The Performance of Optimum Response Surface Methodology Based on MM-Estimator

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1 The Performance of Opmum Response Surface Mehodology Based on MM-Esmaor Habshah Md, Mohd Shafe Musafa, Anwar Frano Absrac The Ordnary Leas Squares (OLS) mehod s ofen used o esmae he parameers of a second-order polynomal response surface mehodology (RSM) model whereby a face-cenered compose desgn of epermen s consdered. The parameers of he model are usually esmaed usng he OLS echnque. Neverheless, he classcal OLS suffers a huge se bac n he presence of a ypcal observaons ha we ofen call oulers. In hs suaon, he opmum response esmaor s no relable as s based on he OLS whch s no ressan o oulers. As an alernave, we propose usng a robus MM-esmaor o esmae he parameers of he RSM and subsequenly he opmum response s deermned. A numercal eample and smulaon sudy are presened o assess he performance of he opmum response-mm based, denoed as Opmum-MM. The numercal resuls sgnfy ha he Opmum-MM s more effcen han he Opmum-OLS. Keywords Response Surface Model (RSM), Ordnary Leas Squares (OLS), Oulers, MM-esmaor. R I. INTRODUCTION esponse Surface Mehodology (RSM) s a well nown ool n process and produc developmen usng desgn of an epermen. The RSM consss of sascal and mahemacal echnques developed n 95s for he purpose of deermnng opmzaon ha are used o mprove esng produc n an ndusry. I s desgned wh a produc of process nvolvng funconal relaonshp beween he values of some measurable response varables, y and a se of epermenal facors (npu varables) denoed by,,...,. The RSM has wde applcaons n a varey of real problems from dversfed areas such as engneerng, food manufacurng, bologcal scences, chemcal scences, ec. The opmum response y s deermned afer a model ha Mohd Shafe Musafa s wh he Insue for Mahemacal Research, Unversy Pura Malaysa, 44 UPM Serdang, Selangor, MALAYSIA (phone: ; fa: ; e-mal: shafe@mah.upm.edu.my). Habshah Md s wh he Depuy Dean of Faculy of Scence, Unversy Pura Malaysa, 44 UPM Serdang, Selangor, MALAYSIA (e-mal: habshah@scence.upm.edu.my). Anwar Frano s wh he Mahemacs Deparmen, Faculy of Scence, Unversy Pura Malaysa, 44 UPM Serdang, Selangor, MALAYSIA (emal: anwarfrano@gmal.com). relae beween he se of ndependen varables and he response varable s esablshed. In general, such a relaonshp s unnown bu can be appromaed by a low-degree polynomal model of he form y = f () β + ε () where = (,,..., ), f ( ) s a vecor funcon of p elemens, β s a vecor of consan coeffcens, and ε s an error erm. Khur and Muhopadhyay () saed ha model () provdes an adequae represenaon of he response. In mos RSM problem, wo mporan models are commonly used eher a frs-order or a second-order model. The regresson coeffcens of he model are ofen esmaed usng he mehod of Ordnary Leas Squares (OLS). The OLS mehod gves good parameer esmaes when he responses are normally dsrbued and no oulers n he daa se. Noneheless, n real pracce, many dsrbuons of he response varable s (consderably) no normal whch s due o he presence of ouler. Oulers occur very frequenly n real daa, and hey ofen go unnoced because nowadays much daa s processed by compuer whou careful nspecon or screenng. Yoha (987) saed ha a small fracon of ouler or even one ouler may have sgnfcan effec on he OLS esmaes. Subsequenly, he deermnaon of he opmum response s no relable as s based on he OLS whch s no ressan o oulers [5, 8]. Robus regresson mehods are recommended o be used, o remedy hs problem (Maronna, 6, Rousseeuw & Leroy, 987). Furhermore, Mongomomery e al. () shown ha robus regresson mehods can help he praconers o denfy possble oulers. The am of hs paper s o nvesgae he effec of oulers on he opmum yeld response. Snce he OLS s no oulers ressan, he alernave robus echnque called MM-esmaor (Maronna, 6) whch has a very hgh breadown pon s used o esmae he model parameers and subsequenly oban he opmum response. The performances of he Opmum-MM and Opmum-OLS echnques are assessed based on numercal eamples and smulaons sudy. II. USING A CENTRAL COMPOSITE DESIGN IN SECOND-ORDER MODEL Issue 6, Volume 6, 757

