FCM & FPCM Algorithm Based on Unsupervised Mahalanobis Distances with Better Initial Values and Separable Criterion

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1 FCM & FPCM Algorth Bsed o Usuervsed Mhlobs Dstes wth Better Itl Vlues d Serble Crtero JENG-MING YIH YUAN-HORNG LIN HSIANG-CHUAN LIU d Ch-Fu Yh Abstrt The fuzzy rtto lusterg lgorths re ost bsed o Eulde dste futo whh oly be used to detet sherl struturl lusters. Gustfso-Kessel (GK) lusterg lgorth d Gth-Gev (GG) lusterg lgorth were develoed to detet o-sherl struturl lusters but both of the bsed o se-suervsed Mhlobs dste eeded ddtol ror forto. A roved Fuzzy C-Me lgorth bsed o usuervsed Mhlobs dste FCM-M ws roosed by our revous wor but t dd t osder the reltoshs betwee luster eters the obetve futo. I ths er we roosed roved Fuzzy C-Me lgorth FCM-MS whh s ot oly bsed o usuervsed Mhlobs dste but lso osderg the reltoshs betwee luster eters d the reltoshs betwee the eter of ll ots d the luster eters the obetve futo the sgulr d the tl vlues robles were lso solved. Three rel dt sets ws led to rove tht the erfore of the FCM-MS lgorth gve ore urte lusterg results th the FCM d FCM-M ethods d the rto ethod whh s roosed by us s the better of the two ethods for seletg the tl vlues. Keywords FCM-MS FCM-M GK lgorths GG lgorths Mhlobs dste. C I. INTRODUCTION Lusterg lys ortt role dt lyss d terretto. It grous the dt to lsses or lusters so tht the dt obets wth luster hve hgh slrty orso to oe other but re very dsslr to those dt obets other lusters. Fuzzy rtto lusterg s brh luster lyss t s wdely used tter reogto feld. The well ow oes suh s C. Bezde s Fuzzy C-Me (FCM) [] re l bsed o Eulde dste futo whh oly be used to detet the dt lsses wth se suer sherl shes. J. M. Yh s wth the Grdute Isttute of Edutol Mesureet d Sttsts d Dertet of Mthets Eduto Ntol Thug Uversty 40 M-Sheg Rd. Thug Cty 403 Tw (orresodg uthor to rovde hoe: ; f: ; e-l: yh@l.tu.edu.tw). Y. H. L. ws wth Dertet of Mthets Eduto Ntol Thug Uversty. He s ow leder of Dertet of Mthets Eduto Ntol Thug Uversty. (e-l: lyh@l.tu.edu.tw). H. C. Lu s wth Dertet of Boforts As UverstyTw). C. F. Yh s wth Dertet of Couter See d Iforto Egeerg Ntol Ch N Uversty Ntou Tw. (e-l: rot758@gl.tu.edu.tw).. Etedg Eulde dste to Mhlobs dste the well ow fuzzy rtto lusterg lgorths Gustfso-Kessel (GK) lusterg lgorth [3] d Gth-Gev (GG) lusterg lgorth [] were develoed to detet osherl struturl lusters but these two lgorths fl to osder the reltoshs betwee luster eters the obetve futo GK lgorth ust hve ror forto of she volue eh dt lss otherwse t oly be osdered to detet the dt lsses wth se volue. GG lgorth ust hve ror robbltes of the lusters. O the other hd Whe y deso of lss s greter th the sle sze of whh lss the estted ovre tr of whh lss y ot be full r t dues the sgulr roble of the verse ovre tr t s ortt ssue wthout geerlly osder bove lgorths. Fousg the bove two fults we dded regultg ftor of ovre tr to eh lss obetve futo d deleted the ostrt of the deterts of ovre tres GK Algorth A roved lgorth bsed o Mhlobs dste Fuzzy C-Me bsed o Mhlobs dste (FCM-M) s roosed by our revous wor [4].Y et l. [5] desrbed eteded obetve futo osstg of fuzzy wth-luster stter tr d ew betwee-luster eters stterg tr. The orresodg fuzzy lusterg lgorth ssures the otess betwee dt ots d luster eters d lso stregthes