J. Sa. Appl. Pro., o., -7 SP Journal of Sascs Applcaons & Proaly --- An Inernaonal Journal @ SP aural Scences Pulsn Cor. On Consrucon of Odd-fraconal Facoral Desns Ike Basl Onukou Deparmen of Sascs, Unversy of Uyo, Akwa Iom Sae, era Emal Address: keaslonukou@yaoo.com Receved: Dec., ; Revsed Fe., ; Acceped Fe. 8, Pulsed onlne: Aprl Asrac: Fraconal desns nvolve selecon from a ven se of expermenal reamens as suse of reamens o makeup a specfed desn measure a as suc sascal properes as alance, relave effcency, D-opmaly ec. For decades sascans ave reled on e use Defnn Conracs DC, and Lan Squares LS o consruc fraconal facoral desns. Bu ese meods are sown o ave very lmed rane of applcaons and somemes produce desns a are snular. Ts paper nroduces e meod of Concenrc Balls for consrucn non-snular fraconal desns. Eac all consss of reamens a are of equal dsance from e cener and usn a se of rules for selecn reamens from a all e meod yelds a small se of admssle desns. Te es memer of s admssle se s e desred desn:{bes n e sense of maxmzn e deermnan of e normalzed nformaon marx or maxmzn e relave effcency of e facoral effecs.}umercal examples sow a e meod covers every rane of expermenal desn condons and can produce fraconal desns a are D-opmal. Keywords: Odd-fracon, concenrc alls, relave effcency Inroducon Consrucon of fraconal facoral desns s a opc a s exensvely reaed n mos sandard exs on desn of expermens; see, e.. Cocran and Cox 957, Anderson and Mclean 974. From n-ndependen, non-socasc varales, were e varae, x appears a s -levels, we e s s s n reamens and consder ree knds of reamen spaces : Te unform or symmerc form; s x, x,, xn; x,,, s, s s sn, Te non-unform or asymmerc ype; A x,, xn; s s. for a leas one par of, Te Irreular ype; ; e.. x, x,, x ; x,,, s, R n Oer eomerc forms can also occur.e. x, x,, xn ; x,,,, n s a produc of connuous nervals; owever, e coverae of s repor does no nclude connuous nervals. As saed earler, e prolem of neres ere s o consruc an -pon desn p,.e. an en e numer of parameers n e response s s s n fraconal facoral desn, p funcon f x. Te fracon s consdered an odd-fracon f s no dvsle y any s, oerwse s a reular fracon. For decades, e pracce as een o consruc fraconal facoral desns usn eer Lan Squares LS or Defnn Conrass DC; see, e.. Anderson and Mclean. Two prolems can arse from s approac: a Te DC and LS meods are napplcale, as n. 7
J. Sa. Appl. Pro., o., -7 Te meods can produce snular or near snular desns as sown n ale..., even wen e relave effcency of e desn s consdered ood. Ts paper nroduces e Concenrc Balls meod of consrucon a as a wde rane of applcaons and can produce an admssle se of equvalen desns, leavn e scens o make a coce. Te meod proceeds as follows: Arrane e suppor pons no H roups or alls, so a suppor pons a are of e same dsance from e cener are n one all. Tus e all, x, x,, xn conans n suppor pons, =,,..., H, xk s an n-componen vecor, k =,,...,n, were, d x xk s e dsance from e cener, and d d d. Paron k H no su-roups accordn o e numer of neave sns and zeros appearn. a e suppor pon xk; see secon ree of s paper. Apply e selecon rules; see secon wo o uld up e requred desn. Tese rules yeld a small se of admssle desns wose deermnans and relave effcences can e easly compared. Applcaon of e dea of roupn of reamens