Applying Grey Relational Analysis to the Decathlon Evaluation Model

Transcription

1 Chg-Lag Chag, Chh-Hug Tsa, Leh Che Applyg Grey Relatoal Aalyss to the Decathlo Evaluato Model Chg-Lag Chag, * Chh-Hug Tsa, ad Leh Che Departmet of Idustral Egeerg ad Maagemet Ta-Hwa Isttute of Techology 1 Ta-Hwa Road, Chug-L Hs-Chu, Tawa, ROC *E-mal: etch@tht.edu.tw Abstract Decathlo, track-ad-feld evets cosst of te separate cotests. Pots are awarded for each evet, ad the overall score determes the wer. However, the class terval ad ts specfed ut for scorg table seem to be ureasoable. To overcome ths defect, Grey relatoal grade deduced by Grey theory [17] wll be used to establsh a complete ad accurate evaluato model for determg who s the best all-aroud athlete amog all cotestats. Ths methodology ot oly wll sgfcatly reduce scorg dsputes but also ca help the atteded team to select the best athletes. A umercal eample for athletc scores rakg a typcal decathlo competto utlzg Grey relatoal aalyss wll also be made ths paper. Ths proposed method may provde the World Games or other sport federatos wth a mproved score awardg method order to determe who, amog all athletes, s the best all-roud athlete. Keywords: Grey theory, Grey relatoal aalyss, Decathlo 1. Itroducto The proposto of Grey theory occurrg the 1990 to 1999 tme perod resulted the uses of Grey theory to each feld, ad the developmet s stll gog o. The maor advatage of Grey theory s that t ca hadle both complete formato ad uclear problems very precsely. It serves as a aalyss tool especally cases whe there s o eough data. It was recogzed that the Grey relatoal aalyss Grey theory had bee largely appled to proect selecto, predcto aalyss, performace evaluato, ad factor effect evaluato due to the Grey relatoal aalyss software developmet. Recetly, ths techque has also appled to the feld of sport ad physcal educato. There are lots of good research results. For eample, the sport techques, Hsu [11] used Grey relatoal aalyss ad determed the effects of strkg ad kckg acto o overall scores. Wag [2] also took several factors affectg the release effect as multple attrbutes to fd how each factor flueces the relato grade rakg for wome avel throw. The process of last force eerto matchg up the acto of left 54

2 Applyg Grey Relatoal Aalyss to the Decathlo Evaluato Model leg s studed order to obta hghest scores. Ye [16] studed the terrelatoshp betwee defesve ad offesve techque ad team-w record through Grey relatoal aalyss. Che [13] had used Grey theory to make predcto for those soccer teams who ca eter roud of eght 2000 World Cup based o the 1998 results of group prelmares. Regardg score fluece factor aalyss for sport competto, She [7] appled Grey relatoal aalyss to aalyze ad obta the factors that fluece Fu-E Lee s overall decathlo scores. He determed the Grey relato grade for overall score ad te dvdual evet. Based o that, the rakg for those factors fluece the overall score s obtaed, ad ca be used as referece for future trag. By usg Grey relato aalyss, Ha [15] studed the effect of score varato each stato o overall score for top 10 athletes Maratho Games the perod betwee 1985 ad 1994 order to uderstad whch stato most affect the overall score. For the body characterstcs, Su [12] took twelve tems of characterstcs from thrtee well-performace PRC athletes, ad foud s ma body characterstcs ad ts rakg order affectg the score by Grey relato aalyss. Sce the wde applcato of Grey relato aalyss to sport competto ad ts advatage beg heret the method, ths paper ams to elmate the ureasoableess regardg the class terval ad pot coverso, ad provde a soud decathlo evaluato model for sport federato s referece. 2. Grey relatoal aalyss The Grey relatoal aalyss uses formato from the Grey system to dyamcally compare each factor quattatvely. Ths approach s based o the level of smlarty ad varablty amog all factors to establsh ther relato. The relatoal aalyss suggests how to make predcto ad decso, ad geerate reports that make suggestos for the veder selecto. Ths aalytcal model magfes ad clarfes the Grey relato amog all factors. It also provdes data to support quatfcato ad comparso aalyss [5]. I other words, the Grey relatoal aalyss s a method to aalyze the relatoal grade for dscrete sequeces. Ths s ulke the tradtoal statstcs aalyss hadlg the relato betwee varables. Some of ts defects are: (1 t must have plety of data; (2 data dstrbuto must be typcal; (3 a few factors are allowed ad ca be epressed fuctoally. But the Grey relatoal aalyss requres less data ad ca aalyze may factors that ca overcome the dsadvatages of statstcs method. The Grey theory ad method are descrbed the followg: 2.1. Ifluece space, measuremet space, ad Grey relatoal space Let P( represet the factor set of a specfc topcs, Q s the fluece relato, the {P(; Q} s fluece space. It must have the followg propertes [6]: 1. Estece of key factors: for eample, the key factors of basketball player are heght, weght, ad reboud. 2. Numbers of factors are lmted ad coutable: for eample each of the heght, weght, ad reboud are coutable. 3. Factor depedablty: each factor must be depedet. 4. Factor epadablty: For eample, besdes the heght, weght, ad 55

