KnE Social Sciences

Published date: May 27 2025

Journal Title: KnE Social Sciences

Issue title: The 4th International Conference on Science, Mathematics, Environment, and Education (ICoSMEE 2023)

Pages: 220 - 234

DOI: 10.18502/kss.v10i11.18744

Putri Kusuma Wardaniputrikusumaw.19@gmail.com Department of Statistics, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Sukolilo

I Nyoman Budiantara Department of Statistics, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Sukolilo

Setiawan Setiawan Department of Statistics, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Sukolilo

A nonparametric regression approach is suitable for the cases in which the shape of the pattern between the response variables and the predictor variables is not known. There are several methods in nonparametric regression, such as spline truncated and Fourier series. In both methods, determining the optimal knot point is crucial. Optimal knot points and oscillation parameters can be selected using the generalized cross-validation (GCV) and unbiased risk (UBR) methods. This study aimed to examine the GCV and UBR methods to select optimal knot point and oscillation parameters on the data on Indonesia’s economic growth rate in 2022. The estimation method used is ordinary least square (OLS). The results obtained used the GCV method because it has MSE value 1.42, which is smaller than MSE of UBR method of 10.614. The coefficient of determination for the GCV method is 89.34%. The optimal number of knot points and oscillation parameters are three and three for nonparametric regression estimator combined of spline truncated and Fourier series.

Keywords: fourier series, generalized cross-validation, nonparametric regression, spline truncated, unbiased risk

[1] Budiantara IN. Splines in Nonparametric and Semiparametric Regression: A Statistical Modeling of the Present and Future. Inauguration Speech for the Position of Professor in the Department of Statistics, Institut Teknologi Sepuluh Nopember Surabaya; 2009.

[2] Budiantara N, Ratna M, Zain I, WibowoW.ModelingthePercentageofPoorPeoplein Indonesia Using Spline Nonparametric Regression Approach. International Journal Basis & Applied Sciences; 2012. pp. 118–24.

[3] Mariati NPAM. Fourier Series and Truncated Spline Regression Modeling in Multivariable Nonparametric Regression (Application: Poverty Data in Papua Province). Surabaya: Institut Teknologi Sepuluh November; 2015.

[4] Afifah N. Mixed Kernel and Fourier Series Estimators in Nonparametric Regression (Case Study: Modeling the Percentage of Poor Population in Indonesia). Surabaya: Institut Teknologi Sepuluh November; 2017.

[5] Gabrela PP. Birespon Nonparametric Regression Modeling Using Mixed Truncated Spline, Kernel, and Fourier Series Estimators (Case Study: Modeling the Percentage of Poor Population and Poverty Depth Index in Papua Province). Surabaya: Institut Teknologi Sepuluh November; 2021.

[6] Sari SUR. Comparison of Multivariable Spline Nonparametric Regression Models Using Generalized Cross Validation (GCV) and Unbiased Risk (UBR) Methods in Selection of Optimal Knot Points (Case Study of Maternal Mortality Rate Data in East Java). Surabaya: Institut Teknologi Sepuluh November; 2016.

[7] Devi AR, Budiantara IN, Ratnasari V. 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. 2018;:1–9.

[8] Dani ATR. Selection of Optimal Knot Points and Optimal Bandwidth Using Cross Validation (CV), Generalized Cross-Validation (GCV), and Unbiassed Risk (UBR) in Nonparametric Regression Modeling Mixed Truncated Spline Estimator and Gaussian Kernel. Surabaya: Institut Teknologi Sepuluh November; 2021.

[9] Amrullah MN, Amalia SH. Comparison of Generalized Cross Validation (GCV) Methods with Cross Validation (CV) to Determine Optimal Knots in Fourier Series Nonparametric Regression (Case Study: poverty Rate in North Sumatra Province). Jurnal Litbang Edusaintech. 2022;3(1):1–10.

[10] Abdullah, Shah T, Sargani GR, Ali A, Siraj W. The Effect of Increase in Population on the Economic Growth of. Developing Country Studies. 2015;5(17):87–94.

[11] Eubank RL. Nonparametric Regression and Spline Smoothing. New York: Marcel Dekker; 1999. https://doi.org/10.1201/9781482273144.

[12] Wu H, Zhang JT. Nonparametric Regression Methods for Longitudinal Data Analysis: Mixed-Effects Modeling Approaches. New York: John Wiley & Sons;