Selection of the Optimal Mathematical Model of Multiple Regression in the Ternary Mixture Experiments

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1 IMK-4 Research & Developemet Heavy Machery 0(04) EN55-60 UDC 6 ISSN Selecto of the Optmal Mathematcal Model of Multple Regresso the Terary Mxture Expermets M. Kolarevć,* - D. Mć - - M. Rajovć - V. Grkovć - Zv. Petrovć Faculty of Mechacal ad Cvl Egeerg Kraljevo, Kraljevo, Serba Faculty of Techcal Sceces, Kosovska Mtrovca, Serba For a three-compoet system, regresso models ca be geerally set the form of polyomals whch are usually defed by the followg Scheffé caocal forms: a) lear model, b) square model, c) complete cube model, d) complete cube model, e) complete quartc model, f) complete quartc model. From a lot of models that meet the adequacy requremet t s ecessary to choose a model wth a ratoal umber of varables for the purpose of easy terpretato ad practcal applcato of the model. The paper presets the crtera for evaluato of the model qualty ad selecto of the optmal model composto wth a ratoal umber of varables. Keywords: Optmal Model, Multple Regresso, Terary Mxture Expermets 0. INTRODUCTION Three-compoet systems ca be graphcally represeted -D space by applyg terary graphs. The ma codto for applcato of terary graphs s: 0 X ; X =. () = X the relatve proporto of a compoet the mxture. From the prevously metoed codtos, t s otceable that the proporto of each compoet the mxture depeds o the proporto of the remag two compoets. x 0.8 x x 0. x x x Fg.. Tragular (trlear) coordate system ad represetato of vertcal sectos ad the drectos of crease the proporto of dvdual compoets Each pot sde the tragle represets a correspodg composto of the three-compoet system. The vertces of the tragle represet pure substaces, whle the pots o the sdes of the tragle represet two-compoet systems. For a pot sde the tragle, the proporto of each compoet s read by drawg les through the gve pot such a way that they are parallel to the sdes of the tragle (Fgure ). For the three-compoet system, regresso models ca be geerally set the form of polyomals whch are defed by the followg caocal or Scheffé forms [] [] [7]: *Corr. Author's Address: Faculty of Mechacal ad Cvl Egeerg Kraljevo, Dostejeva 9,Kraljevo, Serba, kolarevc.m@mfkv.kg.ac.rs

2 IMK 4-Research & Developemet Heavy Machery. Lear q x () = ŷ = β. Quadratc q q q () ŷ= β x + β xx j j = < j j. Specal Qubc q q q q q q (4) ŷ= β x + β xx + β xxx 4. Full Cubc j j jk j k = < j j < j j< q q q q q q q q (5) ŷ= β x + β xx + δ xx(x x) + β xxx j j j j j jk j k = < j j < j j < j j< 5. Specal Quartc q q q q q q q q q q q q j j jk j k jjk j k jkk j k = < j j < j j< < j j< < j j< (6) ŷ= β x + β xx + β x xx + β xx x + β xxx 6. Full Quatrc q q q q q q q q q q j j j j j j j j jk j k = < j j < j j < j j < j j< ŷ= β x + β xx + δ xx(x x) + γ xx(x x) + β xxx q q q q q q q q q q jjkxx j xk jkkxxx j k jklxxx j kxl < j j< < j j< < j j< < l l + β + β + β (7) X = 0.5 X X = 0. X X = 0. X Fg.. Determato of the composto of a alloy the terary system The selecto of the regresso model of the terary system depeds, before all, o the avalable umber of desg pots. The ecessary umber of desg pots for the correspodg umber of compoets ad the requred polyomal degree ca be calculated based o the expresso []: ( + q )! N =. (8)! q ( ) the polyomal degree, q the umber of compoets As there are also complete models of the thrd ad fourth degrees, the umber of regresso coeffcets for them caot be calculated based o the expresso (8), ad therefore the ecessary umber of desg pots for settg up of regresso models of terary systems s preseted Table. The suffcet umber of desg pots for the hghest degree model does ot mea that t wll be the best oe. A too hgh polyomal degree may lead to adopto of a adequate mathematcal model (Fgure ). Besdes, for a more ratoal use ad terpretato, a model wth a degree whch s as low as possble should be selected. I order to carry out the procedure of regresso aalyss of the three-compoet system ad select a adequate regresso model, t s ecessary to respect the followg phases [6]:. Selecto of possble forms of regresso models based o the avalable umber of desg pots. Calculato of regresso coeffcets for all selected models. Checkg the adequacy of selected mathematcal models 4. Selecto of the optmal regresso model 5. Evaluato of the sgfcace of regresso coeffcets of the selected model 6. Calculato of cofdece lmts of regresso coeffcets of the selected model 7. Calculato of cofdece lmts of the selected regresso model 8. Graphcal terpretato of the mathematcal model by cotour ad surface terary graphs. Kolarevć, M. - Mć, D. - Rajovć, M. - Grkovć, V. - Pertovć, Zv.