3 Chg-Lag Chag, Chh-Hug Tsa, Leh Che reboud, the free throw attempt ca be added as a factor. The seres formed by P( s: (0 (0 (0 ( k ( 1 (1,?,??, ( k? where 0, L,m. k 1, L,.? N If the followg codtos are satsfed: 1. Nodmeso: the umerc value for all factors must be odmeso. ; (2 2. If the epectacy s smaller-thebetter (e.g., the cost ad defects, the t ca be epressed by ( ( ma ma ( m 3. If the epectacy s omal-the-best (e.g., the age, ad whe the targeted value s (?? (, o : ma o m the t ca be epressed by 2. Scalg: the factor value for varous seres must be at the same level. 3. Polarzato: f the factor value the seres s descrbed as the same drecto, the seres s comparable. The the measuremet space s epressed as {P(; *(k}, the Grey relatoal space formed by the satsfacto of both factor space ad comparablty s termed by {P(; Γ} Geerato of Grey relato Uder the prcple of seres comparablty, to acheve the purpose of Grey relatoal aalyss, we must perform data processg. Ths processg s called geerato of Grey relato or stadard processg. The epected goal for each factor s determed by Wu [8, 9] based o the prcples of data processg. They are descrbed the followg: (1 1. If the epectacy s larger-the-better (e.g., the beeft, the t ca be epressed by ( ( ma ( m m (3 ( ma 2.3. The Grey relatoal grade The measuremet formula for quatfcato Grey relatoal space s called the Grey relatoal grade. Whe we are determg Grey relato ad takg oly oe seres, 0 (, as a refereced seres, t s called the grade of local Grey relato. If ayoe of the seres, (, s refereced seres, t s called the grade of global Grey relato. Addtoally, the Grey relatoal coeffcet must frst be determed before we obta the Grey relatoal grade. I the Grey relatoal space, {P(; Γ}, there s a seres where ( (1,( 2,?,??,( k? O 0, L,m. k 1, L,.? If the grade of local Grey relato s brought to defe the Grey relatoal coeffcet, γ ( k, ( k, t ca be ( epressed as followg: O N 56

4 Applyg Grey Relatoal Aalyss to the Decathlo Evaluato Model γ ( ( k, (4 ( k 0 m. + ( k + ma. ma. where 0, L,m. k 1, L,.? ; 0 s a refereced seres, s a specfc comparatve seres; ( k ( k : represetg the 0 0 k s absolute value of the dfferece of 0 ad ; m. m. m. 0( k ( k? k ; ma. ma. ma. 0( k ( k? k. After obtag the Grey relatoal coeffcet, we ormally take the average of the Grey relatoal coeffcet as the Grey relatoal grade: 1 Γ γ (,? γ ( ( k, k 1 (5 ( k However, sce real applcato the effect of each factor o the system s ot eactly same, Eq. (5 ca be modfed as:? Γ γ (, β γ ( ( k, where β k k 1 k ( k (6 represets the ormalzed weghtg value of a factor ad β 1 whe equatg both Eq (5 ad (6.? k 1 k The Grey relatoal grade represets the correlato betwee two seres. It s ot mportat a decso-makg. Rather, the rakg order of the relatoal grade s the most mportat formato. Therefore, m s comparatve seres wth ts correspodg Grey relatoal grade s rearraged accordg to the order of the ther magtudes. A Grey relatoal seres s defed as followg: I the Grey relatoal space, {P(; Γ}, refereced seres, 0, ad comparatve seres, ad : 0 ( 0 (k, k 1,..,. ( (k, k 1,.., ;? I ( (k, k 1,.., ;? I If γ ( 0,? γ ( 0,, the stuato dcatg the relatoal grade of vs. 0 s greater tha that of vs. 0, or represeted by Γ 0 >Γ 0. Ths s the relatoal seres for ad [9]. 3. Tradtoal score evaluato model versus Grey relatoal model Decathlo, track-ad-feld evets cosstg of 10 separate pots cotests held o two cosecutve days. Pots are awarded for each evet [3], ad the overall score determes the wer. The athlete receves predetermed pots for reachg certa dstaces, heghts, ad tmes the evets. The cumulatve score from each evet score determes the places of the athlete. The teratoal track-ad-feld rule for Decathlo [4] ca use the followg utlty fucto: 2.4. The Grey relatoal seres (7 TS? S 1 57

