I-Optimal designs for third degree kronecker model mixture experiments

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1 Inernaional Journal of Saisics and Applied Maheaics 207 2(2): 5-40 ISSN: Mahs 207 2(2): Sas & Mahs Received: Acceped: Cheruiyo Kipkoech Deparen of Maheaics and Physical Sciences, Maasai Mara Universiy, Narok, Kenya Koske Joseph Deparen of Saisics and Copuer Science, Moi Universiy, Eldore, Kenya Muiso John Deparen of Saisics and Copuer Science, Moi Universiy, Eldore, Kenya I-Opial designs for hird degree kronecker odel ixure experiens Cheruiyo Kipkoech, Koske Joseph and Muiso John Absrac Mixure experiens are special ype of response surface designs where he facors under sudy are proporions of he ingrediens of a ixure. In response surface designs, he ain ineres of he experiener ay be eiher on precise odel esiaion or precise predicions of responses. The ain ai of I-opialiy is o iniize he average predicion variance hroughou he region of ineres. Alhough ixures experiens are usually inended o predic he response for all possible forulaions of he ixure and o idenify opial proporions for each of he ingrediens, lile research has been done on I-opial hird degree Kronecker designs. I-opial designs were sudied in order o predic he response(s) and he opial forulaions for all possible ingrediens in he siplex cenroid. Weighed Siplex Cenroid Designs (WSCD) and Uniforly Weighed Siplex Cenroid Designs (UWSCD) ixure experiens were obained in order o idenify opial proporions for each of he ingrediens forulaion. Maxial paraeers of ineres for hird degree Kronecker odel were considered. The general equivalence heore for I-opialiy was used o es opialiy of differen ixure forulaions. UWSCD was found o perfor beer han WSCD in ers of average predicion variance wih os forulaions saisfying general equivalen heore for I-opialiy. Keywords: Mixure Experiens, Kronecker Model, WSCD, UWSCD, I-opialiy, General Equivalen Theore. Correspondence Cheruiyo Kipkoech Deparen of Maheaics and Physical Sciences, Maasai Mara Universiy, Narok, Kenya. Inroducion Mixure experiens are associaed wih he invesigaion of he facors, assued o influence he response only hrough proporions in which hey are blended ogeher. I-opial designs iniize he average variance of predicion and herefore ore appropriae for ixure experiens for precise predicions of responses. Average predicion variance provides a easure of he precision of he esiaed response a any poin in he design space. In available lieraure, he I-opial designs crierion is ofen called he V-opialiy crierion, Akinson e al (2007) [], bu he naes IV- or Q-opialiy have been used as well, Borskowski (200) [2]. Laake (975) [6] analyically derived he I-opial weighs for he design poins in case q, assuing ha he design poins are he poins of he ( q,2). In experiens wih ixures, Liu and Neudecker (995) [7] applied Weighed Siplex Cenroid (WSC) o obain V-opial allocaion of observaions which was shown o be an opial design over he enire siplex on Scheffé s polynoial odel using he equivalence heore. Goos e al (206) sudied I-opial designs of ixure experiens for second order, special h cubic and q degree odels. They presened I-opial coninuous designs and conras he wih published resuls. Goos and Syafiri (20 [4] sudied he proble of finding coninuous V-opial ixure designs for he qh degree odel. They provided a criical look a he resuls published in Liu and Neudecker (995) [7] and found ou ha heir designs were no V- opial. The ixure ingrediens, 2,, are such ha i 0 and furher resriced by i. Thus he experienal doain is he probabiliy siplex ~5~

2 Inernaional Journal of Saisics and Applied Maheaics T (,...,,) [0,] : i () i Under experienal condiion T, he response Y is aken o be a real-valued rando variable. In a polynoial regression odel he expeced value E ( Y ) is a polynoial funcion of. The work done by Draper and Pukelshei (998) [] is being exended o polynoial regression odel for hird degree ixure odel. The S-polynoial is given as, (2) E( Y) f( ) i i ij i j ijk i j k i i, j ijk i j and he hoogeneous hird-degree K-polynoial is E( Y ) f ( ) ( ) () ijk i j k i j k in which he Kronecker powers ( ), ( +) vecors, consiss of pure cubic and hree-way ineracions of coponens of in lexicographic order of he subscrips and wih eviden ha hird-degree resricions are for all i, j and k ijk ikl jik jki kij kji All observaions aken in an experien are assued o be of equal unknown variance and uncorrelaed. The oen arix j j j for he hird degree Kronecker odel has all enries hoogeneous in degree six j M ( ) w ( ) f ( ) f () f () d and reflecs he saisical properies of a design. The oen arix can be pariion ino sub oens according o he nuber of ingrediens in a siplex cenroid design as follows M ( ) M( ) 2M( n2) M( n) ( For Uniforly Weighed Siplex Cenroid Designs (UWSCD), he weighs are assued o be disribued uniforly in he sub oens arices. Hence 2... = and heir oen arix is given as, M ( ) M( ) M( n ) M( n ) (5) 2 2. Inforaion Marix Consider he Euclidean uni vecors in denoed by e, e2,..., e and he se for eii j ei ei ej, e ijk ei ej ek for i<j<k, i,j,k={,2,...,} (6) Le K be a k s coefficien arix such ha K ( K K2 K) (7) where ' ' Kee iii i, K2( ) [ ( eiij eiji ejii) ei ] and K ( )( 2) ( eijk) i i, j i, jk, ij ijk The Kronecker odel of he full paraeer vecor considered in his sudy can be wrien as is no esiable. When fiing his odel, he paraeer subsyse ( ) iii i K ' ( ) ( iij iji jii ) for all i, j ( )( 2) ( ijk ) i, j, k i jk (8) where K The paraeer subsyse he paraeer subsyse is given by K of ineres is a axial paraeer syse in he full paraeer odel. The inforaion arix for ~6~

