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

Hongmei Bao; Kaoru Fueda

doi:doi:10.11648/j.acm.20130206.11

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

Hongmei Bao, Kaoru Fueda

Published in Applied and Computational Mathematics (Volume 2, Issue 6)

Received: 28 August 2013 Accepted: 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), 2024. 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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TY  - JOUR
T1  - An Effective Scheme for Estimating a Smoother Parameter in the Method of Regularization
AU  - Hongmei Bao
AU  - Kaoru Fueda
Y1  - 2013/10/20
PY  - 2013
N1  - https://doi.org/10.11648/j.acm.20130206.11
DO  - 10.11648/j.acm.20130206.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 118
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

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). 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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ACS 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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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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - An Effective Scheme for Estimating a Smoother Parameter in the Method of Regularization
AU  - Hongmei Bao
AU  - Kaoru Fueda
Y1  - 2013/10/20
PY  - 2013
N1  - https://doi.org/10.11648/j.acm.20130206.11
DO  - 10.11648/j.acm.20130206.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 118
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  -

Copy | Download