5 Chg-Lag Chag, Chh-Hug Tsa, Leh Che Where TS s the total score, S s the dvdual evet score, ad rage from 1 to 10. Regardg the places determed by above method, there are some scorg dsputes, whch are (1 Accordg to scorg table, each evet reachg certa dstace (meter, heght (meter, ad tme (secod are coverted to evet scores. Total pots are the obtaed by summg up each dvdual evet pots. But whether ths coverso s reasoable s stll questoable. (2 The measuremet ut s same for some evets, but the class terval coverso table are ot the same. For eample, the class terval for Pole Vault Table 1 showg 2 pots or 3 pots dfferece for every 0.01 meter seems to be ureasoable, but 1 pot dfferece for every 0.06 meter Javel. (3 Each evet score represets the athlete possessg some degree of characterstc such as muscular stregth, muscular edurace, cardovascular edurace, etc. Smply covertg the measuremet to pots ad summg up seem to be questoable. (4 There s o way to decde the wer f there s a te-score. O the other had, usg Grey relato aalyss to perform multple attrbutes overall evaluato, the measuremet ut s ot ecessarly the same. It ca easly determe the overall de wthout cosultg utlty fucto. Moreover, the decathlo athlete selecto possesses those above-metoed propertes. Hece, t s approprate to adopt Grey relatoal aalyss to decde the rakg for decathlo athlete. 4. Illustrato of umercal results For showg the sgfcace of Grey relatoal aalyss sport score evaluato ad resolvg the problems of scorg dspute, the method for decathlo rakg usg Grey relatoal aalyss s dscussed here. We assume there are fve cotestats attedg decathlo competto. The score s show Table 2. By utlzg tradtoal method, the results Table 3 showg the rakg order are A, D, B, C ad E. Table 1: Typcal score table for Pole Vault ad Javel Pole Vault Javel Meters Pots Meters Pots , , , , , ,370 Evet 100 M (Secs Table 2: Statstcal data for the decathlo competto Log Shot Put Hgh 400 M (Secs 110 M Hgh Dscus Pole Vault Javel 1500 M (Secs 58

6 Applyg Grey Relatoal Aalyss to the Decathlo Evaluato Model (M (M (M Hurdles (M (M (M Cotestat (Secs A B C D E Table 3: Evet score obtaed by tradtoal method Evet Cotestat 100 M Log Shot Put Hgh 400 M 110 M Hgh Hurdles Dscus Pole Vault Javel 1500 M Total scores Rak A ,733 1 B ,482 3 C ,642 4 D ,546 2 E ,599 5 Moreover, f fve cotestats attedg decathlo competto, the followg procedure dcates how the top of three ca be determed meters, 400 meters, 1500 meters, ad 110 meters hgh hurdles: I ths category, t s obvously that the epectacy s shorter-the-better for the tme, whch ca be determed by: 4.1. Implemetato wth evaluato matr Geerate a evaluato matr by arragg game evet as attrbute colum, ad cotestats as comparatve sequece. I the aalyss, the record for the fve cotestats s used to form evaluato matr as show Table Data ratoalzg Epected goal ca be ratoalzed accordg to each attrbute. A group of assumptos are made for the followg: ( ( ma ma ( m I whch represets the score both at attrbute ad comparatve seres. 2. Log, Shot Put, ad Hgh : The epectacy ths category s loger-thebetter for the dstace, whch ca be determed by: 59

7 Chg-Lag Chag, Chh-Hug Tsa, Leh Che ( ( ( ma m m 4.3. Establshg stadard seres Based o the epectacy of each dvdual evet, the scorg pots for each attrbute are ormalzed to obta the matr table of comparatve seres as show Table 4. I accordace wth our epected goal for each dvdual cotest evet, a deal stadard seres ( 0 1 s establshed the last le Table 4. Table 4: Data ratoalzg Evet Comparatve Seres 100 M Log Shot Put Hgh 400 M 110 M Hgh Hurdles Dscus Pole Vault Javel 1500 M Stadard seres ( Determato of Grey relatoal coeffcet for each cotestat 1. Calculate the mamum ad mmum dfferece by: ma. ma. ma ( k ( K, the?? k resultg mamum dfferece s oe; m m.? m. ( k ( K? k, the resultg mmum dfferece s zero. 2. Calculate the Grey relatoal coeffcet by: k + m ma γ 0(. 0 ( k + ma By substtutg the value of mamum ad mmum dfferece to above equatos, the Grey relatoal coeffcet for each cotestat (dvdual cotest evet s show Table 5. Table 5: The Grey relatoal coeffcet Evet Comparatve 100 M Log Shot Put Hgh 400 M 110 M Hgh Hurdles Dscus Pole Vault Javel 1500 M 60

