Nonparametric Regression with Trapezoidal Fuzzy Data

Transcription

1 Iteratoal Joural o Recet ad Iovato reds Computg ad Commucato ISSN: Volume: 3 Issue: Noparametrc Regresso wt rapezodal Fuzzy Data. Razzaga Departmet of Statstcs Roudee Brac Islamc Azad Uversty Roudee - Ira. Correspodg Autor E-mal: razaga@rau.ac.r S. Daes Departmet of Statstcs Scece ad Researc Brac Islamc Azad Uversty era - Ira. E-mal: s.daes@srbau@rau.ac.r Abstract- s paper s a vestgato to oparametrc fuzzy regresso wt crsp put ad asymmetrc trapezodal fuzzy output. It aalyzes te a oparametrc tecques statstcs amely local lear smootg (L-L-S wt trapezodal fuzzy data to obta te best smootg parameters. I addto t makes a aalyss o oe real-world datasets ad calculates te goodess of ft to llustrate te applcato of te proposed metod. Key Words- Noparametrc Fuzzy Regresso rapezodal Fuzzy Numbers Local Lear Smootg (L-L-S. ***** I. INRODUCION II. PRELIMINARIES Sce te fuzzy regresso was troduced by aaka et al.[] several fuzzy regresso approaces ave bee proposed cludg te matematcal programmg based metods [] least squares based metods [] ad oter metods [3]. I may real-world problems t may be urealstc to predeterme a fuzzy parametrc regresso relatosp especally for a large dataset wt a complcated uderlyg varato tred. Alog ts le of cosderato some oter approaces ave bee developed to adle te fuzzy regresso problems wtout predefg a specfc form of te uderlyg regresso relatosp. For stace Isbus ad aaka [4] ave suggested several fuzzy oparametrc regresso metods by usg te tradtoal back propagato etworks. Also statstcal oparametrc smootg tecques ave aceved sgfcat developmet recet years [5]. ese smootg tecques are especally useful to adle te oparametrc regresso problems ad terefore tere may be oter promsg tools for developg fuzzy oparametrc regresso. I ts aspect Ceg ad Lee [3] ave exteded te k-earest egbor (K-NN ad kerel smootg (K-S metods for te cotext of fuzzy oparametrc regresso. I Wag et al. [6] te local lear smootg metod te specal case of te local polyomal smootg tecque s fuzzfed to adle te fuzzy oparametrc regresso wt crsp put ad LR fuzzy output based o te dstace measure proposed by Damod [7]. Faroos et al. [8] used rdge estmato oparametrc regresso wt tragular fuzzy data. I ts paper we propose to fuzzfy ad aalyze te tree oparametrc regresso tecques statstcal regresso amely local lear smootg (L-L-S te K- earest egbor smootg (K-NN ad te kerel smootg tecques (K-S wt trapezodal fuzzy data. A fuzzy umber A ~ s a covex ormalzed fuzzy subset of te real le R wt a upper sem-cotuous membersp fucto of bouded support [7]. Defto.. A asymmetrc trapezodal fuzzy umber A ~ deoted by ~ ( ( (3 (4 A ( a a a a s defed as: ( a x L ( x a ( ( a a A ( x a x a (3 x a (3 R ( x a (4 (3 a a ( ( (3 ( ( (3 (4 were a a a a are four parameters of te asymmetrc trapezodal fuzzy umber. ~ ( ( (3 (4 Defto.. Suppose tat A ( a a a a ad ( ( (3 (4 B ( b b b b are two trapezodal fuzzy umbers. Damod dstace betwee A ad B ca be expressed as: ( ( ( ( d ( A B ( a b ( a b (3 (3 (4 (4 ( a b ( a b IJRICC Jue 5 ttp:// 386

