DISTANCE MEASURE FOR ORDINAL DATA *

Transcription

1 ARGUMENTA OECONOMICA No (8) 999 PL ISSN Mre Wes DISTANCE MEASURE FOR ORDINAL DATA * The study cosders the proe of costructo esures of srty for ord dt. The ord chrcter of the dt requred the ppcto of specfc esure of the oect s dstce. Wes (993 p ) gves the propos of ew esure of oects srty whch c e pped the stuto whe vres descrg oects re esured o the ord sce. Ths esure ws used order to evute the srtes of oects whch were sed o uers of retos equ to greter th d ser th. The dstce esure tes cre of vres wth equ weghts. We sh descre sght geersto of ths esure so coverg dfferet weghts of vres. The stregths d weesses of the proposed dstce esure re dscussed.. INTRODUCTION Cssfcto utdeso scg d er orderg ethods re portt d frequety pped toos of utvrte sttstc yss. The ppcto of these ethods requres forsto of the ter srty of oects. The use of prtcur costructo of srty esure depeds o the sce o whch the vres re esured. I the esureet theory four sc sces re dstgushed: o ord terv d rto. These were troduced y Steves (959). Aog the four sces of esureet the o s cosdered the weest. It s foowed y the ord sce the terv sce d the rto sce whch s the strogest. The choce of srty esures s rther spe whe the vres descrg eed oects re esured o the se sce. Lterture presets pety of dfferet wys of srty esureet whch c e dopted to vres esured o the sce: rto terv d (or) rto o ( cudg ry vres). A wde rge of srty esures hs ee show : Corc (97); Adererg (973); Evertt (974); Kuf d Rousseeuw (990); Co d Co (994 p. 0 ); Wede d Kur (998 p. 47). * Deprtet of Ecooetrcs d Coputer Scece Wrocłw Uversty of Ecoocs e-: [email protected].

2 M. WALESIAK 68 Wes (993 p ) gves the propos of ew esure of oects srty whch c e pped stuto whe vres descrg those oects re esured oy o the ord sce (see: so Wes Dzechcrz d Bą 998 p ). If we hve set A of oects descred y ord vres the coutg of evets s the oy posse rthetc operto whch c e perfored o these oects. The proposed esure s gve y the foowg foru: d () where: r p r p r p r p f f 0 f for p = ; r = ; =... uer of oect =... uer of ord vre ) ( th ( th th) oservto o th ord vre oect h oservedfor ser t h d retos greter t uer of. oect h oservedfor ser t h d retos greter t uer of Epe. Appcto of dstce () to copute the dstces of oects fro the ptter (de pot). The output resut s vector of dstces.

3 DISTANCE MEASURE FOR ORDINAL DATA 69 Te Dt No. Noteoo Effcecy Equpet Quty Ergoocs Docuetto Cfor Access Cfor Access Cevo Mtsu P-96-3R Cevo Mtsu P-98R Copq Ard 590DT De Lttude CP 66ST Dgt HNote VP Dgt HNote Utr Euroco Futsu LfeBoo 675CDT Futsu LfeBoo 765TCDT Futsu LfeBoo 985CDT GerCo Overdose Epre 8500T Hyud HN IBM ThPd TP380ED Po Tosh Stete Pro 480CDT Tosh Tecr 750DVD Tup Moto Le d 5/ Twhed Arsto FT-9000 DSC Twhed Arsto FT-9000 TFT Twhed Arsto FT-9300T Vos HS LeBoo Advce 66 DSC Vos HS LeBoo Advce 00 TFT Ptter Weghts Source: CHIP 998 o. 4. Te The dstces of oects fro the ptter (de pot) Posto Noteoo Dstce () Posto Noteoo Dstce () Source: ow reserch.

4 70 M. WALESIAK. MODIFICATION OF DISTANCE MEASURE d The dstce esure () tes cre of vres wth equ weghts. We sh descre sght geersto of ths esure so coverg dfferet weghts of vres. Suppose vre weghts w ( =... ) stsfy codtos: w (0; ). () w Three or ethods of vre weghtg hve ee deveoped: pror sed o epert opos procedures sed o forto cuded the dt d coto of these two ethods. Grńs (99) Mg (989) Arhowcz d Ząc (986) d Borys (984) dscuss the proe of vre weghtg utvrte sttstc yss. The proe of whether or ot to weght vres hs cused cotroversy. Ws sys (see: Adederfer d Bshfed 984 p. ) tht weghtg s spy the puto of vue of vre. Seth d So (973) suggest tht the pproprte wy to esure srty s to gve vres equ weght. If vre weghts re ot ufor the dstce esure s defed s (3). w d (3) w w w w Whe vre weghts re equ the foru (3) ecoes dstce esure (). Epe. Appcto of dstce (3) to copute the dstces of oects fro the ptter (de pot). The output resut s vector of dstces. Te 3 w Weghts for vres sed o CHIP epert opos Vre Effcecy Equpet Quty Ergoocs Docuetto Weghts Source: CHIP 998 o. 4.

