A weighted online regularization for a fully nonparametric model with heteroscedasticity

Lei Hu; Lei Hu

doi:10.3934/math.20231381

AIMS Mathematics

2023, Volume 8, Issue 11: 26991-27008. doi: 10.3934/math.20231381

Previous Article Next Article

Research article Special Issues

A weighted online regularization for a fully nonparametric model with heteroscedasticity

Lei Hu ^,

Shenzhen Institute of Information Technology, Shenzhen 518172, China

Received: 21 July 2023 Revised: 20 August 2023 Accepted: 28 August 2023 Published: 22 September 2023
MSC : 41A15, 62G05, 93E24

In this paper, combining B-spline function and Tikhonov regularization, we propose an online identification approach for reconstructing a smooth function and its derivative from scattered data with heteroscedasticity. Our methodology offers the unique advantage of enabling real-time updates based on new input data, eliminating the reliance on historical information. First, to address the challenge of heteroscedasticity and computation cost, we employ weight coefficients along with a judiciously chosen set of knots for interpolation. Second, a reasonable approach is provided to select weight coefficients and the regularization parameter in objective functional. Finally, We substantiate the efficacy of our approach through a numerical example and demonstrate its applicability in solving inverse problems. It is worth mentioning that the algorithm not only ensures the calculation efficiency, but also trades the data accuracy through the data volume.
- B-spline function,
- heteroscedasticity,
- nonparametric identification,
- weight coefficients
Citation: Lei Hu. A weighted online regularization for a fully nonparametric model with heteroscedasticity[J]. AIMS Mathematics, 2023, 8(11): 26991-27008. doi: 10.3934/math.20231381

Related Papers:

Abstract

In this paper, combining B-spline function and Tikhonov regularization, we propose an online identification approach for reconstructing a smooth function and its derivative from scattered data with heteroscedasticity. Our methodology offers the unique advantage of enabling real-time updates based on new input data, eliminating the reliance on historical information. First, to address the challenge of heteroscedasticity and computation cost, we employ weight coefficients along with a judiciously chosen set of knots for interpolation. Second, a reasonable approach is provided to select weight coefficients and the regularization parameter in objective functional. Finally, We substantiate the efficacy of our approach through a numerical example and demonstrate its applicability in solving inverse problems. It is worth mentioning that the algorithm not only ensures the calculation efficiency, but also trades the data accuracy through the data volume.

