A New Method of Construction of Robust Second Order Rotatable Designs Using Balanced Incomplete Block Designs

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Open Journal of Statsts 9-7 http://d.do.org/.6/os..5 Publshed Onlne January (http://www.srp.

1 Open Journal of Statsts Publshed Onlne January ( A ew Method of Construton of Robust Seond Order Rotatable Desgns Usng Balaned Inomplete Blok Desgns Beam Re. torbabu Kottapall Rayalakshm Department of Statsts Aharya agaruna Unersty Guntur Inda Emal: torsugnanam@yahoo.o.n Reeed Deember 8 ; resed Deember ; aepted January ABSTRACT Das [] studed robust seond order rotatable desgns (RSORD) and onstruted seond order rotatable desgns wth orrelated errors (SORDWCE) under the auto orrelated struture usng entral omposte desgn. In ths paper a new method of onstruton of RSORD usng balaned nomplete blok desgns (BIBD) s suggested. In ths method the number of desgn ponts requred s n some ases less than the number requred n Das [] method of onstruton of robust rotatable entral omposte desgns (RRCCD). We may pont out here that ths RSORD usng BIBD has desgn ponts for 7-fators where as the orrespondng RRCCD obtaned by Rayalakshm and torbabu [] needs 57 desgn ponts. Thus the new method leads to a 7-fator RSORD n less number of desgn ponts than the orrespondng RRCCD. Here we also obtaned the arane of the estmated response for the fators 8. Keywords: Response Surfae Desgns; Rotatablty; Seond Order Rotatable Desgns (SORD); Robust SORD; Robustness; Balaned Inomplete Blok Desgns. Introduton In response surfae methodology rotatablty s a natural and hghly desrable property. Ths was ntrodued and deeloped by Bo and Hunter [] assumng the errors to be unorrelated and homosedast. Das and arasmham [5] onstruted seond order rotatable desgns (SORD) through balaned nomplete blok desgns (BIBD). Panda and Das [6] studed frst order rotatable desgns wth orrelated errors. Further Rayalakshm and torbabu [] etended the work of Das [] and onstruted RRCCD for 7. In order to study the nature of robust rotatable desgns rotatablty ondtons for seond order regresson desgns hae been dered assumng the errors to be orrelated. These ondtons hae been further studed under dfferent arane oarane strutures of errors. Here we dere ondtons for rotatablty for a general orrelated errors struture and spealze then to the autoorrelated struture. yˆ In ths paper a new method of onstruton of RSORD usng BIBD s suggested and also obtaned the arane of the estmated response for fators 8.. Condtons of SORDWCE Assumng that the response surfae s of seond order we adopt the model: y e u u u u u where Y s the etor of reorded obseratons on the study arable y β s are the etor of regresson oeffents e u are random errors wth orrelated errors... Condtons of Rotatablty The estmated response at s gen by y ˆ The arane of estmated response at ( ) ˆ ˆ Co ˆ ˆ Co ˆ ˆ ˆ Co ˆ ˆ yˆ () () s gen by ˆ Co Co Co Co sco ss s Co s ltco lt s s lt l t () Copyrght SRes.

2 .. yˆ ss. s. s s s s. lt l t lt l t The arane funton n () wll be a funton of ()... for rotatablty for all f ;. s. ss. ; ; ; ; s < ;. lt. ; l < t ( ) (l t); onstant. a say; ; = onstant = e say; ;.. onstant d say; ; onstant. say; < ; and onstant d ;. Below are gen equalent ondtons for rotatablty n seond order regresson desgns wth orrelated errors mo del () n terms of the elements of the moment matr: (I): ().. l ; < l ( ). ; () (). ; (). l ; < l ().. l ; < l (). lt ; l< t ( ) (l t) (II): (). onstant a say; () onstant. e sa y; (). onstant f say; (III): (). onstant f ; (). onstant ; < (I): ; < (5)..... arane Funton of SORDWCE The estmated response at s gen by yˆ ˆ ˆ (6) The arane funton of a SORDWCE s gen by ˆ yˆ Co( ) ˆ Co ˆ ˆ ˆ ˆ ( ) ae d d d wh h s a funton of d ote: When the errors are homosedast (5) and (7) redue respetely to the usual seond order rotatablty ondtons non-sngularty ondton and arane funton... Condtons for Eat Rotatablty Followng () neessary and suffent ondtons for seond order rotatablty under auto-orrelated struture t s smplfes to. u u u u I* ;. l u lu u lu u u l ;. u u u u u u (7) u u ; u u u u. u u u u u u u u u u ; u u. l u u lu u u lu u u u u lu u u lu u u l ;. l u u lu u u lu u u u l ( u ) u lu u u u u l Copyrght SRes.

