Selection of optimal knot point and best geographic weighted on geographically weighted spline nonparametric regression model

Abstract

This study proposes the development of a nonparametric regression model combined with geographically weighted regression. The regression model considers geographical factors and has a data pattern that does not follow a parametric form to overcome the problem of spatial heterogeneity and unknown regression functions. This study aims to model provincial food security index data in Indonesia with the GWSNR model, so finding the optimal knot point and the best geographic weighting is necessary. We propose the selection of optimal knot points using the Cross Validation (CV) and Generalized Cross Validation (GCV) methods. The optimal knot point will control the accuracy of the regression curve as we also consider the MSE value in showing the ability of the model. In addition, we determine the best geographic weighting and test the significance of the model parameters. We demonstrate the GWSNR model on food security index data. The best GWSNR model uses the Gaussian kernel weighting function and selects the optimal knot point as one-knot point based on the lowest CV and GCV values. Simultaneous and partial parameter test results show that there are 10 area classifications with different effects on each group of classification results. Some of the highlights of the proposed approach are:

• •The method is the development of a nonparametric regression model with geographic weighting, which combines nonparametric and spatial regression in modeling the national food security index.
• •There are three-knot points tested in nonparametric truncated spline regression and three geographic weightings in spatial regression. Then the optimal knot point and best bandwidth are determined using Cross Validation and Generalized Cross Validation.
• •This article will determine regional groupings in Indonesia in 2022 based on significant predictors in modeling the national food security index numbers.

Graphical abstract

Specifications table

Subject area:	Mathematics and Statistics
More specific subject area:	Statistics: Nonparametric Regression, Spatial Regression
Name of your method:	Geographically Weighted Spline Nonparametric Regression (GWSNR) Model
Name and reference of original method:	Sifriyani, I. N. Budiantara, S. H. Kartiko, and Gunardi, A new method of hypothesis test for truncated spline nonparametric regression influenced by spatial heterogeneity and application, Abstract and Applied Analysis. 2018 (2018). https://doi.org/10.1155/2018/9,769,150
Resource availability:	National food security index $\left(y\right)$ and the predictors $\left(x\right)$ from publication of the agricultural data center and information system of the secretariat general of the ministry of agriculture.

Background

Indonesia is committed to realizing sustainable development goals (SDGs), which include eliminating poverty and ending hunger, achieving food security, improving nutrition, and promoting sustainable agriculture. In achieving these targets, the National Food Agency has the task and role of coordinating, establishing, and implementing policies for preventing and handling food and nutrition insecurity [1]. There is an interest in sustainable development and meeting food needs, so researchers must discuss and research the food security index. This research aims to model and find factors influencing the national food security index.

Spatial effects and geographical factors influence food security index data. In statistics, spatial effects occur when the data shows heteroscedasticity and spatial autocorrelation. Regression analysis for geographically and locally oriented data can use the Geographically Weighted Regression (GWR) model. Fotheringham first introduced GWR in 1967 [2]. Research using GWR theory was carried out by Brunsdon et al. [3], Chasco et al. [4], Mennis et al. [5], Sefa mizrak et al. [6], Sifriyani et al. [[7], [8]], Fei Jiang et al. [9], and Abel Kebede Reda, et al. [10]. GWR development has also been carried out by statisticians such as [[11], [12], [13]]. The development of GWR carried out by statisticians is still in linear form. However, in reality, not all data is known to have a clear relationship pattern, or the regression curve is unknown [14]. So, nonparametric regression is an alternative approach to use in such cases.

Several types of non-parametric regression models are often discussed, such as Spline by Wahba, 1990; Green & Silverman, 1994; Budiantara et al., 1997; Budiantara, 2002. Kernel by Hardle, 1990, and Fourier Series by Antoniadis, 1994 and Wavelets by Antoniadis, 2001. Splines are segmented polynomials that have the property of flexibility. Splines are very dependent on their vertices. Truncated Spline is a segmented polynomial model that allows effective adaptation to the local characteristics of the data. So, Sifriyani et al. [15] developed the Geographically Weighted Spline Nonparametric Regression (GWSNR) model to solve spatial analysis problems where the regression curve is unknown, Then find test statistics for goodness of fit and validation of the GWSNR model [15].

Besides being influenced by spatial effects, food security index data has an unknown regression curve. So, the GWSNR model is suitable for use. This research aims to model provincial food security index data in Indonesia using the GWSNR model. Therefore, finding the optimal knot points and the best geographic weights is necessary. We propose optimal knot selection using Cross Validation (CV) and Generalized Cross Validation (GCV) methods. The optimal knot points will control the accuracy of the regression curve, and we also consider the MSE value in indicating the model's ability. Next, determine the best geographic weighting function, then estimate and test the significance of the model parameters. This research will determine regional groupings in Indonesia in 2022 based on significant predictors in modeling the national food security index numbers.

Method details

Nonparametric regression with a truncated spline approach

In nonparametric regression modeling, one of the approaches that can be used to estimate the regression curve is the truncated spline approach [16], (Sifriyani, et al., Spline and kernel mixe estimators in multivariable nonparametric regression for dengue hemorhagic fever model, 2023). The truncated spline in nonparametric regression is a model that has good statistical and visual interpretation [9,17], besides the truncated spline has high flexibility [18], (Sifriyani, D, M, A, & J, [17]). A truncated spline is a segmented polynomial model. These segmented polynomials provide greater flexibility than ordinary polynomials and this property allows the model to adapt more effectively to the local characteristics of the data [19]. Truncated splines of order $m$ with knot points $R_1,R_2,\dots ,R_r$ where $a<R_1<\dots <R_r<z$, with $a$ and $z$ is real konstanta. The function of nonparametric regression with a truncated spline approach presented in Eq. (1).

$$f\left(x_i\right)=\sum _{k=0}^m{\beta }_k{x}_i^k+\sum _{h=1}^r{\beta }_{m+h}{\left(x_i-R_h\right)}_{+}^m$$ (1)

where ${\beta }_k,{\beta }_{m+h}$ is a real constant with $i=1,2,\dots ,n$ is the unit of observation,$k=0,1,2,\dots ,m$ is the number of predictor variables and $h=1,2,\dots ,r$ is the number of knot points. Then the truncated function is given in Eq. (2).

$${\left(x_i-R_h\right)}_{+}^m=\{\begin{array}{c}{\left(x_i-R\right)}^m,x_i\ge R_h\hfill \\ 0,x_i<R_h\hfill \end{array}$$ (2)

Truncated splines have the following functional characteristics [20]:

• 1.Function $f$ is a piece-wise polynomial of degree $m$ on interval $\left[R_h,R_{h+1}\right]$.
• 2.Function $f$ has a continuous derivative of degree $m-1$
• 3.$f^{\left(m\right)}$ is a step function with jump points $R_1,R_2,\dots ,R_r$.

The nonparametric regression model with a truncated spline approach [19] is given in Eq. (3).

$$\begin{array}{ccc}\hfill y_i& =& \sum _{k=0}^m{\beta }_k{x}_i^k+\sum _{h=1}^r{\beta }_{m+h}{\left(x_i-R_h\right)}_{+}^m+{\epsilon }_i,\hfill \\ \hfill i& =& 1,2,\dots ,n\hfill \end{array}$$ (3)

where $y_i$ is the i th response variable, $x_i$ is the i th predictor variable, ${\beta }_k,{\beta }_{m+h}$ are real constant, $R_1,R_2,\dots ,R_r$ are knot points, and ${\epsilon }_i$ are random errors assumed to be identical, independent, and normally distributed with zero mean and variance ${\sigma }^2$ [8,15].

