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An Effective Scheme for Estimating a Smoother Parameter in the Method of Regularization

Received: 28 August 2013     Published: 20 October 2013
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Abstract

We had proposed a scheme for the surface approximation which consists of the estimation by the regularization method and the evaluation by generalized CV with an influence function [1]. We have to decide the value of the optimal smoother parameter which can minimize the value of the evaluation function. Among the models which have suitable parameters, we have to choose the best model using information criteria such as CV or generalized CV with an influence function (GCVIF). However, the method of GCVIF is not practical, because it requires the calculation of the inverse matrix of the hat matrix and the influence function [2]. Those calculations take a large amount of time when n increases. An efficient scheme which will take a small amount of time is required. On the other hand, there are many parameters which we have to decide.Those are the coefficients of the spline functions and the total number of knots, and positions of the parameters and a smoother parameter of the penalized term. The range of the total number of knots is decided by the total number of sample points. The range of the positions of the knots is decided by the area of the surface. However, it is difficult to estimate the range of the value of the smoother parameter. Therefore, we have to estimate it quite roughly. In this paper, we propose an effective method to estimate the range of the smoother parameter and consequently obtain the parameter precisely. We can reduce the calculation time which does not contribute to the selection of the optimal model and we can determine a more accurate and smoother parameter in a small amount of time.

Published in Applied and Computational Mathematics (Volume 2, Issue 6)
DOI 10.11648/j.acm.20130206.11
Page(s) 118-123
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

Spline Interpolation, Penalized Coefficient, Smoother Parameter, Method of Regularization, Cross-Validation

References
[1] Bao, H. and Fueda, K.(2013). "A New Method for the Model Selection in B-spline Surface Approximation with an Influence Function"Science Journal of Applied Mathematics and Statistics(submitted to).
[2] Bao, H. and Fueda, K.(2013). "A Method for Topographical Estimation of Lake Bottoms by B-spline Surface"American Journal of Theoretical and Applied Statistics Mathematics and Statistics2(4): 102-109.
[3] Cox, M.G.(1972). "The numerical evaluation of B-splines", J. Inst. Math. Appl., 10, pp.134-149.
[4] Cox, M.G.(1975). "An algorithm for spline interpolation", J. Inst. Math. Appl.,15, pp.95-108.
[5] de Boor, C.(1972). "On calculation with B-splines", J. Approx. Theory, 6, pp.50-62.
[6] Schoenberg, I. J., Whitney, A.(1953). "On Pólya frequency functions III", Trans. Amer. Math. Soc, Vol. 74. pp. 246-259, pp. 246-259.
[7] Good, I. J. and Gaskins, R.A.(1971). "Non parametric roughness penalties for probability densities", Biometrika, Vol. 58. pp. 255-277.
[8] 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.
[9] Green, P. J., Silverman, B. W.(1994). "Nonparametric Regression and Generalized Linear Models", Chapman and Hall, London.
[10] Umeyama, S. (1996). "Discontinuity extraction in regularization using robust statistics", Technical report of IEICE.,PRU95-217 (1996). pp. 9-16.
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    Hongmei Bao, Kaoru Fueda. (2013). An Effective Scheme for Estimating a Smoother Parameter in the Method of Regularization. Applied and Computational Mathematics, 2(6), 118-123. https://doi.org/10.11648/j.acm.20130206.11

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    Hongmei Bao; Kaoru Fueda. An Effective Scheme for Estimating a Smoother Parameter in the Method of Regularization. Appl. Comput. Math. 2013, 2(6), 118-123. doi: 10.11648/j.acm.20130206.11

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    AMA Style

    Hongmei Bao, Kaoru Fueda. An Effective Scheme for Estimating a Smoother Parameter in the Method of Regularization. Appl Comput Math. 2013;2(6):118-123. doi: 10.11648/j.acm.20130206.11

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  • @article{10.11648/j.acm.20130206.11,
      author = {Hongmei Bao and Kaoru Fueda},
      title = {An Effective Scheme for Estimating a Smoother Parameter in the Method of Regularization},
      journal = {Applied and Computational Mathematics},
      volume = {2},
      number = {6},
      pages = {118-123},
      doi = {10.11648/j.acm.20130206.11},
      url = {https://doi.org/10.11648/j.acm.20130206.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20130206.11},
      abstract = {We had proposed a scheme for the surface approximation which consists of the estimation by the regularization method and the evaluation by generalized CV with an influence function [1]. We have to decide the value of the optimal smoother parameter which can minimize the value of the evaluation function. Among the models which have suitable parameters, we have to choose the best model using information criteria such as CV or generalized CV with an influence function (GCVIF). However, the method of GCVIF is not practical, because it requires the calculation of the inverse matrix of the hat matrix and the influence function [2]. Those calculations take a large amount of time when n increases. An efficient scheme which will take a small amount of time is required. On the other hand, there are many parameters which we have to decide.Those are the coefficients of the spline functions and the total number of knots, and positions of the parameters and a smoother parameter of the penalized term. The range of the total number of knots is decided by the total number of sample points. The range of the positions of the knots is decided by the area of the surface. However, it is difficult to estimate the range of the value of the smoother parameter. Therefore, we have to estimate it quite roughly. In this paper, we propose an effective method to estimate the range of the smoother parameter and consequently obtain the parameter precisely. We can reduce the calculation time which does not contribute to the selection of the optimal model and we can determine a more accurate and smoother parameter in a small amount of time.},
     year = {2013}
    }
    

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    T1  - An Effective Scheme for Estimating a Smoother Parameter in the Method of Regularization
    AU  - Hongmei Bao
    AU  - Kaoru Fueda
    Y1  - 2013/10/20
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    DO  - 10.11648/j.acm.20130206.11
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    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    EP  - 123
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20130206.11
    AB  - We had proposed a scheme for the surface approximation which consists of the estimation by the regularization method and the evaluation by generalized CV with an influence function [1]. We have to decide the value of the optimal smoother parameter which can minimize the value of the evaluation function. Among the models which have suitable parameters, we have to choose the best model using information criteria such as CV or generalized CV with an influence function (GCVIF). However, the method of GCVIF is not practical, because it requires the calculation of the inverse matrix of the hat matrix and the influence function [2]. Those calculations take a large amount of time when n increases. An efficient scheme which will take a small amount of time is required. On the other hand, there are many parameters which we have to decide.Those are the coefficients of the spline functions and the total number of knots, and positions of the parameters and a smoother parameter of the penalized term. The range of the total number of knots is decided by the total number of sample points. The range of the positions of the knots is decided by the area of the surface. However, it is difficult to estimate the range of the value of the smoother parameter. Therefore, we have to estimate it quite roughly. In this paper, we propose an effective method to estimate the range of the smoother parameter and consequently obtain the parameter precisely. We can reduce the calculation time which does not contribute to the selection of the optimal model and we can determine a more accurate and smoother parameter in a small amount of time.
    VL  - 2
    IS  - 6
    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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