ON THE COMPARISON OF BOUNDARY AND INTERIOR SUPPORT POINTS OF A RESPONSE SURFACE UNDER OPTIMALITY CRITERIA. Cross River State, Nigeria
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1 ON THE COMPARISON OF BOUNDARY AND INTERIOR SUPPORT POINTS OF A RESPONSE SURFACE UNDER OPTIMALITY CRITERIA Thomas Adidaume Uge and Stephen Seastian Akpan, Department Of Mathematics/Statistics And Computer Science, University Of Calaar, Calaar, Cross River State, Nigeria ABSTRACT: The performance of the design matrices otained from oundary and interior support points of a response surface is compared within a specified experimental area ased on first order response surface model. This is achieved y determining the values of the desired optimal design criteria with respect to their corresponding information matrices. These optimality criteria are: D- optimality, G- optimality, A optimality and E- optimality. Four designs aritrarily chosen from oth oundary and interior support points were used for the study. The designs otained from oundary support points were found to e etter than the designs otained from interior support points with respect to the aforementioned design optimality criteria considered in this work. KEYWORDS: Boundary Support Points, Optimality Criteria, Information Matrix, response surface, Interior Support Points INTRODUCTION One of the first to state criteria and otain optimum experimental designs for regression prolems was Smith (9). Many years thereafter, Kiefer (99) developed meaningful computational procedures for finding optimum designs in regression prolems of statistical inference. An optimality criterion showed how good a design is and it is maximized or minimized y an optimal design. Numerous papers y { Kiefer (99), Kiefer (96), Kiefer (96) and Kiefer (9) } and { Kiefer and Wolfowitz (99), Kiefer and Wolfowitz (96) and Kiefer and Wolfowitz (96) } and most recently Vrdoljak (), Emmett et al () and Gilmour and Trinca () play a central role- in the development of optimality criteria. In Kiefer and Wolfowitz (99), their theoretical approach to design optimality and their
2 introduction of the D- and E- optimality criteria for the linear regression model laid the ground work for other design criteria; A-, F-, G- and D- optimality criteria. See for example, Yang (). Many of these design criteria were originally developed for the homogeneous variance linear models, ut most have een adopted for use in a non-linear and non-homogeneous variance situation as well. See for example, Uge and Chigu (). There are many optimality criteria, these criteria are: D-, A-, E-, G-, V-, I-, T-, optimality criteria etc. They are sometimes called alphaetical optimality criteria. A comprehensive and accessile description of optimality criteria and their applications is found in Atkinson et al ().The aim of this work is to compare designs from oundary and interior support points of the response surface using the D-, G-, A-, and E-, optimality criteria for first order response surface model. NOTATIONS AND DEFINITIONS NOTATIONS To define each optimality criteria used in this study some notations are needed. Thus, we consider a design,, to e represented y a matrix M, and for the same design,, the information matrix is defined to e ( ) = M ; Mo = M= information matrix, is some optimality characteristic measure function. The determinant of the information matrix, is det( ); the variance of the estimated variance surface at, is d( ) = ( ) ( ) ( ) DEFINITIONS D-optimality criterion: A design that maximizes the determinant of the information matrix, is called a D-optimal design. Symolically, a design is D-optimal if it gives ( )= max det( ) By a D-optimal design, the generalized variance of the est, linear, uniased estimates of the parameters is minimized, regardless of the actual parameterization of the regression function. 9
3 G-optimality criterion: A design that minimizes the maximum variance of the estimated response function over the design region is called a G- optimality criterion. Symolically, i.e min{ max d (, )}. The experimeter optimizing a design according to the G- optimality criterion intends to get a good estimate of all the oserved responses. A-optimality criterion: A design that minimizes the trace of the dispersion matrix,( ) is called an A- optimality criterion. Symolically, a design is A- optimal if it gives min tr( ) E-optimality criterion: A design that minimizes the maximum eigenvalues of the dispersion matrix, ( ) is called an E-optimal design. Symolically, a design is E-optimal if it gives min, where is the largest eigenvalue of information matrix. GENERAL APPROACH OF THE INVESTIGATION Introduction We shall make comparison under (D-,G-, A-, and E-) optimality criteria etween the oundary and interior support points of the response surface for first order response surface model in the closed interval [,]. The design matrix otained from the oundary and interior support points are defined as follows: - design matrix of oundary support points t design matrix of interior support points All the designs are otained from figure Figure Experimental region (Response Surface)
4 First order response surface model is given y f(x,x ) = a o + a x + a x + e. In figure points circled are the oundary support points while points with asterisks are interior support points. Design matrices for oundary support points are: ( Note; four design matrices are aritrary chosen for the study). Design matrix for interior support points are: ( Note; four design matrices are aritrarily chosen for the study).