Geographically weighted regression

The Geographically Weighted Regression (GWR) model was first introduced by Fotheringham [2]. The GWR model is an extension of the classic linear regression model which was developed to model data with continuous response variables and consider spatial or location aspects. The response variable y in the GWR model is predicted by predictor variables, each of which has a regression coefficient depending on the location where the data is observed, so that each observation location has different regression parameter values [17,19]. GWR model of the relationship between the response variable $y$ and predictor variables $x_1,x_2,\dots ,x_p$ at the i th location is given in Eq. (4).

$$y_i={\beta }_0\left(u_i,v_i\right)+\sum _{k=1}^p{\beta }_k\left(u_i,v_i\right)x_{ki}+{\epsilon }_i$$ (4)

$y_i$ is the value of the response variable at the point of the i th observation location, $x_{ki}$ is the k-th predictor variable at the i-location observation, $\left(u_i,v_i\right)$ represents the geographic coordinates of the Longitude and Latitude of the i th observation location, ${\beta }_0\left(u_i,v_i\right)$ is a constant/intercept of GWR, ${\beta }_k\left(u_i,v_i\right)$ is the k-th parameter at the i th location associated with predictor variables $x_{ki}$ and ${\epsilon }_i$ is the error at the i th location which is assumed to be independent, identical and normally distributed with zero mean and variance ${\sigma }^2$ [[8], [20]].

Materials and model specifications

Constructing a geographically weighted spline nonparametric regression model

The Geographically Weighted Spline Nonparametric Regression (GWSNR) model is a development of nonparametric regression for spatial data with local parameter estimators for each observation location. The truncated spline approach is used to solve spatial analysis problems where the shape of the regression curve is unknown. In the regression model, we use the assumption that the errors are normally distributed with zero mean and variance ${\sigma }^2\left(u_i,v_i\right)$ at each location $\left(u_i,v_i\right)$. The coordinate location $\left(u_i,v_i\right)$ is one of the important factors in determining the weights used to estimate the parameters of the model.

The form of the relationship between the response variables $y_i$ and the predictor variables $\left(x_{1i},x_{2i},\dots ,x_{li}\right)$ at the $i$-th location for the GWSNR regression model can be expressed mathematically in Eq. (5).

$$y_i={\beta }_0\left(u_i,v_i\right)+\sum _{p=1}^l\sum _{k=1}^m{\beta }_{pk}\left(u_i,v_i\right)x_{pi}^k+\sum _{p=1}^l\sum _{h=1}^r{\delta }_{p,m+h}\left(u_i,v_i\right){\left(x_{pi}-R_{ph}\right)}_{+}^m+{\epsilon }_i$$ (5)

Eq. (5) is a GWSNR model of degree $m$ with $n$ areas. The components in Eq. (5) described as $y_i$ is the response variable at the i th location where $i=1,2,\dots ,n$, the variable $x_{pi}$ is the $p$-th predictor variable at the$i$-th location where $p=1,2,\dots ,l,$ the symbol $R_{ph}$ denotes the $h$-th knot point on the component of the $p$-th predictor variable where $h=1,2,\dots ,r,$ the symbol ${\beta }_{pk}\left(u_i,v_i\right)$ denotes the component parameter of the polynomial and defined as the k-th parameter of the $p$-th predictor variable in the $i$-th area. ${\delta }_{p,m+h}\left(u_i,v_i\right)$ is the truncated component parameter of the truncated multivariable spline nonparametric regression model in GWR. ${\delta }_{p,m+h}\left(u_i,v_i\right)$ is the $\left(l+h\right)$-th parameter at the $h$-th knot point and the $p$-th predictor variable in the $i$-th area [15].

Geographically weighted spine nonparametric regression model estimation process

The process of estimating the GWSNR model with the Maximum Likelihood Estimation (MLE) with weights is given as follows.

1. The probability density function for $f\left(y_i\right)$ is shown in Eq. (6)

$$\begin{array}{c}f\left(y_i\right)=\frac{1}{\sqrt{2\pi {\sigma }^2\left(u_i,v_i\right)}}exp(-\frac{1}{2{\sigma }^2\left(u_i,v_i\right)}[y_i-({\beta }_0\left(u_i,v_i\right)+\sum _{p=1}^l\sum _{k=1}^m{\beta }_{pk}\left(u_i,v_i\right)x_{pi}^k\hfill \\ {+\sum _{p=1}^l\sum _{h=1}^r{\delta }_{p,m+h}\left(u_i,v_i\right){\left(x_{pi}-R_{ph}\right)}_{+}^m)]}^2)\hfill \end{array}$$ (6)

2. Generating the joint probability distribution function for the jth location is shown in Eq. (7)

$$\begin{array}{c}\left(y_1,y_2,\dots ,y_n\right)=\prod _{i=1}^{n}f\left(y_i|\tilde{\beta }\left(u_j,v_j\right),\tilde{\delta }\left(u_j,v_j\right),{\sigma }^2\left(u_j,v_j\right)\right)=\prod _{i=1}^n[\frac{1}{\sqrt{2\pi {\sigma }^2\left(u_j,v_j\right)}}\text{exp}(-\frac{1}{2{\sigma }^2\left(u_j,v_j\right)}{w}_{i\left(j\right)}\hfill \\ {\left[y_i-\left({\beta }_0\left(u_j,v_j\right)+\sum _{p=1}^l\sum _{k=1}^m{\beta }_{pk}\left(u_j,v_j\right)x_{pj}^k+\sum _{p=1}^l\sum _{h=1}^r{\delta }_{p,m+h}\left(u_j,v_j\right){\left(x_{pi}-K_{ph}\right)}_{+}^m\right)\right]}^2)]\hfill \\ ={\left(2\pi \right)}^{-\frac{n}{2}}{\left({\sigma }^2\left(u_j,v_j\right)\right)}^{-\frac{n}{2}}\text{exp}(-\frac{1}{2{\sigma }^2\left(u_j,v_j\right)}\sum _{i=1}^n{w}_{i\left(j\right)}[y_i-({\beta }_0\left(u_j,v_j\right)+\sum _{p=1}^l\sum _{k=1}^m{\beta }_{pk}\left(u_j,v_j\right)x_{pj}^k\hfill \\ {+\sum _{p=1}^l\sum _{h=1}^r{\delta }_{pm+h}\left(u_j,v_j\right){\left(x_{pi}-K_{ph}\right)}_{+}^m)]}^2)\hfill \end{array}$$ (7)

3. Generate Likelihood Function is shown in Eq. (8)

$$\begin{array}{c}L\left(\tilde{\beta }\left(u_j,v_j\right),\tilde{\delta }\left(u_j,v_j\right),{\sigma }^2\left(u_j,v_j\right)|\tilde{\mathbf{Y}}\right)={\left(2\pi \right)}^{-\frac{n}{2}}{\left({\sigma }^2\left(u_j,v_j\right)\right)}^{-\frac{n}{2}}\text{exp}[-\frac{1}{2{\sigma }^2\left(u_j,v_j\right)}\hfill \\ \sum _{i=1}^n{w}_{i\left(j\right)}{\left[y_i-\left({\beta }_0\left(u_j,v_j\right)+\sum _{p=1}^l\sum _{k=1}^m{\beta }_{pk}\left(u_j,v_j\right)x_{pi}^k+\sum _{p=1}^l\sum _{h=1}^r{\delta }_{p,m+h}\left(u_j,v_j\right){\left(x_{pi}-K_{ph}\right)}_{+}^m\right)\right]}^2]\hfill \end{array}$$ (8)

