, the elapsed tme for k- means and FCM are and seconds, respectvely. Also, f we assme the nmber of clsters as a constant and assme n=150, d=, c= and =0, then the tme complexty of k-means and FCM are 1000 and 4000, respectvely [3]. So, FCM needs more comptaton tme than k-means. Hence, n the revsed verson st k-means s employed to fnd the centrods of GT FCM. Ths change mproves the comptaton tme of the algorthm and conseqently, the new algorthm s exected mch faster than FGTFCM. Ths new algorthm s called GTFCM. The flowchart of GTFCM s depcted n Fg.3. As depcted n Fg.3, frst of all the npt dataset are appled to the k-means algorthm. Then, k-means clsters npt dataset and fnds ther centrods. After that, these centrods are sed by GT FCM as the ntal centrods. In the next step, the type- fzzfer fncton calclates the secondary membershp fnctons based on α-planes and sng (3), (4) and Medm lngstc term for secondary membershp fncton as depcted n Fg.1. We se 10 α-planes. Then, EM 1 algorthm [13] s sed for type redcton and fndng the centrods of α-planes. EM ntrodced by Mendel and W to enhance the comptaton tme of M. EM s 39% faster than M algorthm and can save abot two teratons whle M fnd the answer sally between two to sx teratons [13]. In or proposed method, for type redcton we se (5) whch are based on EM algorthm, to fnd the centrods of 10 α-planes. 1 Enhancement arnk Mendel 0 m yc 1 v C ( y )

4 Dong so, the type- fzzy membershp fncton redces to type-1 fzzy whch s a prmary membershp fncton. To fnd the precse center of each clster, the centrods shold be determned sng (6). The centrods calclated by (6) are checked wth prevos centrods of each clster. If they are dfferent, the algorthm recalclates the secondary membershp fncton sng new centrods and then the followng steps are repeated. Otherwse, the algorthm fnshes (Fg. 3). Start Intalze centrods wth k-means clsterng algorthm and se ts otpt as ntal centrods of GT FCM Type- fzzfer and prepare secondary membershp fncton wth 10 α- planes. Fnd No oad Dataset For type redcton compte the centrod of α- plane (10) v as the centrod of Is the new centrod eqal to last centrod? End Yes Classfcaton algorthm th clster. GT FCM Fg. 3. Flowchart of proposed GTFCM TABE I IST OF DATASETS THAT USED FO EXPEIMENTS Attrbtes Dataset Clsters Sze Irs Wne Pma Indans Shttle 43,500 9 Magc 19,00 10 most nmeros data class whch s 80% of data and the second class (class ) contans the remanng less nmeros data classes whch s the 0% of data. All of the datasets of table I are appled to GT FCM, FGTFCM and GTFCM 50 tmes. The machne sed for dong the experments and smlatons s an Acer 5750G system wth an Intel Core 7-630QM@.00GHz and 6.00 GB AM and rnnng Wndows 7. MATAB software has been sed for mplementng the algorthms. For far comparsons of comptaton tme of the three algorthms, the target accracy has been assmed the same for all the three algorthms n all the experments. All of the three algorthms (.e. GT FCM, FGTFCM and GTFCM) are based on GT FCM [4], and se the same membershp fnctons (.e. the same ntal condtons). Snce ntal centrods of k-means and GT FCM are selected randomly, we rn each algorthm for 50 teratons,.e. wth 50 sets of random ntal centrods, to show that the random ntal centrods have trval effects on the reslts. The comptaton tme mprovement (speedp) of GTFCM compared to GT FCM and FGTFCM s calclated sng eqaton (1). Therefore, when the speedp s greater than one the comptaton tme of GTFCM s less than GT FCM or FGTFCM and when the speedp s less than one the comptaton tme of GTFCM s greater than GT FCM or FGTFCM. Comptaton tme mprovement (speedp) = n tme of (GTFCM or FGTFCM) (1) n tme of GTFCM B. Expermental eslts The 30% of Wne dataset whch have been selected randomly s appled to the three algorthms whle the target accracy s assmed to be 66% for all three. The comptaton tme of three algorthms are shown n Fg.4, Fg.5, Fg.10 and Fg.11. III. SIMUATION ESUTS In ths secton the expermental setp and smlaton reslts are presented. A. Expermental Setp In ths paper, fve standard datasets of UCI are selected, ncldng Irs, Wne, Pma Indans, Shttle and Magc whch have been lsted n table I [18]. The Shttle data has been dvded nto two classes. One class (class 1) ncldes the Fg. 4. Comparson of comptaton tme for 50 teratons of three algorthms wth a target accracy of 66% for 53 data of Wne dataset 1134