5 t t t t. Comparing design from oundary and interior support points under D- optimality Recall now from the definition of D- optimality criterion, ( )= max det( ). In this context, the set of information matrices, and, etween these matrices, the one that gives a maximum determinant is optimal. In other words, we need to estalish numerically that one of the following inequalities, det ( )< det ( ) and det ( ) < det ( ) holds.
6 Thus four different designs for oth oundary and interior support points were used for the illustrations as given earlier. The designs are,,, and t t t and t The design and information matrices from oundary support points, for first order response surface model are: =, ' =.6 N (where N is the no. of design points). N det ' =.9. The corresponding design and information matrices from interior support points, i.e t ' t t =,. N N det t t ' =.9.
7 Oviously, the inequality det < det holds. By the same token, det ' t t ' ' and det ' for the designs,, and given earlier are t t otained. For each of the designs under consideration the maximum values of det ' and det t ' t are given in tale. Just as we have seen in the case of the designs, that the inequality det t t for the designs, and. ' < det ' holds, the inequality det ' < det t t ' also holds Thus, with respect to the D-optimality criterion, designs otained from the oundary support points are etter in fitting a first order response surface model than those of interior support points. Comparing designs from oundary and interior support points under G-optimality criterion Given a design,, let d(,) denote the variance of the estimated regression function of oundary support points designs. And let d( t, ) denote the variance of the estimated regression function of interior support points designs. In comparing the functions under G- optimality criterion, we need to estalished that one of the following inequalities, Max d(, )<Max d( t, ) and Max d( t, )<Max d(, ) holds. For design, the variance of the estimated regression function of oundary support points designs, d(, ), is d(, )= ( )( ) ( ) For design, ( ) =( ) The maximum of d(, )= ( )( ) ( ) is Max d(, )= Max{ ( )( ) ( )}=.9 The corresponding variance of the estimated regression function of interior support points design, d( t,)= ( )( ) ( ), the maximum for design,, ie Max d(, ) = Max { ( )( ) ( )} =.. Oviously, the inequality Max d(, )<Max d( t, ) holds. By the same token d(, ) and d( t, ) for the designs,, and are otained. For each of the designs under consideration the maximum values of d(, ) and d( t, ) are given in
8 tale. Similarly, the inequality holds for the designs, and. Thus, with respect to G- optimality criterion, designs otained from oundary support points are etter in fitting first order response surface model than those from interior support points. Comparing designs from oundary and interior support points under A-optimality criterion. The comparison etween oundary and interior support points designs ased on the A-optimality criterion considers the trace values of the dispersion matrices, ( ) and ( ). Here, we are to estalished that one of the following inequalities tr( ) < tr( ) and tr ( ) < tr( ) holds. For the designs,, and, the values of tr( ) and tr( ) have een otained in tale. We shall continue our illustration here as in the preceding section. Thus, ( ) =( ) Thus, tr( ) = 6.9 For the corresponding interior support points design, the dispersion matrix is ( ) =( ) Thus, tr( ) =.96. oviously, the inequality tr( ) < tr( ) holds. For the rest of the designs adopted for illustration in the work, inspection of tale reveals that the values of ( ) correspond to the values of tr( ) which shows that the inequality tr( ) < tr( ) holds. Thus, with respect to A-optimality criterion, designs otained from oundary support points are etter in fitting first order respond surface model. Comparing designs from oundary and interior support points under E-optimally criterion. We shall use the result in tale to consider the E-optimality criterion. This tale shows maximum eigenvalues taken of the matrix ( ) and the maximum ( ) for the designs;,, and.
