ORTHOGONAL ARRAYS. PRIYA KOHLI M.Sc. (Agricultural Statistics), Roll No I.A.S.R.I, Library Avenue, New Delhi

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1 ORTHOGOAL ARRAYS PRIYA KOHLI MSc (Agriculturl Sttistics), Roll o IASRI, Librry Aeue, ew Delhi- Chirperso: Dr R Srist Abstrct: A orthogol rry (OA) of stregth t is rry, where d deote the umber of rus d fctors respectiely, with the property tht i y of the t colums ll of the t-tuple bsed o m i (i,,, ) symbols ppers eqully ofte The OA with m m m m (sy) is clled symmetric, otherwise, the rry is sid to be symmetric Seerl methods of costructio of symmetric s well s symmetric OAs re ilble i the literture Some importt methods will be discussed here Oe of the pricipl pplictios of the OAs is i the selectio of leel combitios for frctiol fctoril experimets A OA of stregth t is equilet to orthogol resolutio (t+) pl d it c be used to derie orthogol frctiol pls for mixed fctorils Key words: Orthogol Arrys, Frctiol Fctoril Pls, Hdmrd Mtrix, Orthogol Resolutio Pls Itroductio Orthogol rrys (OAs) were itroduced by Ro (96) followed by Bose d Bush (95) Formlly, OA c be defied s follows: Let M be the set of m symbols deoted by,,, m- Defiitio : A rry A with etries from M is sid to be OA with rus, fctors ech with m leels, stregth t (for t ) d idex λ if eery t sub-rry of A cotis ech t-tuple bsed o M exctly λ times s row This is deoted by OA (,, m, t), OA (, m, t), L (m ), L or O λ ( t,, m ) The idex λ is geerlly ot idicted becuse λ The itegers, k, m, t d λ my be t m referred to s the prmeters of the OA, where deotes the umber of rus, deotes umber of fctors, m deotes the umber of symbols, t deotes the stregth d λ is the idex of the OA I prticulr whe λ, the the OA is sid to he idex uity Exmple : Cosider rry with twele rus d elee fctors s show below:

2 Orthogol Arrys From the boe rry pick up y two (sy, first two) colums It c be see tht ech of the possible rows,,, pper exctly three times This property holds if oe picks up y two colums of the rry resultig ito orthogol rry I y two colums the umber of possible rows ppers costt umber of times thus, the rry is of stregth two The OA described boe c be deoted s OA (,,, ) The first describes the leels of ech fctor d the ext describes the stregth of OA Whe the leel of y oe of the fctors is differet, it is termed s mixed or symmetricl OA The geerliztio of the boe defiitio is gie s follows: Defiitio : A symmetricl OA (,m,m,,m,t) is rry, where is the totl umber of fctors, i which the first m colums he symbols from {,,,m }, the ext m colums he symbols from{,,,m }, d so o, with the property tht i y t, sub-rry eery possible t-tuple occurs equl umber of times s row The OA with m m m m (sy) is sid to be symmetric OA Exmple : (Hedyt, et l, 999) A rry with fie fctors i twele rus with the first four fctors t leels ech d the lst fctor with three leels is mixed OA(, 5,, ) d is s show below:

3 Orthogol Arrys OA (,5,, ) Here the umber of possible t-tuples tht occur i the t sub-rry depeds o which t colums re chose For the rry i which t, the possible pirs i the lst two fctors re ech occurs twice d the possible pirs i the first two fctors re ech occurs three times Properties of OA (,, m, t) ) The prmeters of OA stisfy the equlity λ t m b) A OA of stregth t is lso OA of stregth t, t t The idex of OA of t t stregth t is λm c) If A i, i,,,r is OA ( i,, m, ti ) the the rry A obtied from juxtpositio of these r rrys, A A A A r is OA (,, m, t) where r d the stregth is t for some t mi [ t, t,, t r ] Whe m r d ech A i is OA (,, m, t), fter ppedig to ech row of A, to ech row of A d so o, we obti OA (m, +, m, t) d) A permuttio of rus or fctors i OA results i OA with the sme prmeters e) A permuttio of the leels of y fctor i OA results i OA with the sme prmeters f) Ay sub-rry of OA (,, m, t) is lso OA (,, m, t ), where t mi{, t} g) Pick up the rus i OA (,, m, t) tht begi with prticulr symbol sy d omit the first colum The, we get OA ( /s, -, s, t-) A h) Suppose A is OA (,, m, t) d A itself is OA (,, m, t) The A A is OA,, m, t ) with t mi t, ( { } t

