Regularized Minimum Length Method in Scattered Data Interpolation

Guang Y. Sheu

doi:doi:10.11648/j.acm.20140304.17

| Peer-Reviewed

Regularized Minimum Length Method in Scattered Data Interpolation

Guang Y. Sheu

Published in Applied and Computational Mathematics (Volume 3, Issue 4)

Received: 3 August 2014 Accepted: 15 August 2014 Published: 20 August 2014

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In an attempt of accumulating more experiences of interpolating scattered data using the minimum length method, this study chooses new kernel functions from the machine learning technique to implementing this minimum length method. But, consulting with the regularization theory, a regularized minimum length method is created by solving coefficient of it in a penalized least squares approximation problem. The purpose of creating this regularized minimum length method is responding to a pilot observation finding the instability of original minimum length method under dense interpolation points. Testing the regularized minimum length method finds that applying it is time-saving but its performance is comparable to the radial point interpolation with polynomial reproduction. Inverse multiquadric and rational quadric kernel functions are two preferred kernel function to perform the regularized minimum length method. In conclusion, the proposed regularized minimum length method can be a useful scattered data interpolation method.

Published in	Applied and Computational Mathematics (Volume 3, Issue 4)
DOI	10.11648/j.acm.20140304.17
Page(s)	163-170
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), 2014. Published by Science Publishing Group

Keywords

Regularized Minimum Length Method, Machine Learning, Scattered Data Interpolation

References

[1]	M. D. Buhmann, Radial Basis Function: Theory and Implementation, New York: Cambridge, 2009, 272.
[2]	W. Menke, Geophysical Data Analysis: Discrete Inverse Theory, 3rd ed., San Diego: Academic, 2012, 330.
[3]	X. Liu, Q. Zhou, J. Birkholzer, and W. A. Illman, “Geostatistical reduced-order models in underdetermined inverse problems,” Water Resour. Res., vol. 49, iss. 10, pp. 6587-6600, October 2013.
[4]	J. Kamm, Inversion and Joint Inversion of Electromagnetic and Potential Field Data, Ph. D. Thesis, Department of Earth Science, Uppsala University, Swedish, 2014, 108.
[5]	G. R. Liu, K. Y. Dai, and H. Y. Li, “A meshfree minimum length method for 2-D problems,” Computat, Mech., vol. 38, iss. 6, pp. 533-550, November 2006.
[6]	M. G. Genton, “Classes of kernels for machine learning: a statistics perspective,” J. Mach. Learn. Res., vol. 2, pp. 299-312, December 2001.
[7]	C. Micchelli, “Interpolation of scattered data distance matrices and conditionally positive definite functions,” Constr. Approx., vol. 2, iss. 1, no. 1, pp. 11-22, December, 1986.
[8]	E. Gilleland, Object Recogintion, Two-dimensional Kernel Smoothing: Using the R-package “smoothie” NCAR Technical Notes, National Center of Atmospheric Research, Colorado, USA, 24.
[9]	G. F.. Fasshauer. Meshfree Approximation Methods with MATLAB, Singapore, World Scientific, 2007, 520.
[10]	T. Evgeniou, M. Pontil, and T. Poggio, “Regularization networks and support vector machines,” Adv. Comput. Math. vol. 13, iss. 1, pp. 1-50, April 2000.
[11]	F. Girosi, “An equivalence between sparse approximation and support vector machines,” Neural Comput. vol. 10, iss. 6, pp. 1455-1480, August 1998.
[12]	J. A. Suykens, T. V. Gestel, J. D. Brabante, B. D. Moor, and J. A. Vandewalle. Least Squares Support Vector Machines, Singapore, World Scientific, 2002, 308.
[13]	Y.-S. Xiong and D. Saad, “Noise, regularizers, and unrealizable scenarios in online learning from restricted training sets,” Phys. Rev. E, vol. 64, iss. 1, 011919, June 2001.
[14]	K. V. Yuen. Bayesian Methods for Structural Dynamics and Civil Engineering, New York, Wiley, 2010, 320.
[15]	G. R. Liu and Y. T. Gu. A Introduction to Meshfree Methods and Their Programming, Netherland, Springer, 2005, 480.

Cite This Article

Plain Text BibTeX RIS

APA Style

Guang Y. Sheu. (2014). Regularized Minimum Length Method in Scattered Data Interpolation. Applied and Computational Mathematics, 3(4), 163-170. https://doi.org/10.11648/j.acm.20140304.17

Copy | Download

ACS Style

Guang Y. Sheu. Regularized Minimum Length Method in Scattered Data Interpolation. Appl. Comput. Math. 2014, 3(4), 163-170. doi: 10.11648/j.acm.20140304.17

Copy | Download

AMA Style

Guang Y. Sheu. Regularized Minimum Length Method in Scattered Data Interpolation. Appl Comput Math. 2014;3(4):163-170. doi: 10.11648/j.acm.20140304.17