9 Given a design, let denote the maximum eigenvalues of the dispersion matrix, ( ) and let denote the maximum eigenvalues of dispersion matrix, ( ). In comparing oundary and interior support points designs under E-optimality criterion, we need to estalish that one of the following inequalities, < and < holds. For design,, the maximum eigenvalue of of the corresponding dispersion matrix, ( ) is.. The maximum eigenvalue of the corresponding dispersion matrix, ( ) is 9.. Clearly, the inequality < holds. In the same manner, the values of and for the designs,, and are otained. From tale, we oserved that for these designs the inequality < holds. Thus, with respect to E-optimality criterion designs otained from oundary support points are etter in fitting first order response surface model than those from interior support points. Tale D-optimality values for the given design used in comparison Designs numer of points Max det( ) Max det( ) ( ) Tale G-optimality values of the given design used in comparison Designs Numer of points Max d( ) Max d( ) ( )
10 Tale A-optimality values for the given designs used in comparison Designs Numer of points Tr( ) Tr( ) ( ) Tale E-optimality values for the given design used in comparison Designs Numer of points Max Max ( ) CONCLUSION It can e concluded that for each of the optimality criteria considered, designs otained from oundary support points are etter in fitting and estimating first order response surface model than designs otained from interior support points. Furthermore, a design that is D-optimal minimizes the generalized variance of the estimate of the treatment parameters and the one that is G-optimal minimizes the maximum variance of the estimate of the surface or response. It can e deduced that design otained from oundary support points are minimum variance design. Therefore, they are efficient in estimating first order response surface model. REFERENCES Atkinson, A. C., Donev, A. N. and Toias.(). Optimum Experimental Designs, with SAS, Oxford: Oxford University Press. Emmett, M., Goos, P. and Stillman, E.C.(). A Weighted Prediction-Based Selection Criterion for Response Surface Designs. Quality Relia. Eng. Int., 9-9. Gilmour, S.G. and Trinca, L. A. (). Optimum Designs of Experiments for Statistical Inference. Journ. Roy. Statistics. Soc. C. 6,-. Kiefer, J. (99).Optimum Experimental Design. Journ. Roy. Statist Soc. B,, -9.
11 Kiefer, J. (96). Optimum Designs in Regression Prolems. Annal of Maths Statist.,9-. Kiefer, J. (96).Two More Criteria Equivalent to D-optimality Designs. Annal of Maths Statist.,9-96 Kiefer, J. (9).General Equivalence Theory for Optimal Designs (Approximate Theory). Annals of Maths Statist:, 9-9. Kiefer, J. and Wolfowitz, J. (99). Optimum Designs in Regression Prolems. Annals of Maths. Statist, -9. Kiefer, J.and Wolfowitz, J. (96).The Equivalence of Two Extremum Prolems. Canad J. Math., Kiefer, J. and Wolfowitz, J. (96). Optimum Extrapolation and Interpolation Designs, I. Annals Inst. Statist. Math. 6, 9-. Smith, K. (9). On the Standard Deviation of Adjusted and Interpolated Values of an Oserved Polynomial Function and its Constant and the Guidance they give towards a Proper Choice of the Distriution of Oservations. Biometrika., -. Uge, T.A. and Chigu, P. E. (). On D-optimality Criterion of Non-overlapping and Overlapping Segmentation of the Response Surface. Research Journ. of Maths and Statist., -. Vrdoljak, M. (). Optimality Criteria Method for Optimal Designs in Hyperolic Prolems. Mathematical Communications.,-. Yang, M. (). A-optimal Designs for Generalized Linear Models with Two Parameters. Journ. of Statist. Plann. And Inf.,6-6.
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