4 Orthogol Arrys Defiitio : Two OAs re sid to be isomorphic to ech other if oe c be obtied from the other by sequece of permuttios of the colums, the rows, d the leels of ech fctor Exmple : (Hedyt, et l, 999) Followig two OAs (8,,, ) re isomorphic to ech other: d Ro s Iequlity Theorem : Cosider mixed OA (, m, m,, m, t), where The m m m prmeters of the rry stisfy: u m Im() i i i (m ) i (m ) i (m ) i, if t u () u m I m( ) i i i (m ) i (m ) i (m ) i + i i I u() i (m ) i (m ) i (m ) i +, if t u + () for u The boe theorem sets lower boud o the umber of rus i pl represeted by OA For the symmetric OA the lower boud c be obtied by substitutig m m m m i the equtio () d () Defiitio : A OA ttiig Ro s boud ie, equlity i Ro s Iequlity is kow s Complete or Tight Orthogol Arry Exmple : (Hedyt et l, 999) A OA (8,,, ) for which the equlity holds i () is show below:

5 Orthogol Arrys OA (8,,, ) i + Here, t u+ u ( ) + ( ) i Costructio of Symmetric Orthogol Arrys Orthogol Arrys d Hdmrd Mtrices Defiitio : A squre mtrix H of order d etries ± is clled Hdmrd mtrix if H H H H I, where I is idetity mtrix of order A Hdmrd mtrix with its first colum hig oly +s is sid to be i its semiorml form A Hdmrd mtrix of order ( ) is equilet to symmetric two-symbol tight OA of stregth with rows d - colums Suppose tht H is Hdmrd mtrix of order d let H be writte i its semi orml form The, by deletig the first colum from H, ( ) mtrix with etries + d - is obtied It is esy to see tht this mtrix represets OA (, -,, ) Coersely, if there exists OA (, -,, ), the replcig the two symbols d i the rry by + d -, respectiely, d ugmetig the resultt ( ) mtrix by colum of ll + s, Hdmrd mtrix of order c be obtied Theorem : The existece of Hdmrd mtrix of order ( ) is equilet to the existece of OA (, -,, ) Exmple : (Dey d Mukerjee, 999) I the boe theorem for 8, by deletig the first colum of d replcig - by, the OA (8, 7,, ) obtied is s show below: H 8 OA (8,7,, ) 5

6 Orthogol Arrys Foldoer Techique Two-symbol OAs of odd stregth c be costructed from those of ee stregth by followig procedure clled foldoer techique This techique is origilly due to Box d Wilso (95) The omeclture foldoer is deried from the fct tht i the costructio, the first rus re folded oer to get the ext rus The method c be summrized i the theorem stted below: Theorem : For positie iteger m, the existece of o OA (,,, m) is equilet to tht of OA (, +,, m+) Proof: First suppose tht OA (,,, m) exists Deote this rry by A, d let its symbols be d Further let A is the rry obtied by iterchgig the symbols d i A Cosider the rry A A, A where d deote ectors of ll s d ll s respectiely It is show tht OA (, +,, m+) A is Cosider y (m+) sub-rry, sy A, of A Let A be the correspodig (m+) sub-rry of A Sice A is OA (,,, m), there re rows of A which equl either or Hece, if ppers x times s row of A, the ppers times s row of A I geerl, the similr rgumet m x shows tht eery (m+)-tuple ppers s row of A with frequecy x if it cotis odd umber of s d with frequecy, if it cotis ee umber of s m x Therefore ech possible (m+)-tuple ppers times s row of A A This m shows tht the rry formed by the first colums of is OA (,, m+) Sice both A d A re OA (,,, m), it follows tht is OA (, +,, m+) Coersely, if OA (, +,, m+), sy B, exists the deotig its symbols by d d permutig its rows (if ecessry), B c be expressed s B B B The, is OA (,,, m) B Corollry : For, the existece of Hdmrd mtrix of order implies the existece of OA (,,, ) Exmple : (Dey d Mukerjee, 999) A OA (8,,, ) c be costructed s [ H ] H A A m 6

7 Orthogol Arrys Costructio of Mixed Orthogol Arrys Collpsig d Replcemet Procedures The methods of collpsig d replcemet re due to Addelm (96) Suppose OA of stregth two hs colum iolig m symbols, d let m ( ) be positie iteger such tht m m, mely, m (mod m ) The the m-symbol colum c be collpsed ito m -symbol colum by first groupig the m symbols ito m sets of m m symbols ech d the by replcig the symbols belogig to the sme set by commo symbol The resultig rry is OA of stregth Exmple : (Dey d Mukerjee, 999) Cosider the OA (6, 5,, ), s show below (i trsposed form): OA (6, 5,, ) Sice m diides m, we group the four symbols of the first colum, ito two sets, sy,{,} d{, } Replcig both the symbols i the first set by d those i the secod set by, we get mixed OA (6, 5,, ) s show below (i trsposed form): OA (6, 5,, ) ext is the procedure of replcemet Suppose OA A of stregth hs colum iolig m symbols, d let there exists other OA B lso of stregth with m rows The i A the symbols of the m-symbol colum c be replced by the rows of B, usig oe-oe correspodece, without disturbig the orthogolity It c be see tht if A d B both re tight, the this method of replcemet yields tight OA of stregth two The method of replcemet reduces to tht of collpsig i the specil cse where B cosists of sigle colum Exmple : (Dey d Mukerjee, 999) Let A represet the OA (6, 5,, ) (s show i Exmple ) d B is OA (,,, ) The symbols,,, i the first colum of A re replced the rows,,, of B, tight symmetric OA (6, 7,, ) is obtied This is s show below (i trsposed form): 7

