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

Hongmei Bao; Kaoru Fueda

doi:doi:10.11648/j.sjams.20130105.11

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

Hongmei Bao, Kaoru Fueda

Published in Science Journal of Applied Mathematics and Statistics (Volume 1, Issue 5)

Received: 27 August 2013 Accepted: 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), 2024. 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)

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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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TY  - JOUR
T1  - A New Method for the Model Selection in B-Spline Surface Approximation with an Influence Function
AU  - Hongmei Bao
AU  - Kaoru Fueda
Y1  - 2013/10/20
PY  - 2013
N1  - https://doi.org/10.11648/j.sjams.20130105.11
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
SP  - 38
EP  - 46
PB  - Science Publishing Group
SN  - 2376-9513
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

Hongmei Bao

Graduate School of Environmental Science, Okayama University, Okayama, Japan
Kaoru Fueda

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

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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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - A New Method for the Model Selection in B-Spline Surface Approximation with an Influence Function
AU  - Hongmei Bao
AU  - Kaoru Fueda
Y1  - 2013/10/20
PY  - 2013
N1  - https://doi.org/10.11648/j.sjams.20130105.11
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
SP  - 38
EP  - 46
PB  - Science Publishing Group
SN  - 2376-9513
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  -

Copy | Download