A Measure of Inaccuracy between Two Fuzzy Sets

Transcription

1 LGRN DEMY OF SENES YERNETS ND NFORMTON TEHNOLOGES Volum No 2 Sofa 20 Masur of accuracy btw Two Fuzzy Sts Rajkumar Vrma hu Dv Sharma Dpartmt of Mathmatcs Jayp sttut of formato Tchoy (Dmd vrsty) Noda (.P.) da E-mals: rkvr83@gmal.com bhudv.sharma@jt.ac. bstract: Dcso makg rqurs a strog as wll as flbl approach. Flblty mathmatcal laguag ca b trmd as fuzzss. frm or corporato dals wth svral parts ad groups. Th atur of gotatos or rsultg flblts wll chag from party to party. For ampl f a frm s dalg wth a supplr or a cosumr th atur of gotatos wll b dffrt whl th atur of ts ow trsts rmas th sam. For th purpos of such ad may othr smlar dalgs ad gotatos a tllgt dcso makg mathmatcal masur has to b dfd basd o two fuzzy crtra btw th two gotatg parts wth a prm cocr for o of ths. ths papr a way of masurg th accuracy btw two fuzzy sts propos s proposd. Ths accuracy masur has to b basd o fuzzss rathr tha o radomss. Som trstg mathmatcal proprts of ths masur ar aalyzd ad th rlatos btw fuzzy tropy fuzzy accuracy ad masur of fuzzy dvrgc ar also stablshd. Fally a ampl s prstd to llustrat th applcato of th masur proposd. Kywords: Fuzzy sts tropy dvrgc masur accuracy masur fuzzy tropy fuzzy dvrgc masur. 3

2 . troducto Th thory of fuzzy sts proposd by Z a d h [9] 965 has gad qut cosdrabl mportac varous flds of sgal ad mag procssg rct tms [ 3]. Fuzzss a fatur of ucrtaty rsults from th lack of sharp dstcto of bg or ot bg a mmbr of th st. masur of fuzzss usd ad ctd ltratur s th fuzzy tropy also frst mtod by Z a d h [0] 968. Th am tropy was chos du to trsc smlarty wth Shao s tropy [8]. Fuzzy tropy s th masurmt of fuzzss a fuzzy st ad thus t has a mportat posto fuzzy systms such as fuzzy kowldg basd systms fuzzy dcso makg systms fuzzy cotrol systms fuzzy ural twork systms fuzzy pattr rcogto systms ad fuzzy maagmt formato systms. Whl Shao s tropy rvolutozd th commucato thory wth tsv applcatos svral brachs statstcal studs foud gratr us of Kullback-Lblr s masur of dvrgc [6 7] bg a masur of th dstac btw two dstrbutos (obsrvd ad thortcal say) of a radom varabl Yt aothr formato typ of masur proposd by K a r r d g [5] masurs th accuracy of th dstrbuto of a radom varabl wth rspct to aothr dstrbuto udr rfrc. Ths s th so calld masur of accuracy. Hr th formato cotts of a dstrbuto udr study s avragd ovr a prassgd (kow) dstrbuto. Ths masur has clos coctos ad pottal rol statstcal studs mrgg stuatos wh gotatos ad dals ar struck to a succssful sttlmt two parts to achv magful formato. Thr ar probablstc studs o th masur of accuracy ad ts gralzatos but for fuzzy phoma smlar rsarch has ot b do. Ths s th am of ths papr. Lt us frst am stuatos whr th fuzzy accuracy masur studd hr may play a sgfcat rol. osdr a ampl th cas of a corporato havg clts. t s commoly obsrvd that a corporato dalg wth ts clts vstgats ( fact looks for) th mar whch th clt cosdrs of cours a rathr fuzzy way th st of ssus btw thm. Thus gral th corporato ad th clt hav dffrt fuzzy fuctos o th st of thr commo ssus. Th kowldg of th clt s fucto provds a lmt of formato for th corporato whos avragg wth wghts as ts ow valus of th lmts gv a da of th udrlyg accuracy dalg wth ts clt. t s atural for a dyamc corporato ths compttv world to hav a masur of accuracy ad th to mak a attmpt to fd th lackg part of formato that s th ral masur of thr gotatos. Som basc dftos rlatd to probablstc ad fuzzy st thory ar prstd Scto 2. Scto 3 w troduc a masur of accuracy btw two fuzzy sts. Scto 4 som proprts of fuzzy accuracy ar cosdrd. Th rlatos btw fuzzy tropy fuzzy accuracy ad fuzzy dvrgc ar stablshd Scto 5. Scto 6 a ampl s prstd to llustrat th 4