4. Estimator $\widehat{\tilde{\beta }}\left(u_i,v_i\right)$, $\widehat{\tilde{\delta }}\left(u_i,v_i\right)$ and $\widehat{\tilde{f}}$ can be obtained using the MLE method by solving the optimization in Eq. (9)

$$\begin{array}{c}\begin{array}{c}max\\ \tilde{\beta },\tilde{\delta },{\sigma }^2\end{array}\left\{L\left(\tilde{\beta }\left(u_i,v_i\right),\tilde{\delta }\left(u_i,v_i\right),{\sigma }^2\left(u_i,v_i\right)|\tilde{\mathbf{Y}}\right)\right\}=\begin{array}{c}max\\ \tilde{\beta },\tilde{\delta },{\sigma }^2\end{array}\{{\left(2\pi \right)}^{-\frac{n}{2}}{\left({\sigma }^2\left(u_i,v_i\right)\right)}^{-\frac{n}{2}}\hfill \\ exp-\frac{1}{2{\sigma }^2\left(u_i,v_i\right)}\sum _{i=1}^n{w}_{i\left(j\right)}[y_i-({\beta }_0\left(u_i,v_i\right)+\sum _{p=1}^l\sum _{k=1}^m{\beta }_{pk}\left(u_i,v_i\right)x_{pi}^k+\sum _{p=1}^l\sum _{h=1}^r{\delta }_{p,m+h}\left(u_i,v_i\right){{\left(x_{pi}-R_{ph}\right)}_{+}^m)]}^2\}\hfill \end{array}$$ (9)

Estimator $\widehat{\tilde{\beta }}\left(u_i,v_i\right)$, $\widehat{\tilde{\delta }}\left(u_i,v_i\right)$ and $\widehat{\tilde{f}}$ are completely shown in Theorem 1.

Theorem 1

Suppose the regression model Eq. (5) with normally distributed error ${\epsilon }_i$with zero mean and variance ${\sigma }^2\left(u_i,v_i\right)$ and the weighted likelihood function is given by (8) then the MLE estimator for $\widehat{\tilde{\beta }}\left(u_i,v_i\right)$ and $\widehat{\tilde{\delta }}\left(u_i,v_i\right)$ are given by:

$$\widehat{\tilde{\beta }}\left(u_i,v_i\right)=\mathbf{A}\left(\mathbf{K}\right)\tilde{\mathbf{Y}}$$

$$\widehat{\tilde{\delta }}\left(u_i,v_i\right)=\mathbf{B}\left(\mathbf{K}\right)\tilde{\mathbf{Y}}$$

where

$$\mathbf{A}\left(\mathbf{K}\right)=\mathbf{S}{\left({\mathbf{X}}^{\mathbf{T}}\mathbf{W}\left(u_i,v_i\right)\mathbf{X}\right)}^{-1}[{\mathbf{X}}^{\mathbf{T}}-{\mathbf{X}}^{\mathbf{T}}\mathbf{W}\left(u_i,v_i\right)\mathbf{P}{\left({\mathbf{P}}^{\mathbf{T}}\mathbf{W}\left(u_i,v_i\right)\mathbf{P}\right)}^{-1}{\mathbf{P}}^{\mathbf{T}}]\mathbf{W}\left(u_i,v_i\right)$$

$$\mathbf{R}\left(\mathbf{K}\right)=\mathbf{R}{\left({\mathbf{P}}^{\mathbf{T}}\mathbf{W}\left(u_i,v_i\right)\mathbf{P}\right)}^{-1}[{\mathbf{P}}^{\mathbf{T}}-{\mathbf{P}}^{\mathbf{T}}\mathbf{W}\left(u_i,v_i\right)\mathbf{X}{\left({\mathbf{X}}^{\mathbf{T}}\mathbf{W}\left(u_i,v_i\right)\mathbf{X}\right)}^{-1}{\mathbf{X}}^{\mathbf{T}}]\mathbf{W}\left(u_i,v_i\right)$$

Regression curve estimator $\widehat{\tilde{f}}$ contains the polynomial component represented by the matrix $\mathbf{X}$ and the truncated components represented by matrices $\mathbf{P}$. If $\mathbf{P}=0$, then the truncated multivariable spline nonparametric regression curve estimator in the GWR model $\widehat{\tilde{f}}$ will approach the polynomial parametric regression curve estimator in the GWR model.

Furthermore, if $\mathbf{P}=0$ and matrix $\mathbf{X}$ contains a linear function then the nonparametric multivariable spline truncated regression curve estimator in the GWR model $\widehat{\tilde{f}}$ will be a linear parametric regression curve estimator in the GWR model (multiple linear regression in the GWR model) developed by many researchers [[21], [22], [23], [24], [25], [26], [27], [28]].

Generalized cross validation (GCV) for optimal knot point selection

The knot point is the location on the axis $x$where the regression function changes. The optimal knot point is obtained from the minimum GCV [29]. The GCV method formulation is given in Eq. (10).

$$GCV\left(R_1,R_2,\dots ,R_r\right)=\frac{MSE\left(R_1,R_2,\dots ,R_r\right)}{{\left(n^{-1}trace\left[I-A\left(R_1,R_2,\dots ,R_r\right)\right]\right)}^2}$$ (10)

Where,

$MSE\left(R_1,R_2,\dots ,R_r\right):$Mean Square Error of geographically weighted spline nonparametric regression model

$A\left(R_1,R_2,\dots ,R_r\right)$ is a matrix $\mathit{X}{\left({\mathit{X}}^{\mathit{T}}\mathit{X}\right)}^{-1}{\mathit{X}}^{\mathit{T}}$ which is obtained from the equation $\widehat{y}=A\left(R\right)Y$

$I$: the identity matrics

$R_1,R_2,\dots ,R_r$: Knot points

$n$: the number of obseravations.

Geographic weighting function

The geographic weighting function used is the Gaussian kernel function, the bisquare kernel function and the exponential function [30].