the serto betwee luster eters bsed o the serto rtero. The lusterg lgorth solved the reltoshs betwee luster eters questo but they dd ot osder the dste betwee the eter of ll ots d the eter of eh luster. Ths roble ws lso solved d reseted ths er. Moreover I ths er roved fuzzy lusterg lgorth deoted FCM-MS ws develoed bsed o FCM-M to obt better qulty lusterg results wth ew serble rtero d better tl vlue. The roved equtos for the ebersh d the luster eter were derved fro the ltertg otzto lgorth. The dste betwee the eter of ll ots d the eter of eh luster ws osdered by the uthors of ths er the sgulr roble ws lso solved. A rel dt set ws led to rove tht the erfore of the FCM-MS lgorth gve ore urte lusterg results th the Issue Volue

2 FCM-M d FCM ethods d the rto ethod whh s roosed by us s the better of the two ethods for seletg the tl vlues. II. LITERATURE REVIEW Fuzzy C-Me Algorth (FCM) s the ost oulr obetve futo bsed fuzzy lusterg lgorth t s frst develoed by Du [6] d roved by Bezde [].The lgorth of fuzzy C-Mes Algorth re the foudtos of ths study. The lgorth wll be dsussed s follows. A. Fuzzy C-Me Algorth The obetve futo used FCM s gve by Equto (). FCM J U A X d 0 () s the ebersh degree of dt obet luster C d t stsfes the followg ostrt gve by Equto (). C s the uber of lusters s the fuzzfer > whh otrols the fuzzess of the ethod. They re both reters d eed to be sefed before rug the lgorth. d s the squre Eulde dste betwee dt obet to eter.mzg obetve futo () wth ostrt () s o-trvl ostrt oler otzto roble wth otuous reters d dsrete reters. So there s o obvous lytl soluto. Therefore ltertg otzto shee ltertvely otzg oe set of reters whle the other set of reters re osdered s fed s used here. The the udtg futo for d s obted s (3) ~ (5). Ste : Deterg the uber of luster; d -vlue (let =) gve overgg error 0 (suh s 0.00) rdoly hoose the tl ebersh tr suh tht the ebershs re ot ll equl Ste : Fd U () (3)... l B. FCM-M Algorth l l Mhlobs Id sttst trodued ths dste the 930s. The Mhlobs dste s dste usg the verse of the ovre tr s the etr. It s dste the geoetrl sese beuse the ovre tres s well s ts verse re ostve defte tres. [9] We ll lusters usg the Mhlobs dste s ovre lusters. The etr defed by the ovre tr rovdes orlzto of the dt reltve to ther sred. Usg the Mhlobs dste s doe s follows:.the ovre tr of the esured quttes V s detered over lbrtg set..oe oute the verse of the ovre tr V -. 3.The dste of ew obet to the lbrtg set s estted T usg equto dm( ) V ( ). ; f the dste s sller th gve threshold vlue the ew obet s osdered s belogg to the se set. Oe terestg roerty of the Mhlobs dste s tht t s orlzed. Thus t s ot eessry to orlze the dt rovded roudg errors s vertg the ovre tr re et uder otrol. If the dt re roughly dstrbuted ordg to orl dstrbuto the threshold for etg whether obet belog to the lbrtg set be detered fro the dstrbuto. The Mhlobs dste be led ll robles whh esureets ust be lssfed. A good ele s the deteto of os vedg he. Whe os s serted to the he seres of sesors gves severl esureets betwee hdful d doze. The detetor be lbrted usg set of good os forg lbrto set. The o detetor dfferette good os fro the fe os usg the Mhlobs dste outed o the ovre tr of the lbrto set referee the followg Fgure. Ste 3: Ireet ; utl. (4) (5) Issue Volue

