Determination of the best knot and bandwidth in geographically weighted truncated spline nonparametric regression using generalized cross validation

Abstract

This study proposes the development of nonparametric regression for data containing spatial heterogeneity with local parameter estimates for each observation location. GWTSNR combines Truncated Spline Nonparametric Regression (TSNR) and Geographically Weighted Regression (GWR). So it is necessary to determine the optimum knot point from TSNR and determine the best geographic weighting (bandwidth) from GWR by deciding the best knot point and bandwidth using Generalized Cross Validation (GCV). The case study analyzed the Morbidity Rate in North Sumatra in 2020. This study will estimate the model using knot points 1, 2, and 3 and geographic weighting of the Kernel Function, Gaussian, Bisquare, Tricube, and Exponential. Based on data analysis, we obtained that the best model for Morbidity Rate data in North Sumatra 2020 based on the minimum GCV value is the model using knots 1 and the Kernel Function of Bisquare. Based on the GWTSNR model, the significant predictors in each district/city were grouped into eight groups. Furthermore, the GWTSNR is better at modeling morbidity rates in North Sumatra 2020 by obtaining adjusted R-square = 96.235 than the TSNR by obtaining adjusted R-squared = 70.159. Some of the highlights of the proposed approach are:•The method combines nonparametric and spatial regression in determining morbidity rate modeling.•There were three-knot points tested in the truncated spline nonparametric regression and four geographic weightings in the spatial regression and then to determine the best knot and bandwidth using Generalized Cross Validation.•This paper will determine regional groupings in North Sumatra 2020 based on significant predictors in modeling morbidity rates.

Keywords: Geographically Weighted Truncated Spline Nonparametric Regression (GWTSNR); Kernel function; Morbidity rate; Nonparametric regression; Spatial regression.

Graphical abstract

Fig. 1

Morbidity Rates in Indonesia and North Sumatra 2015 – 2020. It can be seen that for a period of six years, from 2015 to 2020, the morbidity rates in North Sumatra Province were consistently below the national figure. All variables ranging from the response variable to the seven predictor variables that are thought to affect the average, variance to the minimum, and maximum values are calculated.

Fig. 2

Description information based on variable data mapping. The lowest Y is in Humbang Hasundutan Regency, and the highest Y is in Batubara Regency. The lowest ${\mathrm{X}}_1$is in Deli Serdang Regency, and the highest ${\mathrm{X}}_1$ is in West Nias Regency. The lowest ${\mathrm{X}}_2$ is in South Nias Regency, and the highest ${\mathrm{X}}_2$ is in Binjai City. The lowest ${\mathrm{X}}_3$ is in Pakpak Bharat Regency, and the highest ${\mathrm{X}}_3$ is in Medan City. The lowest ${\mathrm{X}}_4$ is in Humbang Hasundutan Regency, and the highest ${\mathrm{X}}_4$ is in Gunungsitoli City. The lowest ${\mathrm{X}}_5$is in West Nias Regency, and the highest ${\mathrm{X}}_5$ is in Medan City. The lowest ${\mathrm{X}}_6$ is in Padang Sidimpuan Regency, and the highest ${\mathrm{X}}_6$ is Pematangsiantar City. And the lowest ${\mathrm{X}}_7$ is Nias, and the highest ${\mathrm{X}}_7$ is in Medan City.

Fig. 3

Scatterplot between Morbidity Rate and Predictors. The plot between the variable morbidity rate with all predictors does not form or follow a certain pattern. So that all predictors are included in nonparametric components.

Fig. 4

Mapping Morbidity Rates in North Sumatra 2020 based on Significant Variables.