3 . lt u u lu tu u u lu tu u u l t u u lu tu u u u u II *. lu tu. u u u u a ; ; ; e u u u u u u u. d ; u u u u u u u III *. d ; u u u u ( u u u u u u u u. ; u u u u u u u u u u u... I * ; (8) where.. and. are as n (II)*() and (III)*() (). The ondton for non-sngularty s gen by a ()* : f (9) where and f a are as n (6). ote: The ondtons for eat rotatablty as stpu- ρ s lated aboe are hard to satsfy n prate sne unknown unless speal efforts are made regardng hoe of the underlyng robust desgn. Towards ths tremendous smplfaton obtans wheneer all odd moments of arous types ansh. These are lsted below for ready re- ferene ; u u u u u u u u u u u u u u u ; u u u u u u u u u u u u uu lu u u ; ; u u lu u u lu u u lu u u u l ; u u lu u u lu u u l( u) u u u u u ; u lu l u u lu tu u u lu tu u u u u lu tu u u u u lu tu ; l t. () As to the een order moments (up to order ) mere onstany of suh moments (of arous types) suffes to ensure eat rotatablty. Speally we desre to hae eah of the followng terms ndependent of and : u u u u u u u onstant ; onstant ; onstant ; Usng the aboe relatons for the newly onstruted desgn we get the followng: u u u u onstant onstant ; Copyrght SRes.

4 u u onstant u () Thus a desgn satsfyng the ondtons mentoned aboe may be suessfully utlzed as a rotatable desgn under olaton of the homosedastty assumpton subet to the oarane struture beng of the type (ondtons of eat rotatablty). Suh a desgn s therefore robust where are onstants.. ew Method of Construton of RSORD Usng Balaned Inomplete Blok Desgn Here we start wth usual SORD usng BIBD hang n non-entral desgn ponts nolng -fators. The set of n desgn ponts an be etended to (n + ) ponts by norporatng (n + ) entral ponts n the followng way. One entral pont s plaed n between eah par of nonentral desgn ponts n the sequene resultng thereby n (n ) suh entral ponts. The other two entral ponts are plaed one at the begnnng and one at the end. If the number of entral ponts of the usual SORD wth whh we started s greater than (n + ) the remanng entral ponts are plaed n any manner f the number s less we need to nlude the requste number of addtonal entral ponts. Here we eamne the non-sngularty for the orgnal desgn and the newly onstruted desgn. Let be the number of desgn ponts of the orgnal desgn (SORD) wth whh we started. Out of let n be the number of non-entral desgn ponts an d m be the number of entral ponts.e. = n + m. In general m < n + Let be the number of desgn ponts of the newly onstruted desgn Where = n + >. For the orgnal desgn wth whh we started the followng are the moment relatons: Let ( b r k λ) denote a BIBD. t k denote a fratonal replate of k n ± leels n whh no nteraton wth less than fe fators s onfounded and m be the number of entral ponts n the desgn. The desgn ponts for the newly onstruted desgn are obtaned as follows:.. Method I: When r < λ t ( k ) Here b mn m and = n + where ( ) nb t k. From the desgn ponts generated from the BIBD smple symmetry ondtons () and () are true. Condtons () are true obously. Condtons () are true as follows: u tk u r u tk u r u u u tk Usng the aboe relatons for the newly onstruted desgn we get the followng: u tk u r u tk u r u u u tk The results are gen n Table... Method II: When r = λ () ( ) Here b t k m = n + m and =n + where n= t( k) b. From the desgn ponts generated from the BIBD sm- ondtons () and () are true. Cond- ple symmetry tons () are true obously. Condtons () are true as follows: u u u u tk r tk r u u u tk Usng the aboe relatons for the newly onstruted desgn we get the followng: u u u u tk r tk r tk u u () u The results are gen n Table... Method III: When r > λ ( ) ( ) Here t k t b m = n + m and = n + t( k) t( ) Where n = b From the desgn ponts generated from the BIBD sm- ondtons () and () are true. Cond- ple symmetry tons () are true obously. Condtons () are true as follows: u tk t( ) u r Copyrght SRes.