2 In general, he model () s used o descrbe he response surface f. A polynomal model s usually a suffcen appromaon n a small regon of he response surface. Therefore, dependng on he appromaon of vecor funcon f, eher frs-order or second-order models are used. Furhermore, a second-order model s useful n appromang a poron of he rue response surface wh parabolc curvaure. The second-order ncludes all lnear erms, plus all quadrac erms and all cross-produc erms are gven as y = β + β + j β j j ε = < + = β + β + β + ε where ( β β β ) ε j = (,,..., ), β =,,...,, s a random error and assumed o be ndependen wh N, σ. The number of parameers n model () mus conan a leas p = + + ½(-) dsnc desgn pon, where s a number of conrol varables. In addon, Myers e al. (9) saed ha he desgn mus nvolve a leas hree levels of each desgn varable o esmae he pure quadrac erms. The desgn seng are usually coded of each facor n each epermen so ha zero represens he cener pon, and + and - represen he upper and lower of he facor, respecvely. For he h facor, such coded levels are obaned as where ( ξ ξ ) () = () ξ s he coded un, ξ s he naural value of he h ndependen varable, and ξ s he mean of he naural un. The coded values were calculaed accordng o he followng equaon: ξ = ( hgh level + low level) ( hgh level + low level) / / There are many desgns avalable for fng a second-order model. The cenral compose desgn (CCD) s he mos frequenly used desgn for fng a second-order response surface. I was frs nroduced by Bo and Wlson (95). I consss of facoral pons, aal pons, and cenral pons. In he consrucon of CCD, Khur and Muhopadhyay () poned ou ha hs desgn consss of he followng hree feaures:. A complee (or a fracon of) facoral desgn whose facors levels are coded as -,. Ths s called he facoral poron.. An aal poron conssng of pons arranged so ha wo pons are chosen on he as of each conrol (4) varable a a dsance of α from he desgn cenre (chosen as he pon a he orgn of he coordnaes sysem).. A chosen number, n of cener pons. III. OPTIMIZATION OF A SECOND-ORDER MODEL Consder a second-order response surface equaon model as y ˆ β + β + B (5) = β β β β and B s a symmerc mar of order whose h dagonal elemen s β ( =,,, ) and s (,j)h off-dagonal elemen s where = (,,..., ), = (,,..., ), β j (,j =,,, ; j). The saonary pon s deermned by frs dfferenang ŷ wh respec o as follows: yˆ = b + B ˆ and, se he dervave o be equals o, he saonary pon can be obaned ( ) where; ˆ β ˆ β b =, M ˆ β (6) = Bˆ b (7) ˆ β Bˆ = sym. ˆ β / ˆ β L L O ˆ β q / ˆ β q / M ˆ β qq / b s a (q ) vecor of he frs-order regresson coeffcens and B s a (q q) symmerc mar. In resul, he predced response value a saonary pon can be calculaed as ( ) + β + B y ˆ β (8) = IV. REGRESSION ESTIMATOR BASED ON MM- ESTIMATION The Ordnary Leas Squares (OLS) mehod s ofen used o esmae he parameers of he model. However, can be adversely affeced by oulers [, 6]. As an alernave, MM- Issue 6, Volume 6, 758

3 esmaor whch has a very hgh breadown pon s used o esmae he parameers of he model. The MM-esmaor s one of he robus regresson echnques end o dampen he effec of he oulers. The MM-esmaor was orgnally proposed by Yoha n 987. Yoha (987) defned he MMesmaor as a hree-sage procedure:. an nal regresson esmae s compued whch s conssen robus and wh hgh breadown pon bu no necessarly effcen.. an M-esmae of he errors scale s compued usng resduals based on he nal esmae.. an M-esmae of he regresson parameers based on a proper redescendng ps-funcon, ρ s compued. For a gven ψ funcon equals o ρ, he MM parameer esmaes are defned as any soluon of; where n n = Y X β ψ = ˆ X () σ s Y be he