Estimation of the regression function by Legendre wavelets

Document Type : Research Article

Authors

• M. Hamzehnejad ¹
• M.M. Hosseini ²
• A. Salemi ²

¹ Department of Mathematic, Graduate University of Advanced Technology, Kerman, Iran.

² Department of Applied Mathematics and Mahani Mathematical Research Center, Shahid Bahonar University of Kerman, Kerman, Iran.

10.22067/ijnao.2022.73876.1079

Abstract

We estimate a function f with N independent observations by using Leg-endre wavelets operational matrices. The function f is approximated with the solution of a special minimization problem. We introduce an explicit expression for the penalty term by Legendre wavelets operational matrices. Also, we obtain a new upper bound on the approximation error of a differentiable function f using the partial sums of the Legendre wavelets. The validity and ability of these operational matrices are shown by several examples of real-world problems with some constraints. An accurate ap-proximation of the regression function is obtained by the Legendre wavelets estimator. Furthermore, the proposed estimation is compared with a non-parametric regression algorithm and the capability of this estimation is illustrated.

Keywords

• Legendre wavelet
• Operational matrix
• Wavelet approximation
• Regression function
• Error analysis

Main Subjects

• Numerical Analysis

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References

1. Abraham, C. Bayesian regression under combinations of constraints, J. Statist. Plann. Inference, 142 (2012) 2672–2687.
2. Abraham, C. and Khadraoui, K. Bayesian regression with B-splines under combinations of shape constraints and smoothness properties,Stat. Neerl. 69 (2015), 150–170.
3. Angelini, C., Canditiis, D.D. and Leblanc, F. Wavelet regression estimation in nonparametric mixed effect models, J. Multivariate Anal. 85 (2003) 267–291.
4. Bowman, W., Jones, M.C. and Gijbels, I. Testing monotonicity of regression, J. Comput. Graph Stat. 7 (1998) 489–500.

5. Corlay, S. B-spline techniques for volatility modeling, J. Comput. Finance, 19 (2016) 97–135.
6. Hamzehnejad, M., Hosseini, M.M. and Salemi, A. An improved upper bound for ultraspherical coefficients, Journal of Mathematical Modeling, 10 (2022), 1–11.
7. Khadraoui, K. A smoothing stochastic simulated annealing method for localized shapes approximation, JJ. Math. Anal. Appl. 446 (2017), 1018–1029.
8. Mammen, E., Marron, J., Turlach, B. and Wand, M. A general projection framework for constrained smoothing, Stat. Sci., 16 (2001) 232–248.
9. Meyer, M.C. Inference using shape-restricted regression splines, Ann.Appl. Stat. 2 (2008), 1013–1033.
10. Mohammadi, F. and Hosseini, M.M. A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations, J. Franklin Inst. 348 (2011) 1787–1796.
11. Mohammadi, M. and Bahrkazemi, M. Bases for polynomial-based spaces,J. Math. Model. 7 (2019) 21–34.
12. Polpo, A., Louzada, F.,Rifo, L.L.R., Stern, J.M. and Lauretto, M. Inter-disciplinary Bayesian Statistics, Proceedings of the 12th Brazilian Meet-ing on Bayesian Statistics (EBEB 2014) held in Atibaia, March 10–14, 2014. Springer Proceedings in Mathematics & Statistics, 118. Springer, Cham, 2015.
13. Rasmussen, C.E. and Williams, C.K.I. Gaussian processes for machine learning, Adaptive Computation and Machine Learning. MIT Press, Cambridge, MA, 2006.
14. Raykar, C. and Duraiswami, R. Fast optimal bandwidth selection for kernel density estimation, Proceedings of the Sixth SIAM International Conference on Data Mining, 524–528, SIAM, Philadelphia, PA, 2006.
15. Shen, J., Tang, T. and Wang, L.L. Spectral methods: Algorithms, analysis and applications, Vol. 41. Springer Science & Business Media, 2011.
16. Vidakovic, B. Statistical modeling by wavelets, Wiley Series in Probability and Statistics: Applied Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999.
17. Wahba, G. Spline models for observational data, CBMS-NSF Regional Conference Series in Applied Mathematics, 59. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990.

18. Wand, M. and Ormerod, J. On semiparametric regression with O’Sullivan penalized splines, Aust. N. Z. J. Stat., 50 (2008) 179–198.
19. Wang, H. A new and sharper bound for Legendre expansion of differentiable functions, Appl. Math. Lett. 85 (2018) 95–102.