Brazilian Journal of Probability and Statistics

Some refinements on Fedorov’s algorithms for constructing D-optimal designs

Luai Al Labadi

Full-text: Open access

Abstract

Well-known and widely used algorithms for constructing D-optimal designs are Fedorov’s sequential algorithm and Fedorov’s exchange algorithm. In this paper, we modify these two algorithms by adding or exchanging two or more points simultaneously at each step. This will significantly reduce the number of steps needed to construct a D-optimal design. We also prove the convergence of the proposed sequential algorithm to a D-optimal design. Optimal designs for rational regression are used as an illustration.

Article information

Source
Braz. J. Probab. Stat., Volume 29, Number 1 (2015), 53-70.

Dates
First available in Project Euclid: 30 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1414674775

Digital Object Identifier
doi:10.1214/13-BJPS228

Mathematical Reviews number (MathSciNet)
MR3299107

Zentralblatt MATH identifier
1329.62348

Keywords
D-optimal design Fedorov’s exchange algorithm Fedorov’s sequential algorithm General equivalence theorem

Citation

Al Labadi, Luai. Some refinements on Fedorov’s algorithms for constructing D-optimal designs. Braz. J. Probab. Stat. 29 (2015), no. 1, 53--70. doi:10.1214/13-BJPS228. https://projecteuclid.org/euclid.bjps/1414674775


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References

  • Al Labadi, L. and Zhen, W. (2010). Modified Wynn’s sequential algorithm for constructing D-optimal designs: Adding two points at a time. Communications in Statistics—Theory and Methods 39, 2818–2828.
  • Atkinson, A. C. and Donev, A. N. (1989). The construction of exact D-optimum experimental designs with application to blocking response surface designs. Biometrika 76, 515–526.
  • Atwood, C. L. (1973). Sequences converging to D-optimal designs of experiments. Annals of Mathematical Statistics 1, 342–352.
  • Bartel, R. G. and Sherbert, D. R. (2000). Introduction to Real Analysis. New York: Wiley.
  • Cook, R. D. and Nachtsheim, C. J. (1980). A comparison of algorithm for constructing exact D-optimal designs. Technometrics 22, 315–324.
  • Covey-Crump, P. A. K. and Silvey, S. D. (1970). Optimal regression designs with previous observations. Biometrika 57, 551–566.
  • Dykstra, O. Jr. (1971). The augmentation of experimental data to maximize $|\mathbf{X}'\mathbf{X}|$. Technometrics 13, 682–688.
  • Fedorov, V. V. (1972). Theory of Optimal Experiments. New York: Academic Press. Translated from the Russian and edited by W. J. Studden and E. M. Kilimmo. Probabilitty and Mathematical Statistics 12.
  • Harman, R. and Pronzato, L. (2007). Improvements on removing non-optimal support points in D-optimum design algorithms. Statistics & Probability Letters 77, 90–94.
  • Hebble, T. L. and Mitchell, T. J. (1971). Repairing response surface designs. Technometrics 14, 767–779.
  • John, R. C. S. and Daper, N. R. (1975). D-optimality for regression design: A review. Technometrics 17, 15–23.
  • Johnson, M. E. and Nachtsheim, C. J. (1983). Some guidelines for constructing exact D-optimal designs on convex design spaces. Technometrics 25, 271–277.
  • Kiefer, J. (1959). Optimum experimental designs. Journal of the Royal Statistical Society: Series B 21, 272–319.
  • Kiefer, J. (1961a). Optimum designs in regression problem, II. Annals of Mathematical Statistics 32, 298–325.
  • Kiefer, J. (1961b). Optimum experimental designs V, with applications to rotatable designs. Annals of Mathematical Statistics 32, 381–405.
  • Kiefer, J. (1962). Two more criteria equivalent to D-optimality of designs. Annals of Mathematical Statistics 33, 792–796.
  • Kiefer, J. (1971). The role of symmetry and approximation in exact design optimality. In Statistical Decision Theory and Related Topics (S. S. Gupta and J. Yackel, eds.) 109–118. New York: Academic Press.
  • Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Annals of Mathematical Statistics 2, 849–879.
  • Kiefer, J. and Wolfowitz, J. (1959). Optimum designs in regression problems. Annals of Mathematical Statistics 30, 271–294.
  • Kiefer, J. and Wolfowitz, J. (1960). The equvalence of two extremum problems. Canadian Journal of Mathematics 12, 363–366.
  • Liski, E. P., Mandal, N. K., Shah, K. R. and Sinha, B. K. (2002). Topics in Optimal Design. Lecture Notes in Statistics 163. New York: Springer.
  • Meyer, R. K. and Nachtsheim, C. J. (1995). The coordinate-exchange algorithm for constructing exact optimal experimental designs. Technometrics 37, 60–69.
  • Mitchell, T. J. and Miller, F. L. (1970) Use of “design repair” to construct designs for special linear models. Report ORNL-4661, pp. 130–131. Mathematics Division, Oak Ridge National Laboratory, Oak Ridge.
  • Mitchell, T. J. (1974). An algorithm for the construction of D-optimal designs. Technometrics 16, 203–211.
  • Nguyen, N. K. and Miller, A. J. (1992). A review of some exchange algorithms for constructing discrete D-optimal designs. Computational Statistics & Data Analysis 14, 489–498.
  • Pazman, A. (1974). A convergence theorem in the theory of D-optimum Experimental Designs. The Annals of Statistics 2, 216–218.
  • Pronzato, L. (2003). Removing non-optimal support points in D-optimum design algorithms. Statistics & Probability Letters 63, 223–228.
  • Silvey, S. D., Titterington, D. M. and Torsney, B. (1978). An algorithm for optimal designs on a finite design space. Communications in Statistics—Theory and Methods 14, 1379–1389.
  • Triefenbach, F. (2008). Design of experiments: The D-optimal approach and its implementation as a computer algorithm. Bachelor’s thesis in Information and Communication Technology, Department of Computing Science, UMEA Univ. Sweden, Department of Engineering and Business Sciences, South Westphalia Univ. Applied Sciences, Germany.
  • Tsay, J. Y. (1976). On the sequential construction of D-optimal designs. Journal of the American Statistical Association 71, 671–674.
  • Wynn, H. P. (1970). The sequential generation of D-optimum experimental designs. Annals of Mathematical Statistics 41, 1655–1664.
  • Yu, Y. (2011). D-optimal designs via a cocktail algorithm. Statistics and Computing 21, 475–481.