3 Inernaional Journal of Saisics and Applied Maheaics C ( M ( )) LM ( ) L` NND( s) (9) k where L is he lef inverse of coefficien arix K and is defined by L` ( K` K) K` (0) K` are linear ransforaion of oen arices. Thus he inforaion arices for I-opial design iniizes average or inegraed predicion variance over he experienal region of ineres. Average Variance = r[( ' ) M ] () d The arix M is he oen arix because is eleens are proporional o oens of unifor disribuion on he design region. In he calculaion of average predicion variance (APV), we exploi he following forula Average Variance f '( ) M f ( ) d r[ M f ( ) f ( ) ' d] This expression can be siplified as r[ M f ( ) f '( ) d] d (2) Leing R f() f '() d, he AV is expressed as A r M R d V [ ] For paraeer sub syse of ineres, inverse of inforaion arix (9) as, Ck ( LM( ) L` NND( s) ) (5) Assuing ha he experienal region is he full ( q ) diensional siplex q he forula, ( pi ) ( pi!) p p2 p p q i i R, 2,,..., q d, d2, d,..., dq q q Sq ( q p ) ( p q)! q i i i When he experienal region is he full ( q ) diensional siplex q volue d d s q ( q) ( q)! q i () ( S, he eleens of R can be obained using S, hen is is volue is given by (7) Therefore average predicion variance was obained by Average Variance ( q )! r[ C R] (8) k. Equivalence Theore The general equivalence heore provides a ehodology o check he opialiy of a given coninuous design, for any convex and differeniable design opialiy crierion. Akinson (2007) [] explain ha a coninuous design wih inforaion arix C is I- opial if and only if ' f () C LC f() rc ( L) (9) for each poin in he experienal region according o he general equivalence heore. The equivalence heore designs is no consrucive, bu i can be used o check opialiy of a given designs. 4. Four Facors Mixure Experiens Using siplex resricions, we obained he opial weighs for WSCD,,, and 2 4 respecively. Fro equaion (, we have he oen arix for four facors ixure experiens as, M ( ) 5 M( ) 5 M( n2) 5 M( n) 5 M( n (20) where, M ( ) [ e ( e )' e ( e )' e ( e )' e ( e )'] (2) (6) as 4/5, 6/5, 4/5 and /5 ~7~

4 Inernaional Journal of Saisics and Applied Maheaics M ( ) [( d d d )( d d d ) ' ( d d d )( d d d )' ( d d d )( d d d )' ( d d d )( d d d )' ( d d d )( d d d )' ( d d d )( d d d )'] M ( ) [( f f f )( f f f ) ' ( f f f )( f f f ) ' ( f f f )( f f f )' ( f f f )( f f f )'] M ( ) J (2 also, d ( e e ), d ( e e ), d ( e e ), d ( e e ), d ( e e ), d ( e e ), d ( e e ) f ( e ), f ( e ), f ( e ), f ( e ), e, e 0 0 2, e 0 0 and e The Uniforly Weighed Siplex Cenriod Designs (UWSCD) for four ingrediens were assued o assign unifor weighs o he four eleenary cenroid designs, n, n2, n and n 4, such ha all weighs are equal. Tha is, = 0.25 (25) 2 4 The oen arix for uniforly weighed siplex cenriod designs in (5) for four ingrediens is given by M ( ) 0.25 M( ) 0.25 M( n ) 0.25 M( n ) 0.25 M( n ) (26) 2 where, M ( ), M ( n 2), M ( n ) and M ( n are given in (2), (), (2) and (2 respecively. The coefficien arix K for he four ingrediens paraeer subsyses of ineres in (7) and (8) is given as K K, K, K (27) where 4 2 K e e ' e e ' e e ' e e ' e e ' iii i i K2 9 ( eiij eiji ejii ) ei ' ij i j and 4 K 24 eijk ijk i jk The lef inverse L in (0) for four ingrediens is given as, L' K,6 K, 24 K (28) 2 () (2) 5. Applicaion of I-Opial in Four Ingrediens Using () and (28), we obained he inforaion arix for Weighed Siplex Cenroid Designs (WSCD) as c c2 c C2 c2 c c2 c' c' 2 c (29) where, ~8~