• 1.
  The Gaussian kernel function is stated in Eq. (11).

  $$w_{ij}\left(u_i,v_i\right)=\varnothing \left(\frac{d_{ij}}{\sigma b}\right),j=1,2,\dots ,n$$ (11)

  where $w_{ij}\left(u_i,v_i\right)$ is the geographic weighting, $\varnothing$ is the standard normal density function, $d_{ij}$ is the distance from location-i to location-j and $b$ is the bandwidth value. The bandwidth value is a function smoothing parameter value whose value is always positive. $\sigma$ shows the standard deviation of the distance vector $d_{ij}$.
• 2.
  The bisquare kernel function is stated in Eq. (12).

  $$w_{ij}\left(u_i,v_i\right)=\{\begin{array}{cc}{\left(1-{\left(\frac{d_{ij}}{b}\right)}^2\right)}^2\hfill & ,\text{if}{d}_{ij}\le b\hfill \\ 0\hfill & ,\text{otherwise}\hfill \end{array}$$ (12)

  where $d_{ij}=\sqrt{{\left(u_i-u_j\right)}^2+{\left(v_i-v_j\right)}^2}$ is the Euclidean distance between location $\left(u_i,v_i\right)$ to location $\left(u_j,v_j\right)$ and b are the bandwidth values.
• 3.
  The exponential function is stated in Eq. (13).

  $$w_{ij}\left(u_i,v_i\right)=\sqrt{exp\left(-{\left(\frac{d_{ij}}{b}\right)}^2\right)}$$ (13)

  $d_{ij}$ is the distance from location-i to location-j and b is the bandwidth value or smoothing parameter whose value is always positive. In finding the optimum bandwidth value using the Cross Validation (CV) method which is mathematically defined in Eq. (14).

  $$CV=\sum _{i=1}^n{\left(Y_i-{\widehat{Y}}_{\ne i}\left(b\right)\right)}^2$$ (14)

  where ${\widehat{Y}}_{\ne i}\left(b\right)$ is the estimator $Y_i$ on observations at location$\left(u_i,v_i\right)$ that were omitted from the estimation process. To get the optimum $b$ value, it is obtained from the value of the CV iteration results and the minimum CV value is taken. Additionally, The optimum bandwidth can be determined using GCV defined in Eq. (15).

  $$GCV=\frac{n^{-1}{\sum }_{i=1}^n{\left[y_i-{\widehat{y}}_i\right]}^2}{{\left(1-\frac{tr\left(H\left(h\right)\right)}{n}\right)}^2}$$ (15)

  Where $tr\left(H\left(h\right)\right)$ the sum of the main diagonal elements of the $nxn$ weight matrix. The optimum bandwidth is chosen by finding the smallest GCV. The smallest GCV is generated from the model that has the slightest error.

Geographically weighted spline nonparametric regression fitment test

The purpose of the statistical model fit test is to ascertain whether the Geographically Weighted Spline Nonparametric Regression model is more suitable for data analysis than the nonparametric global regression model [15,31]. The formulation of the GWSNR suitability test hypothesis is as follows:

• $H_0:$${\beta }_{kl}\left(u_i,v_i\right)={\beta }_{kl}\text{and}{\delta }_{k,m+h}\left(u_i,v_i\right)={\delta }_{k,m+h};k=1,2,\dots ,p;l=1,2,\dots ,m;i=1,2,\dots ,n$

  (There is no significant difference between the GWSNR Model and the Global Model of truncated spline in multivariable nonparametric regression)

• $H_1:$$\text{There}\text{is}\text{at}\text{least}\text{one}{\beta }_{kl}\left(u_i,v_i\right)\ne {\beta }_{kl}\text{or}{\delta }_{k,m+h}\left(u_i,v_i\right)\ne {\delta }_{k,m+h}$

  (There is a significant difference between the GWSNR Model and the Global Model of truncated spline in multivariable nonparametric regression)

The rejection area $H_0$ if $F_{count}\left(V\right)>F_{\left(\alpha ,d{f}_1,d{f}_2\right)}$ or $p-value<\alpha$. The test statistic is given in Eq. (16).

$$V=\frac{\frac{{\mathit{y}}^{\mathit{T}}\mathit{Sy}}{n-pm-1}}{\frac{{\mathit{y}}^{\mathit{T}}\mathit{D}\left(u_i,v_i\right)\mathit{y}}{tr\left({\left(\mathit{I}-\xi \right)}^{\mathit{T}}\left(\mathit{I}-\xi \right)\right)}}$$ (16)

where

$V$ is statistics test for geographically weighted spline nonparametric regression fitment test, $\mathit{y}$ is the response variable vector, n is the number of observation units, p is the number of predictor variables, m is the number of truncated spline orders and $\mathit{I}$ is the Identity matrix

$$\begin{array}{c}\mathit{y}=\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_n\end{array}\right],\mathit{S}=\left(I-Q{\left(Q^{T}Q\right)}^{-1}{Q}^T\right)\hfill \\ \mathit{D}\left(u_i,v_i\right)=\left(I-W\left(u_i,v_i\right)Q{\left(Q^{T}W\left(u_i,v_i\right)Q\right)}^{-1}{Q}^T\right)\left(I-Q{\left(Q^{T}W\left(u_i,v_i\right)Q\right)}^{-1}{Q}^{T}W\left(u_i,v_i\right)\right)\hfill \\ \xi =Q{\left(Q^{T}W\left(u_i,v_i\right)Q\right)}^{-1}{Q}^{T}W\left(u_i,v_i\right)\hfill \\ =\left[\begin{array}{c}{q}_1^T{\left(Q^{T}W\left(u_1,v_1\right)Q\right)}^{-1}{Q}^{T}W\left(u_1,v_1\right)\\ q_2^T{\left(Q^{T}W\left(u_2,v_2\right)Q\right)}^{-1}{Q}^{T}W\left(u_2,v_2\right)\\ ⋮\\ q_n^T{\left(Q^{T}W\left(u_n,v_n\right)Q\right)}^{-1}{Q}^{T}W\left(u_n,v_n\right)\end{array}\right]\hfill \\ q_i^T=\left[\begin{array}{cccccccc}1& x_{1i}& x_{1i}^2& \cdots & x_{1i}^m& x_{2i}& \cdots & {\left(x_{li}-R_{lr}\right)}_{+}^m\end{array}\right]\hfill \end{array}$$

Simultaneous GWSNR model parameter significance test

Simultaneous parameter significance test was carried out using the F test or analysis table of variance. The purpose of the test is to determine whether there is a simultaneous or simultaneous influence between the response variables on the predictor variable or not [8,15]. The formulation of the simultaneous test hypothesis is as follows:

$H_0:$ ${\beta }_{11}\left(u_i,v_i\right)={\beta }_{12}\left(u_i,v_i\right)=\cdots ={\beta }_{lm}\left(u_i,v_i\right)={\delta }_{1,m+1}\left(u_i,v_i\right)={\delta }_{1,m+2}\left(u_i,v_i\right)=\dots ={\delta }_{l,m+r}\left(u_i,v_i\right)=0;i=1,2,\dots ,n$

$H_1:$ $\text{There}\text{is}\text{at}\text{least}\text{one}{\beta }_{pk}\left(u_i,v_i\right)\ne 0\text{or}{\delta }_{p,m+h}\left(u_i,v_i\right)\ne 0;p=1,2,\dots ,l;k=1,2,\dots ,m;h=1,2,\dots ,r;i=1,2,\dots ,n$

The rejection area $H_0$ if $\left(V^{*}\right)>F_{\left(\alpha ,d{f}_1,d{f}_2\right)}$ or $p-value<\alpha$. The test statistical significance of simultaneous GWSNR model parameters is given in Eq. (17).

