On the Feasibility of Semi-algebraic Sets in Poisson Regression

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9725)

Abstract

Designing experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. The local optimality approach is to study the regions in parameter space where a given design is optimal. In many situations these regions are semi-algebraic. We investigate regions of optimality using computer tools such as yalmip, qepcad, and Mathematica.

Keywords

Algebraic statistics Optimal experimental design Poisson regression Semi-algebraic sets 

References

  1. 1.
    Abbott, J., Bigatti, A.M., Lagorio, G.: CoCoA-5: a system for doing Computations in Commutative Algebra. http://cocoa.dima.unige.it
  2. 2.
    MOSEK ApS, The MOSEK optimization toolbox for MATLAB. Version 7.1 (revision 28) (2015)Google Scholar
  3. 3.
    Basu, S., Pollack, R.D., Roy, M.-F.: Algorithms in Real Algebraic Geometry, vol. 10. Springer, Heidelberg (2006)MATHGoogle Scholar
  4. 4.
    Brown, C.W.: QEPCAD B: a program for computing with semi-algebraic sets using CADs. ACM SIGSAM Bull. 37(4), 97–108 (2003)CrossRefMATHGoogle Scholar
  5. 5.
    De Loera, J.A., Malkin, P.N., Parrilo, P.A.: Computation with polynomial equations and inequalities arising in combinatorial optimization. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, pp. 447–481. Springer, New York (2012)CrossRefGoogle Scholar
  6. 6.
    Doebler, A., Holling, H.: A processing speed test based on rule-based item generation: an analysis with the Rasch Poisson counts model. In: Learning and Individual Differences (2015). doi:10.1016/j.lindif.2015.01.013
  7. 7.
    Graßhoff, U., Holling, H., Schwabe, R.: Optimal design for count data with binary predictors in item response theory. In: Ucinski, D., Atkinson, A.C., Patan, M. (eds.) Advances in Model-Oriented Design and Analysis, pp. 117–124. Springer, Switzerland (2013)CrossRefGoogle Scholar
  8. 8.
    Graßhoff, U., Holling, H., Schwabe, R.: Optimal design for the Rasch Poisson counts model with multiple binary predictors, Technical report (2014)Google Scholar
  9. 9.
    Graßhoff, U., Holling, H., Schwabe, R.: Poisson model with three binary predictors: when are saturated designs optimal? In: Steland, A., Rafajłowicz, E., Szajowski, K. (eds.) Stochastic Models, Statistics and Their Applications, pp. 75–81. Springer International Publishing, Switzerland (2015)CrossRefGoogle Scholar
  10. 10.
    Grayson, D.R., Stillman, M.E.: Macaulay2, a software systemfor research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
  11. 11.
    Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3.0 – a computer algebra system for polynomial computations. In: Kerber, M., Kohlhase, M. (eds.) Symbolic Computation and Automated Reasoning, The Calculemus-2000 Symposium, pp. 227–233. A. K. Peters Ltd., Natick (2001)Google Scholar
  12. 12.
    Kahle, T., Oelbermann, K.-F., Schwabe, R.: Algebraic geometry of Poisson regression. J. Algebraic Stat. (2015, to appear). arXiv:1510.05261
  13. 13.
    Kiefer, J., Wolfowitz, J.: The equivalence of two extremum problems. Can. J. Math. 12(5), 363–365 (1960)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Löfberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004)Google Scholar
  15. 15.
    Pukelsheim, F.: Optimal design of experiments. In: Classics in Applied Mathematics, vol. 50. SIAM (2006)Google Scholar
  16. 16.
    Wolfram Research, Mathematica 10.4.1 (2016)Google Scholar
  17. 17.
    Richardson, D.G., Krandick, W.: Compiler-enforced memory semantics in the SACLIB computer algebra library. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 330–343. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Otto-von-Guericke Universität MagdeburgMagdeburgGermany

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