L-optimal designs for a trigonometric Fourier regression model with no intercept
DOI:
https://doi.org/10.21638/spbu01.2022.107Abstract
This paper is devoted to the problem of constructing L-optimal designs for a trigonometric Fourier regression model with no intercept. The paper considers diagonal matrices L with a combination of zeros and ones on the main diagonal. It is shown that in the case when L = I (i. e., when the unit matrix is chosen as the matrix L), the L-optimal design coincides with the D-optimal one. In the more general case (when some diagonal elements are equal to zero), it is shown that the dimension of the problem can be reduced if the optimal design is symmetric. The obtained results are illustrated by the example of the problem of constructing two L-optimal designs for the trigonometric model of order 12, which is reduced to the problem of constructing designs for models of order 3 and 4 correspondingly.Keywords:
L-optimal designs, c-optimal designs, optimal designs for estimating the individual coefficients,, trigonometric regression models with no intercept
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Литература
1. Ермаков С.М.,Жиглявский А.А. Математическая теория оптимального эксперимента. Москва, Наука (1987).
2. Atkinson A., Bogacka B., Bogacki M. D- and T-optimum designs for the kinetics of a reversible chemical reaction. Chemometrics and Intelligent Laboratory Systems 43, 185–198 (1998). https://doi.org/10.1016/S0169-7439(98)00046-X
3. Asprey S., Macchietto S. Statistical tools for optimal dynamic model building. Computers and Chemical Engineering 24, 1261–1267 (2000). https://doi.org/10.1016/S0098-1354(00)00328-8
4. Ucinski D., Bogacka B. T-optimum designs for discrimination between two multiresponse dynamic models. Journal of the Royal Statistical Society, Ser. B 67, 3–18 (2005). https://doi.org/10.1111/j.1467-9868.2005.00485.x
5. Atkinson A.C., Fedorov V.V. The designs of experiments for discriminating between two rival models. Biometrika 62, 57–70 (1975). https://doi.org/10.2307/2334487
6. Atkinson A.C., Fedorov V.V. Optimal design: Experiments for discriminating between several models. Biometrika 62, 289–303 (1975). https://doi.org/10.2307/2335364
7. Wiens D.P. Robust discrimination designs. Journal of the Royal Statistical Society, Ser. B 71, 805–829 (2009).
8. Tommasi C., L´opez-Fidalgo J. Bayesian optimum designs for discriminating between models with any distribution. Computational Statistics & Data Analysis 54 (1), 143–150 (2010). https://doi.org/10.1016/j.csda.2009.07.022
9. Waterhouse T., Woods D., Eccleston J., Lewis S. Design selection criteria for discrimination/ estimation for nested models and a binomial response. Journal of Statistical Planning and Inference 138, 132–144 (2008). https://doi.org/10.1016/j.jspi.2007.05.017
10. Dette H., Melas V.B., Shpilev P. T-optimal designs for discrimination between two polynomial models. Annals of Statistics 40 (1), 188–205 (2012).
11. Dette H.,Melas V.B., Shpilev P. T-optimal discriminating designs for Fourier regression models. Computational Statistics and Data Analysis 113, 196–206 (2017). https://doi.org/10.1214/11-AOS956
12. Dette H., Melas V.B., Shpilev P. Robust T-optimal discriminating designs. Annals of Statistics 41 (4), 1693–1715 (2013). https://doi.org/10.1214/13-AOS1117
13. Gaffke N. Further characterizations of design optimality and admissibility for partial parameter estimation on linear regression. Ann. Statist. 15 (3), 942–957 (1987).
14. Kiefer J.C. General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2, 849–879 (1974). https://doi.org/10.1214/aos/1176342810
15. Pukelsheim F. Optimal Design of Experiments. Philadelphia, SIAM (2006).
16. M¨uller C., P´azman A. Applications of necessary and sufficient conditions for maximin efficient designs. Metrika 48, 1–19 (1998). https://doi.org/10.1007/S001840050001
17. Dette H., Neugebauer H. Bayesian optimal one point designs for one parameter nonlinear models. Journal of Statistical Planning and Inference 52, 17–31 (1996). https://doi.org/10.1016/0378- 3758(95)00104-2
18. Dette H., Neugebauer H. Bayesian D-optimal designs for exponential regression models. Journal of Statistical Planning and Inference 60, 331–349 (1997). https://doi.org/10.1016/S0378- 3758(96)00131-0
19. Dette H. Designing experiments with respect to ”standardized” optimality criteria. Journal of the Royal Statistical Society, Ser. B 59, 97–110 (1997). https://doi.org/10.1111/1467-9868.00056
20. Dette H.,Melas V.B., Shpilev P. Optimal designs for trigonometric regression models. Journal of Statistical Planning and Inference 141 (3), 1343–1353 (2011). https://doi.org/10.1016/j.jspi.2010.10.010
21. Шпилев П.В. Теорема эквивалентности для вырожденных L-оптимальных планов. Вест- ник Санкт-Петербургского университета. Математика. Механика. Астрономия 2, вып. 1, 68– 74 (2015).