5 In another experment, 70% of Irs dataset whch have been selected randomly s appled to the three algorthms whle the target accracy s assmed to be 70% for all three algorthms. Comparng table II and table III, reveals that GTFCM otperforms GT FCM and FGTFCM sgnfcantly for low target accraces. For the experments performed for generatng reslts of table II and III, 70% of each dataset whch selected randomly, were sed. However, for the experments done for generatng reslts of Fg.4 and 5, 30% of each dataset, whch selected randomly, were exploted. Snce, n ths paper we focs on comptaton tme redcton and not accracy, the target accraces are selected close to the maxmm accracy. TABE II COMPAING COMPUTATION TIME (IN SECONDS) OF THEE AGOITHMS WITH OW TAGET ACCUACY Method GTFCM (n seconds) FGTFCM (n seconds) GTFCM (n seconds) Speedp of GT- FCM vs. GT FCM Speedp of GT- FCM vs. FGTFCM a Target accracy Irs Acc a : 50% Wne 50% Pma Indans 65% Shttle 70% e-5.3e e-5 1.7e-5 1.7e-5.37e-4.51e-5 1.5e e-5.31e As llstrated n Fg.4 and Fg.5, the comptaton tme of FGTFCM and GTFCM enhanced sgnfcantly compared to GT FCM. The average speedp of GTFCM compared to GT FCM and FGTFCM are and respectvely for Fg. 4 and are 1509 and 1.04, for Fg. 5, correspondngly. Becase the comptaton tme of GT FCM s mch larger than the two others, the dfferences of FGTFCM and GTFCM comptaton tme s not seen n these fgre. Therefore, two other Fg. 6 and 7, whch are zoomed n of Fg. 4 and 5, have been added for comparng only FGTFCM aganst GTFCM. The same as Fg.4 and 5, n Fg. 6 and 7 the reslts of FGTFCM and GTFCM are very smlar. Therefore to clarfy the dfference of these clsterng algorthms, n Fg. 7 and Fg.8, the same reslts are redrawn st for 35 teratons. TABE III COMPAING COMPUTATION TIME OF THEE AGOITHMS (IN SECONDS) ON DIFFEENT DATASETS WITH DIFFEENT TAGET ACCUACY Method GTFCM (n seconds) FGTFCM (n seconds) GTFCM (n seconds) Speedp of GT- FCM vs. GT FCM Speedp of GT- FCM vs. FGTFCM Irs Acc a : 86% Wne 71% Pma Indans 68% Shttle 74.7% e e e e e-5.30e e a Target accracy Fg. 5. Comparson of comptaton tme for 50 teratons of three algorthms wth a target accracy of 70% for 105 data of Irs dataset Fg. 6. Comparson of comptaton tme for 50 teratons of two algorthms wth a target accracy of 70% for 105 data of Irs dataset 1135

6 Fg. 7. Comparson of comptaton tme between two algorthms to reach to the 66% accracy n 50 tmes teratons for 53 data of Wne dataset Fg. 9. Comparson of comptaton tme between two algorthms to reach to the 66% accracy n 35 tmes teratons for 53 data of Wne dataset Fg. 8. Comparson of comptaton tme for 35 teratons of two algorthms wth a target accracy of 70% for 105 data of Irs dataset Fg. 10 shows the comptaton tme for GT FCM, FGTFCM and GTFCM, when the accracy s 60% and 30% of Pma Indans dataset s selected. The average comptaton tme mprovement of GTFCM compared to GT FCM and FGTFCM for ths dataset are 344 and 1911, respectvely. The same experments have been done sng 70% of Irs dataset assmng 75% target accracy. The reslts have been depcted n Fg. 11. The average comptaton tme mprovement of GTFCM compared to GT FCM and FGTFCM for ths dataset are and 0.55, respectvely. The reason for performance degradaton of GTFCM compared to FGTFCM s de to the last fve rns of FGTFCM whch have long rntme. Fg. 10. Comparson of comptaton tme for 50 teratons of three algorthms wth a target accracy of 60% for 30 data of Pma Indans dataset De to the large dfferences between execton tme of the last 15 rns of FGTFCM and the other 35 rns, t seems that crves of FGTFCM and GTFCM are very close to each other for the frst 35 rns. However, the real dfferences of FGTFCM and GTFCM comptaton tme are not llstrated n Fg. 11, clearly. To clarfy ths, Fg. 1, whch s zoomed n of Fg. 11, has been added for comparng FGTFCM and GTFCM for the frst 35 rns. Consderng only the frst 35 rns, the average execton tme mprovement of GTFCM compared to FGTFCM for ths dataset s The averages of comptaton tme (n seconds) of 50 teratons of each algorthm on a specfc dataset wth specfc target accracy are avalable n table II and table III. Also, table II and III show the speedp of GTFCM compared to FGTFCM and GT FCM for each specfc dataset wth specfc target accracy. The npt data sed for these tables are 70% of data of each dataset that were selected randomly. 1136