response varable and X he p-vecor of covaraes observed for =,,,n. In he followng, a sep-by-sep procedure for opmum-mm and opmum-ols are presened based on RSM: Sep : Buldng an approprae second-order response surface model for each response and compue he regresson coeffcens of he second-order model () usng he RSM based on OLS and RSM based on MM-esmaor. Sep : Ne, he adequacy of he second-order model s esed, whch can be assessed from he analyss of varance (ANOVA) able. Sep : Perform he canoncal analyss. The canoncal analyss s performed o deermne he locaon and he naure of he saonary pon of he second-order model. The saonary pon s hen be referred as he cener pon of he new regon of neres. V. NUMERICAL EXAMPLES In hs secon, a numercal eample s presened o nvesgae he performance of he proposed opmum-mm. The opmzaon of anhan gum producon by X. campesrc n 8 baches of epermens was obaned from an epermen conduced by Psomas e al. (7). There are wo response varables observed n hs epermen whch are anhan gum ( ) and bomass ( ) producon whle he predcor varables or npu facors are agaon rae ( ), emperaure ( ), and me of culvaon ( ), usng a face cenered compose desgn of epermens. In hs arcle, we only focus on anhan gum producon (y) as a response varable. To see he effec of oulers on he opmum-ols and he opmum-mm, we purposely modfy (conamnaed) he anhan daase n hs epermen. The objecve of hs epermen s o search for an opmal seng and opmal yeld (response) ha can acheve he arge wh mamum opmum of anhan gum producon wh and whou conamnaed daa. Table : The Xanhan Gum Producon Daa Run Agaon rae ( ) Temperaure ( ) Tme ( ) y.78 (7.8) (69.9).5.6 (.6) (4.8).5.5 (5.) (47.5) Issue 6, Volume 6, 759

4 Table presens he daa se aen from Psomas e al. (7) epermens whch conans hree ndependen varables n coded and uncoded form accordng o he epermenal desgn and he response (anhan gum producon) for all epermens. The coded values of ndependen facors were calculaed as follows; Agaon rae: Temperaure: Tme: In hs arcle, he second order response surface models (OLS and MM-esmaor) are fed usng S-Plus 6. Professonal sofware. A. The Esmaed Coeffcens A second order polynomal model was fed o he producon of anhan gum daa. Table presens he esmaed regresson coeffcens, neracve erms, quadrac erms and probably values (p-values) based on he OLS and he MM-esmaor. The analyss was done usng he coded values. Psomas e al. (7) saed ha when he regresson model s deermned wh coded values of he varables, he sze of each coeffcen gves a drec measuremen of he mporance of each effec. Thus, he second-order model s approprae for anhan daase. Coeffcens wh p-values larger han.5 are consdered no sgnfcan and are no ncluded n he model. I s mporan o pon ou ha when a hgher order (square and neracon facor) was sgnfcan, he lnear facor mus follow he laer (Mongomery e al, ). The resuls of Table show ha a α =.5, all coeffcens based on OLS are sgnfcan. Two neracon facors of MMesmaor are no sgnfcan. Bu, afer he deleon of facor and and recompued he esmaes for he second me, all he coeffcens are sgnfcan. Table : Esmaed coeffcen for Xanhan daa usng OLS (A) and MM (B) Term Coeffcen SE coeffcen -value p-value (A) OLS esmaor Consan (B) MM-esmaor Consan Issue 6, Volume 6, 76

5 (a) Conour Plo (b) Response Surface Fgure : The effec on Agaon Rae versus Temperaure and Ther Muual effec on he Xanhan Daase (a) Conour Plo (b) Response Surface Fgure : The effec on Agaon Rae versus Tme and Ther Muual effec on he Xanhan Daase (a) Conour Plo (b) Response Surface Fgure : The effec on Temperaure versus Tme and Ther Muual effec on he Xanhan Daase Issue 6, Volume 6, 76

