Nonparametric Regression Mixed Estimators of Truncated Spline and Gaussian Kernel based on Cross-Validation (CV), Generalized Cross-Validation (GCV), and Unbiased Risk (UBR) Methods

Vita Ratnasari, I Nyoman Budiantara, Andrea Tri Rian Dani


Nowadays, most nonparametric regression research involves more than one predictor variable and generally uses the same type of estimator for all predictors. In the real case, each predictor variable likely has a different form of regression curve so that if it is forced, it can produce an estimation form that does not match the data pattern. Thus, it is necessary to develop a regression curve estimation model under the data pattern, namely the mixed estimator. The focus of this study is an additive nonparametric regression model, a mix of the Truncated Spline and Gaussian Kernel. There is a knot point in the Truncated Spline, while in the Gaussian Kernel, there is bandwidth. To choose the optimal knot point and bandwidth in a mixed estimator model, various methods can be used, including Cross-Validation (CV), Generalized Cross-Validation (GCV), and Unbiased Risk (UBR). This research proposes the optimal knot point and bandwidth estimation on the mixed estimator Truncated Spline and Gaussian Kernel model. Furthermore, the comparison between CV, GCV, and UBR is used to validate the proposed method. The simulation study was carried out by generating the Truncated Spline function and the Gaussian Kernel on a combination of sample size variations and variances. The simulation results show that the GCV method provides a higher coefficient of determination (R2) value and better accuracy for each combination of sample sizes and variance variations.


Cross-validation; generalized cross-validation; mixed estimators; unbiased risk.

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A. T. R. Dani, V. Ratnasari, and I. N. Budiantara, “Optimal Knots Point and Bandwidth Selection in Modeling Mixed Estimator Nonparametric Regression,†IOP Conf. Ser. Mater. Sci. Eng., vol. 1115, no. 1, p. 012020, 2021, doi: 10.1088/1757-899x/1115/1/012020.

R. L. Eubank, Nonparametric Regression and Spline Smoothing, New York: Marcel Dekker, 1999.

I. N. Budiantara, V. Ratnasari, M. Ratna, and I. Zain, “The Combination of Spline and Kernel Estimator for Nonparametric Regression and its Properties,†Appl. Math. Sci., vol. 9, no. 122, pp. 6083–6094, 2015, doi: 10.12988/ams.2015.58517.

N. Y. Adrianingsih, I. N. Budiantara, and J. D. T. Purnomo, “Modeling with Mixed Kernel, Spline Truncated and Fourier Series on Human Development Index in East Java,†IOP Conf. Ser. Mater. Sci. Eng., vol. 1115, no. 1, p. 012024, 2021, doi: 10.1088/1757-899x/1115/1/012024.

D. R. Sari Saputro, K. R. Demu, and P. Widyaningsih, “Nonparametric truncated spline regression model on the data of human development index (HDI) in indonesia,†J. Phys. Conf. Ser., vol. 1028, no. 1, pp. 6–10, 2018, doi: 10.1088/1742-6596/1028/1/012219.

N. Chamidah, B. Lestari, A. Massaid, and T. Saifudin, “Estimating mean arterial pressure affected by stress scores using Spline Nonparametric Regression model approach,†Commun. Math. Biol. Neurosci., vol. 2020, pp. 1–12, 2020.

G. Wahba, Spline Models for Observational Data, Pennsylvania: SIAM, 1990.

N. P. A. M. Mariati, N. Budiantara, and V. Ratnasari, “Truncated Spline Estimation of Percentage Poverty Modeling in Papua Province,†ICSA - Int. Conf. Stat. Anal. 2019, vol. 1, pp. 69–82, 2021, doi: 10.29244/icsa.2019.pp69-82.

B. Fatmawati, Budiantara, I N; Lestari, “Comparison of Smoothing and Truncated Spline Estimators in Estimating Blood Pressure Models,†Int. J. Innov. Creat. Chang., vol. 5, no. 3, pp. 685–707, 2019.

N. P. A. M. Mariati, I. N. Budiantara, and V. Ratnasari, “Smoothing Spline Estimator in Nonparametric Regression (Application: Poverty in Papua Province),†Proc. 7th Int. Conf. Res. Implementation, Educ. Math. Sci. (ICRIEMS 2020), vol. 528, no. Icriems 2020, pp. 309–314, 2021, doi: 10.2991/assehr.k.210305.044.

B. Lestari, Fatmawati, and I. N. Budiantara, “Spline estimator and its asymptotic properties in multiresponse nonparametric regression model,†Songklanakarin J. Sci. Technol., vol. 42, no. 3, pp. 533–548, 2020, doi: 10.14456/sjst-psu.2020.68.

L. R. Cheruiyot, “Local linear regression estimator on the boundary correction in nonparametric regression estimation,†J. Stat. Theory Appl., vol. 19, no. 3, pp. 460–471, 2020, doi: 10.2991/jsta.d.201016.001.

R. Hidayat, I. N. Budiantara, B. W. Otok, and V. Ratnasari, “An extended model of penalized spline with the addition of Kernel Functions in nonparametric regression model,†Appl. Math. Inf. Sci., vol. 13, no. 3, pp. 453–460, 2019, doi: 10.18576/amis/130318.