3 lgorth rogresses The etr hges dylly.[0] FIG.. Use the Mhlobs dste to detet the fe os. (Covert fro Besset D. H. 69 FIG..) Aother feld of lto s the deterto of er ells fro bosy.preters of ells be esured utotlly d eressed ubers. The ovre tr be detered usg ether esureets of helthy ells or esureets of lgt ells. Idetfto of erous ells be utoted usg the Mhlobs dste. The fl gol of the obet leetg the Mhlobs dste s to oute the squre Mhlobs dste s defed equto d ( ) V ( ). T M Ileetto of the Mhlobs dste s dtted by ts future reuse luster lyss. There we eed to be ble to uulte esureets whle usg the result of reedg uulto.thus outto of the eter d the verse ovre tr ust be doe see the fgure.eltly wth the ethod outer Preters. There re two wys of retg ew ste. Oe s to sefy the deso of the vetors tht wll be uulted to the obet. The seod sules vetor s the tettve eter. The orlzg roertes of the Mhlobs dste e t del for ths ts. Whe Eulde dste s used the etr res the se ll dretos. Thus the etet of eh luster hs ore or less rulr shes. Wth the Mhlobs dste the ovre tr s uque for eh luster. Thus ovre lusters hve dfferet shes se the etr dts tself to the she of eh luster. As the FIG.. Method for suessve roto. (Covert fro Besset D. H. 8 FIG. 4.4) For rovg the bove two robles our revous wor [4] roosed the roved lgorth FCM-M whh dded l regultg ftor of ovre tr to eh lss obetve futo d deleted M the ostrt of the detert of ovre tres GK Algorth s the obetve futo (6). Usg the Lgrge ultler ethod We ze the obetve futo (6). Costrt (7) wth reset to the reters d we obt the solutos s (0) () d(3). We wt to vod the sgulr roble d to selet the better tl ebersh tr the udtg Issue Volue 3 009

4 futos for d re obted s (8) ~ (3-8). Both of FCM d FCM-M ot elot ll of the ebershs wth the se vlue. FCM s sel se of FCM-M whe ovre tres equl to detty tres by our revous wor [8]. Costrts: ebersh... (7)... s the set of ovre of luster. Ste : Deterg the uber of luster; d -vlue (let =) gve overgg error 0 (suh s 0.00 ). Method : hoose the result ebersh tr of FCM lgorth s the tl oe. Method : let 0... be the result eters of -e lgorth Ad d be 0 dstes betwee dt obet to eter 0. M 0 d d 0 d d d d u M 0 M d d u u... u u u... u u u... u... u u u U u u... u u u... u Ste : Fd s s s s s fs 0 s 0 fs 0 s s s s J U A X F C M M l s s 0 s (6) (8) (9) (0) () () w w l s w s s wsl s where w = Ste 3: Ireet ; utl C. FCM-MS Algorth[3]. (3) The lusterg otzto ws bsed o obetve futos. The hoe of rorte obetve futo s the ot to the suess of the luster lyss.[4] I FCM-M lgorth t dd t osder the reltoshs betwee luster eters the obetve futo ow we roosed roved Fuzzy C-Me lgorth FCM-MS whh s ot oly bsed o usuervsed Mhlobs dste but lso osderg the reltoshs betwee luster eters d the reltoshs betwee the eter of ll ots d the luster eters the obetve futo the sgulr d the tl vlues robles were lso solved. Let { 3 } be set of dt ots rereseted by -desol feture vetors =(... ) R. The dt tr Z hs the luster eter tr A=[ ] << d the ebersh tr U [ ] where s the ebersh vlue of belogg to. V[ v ] eress the weghtg tr d v s the weghtg vlue betwee v d v. The fuzzy eoet s greter th [7]. Thus the roosed obetve futo s J U V A X FCMMS v ( ) l l l l (4) Suh tht [0] 0 where (5) v s defed s yy l yry s r s vl where y = yry s y ry s r s r s (6) The gol of the lusterg lgorth s to detfy the luster eters d the ebersh vlues by solvg otzto roble. Altertg otzto s Issue Volue 3 009