8 Orthogol Arrys ), OA(6,5, A ),, (, OA B The resultt OA is s gie below (i trsposed form): ), OA (6,7, Costructio of Mixed OA usig Hdmrd Mtrix Seerl mixed OA of stregth two c be costructed by the use of the properties of Hdmrd mtrices The followig prelimiries will be helpful: Defiitio : A positie iteger will be clled Hdmrd umber if Hdmrd mtrix,, of order exists (exceptio ) H This method mkes use of the Hdmrd property which c be explied s: Let the iitil colum of H be (the ector of ll oes) A set of three distict colums of ( ) is sid to he Hdmrd property if the Hdmrd product of y two colums i the set equls the third (the Hdmrd product of two ectors d H ( ) ( ) b b b b is defied s ( ) b b b b 8

9 Orthogol Arrys For istce, the sub mtrix of H gie by three such colums hs rows ( ), ( ), ( ), ( ) ech with frequecy Lemm : Let H be Hdmrd mtrix of order ( 8), d suppose tht there is set of three colums of H hig the Hdmrd property Deote these colums by,, The i y sub mtrix of H mde up of the colums i,i,, d c, where c is y colum of H other th d the three with the Hdmrd property, ech of the 8 ectors ( ± ), ( ± ), ( ± ), ( ± ) occurs eqully ofte s row Suppose tht there is Hdmrd mtrix H cotiig set of three colums with the Hdmrd property If the four rows ( ), ( ), ( ), ( ) uder the colums with Hdmrd property re replced by the symbols,,, respectiely, d the iitil colum of H is lso deleted, the the resultig (-) mtrix t represets tight OA (, -,, ) More geerlly, if H icludes t ( ) disjoit sets of colums such tht ech set hs the Hdmrd property, the the techique of deletig d replcig ech set of three colums by sigle four-symbol colum yields tight OA (, -t-,, ) t t Theorem : If d T re Hdmrd umbers stisfyig, T, the there T T T+ exists OA (T, T-T+,, ), which is tight Proof: Let () H [ b M Mb ], H [ M M M ] T T T M, where b, b,, b re the colums of H other th the iitil colum d similrly,,, T re defied Cosiderig H T HT H, it c be see tht H T cotis T- disjoit sets of colums, gie by{ b,, b }, i T, ech set hig the Hdmrd property Hece, the required OA c be costructed Exmple : (Dey d Mukerjee, 999) Let, T (i boe theorem) H H H ; H 9

10 Orthogol Arrys The, followig (), [ ] [ ], b The s i the proof of theorem, if oe forms 8 H H H, replces the triplet of colums by sigle four-symbol colum, d filly deletes the iitil colum of ll oes d replces - by, the rry obtied is s show below: {,, b b } ), OA (8,5, Theorem : If d T re Hdmrd umbers, the there exists tight OA (T, (T-)-s+, x, ), where s mi (-, T-) s ) (T Theorem : Let ( ) d T be Hdmrd umbers The the existece of symmetric OA (T, k, u, ) such tht ) (T k d k implies the existece of OA,),(u) k T T (T, k T T + + Theorem : If ( ) is Hdmrd umber, the the rry OA exists, where T /,),T (, + Exmple : (Dey d Mukerjee, 999) Let i Theorem The, strtig from H

11 Orthogol Arrys we get OA (,, 6, ) s show below(i trsposed form) I the secod colum of the rry, the symbols re coded usig the trsformtio,,, Similrly, i the two symbols colums the trsformtio is doe OA (,, 6 56,) Applictios of Orthogol Arrys Oe of the pricipl pplictios of the OAs is i the selectio of leel combitios for frctiol fctoril experimets Usig OAs orthogol frctiol fctoril pls for certi mixed fctoril c be obtied Orthogol Frctiol Pls for Asymmetricl Fctorils Derible From OAs A OA (,, m, t) is equilet to orthogol resolutio (t+) pl for m fctoril i rus Similrly mixed (,r,m mr, t) of stregth t is equilet to orthogol resolutio (t+) pl for m m m r fctoril i rus The coerse, howeer, is ot lwys true A geerl procedure for costructig orthogol frctiol fctoril pls c be described s follows: Let A be OA (,, m, t), t, d suppose it is possible to prtitio A s A [ A A M M u M A ] () where ech A i (i,,,u ) is mtrix of order r, r, such tht ech A is u i OA ( r,, m, t ) Let d be multiple of u, ie d ku, where k ( ) is iteger d let