F. Yan, Q. S. Xu, M. L. Tang, and Z. Chen, “Kernel density-based likelihood ratio tests for linear regression models,†Stat. Med., vol. 40, no. 1, pp. 119–132, 2021, doi: 10.1002/sim.8765.

N. Chamidah and T. Saifudin, “Estimation of children growth curve based on kernel smoothing in multi-response nonparametric regression,†Appl. Math. Sci., vol. 7, no. 37–40, pp. 1839–1847, 2013, doi: 10.12988/ams.2013.13168.

I. Wayan Sudiarsa, “Simulations Study Combined Estimator Fourier Series and Spline Truncated in Multivariable Nonparametric Regression,†IOP Conf. Ser. Mater. Sci. Eng., vol. 546, no. 5, 2019, doi: 10.1088/1757-899X/546/5/052074.

D. R. S. Saputro, A. Sukmayanti, and P. Widyaningsih, “The nonparametric regression model using Fourier series approximation and penalized least squares (PLS) (case on data proverty in East Java),†J. Phys. Conf. Ser., vol. 1188, no. 1, 2019, doi: 10.1088/1742-6596/1188/1/012019.

A. Prahutama, Suparti, and T. W. Utami, “Modelling fourier regression for time series data - A case study: Modelling inflation in foods sector in Indonesia,†J. Phys. Conf. Ser., vol. 974, no. 1, pp. 0–9, 2018, doi: 10.1088/1742-6596/974/1/012067.

M. F. F. Mardianto, S. M. Ulyah, and E. Tjahjono, “Prediction of national strategic commodities production based on multi-Response nonparametric regression with fourier series estimator,†Int. J. Innov. Creat. Chang., vol. 5, no. 3, pp. 1151–1176, 2019.

I. Nyoman Budiantara et al., “Modeling percentage of poor people in Indonesia using kernel and Fourier series mixed estimator in nonparametric regression,†Investig. Operacional, vol. 40, no. 4, pp. 538–550, 2019, doi: 10.5281/zenodo.3721293.

R. Hidayat, I. N. Budiantara, B. W. Otok, and V. Ratnasari, “A reproducing kernel hilbert space approach and smoothing parameters selection in spline-kernel regression,†J. Theor. Appl. Inf. Technol., vol. 97, no. 2, pp. 465–475, 2019.

R. Hidayat, I. N. Budiantara, B. W. Otok, and V. Ratnasari, “Kernel-Spline Estimation of Additive Nonparametric Regression Model,†IOP Conf. Ser. Mater. Sci. Eng., vol. 546, no. 5, 2019, doi: 10.1088/1757-899X/546/5/052028.

P. Dewanti, I. Nyoman Budiantara, and A. T. Rumiati, “Modelling of SDG’s Achievement in East Java Using Bi-responses Nonparametric Regression with Mixed Estimator Spline Truncated and Kernel,†J. Phys. Conf. Ser., vol. 1562, no. 1, 2020, doi: 10.1088/1742-6596/1562/1/012016.

D. P. Rahmawati, I. N. Budiantara, D. D. Prastyo, and M. A. D. Octavanny, “Mixed Spline Smoothing and Kernel Estimator in Biresponse Nonparametric Regression,†Int. J. Math. Math. Sci., vol. 2021, 2021, doi: 10.1155/2021/6611084.

P. Craven and G. Wahba, “Smoothing noisy data with spline functions - Estimating the correct degree of smoothing by the method of generalized cross-validation,†Numer. Math., vol. 31, no. 4, pp. 377–403, 1978, doi: 10.1007/BF01404567.

Y. Wang, “Smoothing spline models with correlated random errors,†J. Am. Stat. Assoc., vol. 93, no. 441, pp. 341–348, 1998, doi: 10.1080/01621459.1998.10474115.

H. Nurcahayani, I. N. Budiantara, and I. Zain, “The Curve Estimation of Combined Truncated Spline and Fourier Series Estimators for Multiresponse Nonparametric Regression,†Mathematics, vol. 9, no. 10, p. 1141, 2021.

A. R. Devi, I. N. Budiantara, and V. Ratnasari, “Unbiased risk and cross-validation method for selecting optimal knots in multivariable nonparametric regression spline truncated (case study: Unemployment rate in Central Java, Indonesia, 2015),†AIP Conf. Proc., vol. 2021, no. December, 2018, doi: 10.1063/1.5062767.

T. W. Utami, M. A. Haris, A. Prahutama, and E. A. Purnomo, “Optimal knot selection in spline regression using unbiased risk and generalized cross validation methods,†J. Phys. Conf. Ser., vol. 1446, no. 1, 2020, doi: 10.1088/1742-6596/1446/1/012049.

B. A. M. Al-Talib and A. A. Hammodat, “Using Some Wavelet Shrinkage Techniques and Robust Methods to Estimate the Generalized Additive Model Parameters in Non-Linear Models,†Int. J. Adv. Sci. Eng. Inf. Technol., vol. 10, no. 6, p. 2344, 2020, doi: 10.18517/ijaseit.10.6.12767.



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