Relationship Between the Fixed Point Theorem and the EM Algorithm

Authors

  • Ahsene Lanani Department of Mathematics, Faculty of Exact Sciences, University of Freres Mentouri Constantine 1, 25000

DOI:

https://doi.org/10.26713/jims.v10i4.1064

Keywords:

EM algorithm, Fixed point, Linear model, Nonlinear equation

Abstract

When we are confronted with solving nonlinear equations which do not admit explicit solutions, we must use approximate methods based on iterative processes or algorithms. One of the best known iterative methods is the fixed point theorem, often applied in analysis or algebra. In our case, we will apply this method in a stochastic context. By means of this application, we show the relationship between this method and the EM algorithm, which is an iterative process, often applied in statistics.

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References

R. P. Agarwal, M. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press (2001).

A. P. Dempster, N. M. Laird and D. B. Rubin, Maximum likelihood with incomplete data via the E-M algorithm, J. Roy. Stat. Soc. B39 (1977), 1 – 38.

J. Dugundji and A. Granas, Fixed Point Theory, Springer-Verlag (2003).

V. Florin, Parameter convergence for EM and MM algorithms, Statistica Sinica 15 (2005), 831 – 840.

J. L. Foulley and D. A. Van Dyk, The PX EM algorithm for fast fitting of Henderson's mixed model, Genetics Selection Evolution 32 (2000), 143 – 163.

J. L. Foulley, F. Jaffrézic and C. R. Granié, EM-REML estimation of covariance parameters in gaussian mixed models for longitudinal data analysis, Genetics Selection Evolution 32 (2000), 129 – 141.

F. W. Homer and N. Peng, Anderson acceleration for fixed-point iterations, SIAM J. Num. Anal. 49(4) (2011), 1715 – 1735.

W. A. Kirk and M. A. Khamsi, An Introduction to Metric Spaces and Fixed Point Theory, John Wiley, New York (2001).

W. A. Kirk and S. Brailey, Handbook of Metric Fixed Point Theory, Springer-Verlag (2001).

K. Kubjas, E. Robeva and B. Sturmfels, Fixed points of the EM algorithm and nonnegative rank boundaries, The Annals of Stat. 43(1) (2015), 422 – 461.

N. M. Laird and J. H. Ware, Random effect models for longitudinal data, Biometrics 38 (1982), 963 – 974.

N. M. Laird, N. Lange and D. Stram, Maximum likelihood computations with repeated measures: application of The EM algorithm, J. Amer. Statist. Assoc. 82(397) (1987), 97 – 105.

M. J. Lindstrom and D. M. Bates, Newton-Raphson and EM algorithm for linear mixed effect models for repeated-measures data, J. Amer. Statist. Assoc. 83(404) (1988), 1014 – 1022.

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Published

2018-12-31
CITATION

How to Cite

Lanani, A. (2018). Relationship Between the Fixed Point Theorem and the EM Algorithm. Journal of Informatics and Mathematical Sciences, 10(4), 697–702. https://doi.org/10.26713/jims.v10i4.1064

Issue

Vol. 10 No. 4 (2018)

Section

Research Articles

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