7 Accordng to table II the maxmm speedp for GTFCM vs. FGTFCM s and for GTFCM vs. GT FCM s and for table III are and 306, respectvely. Totally, we tred all of these datasets for fve dfferent accraces. Bt, de to page lmtaton we only presented two grops of the reslts n tables II and III and the rest of the reslts were not presented. The average speedp for all of the reslts obtaned for GTFCM vs. FGTFCM and GT FCM, are 1818 and 7456, respectvely. Accordng to table II and table III, FGTFCM obtans better reslts for Irs dataset. The reason s that the combnaton of k-means and FCM prodces better ntal centrods compared to k-means, whch shortens GT FCM execton tme. However, the combnaton of k-means and FCM prodces worse ntal centrods compared to the proposed method for Wne dataset whch lengthens GT FCM execton tme. The proposed method obtans better reslts for 80% of case stdes sed for tables II and III. The best reslts are bolded n table II and table III. Fg. 11. Comparson of comptaton tme for 50 teratons of three algorthms wth a target accracy of 75% for 105 data of Irs dataset Fg. 1. Comparson of comptaton tme for 35 teratons of two algorthms wth a target accracy of 75 % for 105 data of Irs dataset In table III, for the Shttle dataset, the GT FCM performs better compared to FGTFCM and GTFCM. The reason s that GT FCM reaches to the target accracy, n the frst teraton. So, n both FGTFCM and GTFCM algorthms, k-means and FCM algorthms whch are sed to ntalze the centrod of GT FCM, reslts n longer comptaton tme for FGTFCM and GTFCM. Therefore, these algorthms rn slower than GT FCM. Also, n table III, for Irs dataset, FGTFCM s faster than GTFCM. The reason of ths phenomenon s that for teratons from 50 teratons of GTFCM are very tme consmng. The effects of these slow teratons cases the average tme of GTFCM algorthm to be lower than FGTFCM. Consderng k- means, n whch ts ntal centrods are selected randomly, reveals that f the random centrods are not selected properly, t reslts n poor accracy and therefore cases GT FCM to need more teraton and hence, makes GTFCM slower. The same problem happens n fgres 6 and 7. Accordng to table II and III, althogh, n several cases GTFCM s slower than GT FCM and FGTFCM, the comptaton tme dfferences of GTFCM n these cases s very small and are approxmately 4.0e-7 seconds compared to GT FCM and FGTFCM. IV. CONCUSION ecently, several works have sed clsterng and classfcaton n seqental strctres to mprove the performance of classfcaton algorthms. As they ndcated, the effcency of classfcaton learnng s enhanced f the npt data s frst clstered and then sed for classfcaton. However, there s a trade-off between comptaton tme and accracy of clsterng algorthms. In ths paper, a new clsterng method s ntrodced to mprove the comptaton tme of a classfcaton algorthm by preprocessng classfcaton dataset. To address the conflct of hgh comptaton tme and hgh accracy of clsterng algorthm, we propose a hybrd clsterng algorthm called GTFCM. Ths algorthm s a combnaton of hgh accracy general type- fzzy C-means (GT FCM) that can deal wth ncertanty va sng α-planes wth low comptaton tme k- means algorthm for npt data preprocessng of classfcaton algorthms. The proposed algorthm mproves the speed of GT FCM and rn on fve datasets of UCI for clsterng wth dfferent target accracy. GTFCM has better effcency compared to GT FCM for almost all cases. The reason s that GTFCM prodces better ntal centrods for GT FCM. The average speedp of GTFCM compared to GT FCM on fve datasets ncldng Irs, Wne, Pma Indans, Shttle and Magc, s In 86% of case stdes; GTFCM s faster than FGTFCM. The reason s that for these cases combnaton of k-means and FCM takes long tme to converge to good ntal centrods for GT FCM, however GTFCM does not have the overhead of FCM. In the remanng 14% cases, becase the combnaton of k-means and FCM prodces better ntal centrods for GT FCM, FGTFCM has better speedp compared to GTFCM. Fnally, the average speedp of GTFCM compared to FGTFCM on fve datasets ncldng Irs, Wne, Pma Indans, Shttle and Magc, s In conclson, the proposed method 1137

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