8 Applyg Grey Relatoal Aalyss to the Decathlo Evaluato Model seres Determato of the relatoal grade for each cotestat problems that tradtoal method could ot overcome such as te score dspute. The relato grade for each comparatve seres s determed by averagg the Grey relato coeffcet of each dvdual cotest evet. The Grey relato grade ca be epressed by: Γ 1 0? k 1 where 10 γ ( ( k, ( k, 0 Substtutg the coeffcet of Grey relato to above equato, we ca get each cotestat s Grey relato grade, whch are Γ ; Γ ; Γ ; 5. Coclusos Based o our study ths paper, the Grey relatoal aalyss ca be appled aalyzg sport techology, selecto of coach, ad evaluato of overall performace for decathlo. Through quattatve aalyss of Grey relato, t provdes more accurate ad subectve data. The method s formulated tabulated form so that t ca be coveetly used for other sport evaluatos. I addto, there s o eed to make score coverso or cosultg utlty fucto. We beleve ths method ca provde a soud aalytcal approach to the decathlo evaluato model. Γ ; Γ Ths Grey relato grade s the overall performace for the decathlo Obtag the rakg 03 > Sce we get Γ 01 >Γ 04 >Γ 02 >Γ Γ 05, the rakg order for these fve cotestats s A, D, B, C ad E. Although Grey relatoal aalyss does agree well wth the tradtoal method, t possesses a overwhelmg advatage to solve the Refereces [1] Wag, Y.T., The Cotrol Theory, Iformato Theory, System Scece, ad Phlosophy, Cha Rem Uversty Press, [2] Wag, L, The Study of Last Force Eerto o the Left Leg for PRC Javel Athlete, Pekg Sport Uversty Press, Vol. 3, Book 3, pp.73-78,

9 Chg-Lag Chag, Chh-Hug Tsa, Leh Che [3] Chese Tape Track ad Feld Assocato, Scorg Table for Decathlo Competto, Tape, [4] Republc Sport Federato, Rules for Iteratoal Track ad Feld, Tape, Ifluece of Body Characterstcs o PRC Hammer Player, Pekg Sport Uversty Press, Vol. 20, Part 4, pp.90-94, [13] Che, P.R., Applcato of Grey Relatoal Aalyss to the Predcto of FIFA World Cup, pp , [5] Sh, K.C., Grey Iformato Relato, Chu-Hwa Book Compay, Tape, [6] Chag, K.S., The Itroducto of Grey Theory, Gaul Publshg Co., Tape, [7] She, C.B., Grey Theory Applcato to the Ivestgato of Cotrbuto Factors ad Performace Predcto for Fu-e Lee s Decathlo Records, Physcal Educato ad Sport, Vol. 73, pp.32-45, [8] Wu, H.H., The Itroducto of Grey Aalyss, Gaul Publshg Co., Tape, [9] Wu, H.H., The Method ad Applcato of Grey Decsos, Che-Kuo Isttute of Techology, Chag-Hwa, [10] Chow, C.C., The Study of Table Tes Player Selecto Tape ad Malad, College Research Quarterly for Eercse ad Sport, [11] Hsu, K.M., The Method of Swmmg Techque ad Its Applcato The Applcato of Grey Relatoal aalyss, Tape Sport College Press, Vol. 3, pp , [12] Su, Y.P., The Ivestgato of the 62

10 Applyg Grey Relatoal Aalyss to the Decathlo Evaluato Model [14] Cheg, C.Y., The Study of Youg Soccer Athlete Selecto Crtera Ivolvg Body Structure, Physcal Mechasm, ad Body Characterstcs for Chese Korea Ethc Morty Group, Cha Physcal Educato Scece ad Techology, Vol. 32, Part 2, [15] Ha, D., The Grey Relatoal Aalyss for Dstgushed Wome Maratho, Cha Physcal Educato Scece ad Techology, Vol. 32, Part 9, [16] Ye, M.Y., Usg Grey Relatoal Aalyss to Study the Defese ad Offese Techques ad Establshmet of Trag Goal for Ku-Ta Basketball Team, Physcal Educato ad Sport, Vol. 88, pp.38-45, [17] Deg, J.L, The Itroducto to Grey System Theory, The Joural of Grey System, Vol. 1, No.1, pp.1-24,