$$V^{*}=\frac{\left(\frac{{\mathit{y}}^{\mathit{T}}\mathit{M}\left(u_i,v_i\right)\mathit{y}}{tr\left({\left(\mathit{I}-{\mathit{B}}_{\mathit{w}}\right)}^T\left(\mathit{I}-{\mathit{B}}_{\mathit{w}}\right)\right)}\right)}{\left(\frac{{\mathit{y}}^{\mathit{T}}\mathit{D}\left(u_i,v_i\right)\mathit{y}}{tr\left({\left(\mathit{I}-\mathit{\xi }\right)}^T\left(\mathit{I}-\mathit{\xi }\right)\right)}\right)}$$ (17)

where

$$\begin{array}{ccc}\hfill \mathit{M}\left(u_i,v_i\right)& =& {\left(I-B_{\omega }\right)}^T\left(I-B_{\omega }\right)\hfill \\ \hfill \mathit{D}\left(u_i,v_i\right)& =& \left(I-W\left(u_i,v_i\right)Q{\left(Q^{T}W\left(u_i,v_i\right)Q\right)}^{-1}{Q}^T\right)\left(I-Q{\left(Q^{T}W\left(u_i,v_i\right)Q\right)}^{-1}{Q}^{T}W\left(u_i,v_i\right)\right)\hfill \\ \hfill {\mathit{B}}_{\mathit{\omega }}& =& \left[\begin{array}{cccc}\frac{w_{1\left(1\right)}}{{\sum }_{j=1}^n{w}_{j\left(1\right)}}& \frac{w_{2\left(1\right)}}{{\sum }_{j=1}^n{w}_{j\left(1\right)}}& \cdots & \frac{w_{n\left(1\right)}}{{\sum }_{j=1}^n{w}_{j\left(1\right)}}\\ \frac{w_{1\left(2\right)}}{{\sum }_{j=1}^n{w}_{j\left(2\right)}}& \frac{w_{2\left(2\right)}}{{\sum }_{j=1}^n{w}_{j\left(2\right)}}& \cdots & \frac{w_{n\left(2\right)}}{{\sum }_{j=1}^n{w}_{j\left(2\right)}}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{w_{1\left(n\right)}}{{\sum }_{j=1}^n{w}_{j\left(n\right)}}& \frac{w_{2\left(n\right)}}{{\sum }_{j=1}^n{w}_{j\left(n\right)}}& \cdots & \frac{w_{n\left(n\right)}}{{\sum }_{j=1}^n{w}_{j\left(n\right)}}\end{array}\right],\hfill \end{array}$$

Partial parameter significance test

Formulation of the partial parameter test hypothesis of the model in GWSNR.

$H_0:$ ${\beta }_{pj}\left(u_i,v_i\right)=0\text{and}{\delta }_{p,m+h}\left(u_i,v_i\right)=0;$

$$p=1,2,\dots ,l;j=1,2,\dots m;h=1,2,\dots ,r;i=1,2,\dots ,n$$

$H_1:$ $Thereisatleastone{\beta }_{pj}\left(u_i,v_i\right)\ne 0\text{or}{\delta }_{p,m+h}\left(u_i,v_i\right)\ne 0;p=1,2,\dots ,l;j=1,2,\dots ,m;h=1,2,\dots ,r;i=1,2,\dots ,n$

The rejection area $H_0$ if $|t_{count}|>t_{\left(\frac{\alpha }{2},\left(n-1\right)\right)}$ or $p-value<\alpha$. The test statistics [15,32] to be used is given in Eq. (18).

$$t=\frac{\widehat{\tilde{\eta }}\left(u_i,v_i\right)}{SE\left(\widehat{\tilde{\eta }}\left(u_i,v_i\right)\right)}$$ (18)

The component $SE\left(\widehat{\tilde{\eta }}\left(u_i,v_i\right)\right)=\sqrt{g_{kk}}$ and $g_{kk}$ is the ($k+1$)-th diagonal element of matrix ${\left(Q^{T}W\left(u_i,v_i\right)Q\right)}^{-1}{\widehat{\sigma }}^2\left(u_i,v_i\right)$. If the significance level is $\alpha$, then the decision taken will be rejected H₀ if the value $\left|t\right|>t_{\left(\frac{\alpha }{2},\left(n-1\right)\right)}$

Research methods

Food security is a condition where food is met for all people and countries at all times, reflected in food that is nutritious, safe, high quality, diverse, nutritious, and affordable. The Food Security Index is a measure of several indicators used to produce a composite score of food security conditions in a region [1]. In this research, several predictor variables were used, which were thought to affect the food security index in 2022. The variables are followed as follows:

$y$: Food Security Index

$x_1:$Rice Production

$x_2:$Red Chili Production

$x_3:$Shallot Production

$x_4:$Palm Oil Production

$x_5:$Beef Production

$x_6:$Production of chicken meat

$x_7:$Expenditure For Food

$x_8:$Percentage of Poor Population

$x_9:$Percentage of Population According to Inadequate Consumption

$x_{10}:$Percentage of Population with Food Insecurity

The data used in this research is secondary data. Food Security Index data can be accessed from the official website of the National Food Agency in a publication entitled Food Security Statistics for 2023. All predictors considered to influence the Food Security Index can be accessed from the websites of the Indonesian Central Statistics Agency and the National Food Agency. The research units used were 34 provinces in Indonesia. The Geographically Weighted Spline Nonparametric Regression Model procedures are spatial Mapping, descriptive statistical analysis, testing spatial effects, and GWSNR modeling. The stages in this research include:

• 1.Spatial Mapping based on variable characteristics and observational data
• 2.Descriptive Statistical Analysis.
• 3.Testing the spatial effect with the Breusch-Pagan method.
• 4.GWSNR modeling and determination of geographic weighting
• 5.Calculate the Euclidean distance between observation locations based on geographic location
• 6.Determine the optimum bandwidth based on the CV value of the Gaussian kernel function, the Bisquare kernel function and the Exponential kernel function for each observation location
• 7.Selection of the optimum knot points using Cross Validation (CV) and Generalized Cross Validation (GCV) methods
• 8.Estimation of GWSNR model parameters
• 9.Testing the significance of the simultaneous parameters of the GWSNR model
• 10.Testing the significance of the partial parameters of the GWSNR model
• 11.Testing the suitability of the GWSNR model
• 12.Area mapping based on significant variables.
• 13.GWSNR Model Goodness and Accuracy Measures
• 14.Interpretation of the model GWSNR. model significance test was carried out in three stages, namely Simultaneous model parameter test using analysis of variance or F test, Partial model parameter test using T test and Model suitability test aimed at testing differences in the GWSNR model and the Global Spline Nonparametric Regression model.

Spatial distribution of food security indices and predictor variables

The spatial distribution of each variable is shown in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10, Fig. 11. The value of each variable is displayed at the value intervals provided by the figure.

Fig. 1.

Spatial Distribution of Food security index in 2022 $\left(y\right).$

Fig. 2.

Spatial Distribution of Rice production in 2022 $\left(x_1\right).$

Fig. 3.

Spatial Distribution of Red chili production in 2022 $\left(x_2\right).$

Fig. 4.

Spatial Distribution of Shallot production $\left(x_3\right).$

Fig. 5.

Spatial Distribution of Palm oil production $\left(x_4\right).$

Fig. 6.

Spatial Distribution of Beef production $\left(x_5\right).$

Fig. 7.

Spatial Distribution of Production of laying chicken meat $\left(x_6\right).$

Fig. 8.

Spatial Distribution of Average monthly food expenditure per capita $\left(x_7\right).$

Fig. 9.

Spatial Distribution of Percentage of poor population $\left(x_8\right).$

Fig. 10.

Spatial Distribution of Percentage of population by status of inadequate food consumption $\left(x_9\right).$

Fig. 11.