5 Table. arane of the estmated response for the fators 8 when r < λ. y ˆ y ˆ ρ ( b r k ) ( br k ) = 7 = n = 8 m = 6 = 8 = 8 n = m = 8 yˆ ( 5 b5 r k ) = 8 = n = 9 m =.9.55σ.9σ d +.77σ d.98σ.7σ d +.σ d.9σ.7σ d +.9σ d.8.85σ.56σ d +.6σ d.8σ.96σ d +.85σ d.5σ.7σ d +.5σ d.7.86σ.96σ d +.9σ d.6σ.657σ d +.9σ d.67σ.σ d +.6σ d.6.799σ.96σ d +.σ d.78σ.8σ d +.98σ d.6σ.76σ d +.7σ d.5.75σ.8σ d +.8σ d.88σ.56σ d +.87σ d.5σ.σ d +.78σ d..6σ.5σ d +.6σ d.8σ.σ d +.86σ d.σ.7σ d +.8σ d..58σ.6σ d +.σ d.87σ.7σ d +.89σ d.σ +.σ d +.9σ d..5σ +.σ d +.8σ d.6σ +.65σ d +.9σ d.σ +.5σ d +.9σ d..5σ +.σ d +.σ.7σ +.6σ d +.9σ d.8σ +.7σ d +.96σ d d.55σ +.65σ d +.5σ d.σ +.67σ d +.86σ d.7σ +.8σ d +.95σ d..59σ +.6σ d +.8σ d.8σ +.89σ d +.7σ d.σ +.88σ d +.9σ d..56σ +.78σ d +.986σ d.6σ +.9σ d +.5σ d.7σ +.87σ d +.85σ d σ +.69σ d σ +.86σ d..6 σ d +.σ d.σ +.8σ d +.76σ d..7σ +.σ d +.7σ d. σ +.67σ d +.9σ d.5σ +.7σ d +.65σ d σ +.7σ d +.6σ d.σ. σ d +.58σ d.8σ +.6σ d +.5σ d.6.55 σ +.σ d +.75σ d.5σ +. σ d +.σ d.σ +.9σ d +.σ d.7.96σ +.5σ d +.σ d.68σ +.8σ d +.9σ d.σ +.6σ d +.σ d.8.9σ +.9σ d +.σ d.σ +.55σ d +.57σ d.8σ +.σ d +.σ d.9.6σ +.76σ d +.5σ d.9σ +.7σ d +.7σ d.957σ +.σ d +.9σ d ρ y ˆ ( 6 b r 5 k ) = 85 = n = 9 m = y ˆ ( 7 b7 r k ) = 5 = n = 6 m = y ˆ ( 8 b r 7 k ) = 8 = 57 n = m = σ.8σ d +.σ d.687σ.67σ d +.85σ d.σ.σ d +.6σ d.8.σ.68σ d +.68σ d.7σ.σ d +.5σ d.57σ.6σ d +.59σ d.7.8σ.σ d +.9σ d.6σ.7σ d +.σ d.σ.8σ d +.58σ d.6.88σ.5σ d +.7σ d.66σ.9σ d +.σ d.8σ.σ d +.6σ d.5.59σ.6σ d +.6σ d.σ.σ d +.σ d.67σ.σ d +.7σ d..8σ +.σ d +.5σ d.8σ +.9σ d +.σ d.56σ +.σ d +.78σ d..σ +.59σ d +.6σ d.9σ +.σ d +.σ d.9σ +.σ d +.86σ d..σ +.σ d +.78σ d.85σ +.69σ d +.8σ d.5σ +.6σ d +.9σ d..8σ +.σ d +.86σ d.8σ +.88σ d +.σ d.σ +.56σ d +.96σ d.6σ +.5σ d +.86σ d.79σ +.σ d +.σ d.σ +.6σ d +.96σ d..9σ +.6σ d +.76σ d.8σ +.σ d +.9σ d.σ +.65σ d +.9σ d..6σ +.6σ d +.58σ d.85σ +.σ d +.σ d.5σ +.6σ d +.88σ d..8σ +.5σ d +.σ d.9σ +.95σ d +.7σ d.5σ +.58σ d +.79σ d σ +.σ d +.σ.8σ +.8σ d +.σ d.57σ +.5σ..8 d d +.68σ d.5.78σ +.σ d +.65σ d.σ +.7σ d +.8σ d.69σ +.σ d +.56σ d.6.6σ +.89σ d +.9σ d.66σ +.56σ d +.65σ d.88σ +.σ d +.σ d.7.8σ +.65σ d +.9σ d.7σ +.σ d +.8σ d.σ +.5σ d +.σ d.8.7σ +.σ d +.6σ d.9σ +.6σ d +.σ d.86σ +.6σ d +.σ d.9.98σ +.σ d +.9σ d.7σ +.σ d +.5σ d.8σ +.8σ d +.σ d Copyrght SRes.