owards consrucon of D-opmal exac desns ave een employed y Onukou and Iwundu 7; and for D-opmals desns for -level facoral models and auoreressve error y Ye and Huan5. Consrucon and analyss of fraconal facoral desns on a wder plaform as een consdered y Guns and Mason 9.A rane of ecnques for consrucon of asymmerc fraconal facorals as well as condons for nonexsence of e desns ave een ven y Dey and Raul 999. A way as offered y Oludua and Madukafe 9 for serean fraconal facoral desns on e ass of er D-opmal and loss of nformaon values. As lon as neres n a facoral expermen s resrced o a lmed numer of parameers facoral effecs researc n fraconal desns wll connue o flours. In wa follows, e asc alera for e ecnque s dscussed n secon wo, wle numercal llusraons are ven n secon ree.. Alerac Bass Te expermenal space wll e represened y e rple, F, ; x, x,, x ; x,, s,,, n s a connuous, compac, merc space of rals, x x n Fx f x; x s a se of connuous, dfferenale funcons. x x; x s a se of connuous, non-neave error funcons. Eac se of e rple s consdered fne and oeer ey form a ass for n-dep sudy of e sujec of desn of expermens; see, e.. Pazman 987, Aknson and Donev 99, Onukou 997. Le f x e a frs-order neracve funcon defned y. f x e = x j s an p n n p exended desn marx; e p parameers comprsn e lnear and neracve erms,, s an lock ncdence marx; da k, k,, k ; k j en e sze of e j lock, s a p-parameer vecor of reamen effecs e s an -componen vecor of random error Te deermnan of e nformaon marx n. equals, k j de de I R; R j s e marx of loss of nformaon..
Pranes Kumar: Sascal Dependence: Copula Funcons A eomerc meann of loss of nformaon as cos. of e anle of nclnaon of a reamen effec on e locks as een reaed y Onukou. ow, for an -pon desn n one lock =, R rr ; r r, r,, r ; r,,,, p.; r s e loss of nformaon on e reamen effec. Hence, e eomerc mean, p. r r p p ves a measure of e overall effcency of e desn relave o a complee lock desn. oce a for r, r. s maxmzed wen e desn s alanced; e. wen r r r. Bu r. does no p ake no accoun e deermnan, de, and erefore can ake non-zero values for snular desns. Bu y ncludn e deermnan, we e e creron for comparn desns:.4 d m r ; m de / To maxmze.4 e follown selecon rules are o e appled wen makn-up e desn measure : max x j mn xj mn xj xj, j,,, p, j j We recall a x s e exended reamen marx. j Realsn a e numer of suppor pons n n s n p n n ; en, for e response funcon.,e opmal -pon desn s consruced from no e case for a complee quadrac funcon, n.5 f x a a x a x x a x e n only. Bu, s s Te sarn pon of e procedure s dependen on e relave values of and p. Wen = p, e procedure oans e desn as follows: a A e nal sep ake p n suppor pons from and e res of p + n + from, y applcaon e rules n.4 aove. Ts yelds a se of r admssle desns S A.,, r m max m ; S A Compue c A e k sep ake p + k n suppor pons from k k m k max m ; SA d Sop, f k k k m m m and e res from, and compue For >> p, e aove sequence can en y akn p or p+ pons from. I seems a y properly relan e numer of pons o e aken from o e rao p/, sould e possle o develop a nonerave procedure for consrucn opmal fraconal facoral desns for quadrac response funcons.. umercal Examples Gven are rvarae frs-order neracve response surface,. f x, x, x a ax ax ax axx axx axx e and a cuc surface, x, x, x ; x,,,,,.
