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Unformatted text preview: Pointwise Bias Error Bounds for Response Surface Approximations and Min-Max Bias Design MELIH PAPILA Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611-6250 [email protected] RAPHAEL T. HAFTKA Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611-6250 [email protected] LAYNE T. WATSON Departments of Computer Science and Mathematics, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061-0106 [email protected] Abstract. Two approaches addressing response surface approximation errors due to model inad- equacy (bias error) are presented, and a design of experiments minimizing the maximal bias error is proposed. Both approaches assume that the functional form of the true model is known and seek, at each point in design space, worst case bounds on the absolute error. The first approach is implemented prior to data generation. This data independent error bound can identify locations in the design space where the accuracy of the approximation fitted on a given design of experiments may be poor. The data independent error bound can easily be implemented in a search for a design of experiments that minimize the bias error bound as it requires very little computation. The second approach is to be used posterior to the data generation and provides tightened error bound consistent with the data. This data dependent error bound requires the solution of two linear programming problems at each point. The paper demonstrates the data independent error bound for design of experiments of two-variable examples. Randomly generated polynomials in two variables are then used to validate the data dependent bias-error bound distribution. 1 Nomenclature A Alias matrix b Vector of estimated coefficients of basis functions b j Estimated coefficients of basis functions c (2) Vector of bounds for coefficients of the basis functions E b Vector of true prediction errors due to bias at the data points e b ( x ) True bias error at design point x | e D b ( x ) | Data dependent worst case bias error bound at design point x | e I b ( x ) | Data independent worst case bias error bound at design point x fl fl e I b fl fl av Average of data independent bias error bounds over the design domain fl fl e I b fl fl max Data independent maximum absolute bias error bound over the design domain e es ( x ) Estimated standard error at design point x [¯ e es ] max Maximum of normalized estimated standard error over the design domain F ( x ) ,F (1) ( x ) ,F (2) ( x ) Vectors of basis functions at x f j ( x ) Basis functions N Number of data points n 1 Number of basis functions in the regression model n 2 Number of missing basis functions in the regression model s 2 Error mean square x Design point X,X (1) ,X (2) Gramian (design) matrices x 1 ,x 2 ,...,x n Design variables x ( i ) 1 ,x ( i ) 2 ,...,x ( i ) n Design variables for i th design point y Vector of observed responses ˆ y ( x ) Response surface approximation at a design point...
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This note was uploaded on 06/06/2011 for the course EAS 4240 taught by Professor Peterifju during the Spring '08 term at University of Florida.
- Spring '08