5 Inernaional Journal of Saisics and Applied Maheaics c x I y J x , y c x I y J x y c x I y J x y , , c x x c x I x c x x Using (7) and (28), we obained he inforaion arix for unifor weighed siplex cenroid designs as, c c2 c C2u c2 c c2 c' c' 2 c where c, c2, c, c, c2 and c are given as 4 4 c xi4 yj4 x , y c2 c' 2 x2i4 yj4 x , y c xi4 yj4 x , y c c' x 4 x c2 c' 2 x2 4 x c x x In four ingrediens designs, we obained he inverse of he inforaion arix (29) for paraeer sub syse of ineres as, ( I 0.002( J 4.6( I ( J.5628( C2 4.6( I ( J ( I ( J ( ().5628(' (' We now obained he arix L hrough inegraion of he paraeer subsyse of ineres of he Kronecker odel as given in (6). Thus, L becoes, 0.0( I ( J ( I ( J ( L ( I ( J 0.000( I ( J ( (2) (' (' Now using () and (2), we obained C2 L as ( I ) ( J ) ( I ) ( J ) ( ) C L ( I ) ( J ) ( I ) ( J ) ( ) () (' ) (' ) Therefore, he average predicion variance is given by he race of (). Hence, APV r[ C L] ( 2 Siilarly, we obained he inverse of he inforaion arix (0) for he paraeer subsyse of ineres as, ( I ( J ( I ( J.9827( C2u ( I ( J ( I ( J 2.709( (5).9827(' 2.709(' Using (2) and (5), we obained he arix C2u L, for he I-opial design, such ha, I J I J ( ) I J 0.054I J ( ) C L (6) 2u (' ) (' ) Thus, he average predicion variance becoes, r[ C L] (7) 2u (0) ~9~

6 Inernaional Journal of Saisics and Applied Maheaics Coparing ( and (7), WSCD perfored beer han UWSCD due o is saller average predicion variance leading o ore accurae predicion of responses in ixure experiens. 6. Equivalence Theore for Weighed Siplex Cenroid Designs For four ingrediens, he design is said o be I-opial, if i saisfy he following inequaliy a a given design poins of he siplex cenroid. ' f ( ) C LC f( ) (8) 2 2 BLENDS Table : Equivalence Theore for Four Ingrediens (WSCD). Average Predicion Variances ' () () r[ C L] f C LC f Opialiy, 0, 0, > No I Opial ½, ½, 0, < I-Opial /, /, /, < I-Opial ¼, ¼, ¼, ¼ < I-Opial 7. Equivalence Theore for Unifor Weighed Siplex Cenroid Kronecker For four ingrediens, he design is said o be I-opial if and only if i saisfy, ' f ( ) C2u LC2u f( ) (9) a a given design poins of he siplex cenroid designs. BLENDS Table 2: Equivalence Theore for Four Ingrediens (WSCD) Average Predicion Variances ' () 2 2 () r[ C2 L] f C LC f Opialiy, 0, 0, > No I Opial ½, ½, 0, < I-Opial /, /, /, < I-Opial ¼, ¼, ¼, ¼ < I-Opial I was observed ha he pure blends in boh WSCD and UWSCD did no saisfy he general equivalence heore for I-opialiy (8) and (9) respecively. The binary, ernary and quaernary blends were found o saisfy he general equivalence heore, herefore, I-opial. However, cenre poin ixures (¼, ¼, ¼, ¼) were beer han all he binary and ernary ixures due o heir saller average predicion variance, herefore, predicion of responses a his poin was accuraely achieved han any oher poin in he siplex cenroid designs. 8. Conclusion The average predicion variances for WSC designs were slighly lower han hose of UWSC designs, herefore yielding ore accurae resuls. All ixures forulaions excep pure blend saisfied he general equivalence heore for I-opialiy a differen poins of he siplex cenroid designs. On ixure forulaions, he cenroid poin also yield accurae predicion han any oher poin in he siplex. 9. References. Akinson AC, Donev AN, Tobias RD. Opiu Experienal Designs, wih SAS, Oxford Universiy Press Borkowski JJ. A Coparison of Predicion Variance Crieria for Response Surface Designs. Journal of Qualiy Technology (): Draper NR, Pukelshei F. Mixure odels based on hoogeneous polynoials. J. Sais. Plann. Inference : Goos P, Syafiri U. V-opial ixure designs for he qh degree odel. Cheoerics and Inelligen Laboraory Syses, 204 6: Goos P, Jones B, Syafiri U. I-opial Mixures designs. Journal of he Aerican Saisical Associaion. 206 (5: Laake P. On he opial allocaion of observaion in experiens wih ixures, Scandinavian Journal of Saisics, 975 2: Liu S, Neudecker H. A V-Opial design for Scheffé s Polynoial odel. Saisics and Probabiliy leers : ~40~