Spatial Distribution of Percentage of population with moderate or severe food insecurity, scale of experience of food security $\left(x_{10}\right).$

Method validation

Descriptive statistics

Descriptive statistical data on research variables consist of average, minimum value, and maximum value. The results of descriptive statistical calculations are presented in Table 2.

Table 2.

Descriptive Statistics of Research Data.

Variable	Minimum	Maximum	Mean
$y$	35.48	83.82	72.43
$x_1$	855	9789,588	1600,450
$x_2$	1	343,067	40.017
$x_3$	2	564,255	58,958.6
$x_4$	1	8785,327	1378,073
$x_5$	627	93,303	12,876
$x_6$	10	38,874	4267.8
$x_7$	453,031	923,933	634,229
$x_8$	4,45	26,56	10,243
$x_9$	1,78	37,37	11,324
$x_{10}$	2,87	15,31	6,04

The average Food Security Index (IKP) in Indonesia is 72.43 %, with the lowest Food Security Index being 35.48 % and the highest being 83.82 %. The province with the highest IKP is Bali Province and the lowest IKP is in Papua Province. Descriptive statistics for the 2$x_1-x_{10}$ variables are given in Table.

Spatial effect testing

The spatial heterogeneity test aims to determine whether there is a spatial effect on the food security index variable. If the test results detect a spatial effect, spatial modeling is the right step to use in the analysis of the food security index. The results of the spatial effect test are given in Table 3.

Table 3.

Breusch-Pagan Test.

Breusch Pagan	Degrees of Freedom	p-value	Decision
12.14	20	0.0091	spatial heterogeneity effect

Based on the results of the Breusch-Pagan test, it was found that p-value (0.0091) < $\alpha$ (0.05) then decided to refuse $H_0$ with the understanding that there is a spatial heterogeneity effect on the food security index

Geographically weighted spline nonparametric regression modeling

Stages and results of Geographically Weighted Spline Nonparametric Regression Modeling starts with the process of estimating the model with the details of selecting geographic weights, determining the optimal knot point according to the observed data pattern used.

Geographic weighting selection

The steps for finding the GWSNR model begin with selecting the best geographic weighting. Geographical weighting is determined based on the optimal bandwidth value. The geographic weights used in this study are the Gaussian kernel function, the Bisquare kernel function and the Exponential kernel function. The following Table 4 results of calculations for the weighting function.

Table 4.

Geographic Weighting Function.

Geographic Weighting Function	Bandwidth	CV	GCV
Kernel Gaussian Function	19.34	1214.49	161.30
Kernel Bisquare Function	42.29	1255.29	166.72
Kernel Exponential Function	31.28	1215.06	164.02

Based on the results of the geographic weighting calculations shown in Table 4, the best weighting function is the Gaussian kernel function with a bandwidth of 19.34 and has the smallest CV value of 1.214,49 and the smallest GCV of 161.30.

Optimum knot point selection

Determination of the optimum knot point is obtained based on the minimum GCV value. The following is the minimum GCV value using 1 knot point, 2 knot point, and 3 knot point.

Based on Table 5, the best knot point is obtained by using one knot point with a minimum GCV value of 27.333. The optimum knot points with one knot point are as follows:

$k_{1,1}$ = 98,742	$k_{6,1}$ = 398.64
$k_{2,1}$ = 3431.7	$k_{7,1}$ = 457,740
$k_{3,1}$ = 5644.5	$k_{8,1}$ = 4.67
$k_{4,1}$ = 87,854	$k_{9,1}$ = 2.14
$k_{5,1}$ = 1553.8	$k_{10,1}$ = 2.99

Table 5.

Optimum Knot Point.

Knot Point(s)	CV	GCV
1	273.37	27.333
2	330.25	27.546
3	440.10	29.953

The knot point value is used for GWSNR modeling.

Geographically weighted spline nonparametric regression model estimation

The optimal knot point result obtained by the lowest GCV is at one knot point, so the general GWSNR model is given in Eq. (19).

$$y_i={\beta }_0\left(u_i,v_i\right)+\sum _{g=1}^{10}\sum _{j=1}^1{\beta }_{gj}\left(u_i,v_i\right)x_{gi}^j+\sum _{g=1}^{10}\sum _{h=1}^1{\delta }_{g,m+h}\left(u_i,v_i\right){(x_{gi}-K_{gh})}_{+}+{\epsilon }_i$$ (19)

The results of the GWSNR model parameters are inputted to each model. The modeling results obtained are 34 models according to 34 provinces of observational data. The GWSNR model for West Sumatra Province is given in Eq. (20), the GWSNR Model for East Java Province is given in Eq. (21), and the GWSNR Model for South Sulawesi Province is given in Eq. (22) below.

GWSNR model for West Sumatra Province

$$\begin{array}{ccc}\hfill {\widehat{y}}_3& =& -0.55+1.44\times {10}^{-4}{x}_{1,3}-3.77\times {10}^{-3}{x}_{2,3}+2.87\times {10}^{-4}{x}_{3,3}-1.36\times {10}^{-5}{x}_{4,3}-8.77\times {10}^{-3}{x}_{5,3}\hfill \\ & & +5.73\times {10}^{-3}{x}_{6,3}+4.55\times {10}^{-4}{x}_{7,3}-2.43x_{8,3}+1.58x_{9,3}-33.46x_{10,3}-1.44\times {10}^{-4}\left(x_{1,3}+98,742\right)\hfill \\ & & +3.76\times {10}^{-3}\left(x_{2,3}+3,431.7\right)-2.81\times {10}^{-4}\left(x_{3,3}+5,644.5\right)+1.26\times {10}^{-5}\left(x_{4,3}+87,854\right)\hfill \\ & & +8.69\times {10}^{-3}\left(x_{5,3}+1,553.8\right)-5.63\times {10}^{-3}\left(x_{6,3}+398.64\right)-4.75\times {10}^{-4}\left(x_{7,3}+457,740\right)+1.52\left(x_{8,3}+4.67\right)\hfill \\ & & -2.10\left(x_{9,3}+2.14\right)+32.88\left(x_{10,3}+2.99\right)\hfill \end{array}$$ (20)

GWSNR model for East Java Province

$$\begin{array}{ccc}\hfill {\widehat{y}}_{15}& =& -0.98+1.55\times {10}^{-4}{x}_{1,15}-3.62\times {10}^{-3}{x}_{2,15}+2.19\times {10}^{-4}{x}_{3,15}-1.63\times {10}^{-5}{x}_{4,15}-9.10\times {10}^{-3}{x}_{5,15}\hfill \\ & & +4.44\times {10}^{-3}{x}_{6,15}+4.76\times {10}^{-4}{x}_{7,15}-0.63x_{8,15}+5.60x_{9,15}-41.99x_{10,15}-1.55\times {10}^{-4}\left(x_{1,15}+98.742\right)\hfill \\ & & +3.61\times {10}^{-3}\left(x_{2,15}+3,431.7\right)-2.12\times {10}^{-4}\left(x_{3,15}+5,644.5\right)+1.53\times {10}^{-5}\left(x_{4,15}+87,854\right)\hfill \\ & & +9.00\times {10}^{-3}\left(x_{5,15}+1,553.8\right)-4.29\times {10}^{-3}\left(x_{6,15}+398.64\right)-4.95\times {10}^{-4}\left(x_{7,15}{x}_{7,3}+457,740\right)-0.27\left(x_{8,15}+4.67\right)\hfill \\ & & -6.25\left(x_{9,15}+2.14\right)+41.51\left(x_{10,15}+2.99\right)\hfill \end{array}$$ (21)