References
1. Ermakov S.M., Zhiglyavskii A.A. Mathematical theory of optimal experiment. Moscow, Nauka Publ. (1987). (In Russian)
2. Atkinson A., Bogacka B., Bogacki M. D- and T-optimum designs for the kinetics of a reversible chemical reaction. Chemometrics and Intelligent Laboratory Systems 43, 185–198 (1998). https://doi.org/10.1016/S0169-7439(98)00046-X
3. Asprey S., Macchietto S. Statistical tools for optimal dynamic model building. Computers and Chemical Engineering 24, 1261–1267 (2000). https://doi.org/10.1016/S0098-1354(00)00328-8
4. Ucinski D., Bogacka B. T-optimum designs for discrimination between two multiresponse dynamic models. Journal of the Royal Statistical Society, Ser. B 67, 3–18 (2005). https://doi.org/10.1111/j.1467-9868.2005.00485.x
5. Atkinson A.C., Fedorov V.V. The designs of experiments for discriminating between two rival models. Biometrika 62, 57–70 (1975). https://doi.org/10.2307/2334487
6. Atkinson A.C., Fedorov V.V. Optimal design: Experiments for discriminating between several models. Biometrika 62, 289–303 (1975). https://doi.org/10.2307/2335364
7. Wiens D.P. Robust discrimination designs. Journal of the Royal Statistical Society, Ser. B 71, 805–829 (2009).
8. Tommasi C., L´opez-Fidalgo J. Bayesian optimum designs for discriminating between models with any distribution. Computational Statistics & Data Analysis 54 (1), 143–150 (2010). https://doi.org/10.1016/j.csda.2009.07.022
9. Waterhouse T., Woods D., Eccleston J., Lewis S. Design selection criteria for discrimination/ estimation for nested models and a binomial response. Journal of Statistical Planning and Inference 138, 132–144 (2008). https://doi.org/10.1016/j.jspi.2007.05.017
10. Dette H., Melas V.B., Shpilev P. T-optimal designs for discrimination between two polynomial models. Annals of Statistics 40 (1), 188–205 (2012).
11. Dette H.,Melas V.B., Shpilev P. T-optimal discriminating designs for Fourier regression models. Computational Statistics and Data Analysis 113, 196–206 (2017). https://doi.org/10.1214/11-AOS956
12. Dette H., Melas V.B., Shpilev P. Robust T-optimal discriminating designs. Annals of Statistics 41 (4), 1693–1715 (2013). https://doi.org/10.1214/13-AOS1117
13. Gaffke N. Further characterizations of design optimality and admissibility for partial parameter estimation on linear regression. Annals of Statistics 15 (3), 942–957 (1987).
14. Kiefer J.C. General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2, 849–879 (1974). https://doi.org/10.1214/aos/1176342810
15. Pukelsheim F. Optimal Design of Experiments. Philadelphia, SIAM (2006).
16. M¨uller C., P´azman A. Applications of necessary and sufficient conditions for maximin efficient designs. Metrika 48, 1–19 (1998). https://doi.org/10.1007/S001840050001
17. Dette H., Neugebauer H. Bayesian optimal one point designs for one parameter nonlinear models. Journal of Statistical Planning and Inference 52, 17–31 (1996). https://doi.org/10.1016/0378- 3758(95)00104-2
18. Dette H., Neugebauer H. Bayesian D-optimal designs for exponential regression models. Journal of Statistical Planning and Inference 60, 331–349 (1997). https://doi.org/10.1016/S0378- 3758(96)00131-0
19. Dette H. Designing experiments with respect to ”standardized” optimality criteria. Journal of the Royal Statistical Society 59, 97–110 (1997). https://doi.org/10.1111/1467-9868.00056
20. Dette H.,Melas V.B., Shpilev P. Optimal designs for trigonometric regression models. Journal of Statistical Planning and Inference 141 (3), 1343–1353 (2011). https://doi.org/10.1016/j.jspi.2010.10.010
21. Shpilev P.V. Equivalence theorem for singular L-optimal designs. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 2, iss. 1, 68–74 (2015). (In Russian) [Eng. transl.: Vestnik StPetersburg University. Mathematics 48 (1), 29–34 (2015). https://doi.org/10.3103/S1063454115010094].
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Published
2022-04-10
How to Cite
Melas, V. B., & Shpilev, P. V. (2022). L-optimal designs for a trigonometric Fourier regression model with no intercept. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 9(1), 64–75. https://doi.org/10.21638/spbu01.2022.107
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