5 oulr thetl tool for the regulr obetve futo-bsed fuzzy lusterg lgorths. The otl udte equtos be obted usg the Lgrge ethod by settg the rtl dervtve of the Lgrge wth reset to v d wth reset to equl to zero. Settg J / equl to zero gves the udte equto for. The ew fuzzy lusterg lgorth be surzed the followg stes: Ste : Deterg the uber of luster; d -vlue (let =) gve overgg error 0 (suh s 0.00 ). Method : hoose the result ebersh tr of FCM lgorth s the tl oe. Method : let -e lgorth d obet to eter 0 Ste : Fd 0... wl wrs r s v w w l rs rs r s r s be the result eters of d be dstes 0 betwee dt. M 0 0 d d d d d d u M M d d 0 ( )( ) ( )( 0 ) where w rs r s (8) (0) v l I ( ) l... ( ) v l l ( ) l ( ) s s s s s s f s 0 s 0 f s 0 s s s s s. s s 0 l s s s s l s (7) (9) () () (3) (4) (5) Ste 3: Ireet ; utl. D..4 Fuzzy Possblt y C-M e Algorth[] Cobg FCM d PCM the roved fuzzy rtto lusterg lgorths Fuzzy Posblty C-Me (FPCM) ws roosed J FPCMU T A X t (6) ostrts:ebersh... (7) tylty t... (8) Ste : Deterg the uber of luster; d -vlue (let =) 3 gve overgg error 0 (suh s 0.00 ) hoose the result ebersh tr of FCM lgorth s the tl oe d the result tylty tr of PCM lgorth s the tl oe resetvely; (9) U Ste : Fd t t... t t t... t t t... t... t t t T t t... t t t... t t t... (3) l l l t (33) l l l Ste 3: Ireet ; utl E. FPCM-M Algorth (30) (3) Issue Volue

6 Now for rovg the bove robles we dded regultg ftor of ovre tr l to eh lss obetve futo. The roved ew lgorth Fuzzy Possblty C-Me bsed o Mhlobs dste (FPCM-M) s obted. Usg the Lgrge ultler ethod to ze the obetve futo (34) wth ostrt (35) reset to reters t we obt the solutos s (38) (39) (4)d(4)To vod the sgulr roble d to selet the better tl ebersh tr the udtg futos for d re obted s (36)~ (4).Note () All of the fuzzy rtto lusterg lgorths ot elot ll of the ebershs wth the se vlue () FPCM s sel se of FPCM-M whe Addg regultg ftor of eh lusters ovre obetve futo we roosed the fuzzy ossblty -e bsed o hlobos dste (FPCM-M) s followg. JFPCM MU T X t l ostrts:ebersh... tylty (34) t... (35) Ste : Deterg the uber of luster; d -vlue (let =) 3 Gve overgg error 0 (suh s 0.00 ) hoose the result ebersh tr of FPCM-M lgorth s the tl oe d the orlzed result tylty tr of PCM-M lgorth s the tl oe resetvely; (36) U t t... t t t... t t t... t... t t t T t t... t t t... t Ste : Fd t t... (37) (38) t t... s s s s s f s 0 s 0 f s 0 s s s s s s 0 s l s s s s l (40) (4) l t s s s l (4) Ste 3: Ireet ; utl F. The G-K Algorth (39) The well-ow Gustfso & Kessel lgorth (G-K lgorth) ws roosed by Gustfso & Kessel (979). It s fuzzy rtto lusterg lgorths bsed o Mhlobs dste d eteso of the fuzzy -es lgorth o dtve or whh wll rovde forto bout the lusters of vrous shes dt set. Eh luster s hrterzed by ts orlzto tr M M. The tr M s led s the otzto of vrbles the -es futol. Eh luster s ble to dt ts ow or orde wth toology dt of sef rego. The obetve futo s defed s the equto of (43). J U A M X M (43) GK ostrts:ebersh... (44) Eh grous of the deteret of stdrdzto ovre tr of luster I M... (45) Issue Volue