2 Iteratoal Joural o Recet ad Iovato reds Computg ad Commucato ISSN: Volume: 3 Issue: fuctos for dervg oparametrc regresso model based o te smootg parameters. s dstace measures te closeess betwee two trapezodal fuzzy membersp fuctos we ( d A B. It meas tat te membersp fuctos of A ad B are equal. Let F : ( y ( y ( y (3 y (4 be a set of all trapezodal fuzzy umbers. e followg uvarate fuzzy oparametrc regresso model s cosdered by F ( x. I ts model X s a crsp depedet varable (put ad s a symmetrc trapezodal fuzzy depedet varable (output. s a error term ad s a operator wose defto depeds o te fuzzy rakg metod used. I ts paper for te oparametrc regresso tecques K-N-N ad K-S are based o te cocept of local averagg. I oter words te estmated value of te regresso surface at pot k s te wegted average of te resposes of te observatos te egborood of k. Defto.3. Let K were te dex s ascedg order te te smootg fucto based o local averagg ca be represeted as: S ( K K AVE ( k j k ( ( (3 (4 j j j j AVE ( y y y y k j k were AVE deotes te mea meda or ay wegted average. III. Smootg metods for trapezodal fuzzy umbers e basc dea of smootg s tat f a fucto f s farly smoot te te observatos made at ad ear sould cota formato about value of. us t sould be possble to use local averagg of te data to costruct a estmator for Fx ( wc s called te smooter. ere are several smootg tecques. We proposed K-earest egbor smootg (K-NN kerel-smootg (K-S ad local lear smootg (L-L-S metods for trapezodal varable ts secto. I te followg dscusso asymmetrc trapezodal fuzzy umbers are appled as asymmetrc trapezodal membersp j ese models are cosdered uvarate fuzzy oparametrc regresso model as: IJRICC Jue 5 ttp:// F x { } ( 3 4 x x x x { } were s a trapezodal fuzzy depedet varable as output. x s a crsp depedet varable as put x ad x doma s assumed to be. Fx ( s a mappg D F. e defto of te smootg metod for trapezodal fuzzy varables s as follows: - Local lear smootg metod (L-L-S I te followg dscusso Razzaga et al. [9] proposed te frst lear regresso aalyss wt trapezodal coeffcets. Asymmetrc trapezodal fuzzy umbers are appled as asymmetrc trapezodal membersp fuctos for dervg bvarate regresso model. A uvarate regresso model ca be expressed as: 3 4 A A X a a a a 3 4 a a a a X ( s model ca be rewrtte as 3 ( a a X a a X a a X a a X were ad s te sample sze. ( ( (3 (4 ad s a observed value for. So. Lad R. are te left boud ad rgt boud of te predcted at membersp level. Also L ad R are left boud ad rgt bouds of observed level. ereupo at membersp 387

3 Iteratoal Joural o Recet ad Iovato reds Computg ad Commucato ISSN: Volume: 3 Issue: L a a X 3 3 x x x ' 3 x ( x x 5 ( a ( a X x x x 3 3 R a a X ' 4 x ( x x ( a ( a X ( ( (3 were ( x ( x ( x ad ( ( L ( (3 (4 R ( Let. X be a sample of te observed crsp puts ad trapezodal fuzzy outputs wt uderlyg fuzzy regresso fucto of model (. Fx ( s estmated at ay x D based o x for. We te local lear smootg tecque s used we sall estmate 3 x x x ad eac x D (4 ( x for by usg te dstace proposed by Damod [7] as a measure of te ft ( Defto.. s dstace s used to ft te fuzzy oparametrc model (. Let 3 x x x ad (4 ( x ave cotuous dervatves te doma x D. e for a gve x D ad aylors expaso 3 x x x ad (4 ( x ca be locally approxmated egborood of x respectvely by te followg lear fuctos: ( x x x ' x ( x x (3 x x x ' ( x x x 4 (4 ( x are respectvely te dervatves of 3 x x x ad 4 x based o Damod dstace (Defto. ad te local lear smootg metod s estmated at x F x ( x x 3 4 x x by mmzg 3 4 d d ( (7 3 4 K ( x x 3 4 Wt respect to ad 3 4 for te gve kerel k(. ad smootg parameter were K x x k x x are a sequece of wegts at x.wo commoly used kerel fuctos are parabolc sape fuctos: k ( x x f x.75( ad Gaussa fucto: oterwse for IJRICC Jue 5 ttp:// 388

4 Iteratoal Joural o Recet ad Iovato reds Computg ad Commucato ISSN: Volume: 3 Issue: k x Equato (8 as egt ukow parameters x exp( 3 4 x x x x Also by substtutg (3 (4 (5 ad (6 at (7 te followg ca be obtaed 3 4 x x x x to derve a formula for te ukow parameters oparametrc regresso based o mmzg ts dstace d te dervatves (8 wt respect to te egt ukow parameters eed to be derved set to zero ad solve te egt ukow parameters. 3 4 d ( 3 4 K ( x x ' ( x x x x K ( x x ' ( x x x x K ( x x 3 3 ' 3 ( x x x x K ( x x 4 4 ' 4 x x x x K ( x x (8 By solvg ts wegted least-squares problem te followg ca be obtaed 3 4 x x x x 3 4 x x x x at x. So te estmato Fx ( at x s: x ( x x 3 4. x x Accordg to te prcple of te wegted least-squares ad utlzg matrx otatos we ca obta ' ( x x ( X x (9 ( ; ( W ( ; W x X x X x x ' ( x x ( X x ( ( ; ( W ( ; W x X x X x x 3 ' 3 ( x x ( X x ( (3 ; ( W ( ; W x X x X x x 4 ' 4 ( x x ( X x ( (4 ; ( W ( ; W x X x X x x IJRICC Jue 5 ttp:// were ( x x ( ( X x x x ( x x ( (3 ( (3 ( (3 ( (3 (4 (4 (4 (4 389

