Parameter estimation in mixed estimator nonparametric regression-spline truncated and fourier series (MENR-SF) for behavioral factors of prevalence of heart disease

Budiantara, I.; Chamidah, N.; Dani, A.; Anshari, M.

Abstract: This study aims to develop and apply a Mixed Estimator Nonparametric Regression–Spline Truncated and Fourier Series (MENR–SF) to model the nonlinear relationships between behavioral factors and the prevalence of heart disease in Indonesia. The proposed approach simultaneously combines spline truncated estimators and Fourier series within a unified nonparametric regression framework, allowing each predictor variable to be modeled according to the specific characteristics of its relationship with the response variable. Parameter estimation is conducted using the Least Squares method, while the optimal number of spline knots and Fourier oscillations is determined based on the Generalized Cross-Validation (GCV) criterion. The application of the MENR–SF model to data from the 2023 Indonesian Health Survey (Survei Kesehatan Indonesia, SKI), with 38 provinces as the units of analysis, indicates that the best-performing model is obtained when the prevalence of daily smoking, the proportion of insufficient physical activity, and habitual consumption of fatty foods are modeled using spline truncateds, whereas the proportion of hypertension control is modeled using a Fourier series. The optimal combination, with three spline knots and three Fourier oscillations, yields a minimum GCV value of 1.197, low prediction error, and a coefficient of determination of 0.94, indicating an excellent ability of the model to explain variations in heart disease prevalence. These findings conclude that MENR–SF is a flexible and accurate approach for modeling complex nonlinear relationships in health data. The model offers enhanced flexibility and richer interpretability regarding the effects of behavioral factors, thereby holding strong potential to support data-driven health analysis and policy formulation.

How to cite this paper

 Budiantara, I., Chamidah, N., Dani, A & Anshari, M. (2026). Parameter estimation in mixed estimator nonparametric regression-spline truncated and fourier series (MENR-SF) for behavioral factors of prevalence of heart disease.International Journal of Data and Network Science, 10(2), 597-608.