4 Pranes Kumar: Sascal Dependence: Copula Funcons Usn eac of e ree meods, we consder e consrucon of wo fraconal facoral desns: 9 a, for a reular fracon, and 7, for an odd-fracon. 7 Te reamen ale s ven y. Tale of Treamens For x,, x,, x,, cuc surface Usn for nsance xx x as defnn conras; see, e.. Anderson and Mclean, for deals, e DC meod produces e desn 9 DC, wereas e Lan square meod ves 9 LS Te roups and suroups requred for e meod are: - - - - - + - + - + - - + + - + - + - + + + + + - - - - - - - + + - - + + - - + + - + + + + + + 4 Applcaon of e rules under.4 ves wo equvalen desns, Te deermnans of ese desns are repored n Tale. 4 9, 9.. Deermnans, De. And Relave Effcences, RE. For Tree Meods of Consrucn Fraconal Desns for Frs-Order Ineracve Funcons - - - + + +
Pranes Kumar: Sascal Dependence: Copula Funcons 5 Seral umer Fraconal Desn METHOD OF COSTRUCTIO DC LS De. RE. De. RE. De. RE. 9.546.98.546.98 4.4 7 x E-5 x E-5 x E- A A A A 69.95 7 x E- 4 A A.9. 7.757 5 x E- x E- 4 7 A A A A 9.5486 5 x E-.99985.9988.99875.9999 A means o Applcale. Smlarly, for e odd-fracon e meod produces wo equvalen desns: 7, Bo e DC and LS meods are napplcale n s case of odd-fracon. If e space of rals s non-unform asymmerc r-varae surface, x, x, x; x, x,, x,,,,, We consder e consrucon of wo desns: c 4 5 for reular fracon, and d 7 5 for odd-fracon. Jus as n ale., a correspondn reamen ale s se-up and snce 4 s dvsle y, e Lan- Square meod can e appled. Te frs sep s o se-up a Paral Lan Square PLS, L usn e leers a and. ex, supermpose L on e frs ac of 5 reamens were x, and en on e oer ac of 5 reamen a x, a L a a a a a a a Fnally, e reamens a concde w e leer are rouped oeer o form e desn: 4 PLS, de. =.9 x E - On e conrary e meod produces an opmal desns;
Pranes Kumar: Sascal Dependence: Copula Funcons 6 4 w de. = 7.759 x E - Te consrucon of 5 7 also ves wo equvalen desns: 7 and 7 w de. = 9.5486 x E - ; oce a all e desns for e frs-order neracve funcons are consruced from e frs all for all values of. Ts owever s no e case for quadrac response funcon defned n.5. Apparenly, wo concenrc alls are requred o consruc a desn for quadrac funcons. For example, wo equvalen desns are oaned y e meod for a ; 7 namely, and w 4588.. m x E -4. Smlarly, for e asymmerc surface, e meod ves wo equvalen opmal desns for 5; namely, ; 4 4 and 4 4,, s ven aove. Summary and Concluson Te paper as sown a DC meod can e used o consruc fraconal facoral desns only wen e facor levels are unform and even a s, e relave effcency of s meod s comparavely nferor. On e oer and, e Lan square meod can e appled o for unform and non-unform levels provded only a e fracon s reular. Of e ree meods s only e
Pranes Kumar: Sascal Dependence: Copula Funcons 7 meod a can consruc odd-fraconal facoral desns; desns w e es level of effcency; desns a are requred o e alanced n one replcaon as well as desns a are D-opmal. Acknowledemen We are raeful o e Revewer and e Edor-n-Cef for e references numers and 4. References []. Anderson, V.L. and Mclean, R.A. 974: Desn of Expermens: A Realsc Approac, Marcel Dekker. []. Cocran, W.G. and Cox, G.M. 957: Expermenal Desns, nd Edon, J.Wley []. Dey, Aloke and Mukerjee, Raul 999: Fraconal Facoral Plans; Jon Wley and Sons, ew York. [4]. Guns, Rcard, F. and Mason, Roer, L 9: Fraconal Facoral Desn; Wley Inerdsplnary Revews: Compuaonal Sascs, Vol., Issue, Pae 4 44 [5]. Oludua, A.V. and Madukafe, M.S. 9: D-Opmaly and D L -Opmaly crera for Incomplee Block Desns; Gloal Journal of Maemacal Scences, Vol. 8, o. Pae 7 5. [6]. Onukou, I.B. 997: Foundaons of Opmal Exploraon of Response Surfaces, Epraa Press, sukka, era. [7]. Onukou, I.B. and Iwundu, M.P. 7: A Comnaoral Procedure for Consrucn D-Opmal Exac Desns; Sasca Rvsa Vol.67 Pae 45 4. [8]. Pazman, A. 987: Foundaon of Opmum Expermenal Desn, Redel Pulsn Company. [9]. Ye, Hon-Gwa and Huan, Mon-a Lo 5: On Exac D-Opmal Desns w wo-level facors and n auocorrelaed