5 Iteratoal Joural o Recet ad Iovato reds Computg ad Commucato ISSN: Volume: 3 Issue: as ts mmzato gves te optmal value. W x ; Dag( K x x ad s a K x x K x x dagoal matrx wt ts dagoal elemets beg K ( x x for ad symbol s o traspose of a matrx. If we suppose e ( ; ( ; ; H x X x W x X x X x W x e estmate of Fx ( at x s x ( x x 3 4 x x ( e H x ; e H x ; 3 4 e H x ; e H x ; (3 - Smootg parameters selecto ad CV CV m o I fact we may compute to searc for. CV So selected optmal value of by te for a seres of value of CV early depeds o te degree of smootess of L ad R. Large value of leads to lack-of-ft ad small value of makes over-ft. IV. Numercal Example I ts secto tere are a example wc te put s a crsp umber ad te output s a trapezodal fuzzy umber. We estmate te values by usg tree smootg metods. e tese metods ca be compared wt eac oter ad for ts purpose ter GOF ad ter carts are used. Example : s example s a geerated dataset te same way as tat Ceg ad Lee [3]. e followg fucto s x x cosdered f x e 5 e most mportat aspect for averagg tecques ad local lear smootg metod s selectg te sze of egborood to average k ad parameter. ere are dfferet metods for selectg parameter suc as te cross-valdato metod ad geeralzed cross valdato wc are used to obta parameter. Let x ( x x 3 4 x x e fuzzfed cross-valdato procedure (CV for selectg parameter local lear smootg metod based o Damod dstace s defed as: CV d (( ( ( 3 (3 ( ( 4 (4 (( (6 So s uformly geerated wt te terval [ ] ad = ( ( (3 (4 ( y e y e y e y e 3 3 So [.5.5] y f X rad ad e / 4 f X rad []. Local Lear smootg metod s appled to te fttg model. So Gauss ad Parabolc sape kerel are used to produce te wegt sequece for local lear smootg able 3 sows smootg parameter selected by crossvaldato procedure results from dfferet metods. Fgures 4 5 ad 6 sow te results of tree metods. ese results ca be compared usg fgure 3 ad table 3. Lke te prevous example L-L-S metod s better ta K-NN ad K- S metods. I table 3 GOF of L-L-S metod s lower ta K-NN K- S metods. IJRICC Jue 5 ttp:// 383

6 Iteratoal Joural o Recet ad Iovato reds Computg ad Commucato ISSN: Volume: 3 Issue: able e obtaed results of dfferet metods for sample metod kerel Smootg parameter GOF [9]. Razzaga E. Pasa E. Korram A. Razzaga "Fuzzy lear regresso aalyss wt trapezodal coeffcets" Frst Jot Cogress O Fuzzy Ad Itellget Systems 7 Aug. 9-3 Masad Ira. LLS Gauss Parabolc sape _ Fgure: Obtaed results by L-L-S metod wt Gausa kerel for =.43 REFERENCES [] H. aaka S. Uejma K. Asa "Lear regresso aalyss wt fuzzy model" IEEE rasactos o Systems Ma ad Cyberetcs 98 pp [] P.. Cag E. S. Lee A geeralzed fuzzy wegted least-squares regresso Fuzzy Sets ad Systems 8 ( [3] C. B. Ceg E. S. Lee "Noparametrc fuzzy regresso K-NN ad Kerel Smootg tecques" Computers ad Matematcs wt Applcatos pp 39-5 [4] H. Isbus H. aaka" Fuzzy regresso aalyss usg eural etworks" Fuzzy Sets ad Systems 5 99 pp [5] W. Hardle "Appled Noparametrc Regresso" Cambrdge Uversty Press New ork 99. [6] N.Wag W.X. Zag ad C.L Me "Fuzzy oparametrc regresso based o local lear smootg tecque" Iformato Sceces 77 7 pp [7] P. Damod Fuzzy least squares Iformato Sceces pp [8] R. Faroos J. Gasema ad o. Solayma Fard A modfcato o rdge estmato for fuzzy oparametrc regresso Iraa Joural of Fuzzy systems 9 pp IJRICC Jue 5 ttp:// 383