5 DISTANCE MEASURE FOR ORDINAL DATA 7 Te 4 The dstces of oects fro the ptter (de pot) Posto Noteoo Dstce (3) Posto Noteoo Dstce (3) Source: ow reserch. 3. THE STRENGTHS AND WEAKNESSES OF THE DISTANCE MEASURE d Dstce esure d : c e pped stuto whe vres descrg oects re esured oy o the ord sce eeds t est oe pr of o-detc oects A ot to hve zero the deotor Ked s de of correto coeffcet for ord vres ws used for the esure d costructo (see: Ked 955 p. 9) dstce d ssues vues fro the [0; ] terv. Vue 0 dctes tht for the copred oects etwee correspodg oservtos of ord vres oy retos equ to te pce. Vue dctes tht for the copred oects etwee correspodg oservtos o ord vres retos greter th te pce or retos greter th d retos equ to f they re hed for other oects (.e. oects uered =... ; where ) dstce d stsfes codtos: d 0 d 0 d d (for =... ) suto yss proves tht dstce d ot wys stsfes the trge equty trsforto of ord dt y y strcty cresg fucto does ot chge the vue of d dstce.

6 7 M. WALESIAK 4. CONCLUDING REMARKS The use of vres esured o ord sce s retvey rre the terture. Specfc ytc toos re eeded for such forto. The proposed dstce esures () d (3) re pproprte such stutos. Whe vre weghts re equ foru (3) ecoes dstce esure (). The ddto resut of ths study s coputer progr whch ows coputg dstces etwee oects (see Apped). APPENDIX The coputer code the C++ guge coputg the vue of esure (3) of the dstce cosdered s ve t Wrocłw Uversty of Ecoocs the Dept of Ecooetrcs d Coputer Scece Ths verso of the progr ows to copute dstces etwee oects (the output resut s syetrc dstce tr) d so ccuto of the dstces of oects fro the ode or de pot (the output resut s vector of dstces). Ths tr y e used the herrchc ggoertve ethods of the cssfcto for the dvso of set of oects to csses. Ths tr c so e used for further coputtos the SPSS for Wdows pcge. Acowedgeets: The reserch preseted the pper ws supported y the proect KBN H0B 0 6. REFERENCES Arhowcz M. Ząc K. (986): Metod wże zeych w tsoo uerycze procedurch porządow owego [Vre Weghtg Agorth Nuerc Tooy d Ler Orderg Procedures]. Wrocłw Uversty of Ecoocs Reserch Ppers o. 38 pp Adederfer M. S. Bshfed R. K. (984): Custer Ayss Sge Bevery Hs. Adererg M. R. (973): Custer Ayss for Appctos. Acdec Press New Yor S Frcsco Lodo. Borys T. (984): Ktegor ośc w sttystycze ze porówwcze [Ctegory of Quty Sttstc Coprtve Ayss]. Wrocłw Uversty of Ecoocs Reserch Ppers o. 84. Corc R. M. (97): A Revew of Cssfcto (wth Dscusso) Jour of the Roy Sttstc Socety seres: A (3) pp Co T.F. Co M.A. A. (994): Mutdeso Scg Chp d H Lodo. Evertt B. S. (974): Custer Ayss Hee Lodo. Grńs T. (99): Metody tsooetr [Tooetrc Methods] Crcow Uversty of Ecoocs Krów.

7 DISTANCE MEASURE FOR ORDINAL DATA 73 Kuf L. Rousseeuw P. J. (990): Fdg Groups Dt: Itroducto to Custer Ayss Wey New Yor. Ked G. (955): R Correto Methods Grff Lodo. Mg G. W. (989): A Vdto Study of Vre Weghtg Agorth for Custer Ayss Jour of Cssfcto o. pp Seth P. H. A. So R. R. (973): Nuerc Tooy W.H. Free d Co. S Frcsco. Steves S. S. (959): Mesureet Psychophyscs d Utty : Church C. W. d Rtoosh P. (eds.): Mesureet; Deftos d Theores. Wey New Yor. Wes M. (993): Sttystycz z weowyrow w dch retgowych [Mutvrte Sttstc Ayss Mretg Reserch]. Wrocłw Uversty of Ecoocs Reserch Ppers o Wes M. (996): Metody zy dych retgowych [Methods of Mretg Dt Ayss]. PWN Wrszw. Wes M. Dzechcrz J. Bą A. (998): Ord Vres the Segetto of Advertseet Recevers : Rzz A. Vch N. Boc H. H.: Advces Dt Scece d Cssfcto Proc. 6th Cof. Iterto Federto of Cssfcto Socetes Roe. Sprger Hedeerg pp Wede M. Kur W. A. (998): Mret Segetto. Coceptu d Methodoogc Foudtos Kuwer Bosto Dordrecht Lodo. Receved: ; revsed verso