7 If there s o ror forto bout the.... The lgorth s desrbed s follows. Ste : Deterg the uber of luster -vlue (let =) d the overgg error 0 (suh s 0.00 ) d hoosg the tl ebersh tr U (46) Ste : To lulte... F (47) (48) d e t (49) M F F M (50) l l M l Ste 3: Ireet ; utl III. DATA RESOURCE. We hve two rel dt sets ws led to rove tht the erfore of the FCM-MS lgorth gve ore urte lusterg results th the FCM d FCM-M ethods d the rto ethod whh s roosed by us s the better of the two ethods for seletg the tl vlues. A. Eeret of Mthets Tehg Dt A rel dt set of studets wth sle sze 493 fro eleetry shools ws seleted. These dt luded the deedet vrbles test sores of four thets oets (dvso orderg ultlto d le vlue) d 0 questos. The sles were ssged to 4 lusters. The results were show Tble. Tble. The Chrtersts of 4 Clusters Averge Cluster Sle Mthets dste sze oets fro ots to eter 5 dvso orderg ultlto le vlue Eh 5 sle ots were rdoly drw fro Cluster luster d luster 3 resetvely d 5 fro luster 4. How to selet the better tl vlue to rove the luster ury s ortt ssue. I order to test the FCM-M lgorth develoed by the uthors of ths er the four.5 were seleted s tl vlue. After lultg the results were foud tht the ebershs were ll equl to.5 too. Ths evdee dslyed tht the FCM-M lgorth ws wor orretly. There were ethods (Rto Rdo) to lulte the Norlzed tl uber. whh stsfed the Equto (). The stes of Rto Method were s follows. Ste : The dste betwee observg vlue d every luster eter of every Pot sy d. Coute the verge dste of lusterg result rg grou. 0 d d uber of Result Mrg Grou Ste : Coute the dfferee of d d the verge dste of lusterg result rg grou l d d Ste 3: Fd the vlues of u d u M l l Ste 4: Coute the tl ebersh Dfferee of every Pot ( Ml )( M ). The stes of Rdo Method were s follows. Choose y 4 rdo ubers r r r 3 r 4. Tble.Clssfto Aures of Testg Sles Choosg the tl ebersh Coutg dste Clssfto Aures (%) FCM-MS 54 Rto FCM-M 50 FCM 40 FCM-MS 50 Rdo FCM-M 30 FCM 6 Issue Volue

8 Fro the dt of Tble we foud tht the FCM-MS lgorth reseted the best lusterg ures u to 54% d the Rto ethod of FCM-MS ould obt the better results. lyss usg the -e lusterg of SPSS for Wdows 0.0. The results were show Tble 6. B. Eeret of Tehg Frto Dt Aother rel dt set of studets wth sle sze 46 fro eleetry shools ws seleted. The ftors of the dt were lulted by usg ftor lyss. Aordg to the ftors the sles were ssged to 4 lusters bsed o the lusterg lyss. The results were show Tble 4. Fro the dt of Tble 5 we foud tht the Rto ethod ould obt the best results. A rel dt set ws led to rove tht the erfore of the FCM-MS lgorth gve ore urte lusterg results th the FCM-M d FCM ethods. Tble 4. the hrtersts of 4 lusters Cluster Sle sze thets oets verge dste of the ots fro eter of luster 36 Prtto Ut Frto Uow ut Eh 5 sle ots were rdoly drw fro Cluster luster d luster 3 resetvely d 5 fro luster 4. The lssfto ures of testg sles were show Tble 5 Tble5.Clssfto Aures of Testg Sles Choosg the tl ebersh Coutg dste Clssfto Aures (%) FCM-MS 56 Rto FCM-M 38 FCM 36 FCM-MS 44 Rdo FCM-M 30 FCM 4 Fro the dt of Tble 5 we foud tht the FCM-MS lgorth reseted the best lusterg ures u to 56% d the Rto ethod of FCM-MS ould obt the better results. C. Eeret of Tehg Geoetry Dt A rel dt set of sle sze 968 studets fro eleetry shools ws seleted. These dt luded the 0 thets questos. At frst the ftors of 968 dt were lulted by usg ftor lyss. Net ordg to the ftors the sles were ssged to 4 lusters bsed o the lusterg Tble 6 The hrtersts of 4 lusters Cluster Sles sze Grde verge dste of the ots fro eter of luster Fro Cluster 5 sles rdoly were seleted 5 fro luster 5 fro