6 ρ Table. arane of the estm ated response for the fators 8 when r = λ. y ˆ ( b6 r k ) = 5 = 7 n = 5 m = y ˆ ( 7 b7 r k ) = = 57 n = 56 m =.9.σ.9σ d +.5σ d. σ.8σ d +.677σ d.8.σ.6σ d +.6σ d.75 σ.89σ d +.5σ d.7.87σ.66σ d +.85σ d.9σ.5σ d +.96σ d.6.9σ.5σ d +.98σ d. 65σ.6σ d +.5σ d.5.68σ.σ d +.σ d.89σ.σ d +.7σ d..σ.σ d +.65σ d.σ +.σ d +.78σ d..σ +.69σ d +.σ d.9σ +.σ d +.5σ d..8σ +.σ d +.5σ d.9σ +.9σ d +.55σ d..7σ +.7σ d +.55σ d.79 σ +.66σ d +.57σ d.7σ +.6σ d +.5σ d.75 σ +.σ d +.575σ d..8σ +.5σ d +.8σ d.79σ +.σ d +.557σ d..8σ +.58σ d +.σ d.9σ +.7σ d +.5σ d..5σ +.9σ d +.89σ d.9σ +.85σ d +.68σ d..5σ +.σ d +.767σ d.σ +.5σ d +.5σ d.6σ +.7 d +.6σ d.5 6σ.89σ +.σ d +.5σ d.6.78σ +.σ d +.9σ d.66σ +.67σ d +.6σ d.7.5σ +.σ d +.58σ d.96σ +.σ d +.9σ d.8.5σ +.5σ d +.9σ d.75σ +.79σ d +.σ d.9.759σ +.7σ d +.9σ d.5σ +.8σ d +.58σ d u u tk t( ) r u u u tk t( ) Usng the aboe relatons for the newly onstruted desgn we get the followng: u tk t( ) u r tk t( ) u r u tk t( ) u u ( ) u The results are gen n Table. (These epresson follow easly from the defnton of ponts sets generated from BIBD and ther onsequent multplaton wth fatoral ombnatons as eplaned n Raghaarao [7] pp. 98-) Usng (5) and () () () the desgn parameters of the newly onstruted desgn are the followng: a ( ) ( ) f e Usng (5) and notng the epresson ( ) ( ) a f (5) smplfes to Hene the non-sngularty ondton of the aboe desgn s (6) Let f ( ρ) = Copyrght SRes.

