Skip to main content
SpringerLink
Log in

Constructions of Φ p -Optimal Rotatable Designs on the Ball

  • Published:
Sankhya A Aims and scope Submit manuscript

Abstract

We discuss two fundamental methods for constructing Φ p -optimal approximate designs of order 3 on the unit ball. One construction is based on the theory of harmonic polynomials invariant under the Weyl group of type B, and another uses statistical tools such as block designs and orthogonal arrays. The paper provides a systematic treatment of a group-theoretic approach for constructing such designs, which has been traditionally used by Gaffke and Heiligers (1995a, b, c) and other researchers in design of experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bajnok, B. (2007). Orbits of the hyperoctahedral group as euclidean designs. J. Algebraic Combin., 25, 375–397.

    Article  MATH  MathSciNet  Google Scholar 

  • Bannai, E. and Bannai, E. (2006). On optimal tight 4-designs on 2 concentric spheres. European J. Combin., 27, 179–192.

    Article  MATH  MathSciNet  Google Scholar 

  • Broubaki, N. (2002). Lie Groups and Lie Algebras, chap 4–6. In Elements of Mathematics. Springer.

  • Delsarte, P. and Seidel, J.J. (1989). Fisher type inequalities for euclidean t-designs. Linear Algebra Appl., 114-115, 213–230.

    Article  Google Scholar 

  • Dette, H., Melas, V.B. and Pepelyshev, A. (2007). Optimal designs for statistical analysis with zernike polynomials. Statistics, 41, 453–470.

    Article  MATH  MathSciNet  Google Scholar 

  • Farrell, R.H., Kiefer, J. and Walbran, A. (1967). Optimum multivariate designs. In Proceedings of the Berkeley Symposium, vol 1, pp. 113–138.

  • Gaffke, N. and Heiligers, B. (1995a). Algorithms for optimal design with application to multiple polynomial regression. Metrika, 42, 173–190.

    Article  MATH  Google Scholar 

  • Gaffke, N. and Heiligers, B. (1995b). Computing optimal approximate invariant designs for cubic regression on multidimensional balls and cubes. J. Statist. Plann. Inference, 47, 347–376.

    Article  MATH  Google Scholar 

  • Gaffke, N. and Heiligers, B. (1995c). Optimal and robust invariant designs for cubic multiple regression. Metrika, 42, 29–48.

    Article  MATH  Google Scholar 

  • Gaffke, N. and Heiligers, B. (1998). Minimum support invariant designs for multiple cubic regression. J. Statist. Plann. Inference, 72, 229–245.

    Article  MATH  MathSciNet  Google Scholar 

  • Goethals, J.M. and Seidel, J.J. (1981). Cubature formulae, polytopes, and spherical designs. In The geometric vein. Springer, New York-Berlin.

  • Hedayat, A., Sloane, N.J.A. and Stufken, J. (1999). Orthogonal Arrays: Theory and Applications. Springer-Verlag, New-York.

    Book  MATH  Google Scholar 

  • Hughes, D.R. (1965). On t-designs and groups. Amer. J. Math., 87, 761–778.

    Article  MATH  MathSciNet  Google Scholar 

  • Karlin, S. and Studden, W.J. (1966). Tchebycheff systems: With applications in analysis and statistics. John Wiley & Sons, New York-London-Sydney.

    MATH  Google Scholar 

  • Khosrovshahi, G.B. and Laue, R. (2007). t-designs with t ≥ 3. In Handbook of Combinatorial Designs, 2nd end (C.J. Colbourn and J.H. Dinitz eds.). CRC Press, Boca Raton, USA, pp. 79–101.

  • Kiefer, J. (1960). Optimum experimental designs, V, with applications to systematic and rotatable designs. In Proceedings of the Berkeley Symposium vol 11, pp. 381–405.

  • Kôno, K. (1962). Optimal design for quadratic regression on k-cube. Memoirs of the Faculty of Science, Kyushu University, Series A 16, 114–122.

    Article  MathSciNet  Google Scholar 

  • Lau, T.S. (1988). D-optimal designs on the unit q-ball. J. Statist. Plann. Inference, 19, 299–315.

    Article  MATH  MathSciNet  Google Scholar 

  • Mukerjee, R. and Huda, S. (1990). Fourth-order rotatable designs: A-optimal measures. Statist. Probab. Lett., 10, 111–117.

    Article  MATH  MathSciNet  Google Scholar 

  • Neumaier, A. and Seidel, J.J. (1988). Discrete measures for spherical designs, eutactic stars and lattices. Nederl Akad Wetensch Indag Math 50, 321–334.

    Article  MATH  MathSciNet  Google Scholar 

  • Neumaier, A. and Seidel, J.J. (1992). Measures of strength 2e and optimal designs of degree e. Sankhyâ Series A, 54, 299–309.

    MATH  Google Scholar 

  • Nozaki, H. and Sawa, M. (2012). Note on cubature formulae and designs obtained from group orbits. Canad. J. Math., 64, 1359–1377.

    Article  MATH  MathSciNet  Google Scholar 

  • Pesotchinsky, L. (1978). ϕ p -optimal second order designs for symmetric regions. J. Statist. Plann. Inference, 2, 173–188.

    Article  MATH  MathSciNet  Google Scholar 

  • Pukelsheim, F. (2006). Optimal design of experiments. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

    Book  MATH  Google Scholar 

  • Sawa, M. and Xu, Y. (2013). On positive cubature rules on the simplex and isometric embeddings. To appear in Math Comp. arXiv:11083385v1.

  • Sobolev, S.L. (1962). Cubature formulas on the sphere which are invariant under transformations of finite rotation groups. Doklady Akademii Nauk SSSR, 146, 310–313.

    MathSciNet  Google Scholar 

  • Studden, W.J. (1982). Some robust type D-optimal designs in polynomial regression. J. Amer. Statist. Assoc., 77, 916–921.

    MATH  MathSciNet  Google Scholar 

  • Tchakaloff, V. (1957). Formules de cubatures mcaniques coefficients non ngatifs. Bull. des Sciences Mathématiques, 81, 123–134.

    MATH  MathSciNet  Google Scholar 

  • Victoir, N. (2004). Asymmetric cubature formulae with few points in high dimension for symmetric measures. SIAM J. Numer. Anal., 42, 209–227.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Aichi Prefectural University, Aichi, Japan

    Masatake Hirao

  2. Kobe University, Nada-ku, Japan

    Masanori Sawa & Masakazu Jimbo

Authors
  1. Masatake Hirao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masanori Sawa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Masakazu Jimbo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masatake Hirao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hirao, M., Sawa, M. & Jimbo, M. Constructions of Φ p -Optimal Rotatable Designs on the Ball. Sankhya A 77, 211–236 (2015). https://doi.org/10.1007/s13171-014-0053-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13171-014-0053-4

Keywords and phrases.

AMS (2000) subject classification.

Search

Navigation