luster 3 d 5 fro luster 4.The obto the ethod of hoosg the tl ebersh wth dstt outg dste ws show Tble 7. Tble 7 Dt Cluster d Sle szed Cluster Nuber of Sles Fro the dt of Tble 8 we foud tht the lgorth bsed o usuervsed Mhlobs dste of FCM-M s better lssfto ures th bsed o Eulde dste of FCM u to 5%. Slrly Preseted the best lssfto ures 58% s lso bsed o usuervsed Mhlobs dste of FPCM-M[3] u to 58%. Tble 8 Clssfto ures of testg sles. Coutg dste Clssfto Aures (%) FCM 3 FCM-M 5 FPCM 30 FPCM-M 58 IV. CONCLUSIONS Etedg Eulde dste to Mhlobs dste Gustfso-Kessel (GK) lusterg lgorth d Gth-Gev (GG) lusterg lgorth re develoed to detet o-sherl struturl lusters but both of the bsed o se-suervsed Mhlobs dste these two lgorths fl to osder the reltoshs betwee luster eters the obetve futo eedg ddtol ror forto. Whe soe trg luster sze s sll th ts desolty t dues the sgulr roble of the verse ovre tr. It s ortt ssue. The other ortt ssue s how to selet the better tl vlue to rove the luster ury[5]. I ths er fousg tteto to bove Issue Volue

9 two robles roved ew fuzzy lusterg lgorth FCM-MS s develoed to obt better qulty of fuzzy lusterg results. The obetve futo ludes fuzzy wth-luster stter tr ew betwee-rototyes stter tr the regultg ters bout the ovre tres d the regultg ters bout the reltoshs betwee luster eters the reltoshs betwee the eter of ll ots d the luster eters. The udte equtos for the ebershs d the luster eters d the ovre tres re dretly derved fro the Lgrge s ethod whh s dferet fro the GK d GG lgorths. The sgulr roble d the seletg tl vlues roble re roved by the Egevlue ethod d the Rto ethod. Flly uerl ele shows tht the ew fuzzy lusterg lgorth FCM-MS gves ore urte lusterg results th the FCM d FCM-M lgorths for rel dt set the rto ethod whh s roosed by us s the better of the two ethods for seletg the tl vlues. APPENDIX Proof the tl ebershs of FCM-M Algorth d FCM Algorth ot be ll equl. [Proosto]The tl ebershs of FCM-M Algorth d FCM Algorth ot be ll equl. [ roof:] () I FCM-M Algorth Let We get u s The roof s oleted. () I FCM Algorth The roof s slr s bove. l l ACKNOWLEDGMENT Ths er s rtlly suorted by the Ntol See Coul grt ( NSC-97-5-S -4-00). REFERENCES [] J. C. Bezde Ptter Reogto wth Fuzzy Obetve Futo Algorths Pleu NY 98. [] I. Gth d A. B. Gev Usuervsed Otl Fuzzy Clusterg IEEE Trs. Ptter Al. Mhe Itell [3] D. E. Gustfso d W. C. Kessel Fuzzy Clusterg wth Fuzzy Covre Mtr Pro. IEEE Cof. Deso Cotr. S Dego CA 979. [4] H. C. Lu J. M. Yh d S. W. Lu Fuzzy C-e Algorth Bsed O Mhlobs Dstes d Better tl vlues th Itertol Coferee o Fuzzy Theory & Tehology JCIS Slt Le Cty Uth 007. [5] Z. H. Y Y. G. Tg Yugg F. C. Su d Z. Q. Su Fuzzy Clusterg wth Novel Serble Crtero Tsghu See d Tehology Vol [6] J. C. Du A Fuzzy Reltve of the ISODATA Proess d Its Use Detetg Cot Well-serted Clusters J. Cyber Vol [7] J. Yu Q. Cheg d H. Hug Alyss of the Weghtg Eoet the FCM IEEE Trs. Systes M d Cyberets Prt B Vol [8] H. C. Lu J. M. Yh T.W. Sheu d S.W. Lu A New Fuzzy Possblty Clusterg Algorths Bsed O Usuervsed Mhlobs Dstes 007 Itertol Coferee o Mhe Lerg d Cyberets Hog Kog [9] Soesh Ds Gut. The evoluto of the D-sttsts of Mhlobs Id J. Pure Al. Mth.6(995) o [0] Beset Dder H. Obet Oreted Ile-etto of Nuerl Methods.Morg Kuf Publsfers(00). U. S. A. ISBN: [] H. C. Lu J. M. Yh T-We Sheu Sh-Wu Lu (007). A ew Fuzzy Possblty Clusterg