Optimal designs for statistical analysis with Zernike polynomials
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Date
2006-08-07T12:11:16Z
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Abstract
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
Φp-optimal designs in the sense of Kiefer (1974) 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 Φp-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.
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Keywords
D-optimality, E-optimality, Image analysis, Optimal design, Zernike polynomials