References

[1]	J. Cheng, X. Z. Jia, Y. B. Wang, Numerical differentiation and its applications, Inverse Probl. Sci. En., 15 (2007), 339–357. https://doi.org/10.1080/17415970600839093 doi: 10.1080/17415970600839093
[2]	M. Hanke, O. Scherzer, Inverse problems light: numerical differentiation, Am. Math. Mon., 108 (2001), 512–521. https://doi.org/10.1080/00029890.2001.11919778 doi: 10.1080/00029890.2001.11919778
[3]	D. H. Xu, Y. H. Xu, M. B. Ge, Q. F. Zhang, Models and numerics for differential equations and inverse problems, Beijing: Science Press, 2021.
[4]	P. Craven, G. Wahba, Smoothing noisy data with spline functions, Numer. Math., 31 (1978), 377–403. https://doi.org/10.1007/BF01404567 doi: 10.1007/BF01404567
[5]	D. L. Ragozin, Error bounds for derivative estimates based on spline smoothing of exact or noisy data, J. Approx. Theory, 37 (1983), 335–355. https://doi.org/10.1016/0021-9045(83)90042-4 doi: 10.1016/0021-9045(83)90042-4
[6]	J. P. Kaipio, E. Somersalo, Statistical and computational inverse problems, New York: Springer, 2005. https://doi.org/10.1007/b138659
[7]	L. Wasserman, All of nonparametric statistics, New York: Springer, 2006. https://doi.org/10.1007/0-387-30623-4
[8]	G. Claeskens, T. Krivobokova, J. D. Opsomer, Asymptotic properties of penalized spline estimators, Biometrika, 96 (2009), 529–544. https://doi.org/10.1093/biomet/asp035 doi: 10.1093/biomet/asp035
[9]	P. H. C. Eilers, B. D. Marx, Flexible smoothing with b-splines and penalties, Statist. Sci., 11 (1996), 89–121. https://doi.org/10.1214/ss/1038425655 doi: 10.1214/ss/1038425655
[10]	J. Zhang, J. Cheng, M. Zhong, A tikhonov regularization based algorithm for scattered data with random noise, arXiv: 2105.00747. https://doi.org/10.48550/arXiv.2105.00747
[11]	J. A. Fessler, Penalized weighted least-squares image reconstruction for positron emission tomography, IEEE T. Med. Imaging, 13 (1994), 290–300. https://doi.org/10.1109/42.293921 doi: 10.1109/42.293921
[12]	K. R. Ridgway, J. R. Dunn, J. L. Wilkin, Ocean interpolation by four dimensional weighted least squares-application to the waters around Australasia, J. Atmos. Ocean. Tech., 19 (2002), 1357–1375. https://doi.org/10.1175/1520-0426(2002)019 doi: 10.1175/1520-0426(2002)019
[13]	J. M. Wooldridge, Introductory econometrics: a modern approach, Boston: Cengage Learning, 2012.
[14]	Q. Feng, J. Hannig, J. S. Marron. A note on automatic data transformation, Stat., 5 (2016), 82–87. https://doi.org/10.1002/sta4.104 doi: 10.1002/sta4.104
[15]	J. Kalina, On heteroscedasticity in robust regression, International Days of Statistics and Economics, 41 (2011), 228–237. https://doi.org/10.1111/j.1467-9310.2011.00660.x doi: 10.1111/j.1467-9310.2011.00660.x
[16]	B. Sun, L. Ma, T. Shen, R. Geng, Y. Zhou, Y. Tian, A robust data-driven method for multiseasonality and heteroscedasticity in time series preprocessing, Wirel. Commun. Mob. Com., 2021 (2021), 6692390. https://doi.org/10.1155/2021/6692390 doi: 10.1155/2021/6692390
[17]	M. Marzjarani, A comparison of a general linear model and the ratio estimator, International Journal of Statistics and Probability, 9 (2020), 54–65. https://doi.org/10.5539/ijsp.v9n3p54 doi: 10.5539/ijsp.v9n3p54
[18]	A. Bashan, N. M. Yagmurlu, Y. Ucar, A. Esen, A new perspective for the numerical solution of the modified equal width wave equation, Math. Method. Appl. Sci., 44 (2021), 8925–8939. https://doi.org/10.1002/mma.7322 doi: 10.1002/mma.7322
[19]	A. Bashan, N. M. Yagmurlu, Y. Ucar, A. Esen, Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation, Numer. Meth. Part. D. E., 37 (2021), 690–706. https://doi.org/10.1002/num.22547 doi: 10.1002/num.22547
[20]	A. Bashan, A. Esen, Single soliton and double soliton solutions of the quadratic-nonlinear Korteweg-de Vries equation for small and long-times, Numer. Meth. Part. D. E., 37 (2021), 1561–1582. https://doi.org/10.1002/num.22597 doi: 10.1002/num.22597
[21]	Y. Ucar, N. M. Yagmurlu, A. Bashan, Numerical solutions and stability analysis of modified burgers equation via modified cubic b-spline differential quadrature methods, Sigma J. Eng. Nat. Sci., 37 (2019), 129–142.
[22]	A. Bashan, N. M. Yagmurlu, A mixed method approach to the solitary wave, undular bore and boundary-forced solutions of the regularized long wave equation, Comp. Appl. Math., 41 (2022), 169. https://doi.org/10.1007/s40314-022-01882-7 doi: 10.1007/s40314-022-01882-7
[23]	A. Bashan, N. M. Yagmurlu, Y. Ucar, A. Esen, Numerical approximation to the MEW equation for the single solitary wave and different types of interactions of the solitary waves, J. Differ. Equ. Appl., 28 (2022), 1193–1213. https://doi.org/10.1080/10236198.2022.2132154 doi: 10.1080/10236198.2022.2132154
[24]	G. Micula, S. Micula, Handbook of splines, Dordrecht: Springer, 1999. https://doi.org/10.1007/978-94-011-5338-6
[25]	A. Koppel, G. Warnell, E. Stump, A. Ribeiro, Parsimonious online learning with kernels via sparse projections in function space, J. Mach. Learn. Res., 20 (2019), 83–126.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)