6 The geomerc naure of he second-order funcon s dsplayed n fgs. -. The response surfaces of fgure shows ha an ncrease of agaon rae and emperaure smulaneously yelds n a mnma producon of anhan gum producon for boh esmaors. On he oher hand, he effec of agaon rae and me n fgure ndcaes ha he mamum yelds producon of anhan. Furhermore, he ncrease of emperaure and me shows ha he conour of response surface s a mnma effec of anhan producon. These ndcaons for boh he RSM based OLS and he RSM based MM-esmaor suppored by he numercal resuls n he able 4. Table summarzes he correspondng analyss of varance resuls. Of some concern was he sze of he F sascs for he lac of f es. When usng he RSM based on OLS, all he conrbuon of facors are sgnfcan and here s no ndcaon of lac-of- f, evdenced by large p values for he lac-of-f es, whch equals o.555. Smlar o he OLS mehod, he RSM based on MMesmaor also suggesng no evdence of lac-of-f n he model. Table : Analyss of varance for anhan producon usng OLS (A) and MM (B) Source df SS MS F p (A) OLS esmaor Regresson Resdual error Lac-of-f Pure error..4 Toal 7.47 (B) MMesmaor Regresson Resdual error Lac-of-f Pure error..4 Toal 7.47 The opmum response based on MM and OLS are hen compued and he resuls are ehbed n Table 4. The resuls of Table 4 shows ha when here s no ouler, he OLS mehod leads o he opmal seng = (.8744,-.5,.498), whch resuls n opmum yelds equals o.65. In hs suaon, he performance of he MM esmaor s farly closed o he OLS, wh opmal seng = (.894,.7,.5) and opmal yeld equals o.646. Table 4: The opmum response for anhan daa Facor (Agaon rae) (Temperaure) (Tme) Opmum Response Opmum-OLS seng B. Analyss on modfed (conamnaed) anhan daase To see he effec of oulers on he Opmum-MM and Opmum-OLS, we purposely modfy (conamnaed) he response of anhan gum producon daa. Three desgn pons (facoral pon, aal pon, and cenral pon) each wh wo conamnaed pon (n parenhess and bold) are shown n Table. The varables,, and, are cenred and rescaled smlar o he earler daa. As menoned n secon A, coeffcens wh p-values larger han.5 are consdered no sgnfcan and are no ncluded n he model. Thus, for conamnaed daa, all he coeffcens usng he RSM based on OLS are no sgnfcan. However, he coeffcens usng he RSM based on MM are sgnfcan. The RSM based on OLS and he RSM based on MMesmaor were hen appled o he daa and he resuls of he opmum responses are ehbed n Table 5. I can be observed from Table 5 ha n he presence of oulers n he daa se, he OLS based mehod faled o deermne he opmal sengs and opmal response due o he nsgnfcan of all varables n he model. In he even ha he opmal response s obaned, he resul s very msleadng. However, usng MM-esmaor, he resuls are closed o he resuls as n he clean daase. I s neresng o noe ha he MM based mehod s only slghly affeced by he oulers. Oher resuls eamples are conssen and no repored here due o space consran. VI. SIMULATION STUDIES Opmum-MM seng In hs secon, a seres of Mone Carlo smulaon sudy are performed for comparson of he Opmum-MM and he Opmum-OLS wh and whou conamnaed daa usng a face-cenered compose desgn of epermen. A second-order polynomal model was fed and he opmum condons were esmaed. The responses are randomly generaed based on he followng funcon Issue 6, Volume 6, 76

7 Y = β + β ( X ) β ( X β ( X X ) + β ( X ) + β ( X ) + β ( X X ) + β ( X ) + β ( X ) ) + β ( X + + X ) + ε () To generae he Y values, frs we fed he values of varables,, and, are cenred and rescaled from he, and he parameer naural varables so ha [ ] coeffcen, β equal o. The errors, ε are generaed from a normal dsrbuon wh mean zero and varance.. In hs regards, smulaed values were generaed for each model wh 8 desgn pons. In order o chec how he presence of conamnaed daa affecs he esmaors, wo conamnaed daa ( Y 4,, Y, ) were observed. The 4h and h daa pons are replaced wh her correspondng y values ncreased by uns. The opmum responses for anhan gum producon daa wh and whou conamnaon are shown n Table 6 and Table 7. As can be epeced, for clean daa, he resuls (Table 6) clearly ndcae ha he RSM based on OLS mehod performed beer han he RSM based on