References

Amri, I. F., Chamidah, N., Saifudin, T., Purwanto, D., Fadlurohman, A., Fitriyana Ningrum, A., & Amri, S. (2024). Prediction of extreme weather using nonparametric regression approach with Fourier series estimators. Data and Metadata, 4, 1–12. https://doi.org/10.56294/dm2024319
Bilodeau, M. (1992). Fourier smoother and additive models. The Canadian Journal of Statistics, 20(3), 257–269.
Bloomfield, P. (2000). Fourier Analysis of Time Series An Introduction (2nd ed.). John Wiley & Sons, Inc.
Budiantara, I. N. (2019). Regresi Nonparametrik Spline Truncated (1st ed.). ITS Press.
Budiantara, I. N., Ratnasari, V., Ratna, M., & Zain, I. (2015). The Combination of Spline and Kernel Estimator for Nonparametric Regression and its Properties. Applied Mathematical Sciences, 9(122), 6083–6094. https://doi.org/10.12988/ams.2015.58517
Chamidah, N., Lestari, B., Susilo, H., Dewi, T. K., Saifudin, T., Al Akhwal Siregar, N. R., & Aydin, D. (2025). Modeling coronary heart disease risk based on age, fatty food consumption and anxiety factors using penalized spline nonparametric logistic regression. MethodsX, 14. https://doi.org/10.1016/j.mex.2025.103320
Chen, H. (1991). Polynomial splines and nonparametric regression. Journal of Nonparametric Statistics, 1(1–2), 143–156. https://doi.org/10.1080/10485259108832516
Chu, C.-K., & Marron, J. S. (1991). Choosing a Kernel Regression Estimator. Statistical Science, 6(4), 404–436.
Cui, W., & Wei, M. (2013). Strong Consistency of Kernel Regression Estimate. Open Journal of Statistics, 03(03), 179–182. https://doi.org/10.4236/ojs.2013.33020
Dani, A. T. R., Ratnasari, V., & Budiantara, I. N. (2021). Optimal Knots Point and Bandwidth Selection in Modeling Mixed Estimator Nonparametric Regression. IOP Conference Series: Materials Science and Engineering, 1115(1), 012020. https://doi.org/10.1088/1757-899x/1115/1/012020
Eilers, P. H. C., & Marx, B. D. (2010). Splines, Knots, and Penalties. Wiley Interdisciplinary Reviews: Computational Statistics, 2(6), 637–653. https://doi.org/10.1002/wics.125
Eubank, R. L. (1999). Nonparametric Regression and Spline Smoothing. Marcel Dekker.
Fatmawati, F., Nyoman Budiantara, I., & Lestari, B. (2019). Comparison of Smoothing and Truncated Spline Estimators in Estimating Blood Pressure Models. International Journal of Innovation, Creativity, and Change, 5(3), 1177–1199. www.ijicc.net
Ganesh, A., Balasubramanian, G., Jena, S. K., & Pradhan, N. (2011). Fourier Approach to Function Approximation. International Journal of Mathematical Archive, 4, 617–624.
Greblicki, W., & Pawlak, M. (1985). Fourier and Hermite series estimates of regression functions. Annals of the Institute of Statistical Mathematics, 37(1), 443–454. https://doi.org/10.1007/BF02481112
Hackshaw, A., Morris, J. K., Boniface, S., Tang, J. L., & Milenkovi, D. (2018). Low cigarette consumption and risk of coronary heart disease and stroke: Meta-analysis of 141 cohort studies in 55 study reports. In BMJ (Online) (Vol. 360). BMJ Publishing Group. https://doi.org/10.1136/bmj.j5855
Hajduk, A. M., & Chaudhry, S. I. (2016). Sedentary Behavior and Cardiovascular Risk in Older Adults: a Scoping Review. In Current Cardiovascular Risk Reports (Vol. 10, Issue 1, pp. 1–11). Current Medicine Group LLC 1. https://doi.org/10.1007/s12170-016-0485-6
Hardle, W. H., & Kelly, G. K. (1987). Nonparametric Kernel Regression Estimation-Optimal Choice of Bandwidth. Statistics, 18(1), 21–35. https://doi.org/10.1080/02331888708801986
Huang, M., Li, R., & Wang, S. (2013). Nonparametric mixture of regression models. Journal of the American Statistical Association, 108(503), 929–941. https://doi.org/10.1080/01621459.2013.772897
Kementerian Kesehatan RI. (2024). SURVEI KESEHATAN INDONESIA (SKI) 2023 DALAM ANGKA. Kementerian Kesehatan Republik Indonesia.
Kementerian Kesehatan RI. (2025). Profil Kesehatan Indonesia 2024. Kementerian Kesehatan Republik Indonesia.
Kim, A. S., & Johnston, S. C. (2011). Global variation in the relative burden of stroke and ischemic heart disease. Circulation, 124(3), 314–323. https://doi.org/10.1161/CIRCULATIONAHA.111.018820
Lestari, B., Fatmawati, F., & Budiantara, I. N. (2020). Spline Estimator and Its Asymptotic Properties in Multiresponse Nonparametric Regression Model. Songklanakarin Journal of Science and Technology, 42(3), 533–548.
Liu, H., Liu, Z., Gong, Y., Guo, J., Liu, X., Sun, Y., Tang, W., Cheng, W., & Jin, W. (2025). Low physical activity-related disease burden, 1990-2021: assessment of global trends and social determinants based on GBD 2021 data. Journal of Global Health, 15, 04314. https://doi.org/10.7189/jogh.15.04314
Ming, W. Y., & Huang, L.-J. (2018). Fourier Series Neural Networks for Regression. Proceedings of IEEE International Conference on Applied System Innovation, 716–719.
Prahutama, A., Suparti, & Utami, T. W. (2018). Modelling fourier regression for time series data - A case study: Modelling inflation in foods sector in Indonesia. Journal of Physics: Conference Series, 974(1). https://doi.org/10.1088/1742-6596/974/1/012067
Ramli, M., Ratnasari, V., & Nyoman Budiantara, I. (2020). Estimation of Matrix Variance-Covariance on Nonparametric Regression Spline Truncated for Longitudinal Data. Journal of Physics: Conference Series, 1562(1). https://doi.org/10.1088/1742-6596/1562/1/012014
Ratnasari, V., Budiantara, I. N., Ratna, M., & Zain, I. (2016). Estimation of Nonparametric Regression Curve using Mixed Estimator of Multivariable Truncated Spline and Multivariable Kernel. Global Journal of Pure and Applied Mathematics, 12(6), 5047–5057. http://www.ripublication.com/gjpam.htm
Ratnasari, V., Budiantara, N., & Dani, A. T. R. (2021). Nonparametric Regression Mixed Estimators of Truncated Spline and Gaussian Kernel based on Cross-Validation (CV), Generalized Cross-Validation (GCV), and Unbiased Risk (UBR) Methods. International Journal on Advanced Science Engineering Information Technology, 11(6), 2400–2406.
Sanchis-Gomar, F., Perez-Quilis, C., Leischik, R., & Lucia, A. (2016). Epidemiology of coronary heart disease and acute coronary syndrome. In Annals of Translational Medicine (Vol. 4, Issue 13). AME Publishing Company. https://doi.org/10.21037/atm.2016.06.33
Wahba. (1990). Spline Models for Observational Data (2nd ed.). SIAM.
Wahba, G., & Wang, Y. (2014). Spline Function. Encyclopedia of Statistical Sciences, 1–27. https://doi.org/10.4135/9781446247501.n3679
Yatchew, A. (1998). Nonparametric Regression Techniques in Economics. Journal of Economic Literature, 36(2), 669–721. http://www.jstor.orgURL:http://www.jstor.org/stable/2565120http://www.jstor.org/page/info/about/policies/terms.jsp
Zhang, X., Ru, J., & Wu, C. (2023). A Nonparametric Regression-Based Multi-Scale Gradient Correlation Filtering Method for Infrared Small Target Detection. Electronics (Switzerland), 12(7). https://doi.org/10.3390/electronics12071562