3 applcato of th accuracy masur proposd ad our coclusos ar prstd Scto Prlmars Frst lt us covr th probablstc part of th prlmars. Lt Δ P ( p p2... p ): p 0 p 2 b a st of -complt probablty dstrbutos. For ay probablty dstrbuto P ( p p2... p ) Δ Shao s tropy [8] s dfd as () H ( P) p p. Kullback-Lblr s [6 7] masur of dvrgc of tru probablty dstrbuto P ( p p2... p ) Δ to a arbtrary probablty dstrbuto Q ( q q... q ) Δ s gv by 2 (2) D( P Q) p p. q d Krrdg s accuracy [5] of dstrbuto Q wth rspct to dstrbuto P s gv by ( P Q) p q. Ths obvously ca b s as th avrag of formato lmts amly q dstrbuto Q ovr a dstrbuto P a ss gralzg Shao s tropy. Nt for fuzzy thory: Lt { 2... } b a dscrt uvrs of dscours. fuzzy st o s charactrzd by a mmbrshp fucto : [ 0]. Th valu of at stads for th dgr of mmbrshp of. Th st of all fuzzy sts o wll b dotd by FS(). Furthr o th st-thortc opratos o fuzzy sts by Zadh ar dfd as follows. Lt ad b two fuzzy sts o. Th uo of ad s dfd by ( ) ma( ) ( or smply ). Th trscto of ad s dfd by ( ) m( ) ( or smply ). Th complmt of s dfd by. 5

4 Th frst attmpt to quatfy th ucrtaty assocatd wth a fuzzy vt th cott of a dscrt probablstc framwork appars to hav b mad by Z a d h [0] who dfd th (wghtd) tropy of a fuzzy st wth rspct to P as H ( P) ( ) p p. D L u c a ad T r m [4] dfd fuzzy tropy for a fuzzy st corrspodg to () as (3) H [ ( ) ( ( ( ( ( ( ]. h a d a r ad P a l [2] proposd fuzzy dvrgc for two fuzzy sts ad corrspodg to (2) gv by [ ( ( ( ) ( ( D. ( ) ( ) 3. accuracy masur of a fuzzy st W procd wth th followg formal dfto Dfto. Lt b th fuzzy st ad aothr fuzzy st dfd o a dscrt uvrs of dscours { 2... } havg mmbrshp valus ( ) 2... ad ( ) 2... rspctvly. W df th masur of accuracy of a fuzzy st wth rspct to fuzzy st as (4) ( ) [ ( ) ( ( ( ( ( ( ]. Ths ca also b wrtt th followg form: ( ) S( f ( ) f ( y s Karrdg s accuracy [5] fucto for two vts. t s trstg to ot that wh th (4) bcoms (3) th masur of fuzzss gv by D L u c a ad T r m [4]. th fuzzy accuracy. whr S( y) y ( ) ( y) th t scto w study som proprts of 4. Proprts of th fuzzss masur of accuracy Th masur of fuzzy accuracy dfd (4) has th followg proprts. Thorm (for a crsp st). ( ) 0 f ad oly f thr ( ) ( ) 0 or ( ) ( )

5 P r o o f: Frst lt ( ) 0 th [ ( ) ( ( ( ( ( ( ] 0 [ ( ) ( ( ( ( ( ( ] Th abov rlato holds oly wh thr ( ) ( ) 0 ( ) ( ) 2... ovrsly lt thr ( ) ( ) 0 or ( ) ( ) ( ) ( ( ( ( ( ( ( ) 0. for all. or ths mpls Ths provs th thorm. Not. Ths mas that zro accuracy mpls a corrct statmt mad wth complt crtaty. FS ad F th most fuzzy st.. Thorm 2. For ay 0.5 for all ( F ). P r o o f: Lt FS( ) from dfto F ( ) [ ( ) ( ) F ( ) F ] ( ) [ ( ) ( 0.5) ( ( ( 0.5)] Ths provs th thorm. FS [ ( ) ( ( ]. Thorm 3. For ( ) ( ) ( ) ( ). P r o o f: Lt { ( ) } { < ( )} whr ad ar th fuzzy mmbrshp fuctos of ad rspctvly. Th w hav (5) [ ( ) ( ( ) ( ( ( ( ( ( ) ( )] F 7

6 8 ad (6) [ ]. ddg (5) ad (6) w obta. Ths provs th thorm. Thorm 4. For FS. P r o o f: Lt us hr tak { } { } < whr c ad ar th fuzzy mmbrshp fuctos of ad rspctvly. Th w hav (7) [ ] ad (8)

7 9 [ ]. ddg (7) ad (8) w gt th rsult. orollary 4.. Lt FS th. P r o o f: t obvous follows Thorms 2 ad 3. Thorm 5. For F. P r o o f: Lt { } { } < whr ad ar th fuzzy mmbrshp fuctos of ad rspctvly. Th rsult s as follows: (9) ad (0). ddg (9) ad (0) w gt th rsult. Thorm 6. For : F (a) (b)