3 IMK 4-Research & Developemet Heavy Machery Table. Necessary umber of desg pots for settg up of the regresso model of the terary system Respose Number of regresso Regresso model Σ subscrpts coeffcets Lear Secod degree 6 j Icomplete thrd degree j 7 jk Complete thrd degree Icomplete fourth degree Complete fourth degree j jk j jk j j jk Fg.. Expermetal values ad the look of the regresso curve of the frst order (dashed le) ad the regresso curve of the 6th order (cotuous le). QUALITY INDICATORS OF THE REGRESSION MODEL For qualty evaluato of the model ad selecto of the "optmal" composto of the model wth a "ratoal" umber of varables, the followg statstcal-aalytcal values are avalable [4]: - the coeffcet of determato R - the adjusted coeffcet of determato R adj - the resdual mea square-varace ˆσ - Mallows' dcator C p - Predcto Sum of Squares Statstc - PRESS - Akake's Iformato Crtero - AIC - Schwartz's Bayesa Crtero - SBC - Bayes' Iformato Crtero - BIC - Amemya's Predcto Crtero PC - Wtcomb Score - WS Selecto of the Optmal Mathematcal Model of Multple Regresso the Terary Mxture Expermets

4 IMK 4-Research & Developemet Heavy Machery. The coeffcet of multple determato The coeffcet of multple determato s the rato betwee the sum of squares of devato of regresso values from ther arthmetc mea ad the sum of squares of devato of values of the depedet varable from ts arthmetc mea []: ( ŷ -y ) SS SS R = =- =, 0 R. (9) R E = SS T SST ( y -y ) = SS R the regresso sum of squares SS E the error (resdual) sum of squares SS T the total sum of squares The coeffcet of multple determato ca have the value betwee zero ad oe. Accordg to ths crtero, the most represetatve model s the oe whose coeffcet of determato s closer to oe. The dsadvatage of ths dcator s that t s a mootoe o-decreasg fucto of the umber of regresso varables, so that ts hghest value s for the model wth all avalable regresso varables, whch may lead to the selecto of a model whose dmesos are too large [4].. The adjusted coeffcet of multple determato The adjusted coeffcet of multple determato s gve by the expresso []: SSE ( k) R ( ) adj = = R, RA R. (0) SST k ( ) the umber of samples k the umber of varables (coeffcets of the regresso model) ν =-k the umber of degrees of freedom The hghest value of ths coeffcet s oe, ad ulke the coeffcet of determato, t ca also be a egatve umber. The value of the adjusted coeffcet of determato s ot a mootoe creasg fucto of the umber of varables, but t depeds o the umber of degrees of freedom,.e. o the model sze ad t s more sutable tha the coeffcet of determato because t esures that the model does ot clude too may varables. The hghest adjusted coeffcet of determato s relevat for the selecto of the regresso model.. Resdual mea square (estmate of varace) of regresso The resdual mea square of regresso s the rato betwee the resdual sum of squares ad the umber of degrees of freedom ν =-k: ( y ˆ ) y () = ˆ. σ = k The statstcal represetato of the model creases wth the decrease of the resdual mea of squares, ad so ths s the crtero for selecto of the model wth the lowest value of ths dcator..4 Malows' crtero Cp Malows' crtero s gve by the expresso: SSE p Cp - - p,. () ˆ p the umber of parameters the regresso model SS E (p) the resdual sum of squares for a model that cludes p parameters ˆσ the resdual mea of squares for a model wth the maxmum umber of regresso varables Most users adhere to oe of the two crtera for selecto of the optmal model: select the model wth a small value of C p select the model wth a small postve (or egatve) dfferece betwee C p ad p..5 PRESS (Predcto Sum of Squares Statstc) PRESS represets the resdual sum of squares whch s calculated a specfc way for every possble regresso model [4]. If k regresso varables are avalable, a model ca be formed wth oe varable or a combato of two, three or more regresso varables. The total umber of all possble regressos s equal to k -. The PRESS value s calculated for each k - regresso. The PRESS statstc represets the sum of squares of PRESS resduals []: ( ) ˆ( ) ˆ ( ) () PRESS = e = y y. ( ) ( ) = = e ˆ =y -y ˆ the -th PRESS resdual. The umercal calculato s smplfed by applyg the dagoal elemets of the h "hat"-matrx Kolarevć, M. - Mć, D. - Rajovć, M. - Grkovć, V. - Pertovć, Zv.