Copy | Download

@article{10.11648/j.acm.20140304.17,
  author = {Guang Y. Sheu},
  title = {Regularized Minimum Length Method in Scattered Data Interpolation},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {4},
  pages = {163-170},
  doi = {10.11648/j.acm.20140304.17},
  url = {https://doi.org/10.11648/j.acm.20140304.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140304.17},
  abstract = {In an attempt of accumulating more experiences of interpolating scattered data using the minimum length method, this study chooses new kernel functions from the machine learning technique to implementing this minimum length method. But, consulting with the regularization theory, a regularized minimum length method is created by solving coefficient of it in a penalized least squares approximation problem. The purpose of creating this regularized minimum length method is responding to a pilot observation finding the instability of original minimum length method under dense interpolation points. Testing the regularized minimum length method finds that applying it is time-saving but its performance is comparable to the radial point interpolation with polynomial reproduction. Inverse multiquadric and rational quadric kernel functions are two preferred kernel function to perform the regularized minimum length method. In conclusion, the proposed regularized minimum length method can be a useful scattered data interpolation method.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Regularized Minimum Length Method in Scattered Data Interpolation
AU  - Guang Y. Sheu
Y1  - 2014/08/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140304.17
DO  - 10.11648/j.acm.20140304.17
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 163
EP  - 170
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140304.17
AB  - In an attempt of accumulating more experiences of interpolating scattered data using the minimum length method, this study chooses new kernel functions from the machine learning technique to implementing this minimum length method. But, consulting with the regularization theory, a regularized minimum length method is created by solving coefficient of it in a penalized least squares approximation problem. The purpose of creating this regularized minimum length method is responding to a pilot observation finding the instability of original minimum length method under dense interpolation points. Testing the regularized minimum length method finds that applying it is time-saving but its performance is comparable to the radial point interpolation with polynomial reproduction. Inverse multiquadric and rational quadric kernel functions are two preferred kernel function to perform the regularized minimum length method. In conclusion, the proposed regularized minimum length method can be a useful scattered data interpolation method.
VL  - 3
IS  - 4
ER  -

Copy | Download

Author Information

Guang Y. Sheu

Department of Accounting and Information Systems, Chang-Jung Christian University, Tainan, Taiwan

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Guang Y. Sheu. (2014). Regularized Minimum Length Method in Scattered Data Interpolation. Applied and Computational Mathematics, 3(4), 163-170. https://doi.org/10.11648/j.acm.20140304.17

Copy | Download

ACS Style

Guang Y. Sheu. Regularized Minimum Length Method in Scattered Data Interpolation. Appl. Comput. Math. 2014, 3(4), 163-170. doi: 10.11648/j.acm.20140304.17

Copy | Download

AMA Style

Guang Y. Sheu. Regularized Minimum Length Method in Scattered Data Interpolation. Appl Comput Math. 2014;3(4):163-170. doi: 10.11648/j.acm.20140304.17

Copy | Download

@article{10.11648/j.acm.20140304.17,
  author = {Guang Y. Sheu},
  title = {Regularized Minimum Length Method in Scattered Data Interpolation},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {4},
  pages = {163-170},
  doi = {10.11648/j.acm.20140304.17},
  url = {https://doi.org/10.11648/j.acm.20140304.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140304.17},
  abstract = {In an attempt of accumulating more experiences of interpolating scattered data using the minimum length method, this study chooses new kernel functions from the machine learning technique to implementing this minimum length method. But, consulting with the regularization theory, a regularized minimum length method is created by solving coefficient of it in a penalized least squares approximation problem. The purpose of creating this regularized minimum length method is responding to a pilot observation finding the instability of original minimum length method under dense interpolation points. Testing the regularized minimum length method finds that applying it is time-saving but its performance is comparable to the radial point interpolation with polynomial reproduction. Inverse multiquadric and rational quadric kernel functions are two preferred kernel function to perform the regularized minimum length method. In conclusion, the proposed regularized minimum length method can be a useful scattered data interpolation method.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Regularized Minimum Length Method in Scattered Data Interpolation
AU  - Guang Y. Sheu
Y1  - 2014/08/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140304.17
DO  - 10.11648/j.acm.20140304.17
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 163
EP  - 170
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140304.17
AB  - In an attempt of accumulating more experiences of interpolating scattered data using the minimum length method, this study chooses new kernel functions from the machine learning technique to implementing this minimum length method. But, consulting with the regularization theory, a regularized minimum length method is created by solving coefficient of it in a penalized least squares approximation problem. The purpose of creating this regularized minimum length method is responding to a pilot observation finding the instability of original minimum length method under dense interpolation points. Testing the regularized minimum length method finds that applying it is time-saving but its performance is comparable to the radial point interpolation with polynomial reproduction. Inverse multiquadric and rational quadric kernel functions are two preferred kernel function to perform the regularized minimum length method. In conclusion, the proposed regularized minimum length method can be a useful scattered data interpolation method.
VL  - 3
IS  - 4
ER  -

Copy | Download