TY - JOUR

T1 - Optimal designs for statistical analysis with Zernike polynomials

AU - Dette, Holger

AU - Melas, Viatcheslav B.

AU - Pepelyshev, Andrey

N1 - Funding Information: The support of the Deutsche Forschungsgemeinschaft (SFB 475, ‘Komplexitätsreduktion in multivariaten Datenstrukturen’) is gratefully acknowledged. The work by V.B. Melas was partly supported by the Russian Foundation of Basic Research (project 04-01-00519). The work of H. Dette was supported in part by a NIH grant award IR01GM072876:01A1. The authors are also grateful to Isolde Gottschlich, who typed parts of this paper with considerable technical expertise. The authors would also like to thank the reviewers for their constructive comments on an earlier version of this manuscript.

PY - 2007/12

Y1 - 2007/12

N2 - The Zernike polynomials arise in several applications such as optical metrology or image analysis on a circular domain. In the present paper, we determine optimal designs for regression models which are represented by expansions in terms of Zernike polynomials. We consider two estimation methods for the coefficients in these models and determine the corresponding optimal designs. The first one is the classical least squares method and p-optimal designs in the sense of Kiefer [Kiefer, J., 1974, General equivalence theory for optimum designs (approximate theory). Annals of Statistics, 2 849-879.] are derived, which minimize an appropriate functional of the covariance matrix of the least squares estimator. It is demonstrated that optimal designs with respect to Kiefer's p-criteria (p-) are essentially unique and concentrate observations on certain circles in the experimental domain. E-optimal designs have the same structure but it is shown in several examples that these optimal designs are not necessarily uniquely determined. The second method is based on the direct estimation of the Fourier coefficients in the expansion of the expected response in terms of Zernike polynomials and optimal designs minimizing the trace of the covariance matrix of the corresponding estimator are determined. The designs are also compared with the uniform designs on a grid, which is commonly used in this context.

AB - The Zernike polynomials arise in several applications such as optical metrology or image analysis on a circular domain. In the present paper, we determine optimal designs for regression models which are represented by expansions in terms of Zernike polynomials. We consider two estimation methods for the coefficients in these models and determine the corresponding optimal designs. The first one is the classical least squares method and p-optimal designs in the sense of Kiefer [Kiefer, J., 1974, General equivalence theory for optimum designs (approximate theory). Annals of Statistics, 2 849-879.] are derived, which minimize an appropriate functional of the covariance matrix of the least squares estimator. It is demonstrated that optimal designs with respect to Kiefer's p-criteria (p-) are essentially unique and concentrate observations on certain circles in the experimental domain. E-optimal designs have the same structure but it is shown in several examples that these optimal designs are not necessarily uniquely determined. The second method is based on the direct estimation of the Fourier coefficients in the expansion of the expected response in terms of Zernike polynomials and optimal designs minimizing the trace of the covariance matrix of the corresponding estimator are determined. The designs are also compared with the uniform designs on a grid, which is commonly used in this context.

KW - D-optimality

KW - E-optimality

KW - Image analysis

KW - Optimal design

KW - Zernike polynomials

UR - http://www.scopus.com/inward/record.url?scp=36749000026&partnerID=8YFLogxK

U2 - 10.1080/02331880701395395

DO - 10.1080/02331880701395395

M3 - Article

VL - 41

SP - 453

EP - 470

JO - Statistics

JF - Statistics

SN - 0233-1888

IS - 6

ER -