8 (c) ( ) ( ) (d) ( ) ( ( ) ( ) whr ad rprst th complmts of fuzzy sts ad rspctvly. P r o o f: (a) t follows vdtly from th rlato of th mmbrshp of a lmt a st ad ts complmt. (b) Lt us cosdr th prsso [ ( ) ( ( ( ( ( )] [ ( ( ( ) ( ) ( ( ] Ths complts th proof. (c) Lt us cosdr th prsso [ ( ) ( ( ( ( ( )] 0. [( ( ( ) ( ) ( ( ] 0. Ths complts th proof. (d) t obvously follows from (a) ad (c). th t scto w propos a rlato btw fuzzy tropy fuzzy accuracy ad fuzzy dvrgc masur. 5. Rlato btw fuzzy tropy fuzzy accuracy ad fuzzy masur of dvrgc Thorm 7. Lt ad ar two fuzzy sts th H ( ) wth qualty f ad oly f.. ( ) ( ). P r o o f: Wthout loss of gralty w us atural arthms. sg th wll-kow qualty () wth qualty f ad oly f. ( ) Puttg quato () w gt ( ) ( ) ( ) (2) wth qualty f ad oly f ( ) ( ). 20

9 2 ga puttg quato () ylds (3) wth qualty f ad oly f. Multplyg (2) by ad (3) by ad summg ovr w obta [ ] 0 wth qualty f ad oly f. Thus [ ] [ ] 0. Ths provs th thorm. Thorm 8. For two fuzzy sts ad D H. P r o o f: From (3) ad (4) w hav: (4) [ ] H

10 . Subtractg (4) from (5) w gt [ ( ( ( ) ( ( D( ). ( ( ( ( Ths provs th thorm. s kpt fd ad varato (5) ( ) [ ( ) ( ( ( ( ( ( ] Thorm 9. f th mmbrshp fucto ( ) ( ) s allowd ( ) attas ts mmum valu wh ( ) ( ). P r o o f: W rcall that ( ) H D( ). Nt th rsult drctly follows from th fact that H 0 D ( ) 0 ad D ( ) 0 f ad oly f. Ths provs th thorm. 6. umrcal ampl Lt us cosdr a vry prtt corporat problm whch corporato dals wth svral clts ad gotats o a crta umbr of ssus. Lt th st of clts ad ssus b { } { Z Z 2 Z3 Z4 Z5 }. Tabl rprsts th wghts of ssus of th corporato trms of whr fuzzy mmbrshps Z t may b otd that Z Tabl. Wghts of ssus of th corporato (Z ) (Z 2 ) (Z 3 ) (Z 4 ) (Z 5 ) dcats th dgr of mportac of ssu Z to th corporato dalg. Tabl 2 rprsts th dgr of mportac of clts ( Z ) o ssu Z j ths dalg whr j Tabl 2. Wghts of ssus of th clts (Z )0.6 (Z 2 )0.7 (Z 3 )0.5 (Z 4 )0.8 (Z 5 ) (Z )0.9 2 (Z 2 )0.8 2 (Z 3 )0.6 2 (Z 4 )0.4 2 (Z 5 ) (Z )0.7 3 (Z 2 )0.6 3 (Z 3 )0.8 3 (Z 4 )0.4 3 (Z 5 ) (Z )0.6 4 (Z 2 )0.8 4 (Z 3 )0.6 4 (Z 4 )0.7 4 (Z 5 ) (Z )0.3 5 (Z 2 )0.7 5 (Z 3 )0.4 5 (Z 4 )0.6 5 (Z 5 )0.5 22

11 a dal th objctv of th corporato bg to choos th bst clt whch wth th dgr of gotato o commo ssus should b mmum (qual to th tropy valu of th corporato). So usg formula (0) w obta th fuzzy accuracy masurs ( ) whr Tabl 3. accuracy masurs btw a corporato ad clts ( ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ccordg to th accuracy masurs prstd Tabl 3 clt2 wll b mor sutabl for ths dal. 6. oclusos ths work w hav proposd a w accuracy masur calld fuzzy accuracy th sttg of fuzzy st thory. Ths masur ca b cosdrd as a gralzd vrso of fuzzy tropy proposd by D Luca ad Trm [4]. llustratv umrcal ampl llustratd th applcatos of ths accuracy masur busss. Paramtrc studs that troduc othr flblty crtra for th sam mmbrshp fuctos of ths masur ar also udr study ad wll b rportd sparatly. R f r c s. crkl Y. T. M. Kara. Nw Fuzzy pproach to Edg Dtcto. Lctur Nots omputr Scc (LNS) h a d a r D. N. R. P a l. Som Nw formato Masur for Fuzzy Sts. formato Sccs h a r a T. Sgmtato sg Fuzzy Dvrgc. Pattr Rcog. Ltt No D L u c a. S. T r m. Dfto of No-Probablstc Etropy th Sttg of Fuzzy St Thory. formato ad otrol K a r r d g D. F. accuracy ad frc. Joural of Royal Statstcal Socty K u l l b a c k S. R.. L b l r. O formato ad Suffccy. als of Mathmatcal Statstcs K u l l b a c k S. formato Thory ad Statstcs. Dovr Publcato Shao. E. Mathmatcal Thory of ommucato. ll Systm Tchcal Joural Z a d h L.. Fuzzy Sts. formato otrol Z a d h L.. Probablty Masur of Fuzzy Evt. Joural of Mathmatcal alyss ad pplcato No