Abstract
Using the characteristic polynomial coefficients of the inverse of the information matrix, design criteria can be defined between A- and D-optimality (López-Fidalgo and Rodríguez-Díaz, 1998). With a slight modification of the classical algorithms, the gradient expression allows us to find some optimal characteristic designs for polynomial regression. We observe that these designs are a smooth transition from A- to D-optimal designs. Moreover, for some of these optimal designs, the efficiencies for both criteria, A- and D-optimality, are quite good.
Nice relationships emerge when plotting the support points of these optimal designs against the number of parameters of the model. In particular, following the ideas developed by Pukelsheim and Torsney (1991), we have considered A-optimality. Another mathematical expression can be given for finding A-optimal support points using nonlinear regression. This could be very useful for obtaining optimal designs for the other characteristic criteria.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bellhouse, R. and Herzberg, A. M. (1984). Equally spaced design points in polynomial regression: a comparison of systematic sampling methods with the optimal design of experiments. Canadian Journal of Statistics 12, 77–90.
Dette, H. (1994). Discrimination designs for polynomial regression on compact intervals. Annals of Statistics 22, 890–903.
Dette, H. and Studden, W. J. (1994). Optimal designs with respect to Elfving’s partial minimax criterion in polynomial regression. Ann. Inst. Statist. Math. 46, 389–403.
Dette, H. and Wong, W.K. (1995). On G-efficiency calculation for polynomial models. Annals of Statistics 23, 2081–2101.
Dette, H. and Wong, W.K. (1996). Robust optimal extrapolation designs. Biometrika 83, 667–680.
Fedorov, V.V. (1972). Theory of Optimal Experiments. New York: Academic Press.
Hoel, P. G. and Levine, A. (1964). Optimal spacing and weighting in polynomial prediction. Ann. Math. Statist. 35, 1553–1560.
Kiefer, J. and Studden, W. J. (1976). Optimal designs for large degree polynomial regression. Annals of Statistics 4, 1113–1123.
Kiefer, J. and Wolfowitz, J. (1959). Optimum designs in regression problems. Ann. Math. Statist. 30, 271–294.
López-Fidalgo, J. and Rodríguez-Díaz, J.M. (1998). Characteristic Polynomial Criteria in Optimal Experimental Design. In MODA 5-Advances in Model-Oriented Data Analysis and Experimental Design Eds A.C. Atkinson, L. Pronzato and H.P. Wynn, pp. 31–38. Heidelberg: Physica-Verlag.
Pázman, A. (1986). Foundations of Optimum Experimental Design. Dordrecht: Reidel.
Pukelsheim, F. and Studden, W. J. (1993). E-optimal designs for polynomial regression. Annals of Statistics 21, 402–415.
Pukelsheim, F. and Torsney, B. (1991). Optimal weights for experimental designs on linearly independent support points. Annals of Statistics 19, 1614–1625.
Studden, W. J. (1980). D s -optimal designs for polynomial regression using continued fractions. Annals of Statistics 8, 1132–1141.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Rodríguez-Díaz, J.M., López-Fidalgo, J. (2001). Optimal Characteristic Designs for Polynomial Models. In: Atkinson, A., Bogacka, B., Zhigljavsky, A. (eds) Optimum Design 2000. Nonconvex Optimization and Its Applications, vol 51. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3419-5_12
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3419-5_12
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4846-5
Online ISBN: 978-1-4757-3419-5
eBook Packages: Springer Book Archive