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A New Method for the Model Selection in B-Spline Surface Approximation with an Influence Function

Received: 27 August 2013     Published: 20 October 2013
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Abstract

In model selection, the most effective method requires much time.The analysis of the bivariate B-spline model with a penalized term has many difficulties.It has many factors and parameters such the number of the knots, the locations of those knots, number of B-spline functions and the value of the smoothing parameter of the penalized term.For the determination of the model we have to compare a large amount of the combinations of those parameters. Various information criteria are considered and the cross validation (CV) criterion is excellent but it requires a large amount of computational costs. The effect of the influence function and the techniques of the generalized cross validation (CV) are considered. The influence function is related to the first term of a Taylor expansion. Some alternative methods are tested and a new method is proposed. For the verification of this method theoretical proof and the computational results are shown.

Published in Science Journal of Applied Mathematics and Statistics (Volume 1, Issue 5)
DOI 10.11648/j.sjams.20130105.11
Page(s) 38-46
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2013. Published by Science Publishing Group

Keywords

B-Spline Surface, Generalized Information Criterion, Influence Function, Generalized Cross-Validation, Cross-Validation, Kullback-Leibler Divergence, Surface Model Selection

References
[1] Konishi, S., Kitagawa, G. (1996). "Generalised information criteria in model selection.",Biometrika, 83, 875–890.
[2] Stone,M., "Cross-validatory choice and assessment of statistical predictions (with discussion)", Journal of the Royal Statistical Society, Series B, 36 (1974), 111–147.
[3] Ueki, M. and Fueda, K. (2010)."Optimal Tuning Parameter Estimaiton In Maximum Penalized Likelihood Method", Annals of the Institute of Statistical Mathematics, 62, 413-438.
[4] Konishi, S., Kitagawa, G. (2008). "Information Criteria and Statistical Modeling", Springer Science+Business Media, LLC.
[5] Good, I. J. and Gaskins, R.A.(1971). "Non parametric roughness penalties for probability densities", Biometrika, Vol. 58. pp. 255-277.
[6] Good, I. J. and Gaskins, R.A.(1980). "Density estimation and bump hunting by the penalized likelihood method exemplified by scattering and meteorite data", Journal of American Standard Association, Vol. 75. pp. 42-56.
[7] Green, P. J., Silverman, B. W.(1994). "Nonparametric Regression and Generalized Linear Models", Chapman and Hall, London.
[8] Umeyama, S. (1996). "Discontinuity extraction in regularization using robust statistics", Technical report of IEICE.,PRU95-217 (1996). pp. 9-16.
[9] Cox, M.G.(1972). "The numerical evaluation of B-splines", J. Inst. Math. Appl., 10, pp.134-149.
[10] Cox, M.G.(1975). "An algorithm for spline interpolation", J. Inst. Math. Appl.,15, pp.95-108.
[11] de Boor, C.(1972). "On calculation with B-splines", J. Approx. Theory, 6, pp.50-62.
[12] Schoenberg, I. J., Whitney, A.(1953). "On Pólya frequency functions III", Trans. Amer. Math. Soc, Vol. 74. pp. 246-259, pp. 246-259.
[13] Ueki, M. and Fueda, K.(2006)."Over close model problem and a modification of information criteria", Proceedings of The 28th symposium of Japanese Society of Applied Statistics,pp.7-10. (in Japanese)
Cite This Article
  • APA Style

    Hongmei Bao, Kaoru Fueda. (2013). A New Method for the Model Selection in B-Spline Surface Approximation with an Influence Function. Science Journal of Applied Mathematics and Statistics, 1(5), 38-46. https://doi.org/10.11648/j.sjams.20130105.11

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    ACS Style

    Hongmei Bao; Kaoru Fueda. A New Method for the Model Selection in B-Spline Surface Approximation with an Influence Function. Sci. J. Appl. Math. Stat. 2013, 1(5), 38-46. doi: 10.11648/j.sjams.20130105.11

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    AMA Style

    Hongmei Bao, Kaoru Fueda. A New Method for the Model Selection in B-Spline Surface Approximation with an Influence Function. Sci J Appl Math Stat. 2013;1(5):38-46. doi: 10.11648/j.sjams.20130105.11

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  • @article{10.11648/j.sjams.20130105.11,
      author = {Hongmei Bao and Kaoru Fueda},
      title = {A New Method for the Model Selection in B-Spline Surface Approximation with an Influence Function},
      journal = {Science Journal of Applied Mathematics and Statistics},
      volume = {1},
      number = {5},
      pages = {38-46},
      doi = {10.11648/j.sjams.20130105.11},
      url = {https://doi.org/10.11648/j.sjams.20130105.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20130105.11},
      abstract = {In model selection, the most effective method requires much time.The analysis of the bivariate B-spline model with a penalized term has many difficulties.It has many factors and parameters such the number of the knots, the locations of those knots, number of B-spline functions and the value of the smoothing parameter of the penalized term.For the determination of the model we have to compare a large amount of the combinations of those parameters. Various information criteria are considered and the cross validation (CV) criterion is excellent but it requires a large amount of computational costs. The effect of the influence function and the techniques of the generalized cross validation (CV) are considered. The influence function is related to the first term of a Taylor expansion. Some alternative methods are tested and a new method is proposed. For the verification of this method theoretical proof and the computational results are shown.},
     year = {2013}
    }
    

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    AU  - Hongmei Bao
    AU  - Kaoru Fueda
    Y1  - 2013/10/20
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    DO  - 10.11648/j.sjams.20130105.11
    T2  - Science Journal of Applied Mathematics and Statistics
    JF  - Science Journal of Applied Mathematics and Statistics
    JO  - Science Journal of Applied Mathematics and Statistics
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    PB  - Science Publishing Group
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    UR  - https://doi.org/10.11648/j.sjams.20130105.11
    AB  - In model selection, the most effective method requires much time.The analysis of the bivariate B-spline model with a penalized term has many difficulties.It has many factors and parameters such the number of the knots, the locations of those knots, number of B-spline functions and the value of the smoothing parameter of the penalized term.For the determination of the model we have to compare a large amount of the combinations of those parameters. Various information criteria are considered and the cross validation (CV) criterion is excellent but it requires a large amount of computational costs. The effect of the influence function and the techniques of the generalized cross validation (CV) are considered. The influence function is related to the first term of a Taylor expansion. Some alternative methods are tested and a new method is proposed. For the verification of this method theoretical proof and the computational results are shown.
    VL  - 1
    IS  - 5
    ER  - 

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Author Information
  • Graduate School of Environmental Science, Okayama University, Okayama, Japan

  • Graduate School of Environmental and Life Science, Okayama University, Okayama, Japan

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