Algorths Bsed o Usuervsed Mhlobs Dstes. Proeedgs of Itertol Coferee o Mhe Lerg d Cyberets 007 (ICMLC 007) Vol.7. No ISBN X. EI 級論文 [Hog Kog Ch August ]. [] J. M. Yh Yu-Horg L Hsg-Chu Lu (008). FCM Algorth Besed o Usuervsed Mhlobs Dstes wth Better Itl Vlues d Serble Crtero. Proeedgs of the 8th WSEAS Itertol Coferee o APPLIED COMPUTER SCIENCE (ACS'08) ISSN: / ISBN: [Vee Itly Noveber ]. [3] N. R. Pl K. Pl d J.C. Bezde A ed -e lusterg odel Proeedgs of the Sth IEEE Iterto l oferee o Fuzzy Systes Vol.. - Jul [4] Seyed Ar Hd Moof Az Bstfrd (008) A Novel Algorth for Geertg Muhd Ptter Bsed o Cellulr Autot Pro eedgs of the 3th WSEAS Itertol Coferee o APPLIED MATHEMATICS (MATH'08) ISSN: ISBN: [5] J. M. Yh Y.H. L Hsg-Chu Lu (008). Clusterg Alyss Method Besed o Fuzzy C-Mes Algorth of PSO d PPSO wth Alto Ige Dt. Proeedgs of the 8th WSEAS Itertol Coferee o APPLIED COMPUTER SCIENCE (ACS'08) ISSN: / ISBN: [Vee Itly Noveber ]. [6] J. M. Yh Y.H. L Hug W. L.(007). Fuzzy Aroh Method for Coet Struture Alyss Bsed o FLMP d ISM wth Alto Cogto Dgoss of Ler Algebr. Iforto Sees 007 Proeedgs of The 0th Jot Coferee & The th Itertol Coferee o Fuzzy Theory & Tehology (FTT 007) EI-er [ Slt Le Cty Uth U. S. A. July ]. [7] J. M. Yh Y.H. L (007). A Fuzzy Bss o Kowledge Struture Alyss for Cogto Dgoss d Alto o Geoetry Coets for Puls. The Thrd Itertol Coferee o Itellgee Issue Volue

10 Iforto Hdg d Multed Sgl Proessg. (IIH-MSP 007) Vol ISBN: EI-er [Kohsug Cty Tw Noveber ]. [8] J. M. Yh Y.H. L (007). A Itegrto of Fuzzy Theory d ISM for Coet Struture Alyss wth Alto of Lerg MATLAB. The Thrd Itertol Coferee o Itellgee Iforto Hdg d Multed Sgl Proessg. (IIH-MSP 007) Vol ISBN: EI-er [Kohsug Cty Tw Noveber ]. [9] Y.H. L J. M. Yh (008). Fuzzy Log Aroh o Cogto Dgoss wth Alto o Nuber Coet for Puls. Proeedgs of 008 Itertol Coferee o Mhe Lerg d Cyberets ISBN: EI-er [Grd Pr Hotel Kug Ch July ] [0] Y.H. L J. M. Yh M-Ye Che (008). Polytoous IRS wth Alto Coets Dgoss d Clusterg o Frto Subtrto for Puls. Seod Itertol Syosu o Itellgetv Iforto Tehology Alto ISBN: EI-er[Shgh Ch Deeber - 008] [] Y.H. L J. M. Yh (008). A Itegrto of Coet Struture Alyss d S-P Chrt wth Alto Equlty Ao Coets Dgoss. Seod Itertol Syosu o Itellgetv Iforto Tehology Alto ISBN: EI-er[Shgh Ch Deeber Hs reserh terests lude tter reogto of tehg ler lgebr d fuzzy lusterg. J. M. Yh s wth the Grdute Isttute of Edutol Mesureet d Sttsts d Dertet of Mthets Eduto Ntol Thug Uversty 40 M-Sheg Rd. Thug Cty 403 Tw d the De of Geerl Affrs of Ntol Thug Uversty I the future We wll osder Fuzzy Aroh Method for Coet Struture Alyss[6]-[]. Hsg-Chu Lu Dertet of Boforts As Uversty Thug 4354 Tw Ch-Fu Yh Dertet of Couter See d Iforto Egeerg Ntol Ch N Uversty Ntou Tw. El: rot758@gl.tu.edu.tw. Hsg-Chu Lu reeved the Ph.D. degree Sttsts fro Ntol Tw Uversty Tw. He s rofessor t the Dertet of Boforts As Uversty Tw se August 00 d lso hoored rofessor t the Grdute Isttute of Edutol Mesureet d Sttsts Ntol Thug Uversty Tw. He ws the Presdet of Ntol Thug Uversty Tw fro 993 to 000. He hs fuded reserh d ublshed rtles the res of Bosttsts Boforts Fuzzy Theory Edutol Mesureet d E-Lerg. Jeg-Mg Yh reeved the B.S. d M.S. degrees fro the Ntol Tw Norl Uversty Te Tw R. O. C. 983 d 986 resetvely.he s urretly Asote Professor wth Dertet of Mthets Eduto t Thug Uversty Thug Tw R.O.C. Issue Volue