GWSNR model for South Sulawesi Province

$$\begin{array}{ccc}\hfill {\widehat{y}}_{27}& =& -0.69+1.76\times {10}^{-4}{x}_{1,27}-3.63\times {10}^{-3}{x}_{2,27}+1.27\times {10}^{-4}{x}_{3,27}-1.34\times {10}^{-5}{x}_{4,27}-9.86\times {10}^{-3}{x}_{5,27}\hfill \\ & & +4.00\times {10}^{-3}{x}_{6,27}+4.97\times {10}^{-4}{x}_{7,27}+2.93x_{8,27}+8.07x_{9,27}-52.40x_{10,27}-1.72\times {10}^{-4}\left(x_{1,27}+98,742\right)\hfill \\ & & +3.61\times {10}^{-3}\left(x_{2,27}+3,431.7\right)-1.32\times {10}^{-4}\left(x_{3,27}+5,644.5\right)+1.27\times {10}^{-5}\left(x_{4,27}+87,854\right)\hfill \\ & & +9.74\times {10}^{-3}\left(x_{5,27}+1,553.8\right)-3.80\times {10}^{-3}\left(x_{6,27}+398.64\right)-5.13\times {10}^{-4}\left(x_{7,27}+457,740\right)-3.83\left(x_{8,27}+4.67\right)\hfill \\ & & -8.79\left(x_{9,27}+2.14\right)+52.03\left(x_{10,27}+2.99\right)\hfill \end{array}$$ (22)

Comparison between estimator $\widehat{y}$ and variable data $y$ given in Fig. 12.

Fig. 12.

Distribution pattern of estimator $\widehat{y}$ and data variable $y.$

In Fig. 12 it shows that the results of the estimator $\widehat{y}$ approach the original data of the variable $y$ (Food Security Index).

Significance test of GWSNR model parameters

GWSNR model significance test was carried out in three stages, namely Simultaneous model parameter test using analysis of variance or F test, Partial model parameter test using T test and Model suitability test aimed at testing differences in the GWSNR model and the Global Spline Nonparametric Regression model.

Simultaneous significance test of GWSNR model parameters

The formula of the hypothesis of the simultaneous parameter significance test of the GWSNR model is

$H_0:$ ${\beta }_{1,1}\left(u_i,v_i\right)=\cdots ={\beta }_{1,10}\left(u_i,v_i\right)={\delta }_{1,11}\left(u_i,v_i\right)=\dots ={\delta }_{1,20}\left(u_i,v_i\right)=0$

$H_1:$$\text{There}\text{is}\text{at}\text{least}\text{one}{\beta }_{pk}\left(u_i,v_i\right)\ne 0$ or ${\delta }_{p,m+h}\left(u_i,v_i\right)\ne 0$

The results of the calculation of the simultaneous significance test of the GWSNR model are given in Table 6.

Table 6.

Analysis of Variance in Simultaneous Significance Test.

Source of Diversity	Sum of Squares	Degree of Freedom	Middle Square	V*	p-value
Regression	2962.30	33	89.77	7.03	4.68 x 10–5
Residual	217.11	17	12.77
Total	3179.40	50

The results of the analysis obtained values V* = 7.03 > $F_{\left(0.05;33;17\right)}$ = 3.36 or p-value (4.68 × 10^–5) < $\alpha$ (0.05). then decided to refuse $H_0$. The results obtained predictor variables $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,$and $x_{10}$ simultaneously have a significant effect on the response variable $y$.

Partial significance test of GWSNR model parameters

The formula of the hypothesis test of partial significance of GWSNR model parameters is

$H_0:$ ${\beta }_{pj}\left(u_i,v_i\right)=0\text{and}{\delta }_{p,m+h}\left(u_i,v_i\right)=0$

$H_1:$ There is at least one${\beta }_{pj}\left(u_i,v_i\right)\ne 0$ or ${\delta }_{p,m+h}\left(u_i,v_i\right)\ne 0$

This test uses a significance level $\alpha$ = 0.05 and rejection criteria $H_0$ is rejected if p-value < $\alpha$. Based on the results of the partial parameter significance test, 10 group classifications based on significant variables were obtained in Table 7.

Table 7.

Significant Variables.

Group Classification	Significant Variable(s)	Province
1	$x_1$, $x_2$, $x_3$, $x_4$, $x_5$, $x_6$, $x_7$, $x_8$, $x_9$, and $x_{10}$	Aceh, West Sumatera, Riau, and East Java
2	$x_1$, $x_2$, $x_4$, $x_5$, $x_6$, $x_7$, $x_8$, $x_9$, and $x_{10}$	Kep. Riau, DKI Jakarta, West Java, and Bali
3	$x_1$, $x_2$, $x_4$, $x_5$, $x_7$, $x_8$, $x_9$, and $x_{10}$	Jambi, Central Java, South Sulawesi, West Sulawesi, Maluku, and Papua
4	$x_1$, $x_2$, $x_5$, $x_7$, $x_8$, $x_9$, and $x_{10}$	North Sumatera, Bengkulu, Kep. Bangka Belitung, Central Kalimantan, and South Kalimantan
5	$x_1$, $x_2$, $x_7$, $x_8$, $x_9$, and $x_{10}$	West Nusa Tenggara Barat and West Kalimantan
6	$x_1$, $x_2$, $x_5$, $x_7$, $x_8$, and $x_9$	West Papua
7	$x_1$, $x_2$, $x_7$, $x_8$, and $x_9$	North Sumatera, DI Yogyakarta, East Nusa Tenggara, North Kalimantan, North Sulawesi, Central Sulawesi, South-east Sulawesi, and North Maluku
8	$x_1$, $x_7$, $x_8$, and $x_9$	Lampung and Gorontalo
9	$x_7$, $x_8$, and $x_9$	Banten
10	$x_7$ and $x_9$	East Kalimantan

The distribution map based on significant variables is given in Fig. 13.

Fig. 13.

Spatial distribution based on significant variables.

In Fig. 13. the areas that have been mapped based on groups of several predictor variables that are significant to food security index are on the same mapping because they have uniform characteristics. This is in accordance with the first law of geography by W Tobler.

GWSNR model fit test

The formulation of the GWSNR fit test hypothesis is

$H_0:$ ${\beta }_{kf}\left(u_i,v_i\right)={\beta }_{kf}\text{and}{\delta }_{k,m+s}\left(u_i,v_i\right)={\delta }_{k,m+s}$

$H_1:$ There is at least one ${\beta }_{kf}\left(u_i,v_i\right)\ne {\beta }_{kf}$ or ${\delta }_{k,m+s}\left(u_i,v_i\right)\ne {\delta }_{k,m+s}$

The significance level used is $\alpha$ = 0.05 and rejection criteria $H_0$ is rejected if p-value < $\alpha$. The results of statistical calculation of the model fit test are given in Table 8.

Table 8.

Model GWSNR Fit Test.

Source of Diversity	Sum of Squares	Degree of Freedom	Middle Square	V	p-value
Regression	56.26	13	4.33	0.34	0.0097
Residual	217.11	17	12.77
Total	273.37	30

Based on Table 8, we obtained that p-value (0.0097) < $\alpha$ (0.05), then we decided to reject $H_0$. The conclusion in this test is that the GWSNR model is more appropriate to use than the global model.