7 5 Table. arane of the estmated response for the fators 8 when r > λ. y ˆ y ˆ y ˆ ρ ( 5 b r k ) ( 6 b5 r 5 k ) ( 7 b r 6 k ) = = 7 n = 56 m = 6 = 5 = n = 6 m = = 8 = 57 n = m = σ.65σ d +.99σ d.76σ.σ d +.768σ d.5σ.6σ d +.68σ d.8.65σ.9σ d +.86σ d.5σ.79σ d +.56σ d.5σ.σ d +.σ d.7.6σ.765σ d +.76σ d.σ.6σ d +.58σ d.8σ.6σ d +.σ d.6.5σ.67σ d +.668σ d.σ.6σ d +.7σ d.7σ.σ d +.σ d.5.8σ.8σ d +.67σ d `.77σ.σ d +.65σ d.9σ.5σ d +.9σ d..8σ.8σ d +.698σ d.7σ.σ d +.97σ d.9σ +.7σ d +.8σ d..8σ +.5σ d +.79σ d.8σ +.7σ d +.5σ d.8σ +.7σ d +.6σ d..89σ +.57σ d +.756σ d.6σ +.7σ d +.558σ d.7σ +.5σ d +.σ d..78σ +.σ d +.769σ d.σ +.85σ d +.57σ d.68σ +.6σ d +.58σ d.75σ +.8σ d +.76σ d.7σ +.σ d +.57σ d.67σ +.65σ d +.6σ d..79σ +.5σ d +.7σ d.σ +.7σ d +.55σ d.68σ +.7σ d +.6σ d..9σ +.6σ d +.679σ d.6σ +.5σ d +.55σ d.7σ +.69σ d +.6σ d..9σ +.88σ d +.68σ d.8σ +.σ d +.6σ d.8σ +.57σ d +.75σ d..σ +.57σ d +.5σ d.8σ +.86σ d +.σ d.9σ +.8σ d +.σ d.5.89σ +.7σ d +.σ d.78σ +.57σ d +.σ d.σ +.6σ d +.68σ d σ +.7σ d +.8σ d.96σ +.7σ d +.5σ d.6σ +.σ d +.58σ d.8σ +.9σ d +.88σ d.σ +.9σ d +.9σ d.9σ +.67σ d +.5σ d.8.75σ +.8σ d +.57σ d.7σ +.59σ d +.σ d.99σ +.σ d +.98σ d.9.5σ +.σ d +.75σ d.98σ +.8σ d +.57σ d.65σ +.σ d +.7σ d It s readly seen that f n We now argue as follows the desgn to start wth s a SORD n the usual sense so that. Ths ond ton ndeed does satsfy the resed ondton (6) dered here for ρ. To take are of all alues of ρ nludng negate alues t s enou gh to demand... arane Funton The arane funton of SORDWCE under the autoorrelated struture onstruted by the aboe method s obtaned by usng (7) and (6) and notng the followng: T d T a ( ) T T Hene the arane funton s gen by yˆ A B d C ( )d say where A (7) ˆ T T B Copyrght SRes.

8 6 ˆ Co λ T ( ) C ˆ where T. Eample: We llustrate the aboe method wth the onstruton of RSORD for 7-fators wth the help of BIBD. Consder the BIBD ( = 7 b = 7 r = k = λ = ). Here we hae = 57 = n = 56 m =. u u u u u u u 8 Plan of BIBD: B B 5 B 6 B 5 7 B B B 7 7 The desgn (denoted by d ) s dsplayed here for ready referene (olumn beng runs). d d d d Copyrght SRes.

9 7 d d As rega rds the non-sngularty ondton for ths de- sgn we note t hat.798 whh eeeds.796 n ( ). Hene the analyss of aran e of estmated response for arous fators 8 and t s ahee d for all alues o f ρ ρ. We m ay p ont out here that ths RSOR D has de- sgn ponts for 7-fators where as the orrespondng RRCCD obtaned by Rayalakshm and torbabu [ ] needs 57 desgn ponts. Thus the new method leads to a 7-fator RSORD n less numb er of desgn ponts th an the orrespondng RRCCD.. Aknowledgements The authors are thankful to the referee a nd the edtor for the alua ble suggestons whh helped n mprong t he qualty of the paper. I) Calutta Statstal Assoaton Bulletn ol pp [] R.. Das Rob ust Seond Order Rotatable Desgns (Part II) Calutta S tatstal Assoat on Bullet n ol pp [] K. Rayalakshm and B. Re torbabu Robust Ro- tat able Central Composte Desgns Paper Communted for the Possble Publaton. [] G. E. P. Bo an d J. S. Hu nter Multf ator Epermental Desgns for Eplorng Response Surfa es Annals of Mathematal Statsts ol. 8 o. 957 pp. 95- a. do:./aoms/77777 [5] M.. Das and. L. arasmh am Construton of Ro- tatable Desgns through Balaned Inomplete Blok Desgns Annals of Mathematal Statsts o l. o. 96 pp do:./aoms/ 7777 [6] R.. Panda an d R.. D as Frst Order Rotatable Desgns wth Correlated Errors Calutta Sta tstal Asso- aton Bulle tn ol. 99 pp. 8-. [7] D. Raghaarao Construton and Combnatoral Problems n Desgn of Eperments John Wley ew York. 97. REFERECES [] R.. Das Robust Seond Order Rotatable Desgns (Part Copyrght SRes.

Homework Math 180: Introduction to GR Temple-Winter (3) Summarize the article:

Homework Math 180: Introduction to GR Temple-Winter (3) Summarize the article: Homework Math 80: Introduton to GR Temple-Wnter 208 (3) Summarze the artle: https://www.udas.edu/news/dongwthout-dark-energy/ (4) Assume only the transformaton laws for etors. Let X P = a = a α y = Y α