12 Orthogol Arrys αi (i,i,,i) for i,,, u-, where i α i, the symbol i ppers r times Cosider the rry B, gie by Sice ech A A A B () αα αu αuαu+ αu α(k)u αku A (i,,,u ) is OA ( r,, m, t ) d A is OA (,, m, t), it i follows tht B is mixed OA ( k, +, d m, t), which i tur implies tht the colums of B costitute the rus of orthogol resolutio (t+) pl for dk experimet i rus This gies the followig result: Theorem : Let there exist OA A (,, m, t) d let it be possible to prtitio A s [ A M A M MA ] A u such tht ech A i (i,,,u ) is OA ( r,, m, t ), where r The it is u possible to costruct mixed OA ( k, m +, d m, t), where d ku, k( ) iteger or equiletly costruct orthogol resolutio (t+) pl for experimet i rus Exmple : Cosider the OA (6,,, ) dm A M M M M M M M M M This rry c be prtitioed ito sub-rrys, ech such sub-rry beig OA (,,, ) The rry B costructed through the proposed method is s show below: M M M B M M M M M M M M M B is mixed OA (6,, orthogol resolutio IV pl for,) d cosequetly, the colums of B costitute the rus of experimet i 6 rus Resolutio III Pl The specil cse t of the theorem hs bee cosidered by Gupt et l (98) Let be multiple of four It is kow tht ll Hdmrd mtrices of order mod, exist Let H be Hdmrd mtrix of order d let its first colum coti ll uities By omittig this colum from H, we obti mtrix D of

13 Orthogol Arrys order ( ) with elemets - d It is kow tht the rows of D costitute the rus of orthogol resolutio III pl for D A J D J experimet i rus A rry A is OA (,,,) where J is -compoet colum ector of ll uities Let d j be the j th colum of D (j,,, ) d A is prtitioed s A [ A M A M MA ] di di where Ai, i,,, Here ech Ai is OA (, +,, ) Further sice A is OA of stregth two s well, it follows tht the rry A B α A α A α where αi (i,i) for i,,, -, is mixed OA (, +,, ) d thus, the colums of rry B costitute the rus of orthogol resolutio III pl for experimet i rus, where the leels of the -leel fctor re coded s,,, - while the -leel fctors re t leels - d Refereces Addelm, S (96) Orthogol mi-effect pls for symmetricl fctoril experimets Techometrics,, -6 Bose, R C d Bush, K A (95) Orthogol rrys of stregth two d three, A Mth Sttist,, 58-5 Box, G E P d Wilso, K B (95) O the experimetl ttimet of optimum coditios J Roy Sttist Soc, B, -5 Dey, A d Mukerjee, R (999) Frctiol fctoril pls ew York: Wiley Gupt, V K, igm, A K d Dey, A (98) Orthogol mi-effect pls for symmetricl fctorils Techometrics,, 5-7 Hedyt, A S, Sloe, J A d Stufke, J (999) Orthogol rrys: theory d pplictios ew York: Spriger Series Ro, CR (96) Hypercube of stregth d ledig to cofouded desigs i fctoril experimets Bull Clcutt Mth Soc, 8, Some Additiol Refereces Dey, A (985) Orthogol frctiol fctoril desigs ew York: Wiley Dey, A (99) Some orthogol rrys with rible symbols J Combi If System Sc 8, 9-5 Dey, A d Agrwl, V (985) Orthogol frctiol pls derible from orthogol rrys Skhy B7, Dey, A () Asymmetric Orthogol Arrys Desig Workshop Lecture otes, ISI, Kolkt, 5-9, 97-5

ECE 608: Computational Models and Methods, Fall 2005 Test #1 Monday, October 3, Prob. Max. Score I 15 II 10 III 10 IV 15 V 30 VI 20 Total 100

ECE 608: Computational Models and Methods, Fall 2005 Test #1 Monday, October 3, Prob. Max. Score I 15 II 10 III 10 IV 15 V 30 VI 20 Total 100 Nme: ECE 608: Computtiol Models d Methods, Fll 005 Test # Mody, Octoer 3, 005! Your em should hve 0 (te pges.! Pge 9 is itetiolly left l.! Pge 0 cotis list of potetilly useful idetities tht you my use.!