5 IMK 4-Research & Developemet Heavy Machery h h... h h h... h H = X( X`X) X` = h h... h (4) whch the values of depedet varables are cotaed the matrx X. The PRESS resdual devatos are gve by the expresso: ˆ ( ˆ ) ê = = (5) e( ) y y ( ), hj ad so the PRESS dcator ca be show by the expresso []: ê PRESS =. = h (6) A model wth the lowest PRESS dcator ad a relatvely small umber of parameters s chose for the optmal regresso model..6 Akake's Iformato Crtero AIC Akake's Iformato Crtero-AIC s gve by the expresso: ê = (7) AIC = l + p. the umber of values (sample sze) eˆ = the resdual sum of squares p the umber of parameters the model. Small values of ths crtero are desrable..7 Schwartz's Bayesa Crtero SBC Schwartz's Bayesa Crtero SBC s smlar to the prevous crtero: ê = (8) SBC = l + p l..8 Bayes' Iformato Crtero BIC Bayes' Iformato Crtero BIC s defed by the expresso: ê = (9) BIC = l + ( p + ) q q, σˆ q =. ê = ˆσ the estmate of varace based o a model wth all varables.9 Amemya'ş Predcto Crtero PC Amemya's Predcto Crtero PC s gve by the expresso: p ê + = PC (0) =. p ( ) The crtera AIC, SBC, BIC ad PC represet a set of smlar crtera ad a mmum value s desrable all of them..0 Wtcomb Score Desg-Expert [6] uses the followg system to score models.. Calculate the values (M) from the sequetal model of the sum of squares. M = f p 0,5 M = 0,05/p f p > 0,5 M = 0 f model s alased. Calculate the values (L) from the Lack-of-ft table: L = f p 0,0 L = p / 0,0 f p < 0,0. Combe the frst two evaluatos wth the statstcs R, whch forms the total evaluato: Score- = (M)(L)(R predcted) Score- = (M)(L)(R adjusted) Select the model wth the maxmum score. If all model scores are less tha or equal to zero, select the mea model R predcted = - SSPRESS/(SSTotal- SSBlocks) Two models are maly proposed. Desg-Expert the coservatvely defaults to the model scored hghest o the bass of predcted r-squared.. CONCLUSION I the process of vestgato of electrcal ad mechacal propertes of alloys, three-compoet systems play a very mportat role. Regresso aalyss provdes a possblty to use expermetal results order to obta the theoretcal depedece of these values o the molar rato of certa compoets of the mxture. Regresso models for a three-compoet system are geerally defed by polyomals from the frst to the fourth degrees, where the selecto of the regresso model of the terary system depeds, before all, o the avalable umber of desg pots. Selecto of the Optmal Mathematcal Model of Multple Regresso the Terary Mxture Expermets

6 IMK 4-Research & Developemet Heavy Machery It s desrable to aalyze several possble regresso models ad, out of the models wth proved adequacy, select the oe whch best descrbes the gve pheomeo. The fact that the hgher degree models are very complcated for terpretato of the observed pheomeo ad that the hghest degree model does ot always have to be the best oe should be cosdered. For qualty evaluato of the model ad selecto of the "optmal" composto of the model wth a "ratoal" umber of varables, there s a multtude of statstcalaalytcal values whch are descrbed the prevous chapter. The qualty of the soluto depeds o the crtero appled for selecto of the "optmal" model. It s dffulct to say whch crtero s the best oe ad hece t s desrable to combe several crtera to select the adequate mathematcal model.. ACKNOWLEDGEMENT The authors would lke to express ther grattude to the Mstry of Educato ad Scece of the Republc of Serba for ther support to ths research through the projects TR700 & OI REFERENCES [] Corell, Expermets wth Mxtures, d ed., Joh Wley&Sos, Ic, New York, 990. [] Lazć Ž. Desg of Expermets Chemcal Egeerg, Wlez-VCH Verlag GmbH&Co.KGaA, Wehem, 004 [] Motgomery D., Desg ad Aalyss of Expermets, 5th edto, Joh Wley&Sos, INC, New York [4] Šošć I. Prmjejea statstka, Školska kjga, Zagreb, 004 [5] M. Kolarevć, M. Vukćevć, B. Radčevć, M. Bjelć, V. Grkovć: A Methodology For Formg The Regresso Model Of Terary System, The Seveth Treal Iteratoal Coferece Heavy Machery HM 0, Faculty of Mechacal Egeerg, Proceedgs, Vrjačka Baja, pp. Е -6, 9 Jue- July 0 [6] Desg Expert v.8 User`s Gude, Stat-Ease, [7] Kolarevć M, Rajovć.M, Bjelć M., Terary graph jegova prmea u regresooj aalz, IMK-4 Istražvaje razvoj, časops sttuta IMK "4 OKTOBAR" - Kruševac, Goda XI, broj (-) -4, Kruševac 005, str. - Kolarevć, M. - Mć, D. - Rajovć, M. - Grkovć, V. - Pertovć, Zv.

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