MM-esmaor snce has less based, smaller SE and RMSE. The OLS and MM based echnque are farly closed o each oher n hs regard. I s neresng o see ha he opmum-mm has smaller RMSE han he opmum-ols n he presence of oulers. On he oher hand, he performance of opmum-ols s very poor. Table 5: Opmum response for conamnaed daa Desgn Pon Mehod Opmal Seng = ( ) OLS (-.698,-.8, Facoral.76) Pon MM (.5585,.96, ) OLS Aal - - Pon MM (.87,-.7, ) OLS Cener - - Pon MM (.9,.49,.66.67) - ndcae ha he opmum canno be esmaed Opmum Response = y Table 6: Esmaon he Opmum response for clean daa Mehod Opmum-OLS Table 7: Esmaon he Opmum response for conamnaed daase VII. CONCLUSION Ths arcle clearly shows ha he performance of he opmum-mm o esmae he opmal seng and opmal response s comparable o he opmum-ols when no ouler(s) n a daa se. However, he opmum-ols s very sensve o he presence of oulers. On he oher hand, he opmum-mm s very relable as s ouler ressan. Hence, n he presence of ouler(s) s recommended o use he opmum-mm o esmae he opmal response. REFERENCES Opmum-MM Bas SE RMSE Mehod Bas SE RMSE Opmum-OLS Opmum-MM [] C. K. Ch'ng, S. H. Quah, and H. C. Low. The MM-Esmaor n Response Surface Mehodology. Qualy Engneerng, 5 7: [] Frano, A., Md, H. Esmang Bas and RMSE of Indrec Effecs Usng Rescaled Resdual Boosrap n Medaon Analyss. WSEAS TRANSACTIONS on MATHEMATICS, 9(6),, pp [] G. E. P. Bo, D. W. Behnen. Some New Three Level Desgns for he Sudy of Quanave Varables. Technomercs, Vol., No. 4, 96, pp [4] G. E. P. Bo, N. R. Draper. Robus Desgn. Bomera, 6,,975, p. 47 [5] Hua, N. K and Md, H. Robus Indvduals Conrol Char for Change Pon Model. WSEAS TRANSACTIONS on MATHEMATICS, 9(7),, pp Issue 6, Volume 6, 76

8 [6] Hampel F.R., Ronche E. M., Rousseeuw P. J., Sahel W.A. Robus Sascs: The Approach Based on Influence Funcon. New Yor: 986. John Wley & Sons, Inc. [7] H. Gonda Neddermejer, G. J. Oormarssen, N. Persma, R. Deer. A framewor for response surface mehodology for smulaon opmzaon. Proceedngs of he Wner Smulaon Conference. pp [8] Md, H., Rana, S., Imon, A.H.M.R. The Performance of Robus Weghed Leas Squares n Presence of Oulers and Heeroscedasc Errors. WSEAS TRANSACTIONS on MATHEMATICS, 8(7), 9 pp [9] Maronna,R.A., Marn,R.D., Yoha, V. J. Robus Sascs: Theory and Mehods. 6. John Wley and Sons. [] Mongomery, D. C., Pec, E. A., and Vnng, G. G. Inroducon o Lnear Regresson Analyss. rd Edon.. John Wley and Sons, Inc. [] Morgenhaler S., Schumacher, M. M. Robus analyss of a response surface desgn. Chemomercs and Inellgen Laboraory Sysem 47, 999, pp [] Myers H. R., Khur I. A., Carer H. W. Response Surface Mehodology. Technomercs, Vol., 989, pp [] Myers H. R., Khur A. I., Carer, W. H. Jr. Response Surface Mehodology: Technomercs, 989, Vol., No., pp [4] Myer, R. H., Mongomery, D. C., Anderson- Coo, C. M. Response Surface Mehodology: Process and Produc Opmzaon Usng Desgned Epermens. nd Edon. Canada: 9. John Wley and Sons, Inc. [5] Psomas, S.K., Laopoulou-Kyrades, M., and Kyrads, D.A. Opmzaon Sudy of Xanhan Gum Producon Usng Response Surface Mehodology. Bochemcal Engneerng Journal, 7, Vol.5, pp [6] Raml, N.M., Md, H., Imon, A.H.M.R. Esmang Regresson Coeffcens usng Weghed Boosrap wh Probably. WSEAS TRANSACTIONS on MATHEMATICS, 8(7), 9, pp.6-7. [7] Smpson J. R., Mongomery D. C. A performance-based Assessmen of Robus Regresson Mehods. Comm. In Sa.-Smulaon and Compuaon, 7(4), 998, [8] Rousseeuw, R. J. & Leroy, A. M. Robus Regresson and Ouler Deecon John Wley and Sons, Inc. [9] Vdmar, T. J., Mcean, J. W. A Mone Carlo Sudy of Robus and Leas Squares Response Surface Mehods. Journal of Sascal Compuaon and Smulaon, 996, Vol.54, pp. -8. [] W.N. Venables, B.D. Rpley. Modern Appled Sascs wh S-Plus rd Edon Sprnger Verlag, New Yor. [] Yoha V. J. Hgh Breadown-Pon and Hgh Effcency Robus Esmaes For Regresson. The Annals of Sascs, 987, Vol. 5, No., Issue 6, Volume 6, 764