GWSNR model goodness-of-fit and accuracy measures

The measure of the goodness-of-fit and accuracy of the model used in this study is the coefficient of determination and Root Mean Square Error (RMSE) the results of which can be seen in Table 9 below.

Table 9.

Model Goodness-of-Fit and Accuracy Measures.

Model	$R^2$ Value	RMSE
Nonparametric Regression	24.02	4.28
GWR	92.78	3.41
GWSNR	95.16	2.57

The coefficient of determination for the GWSNR model is 95.16 % which indicates that the GWSNR model is for variables $x_1$, $x_2$, $x_3$, $x_4$, $x_5$, $x_6$, $x_7$, $x_8,x_9$and $x_{10}$ can explain the diversity of the Food Security Index of 34 provinces in Indonesia of 95.16 % with an RMSE value of 2.57.

GWSNR model interpretation

The interpretation of the GWR model for the province of West Sumatra is explained in the following description:

• 1.
  GWSNR model interpretation for variable $x_1$ (rice production) and other variables are considered constant. The effect of rice production on the Food Security Index can be interpreted in the Equation Model (23).

  $$\begin{array}{ccc}\hfill {\widehat{y}}_3& =& -0.55+1.44\times {10}^{-4}{x}_{1,3}-1.44\times {10}^{-4}\left(x_{1,3}+98,742\right)\hfill \\ \hfill {\widehat{y}}_3& =& \{\begin{array}{cc}-0.55+1.44\times {10}^{-4}{x}_{1,3}\hfill & x_{1,3}<98,742\hfill \\ -14.77\hfill & x_{1,3}\ge 98,742\hfill \end{array}\hfill \end{array}$$ (23)

  The interpretation of this model is that when rice production is less than 98,742 tons, if rice production increases by 1 ton, the Food Security Index in Indonesia will increase by 1.44 $\times$ 10^–4.

• 2.
  GWSNR model interpretation for variable $x_2$ (red chili production) and other variables are considered constant. The effect of red chili production on the Food Security Index can be interpreted in the Equation Model (24)

  $$\begin{array}{ccc}\hfill {\widehat{y}}_3& =& -0.55-3.77\times {10}^{-3}{x}_{2,3}+3,76\times {10}^{-3}\left(x_{2,3}+3,431.7\right)\hfill \\ \hfill {\widehat{y}}_3& =& \{\begin{array}{cc}-0.55-3.77\times {10}^{-3}{x}_{2,3}\hfill & x_{2,3}<3,431.7\hfill \\ 12.35-0.1\times {10}^{-4}{x}_{2,3}\hfill & x_{2,3}\ge 3,431.7\hfill \end{array}\hfill \end{array}$$ (24)

  The interpretation of this model is that when red chili production is less than 3431.7 tons, if red chili production increases by 1 ton, the Food Security Index in Indonesia will decrease by 3.77 $\times$ 10^–3.

• 3.
  GWSNR model interpretation for variable $x_3$ (shallot production) and other variables are considered constant. The effect of shallot production on the Food Security Index can be interpreted in the Equation Model (25).

  $$\begin{array}{c}{\widehat{y}}_3=-0.55+2.87\times {10}^{-4}{x}_{3,3}-2.81\times {10}^{-4}\left(x_{3,3}+5,644.5\right)\hfill \\ {\widehat{y}}_3=\{\begin{array}{cc}-0.55+2.87\times {10}^{-4}{x}_{3,3}\hfill & x_{3,3}<5,644.5\hfill \\ -2.14+0.06\times {10}^{-4}{x}_{3,3}\hfill & x_{3,3}\ge 5,644.5\hfill \end{array}\hfill \end{array}$$ (25)

  The interpretation of this model is, when shallot production is less than 5.644,5 tons, then if shallot production increases by 1 ton, the Food Security Index in Indonesia will increase by $2.87\times {10}^{-4}$.

• 4.
  GWSNR model interpretation for variable $x_4$ (oil palm production) and other variables are considered constant. The effect of palm oil production on the Food Security Index can be interpreted in the Equation Model (26).

  $$\begin{array}{c}{\widehat{y}}_3=-0.55-1.36\times {10}^{-5}{x}_{4,3}+1.26\times {10}^{-5}\left(x_{4,3}+87,854\right)\hfill \\ {\widehat{y}}_3=\{\begin{array}{cc}-0.55-1.36\times {10}^{-5}{x}_{4,3}\hfill & x_{4,3}<87,854\hfill \\ 0.56-0.01\times {10}^{-4}{x}_{4,3}\hfill & x_{4,3}\ge 87,854\hfill \end{array}\hfill \end{array}$$ (26)

  The interpretation of this model is when the palm oil production is less than 87,854 tons, if palm oil production increases by 1 ton, then the Food Security Index in Indonesia will decrease by 1.36 $\times$ 10^–5.

• 5.
  GWSNR model interpretation for variable $x_5$ beef production and other variables are considered constant. The effect of beef production on the Food Security Index can be interpreted in the Equation Model (27).

  $$\begin{array}{c}{\widehat{y}}_3=-0.55-8.77\times {10}^{-3}{x}_{5,3}+8.69\times {10}^{-3}\left(x_{5,3}+1,553.8\right)\hfill \\ {\widehat{y}}_3=\{\begin{array}{cc}-0.55-8.77\times {10}^{-3}{x}_{5,3}\hfill & x_{5,3}<1,553.8\hfill \\ 12.95-0.8\times {10}^{-4}{x}_{5,3}\hfill & x_{5,3}\ge 1,553.8\hfill \end{array}\hfill \end{array}$$ (27)

  The interpretation of this model is that when beef production is less than 1553.8 tons, if beef production increases by 1 ton, the Food Security Index in Indonesia will decrease by 8.77 $\times$ 10^–3.

• 6.
  GWSNR model interpretation for variable $x_6$ chicken meat production and other variables is considered constant. The effect of chicken meat production on the Food Security Index can be interpreted in the Equation Model (28).

  $$\begin{array}{c}{\widehat{y}}_3=-0.55+5.73\times {10}^{-3}{x}_{6,3}-5.63\times {10}^{-3}\left(x_{6,3}+398.64\right)\hfill \\ {\widehat{y}}_3=\{\begin{array}{cc}-0.55+5.73\times {10}^{-3}{x}_{6,3}\hfill & x_{6,3}<398.64\hfill \\ -2.79+1.0\times {10}^{-4}{x}_{6,3}\hfill & x_{6,3}\ge 398.64\hfill \end{array}\hfill \end{array}$$ (28)

  The interpretation of this model is, when chicken meat production is less than 398.64 tons, if chicken meat production increases by 1 ton, the Food Security Index in Indonesia will increase by 5.73 $\times$ 10^–3.

• 7.
  GWSNR model interpretation for variable $x_7$ (expenditure on food) and other variables is considered constant. The effect of expenditure on food on the Food Security Index can be interpreted in the Equation Model (29).

  $$\begin{array}{c}{\widehat{y}}_3=-0.55+4.55\times {10}^{-4}{x}_{7,3}-4.75\times {10}^{-4}\left(x_{7,3}+457,740\right)\hfill \\ {\widehat{y}}_3=\{\begin{array}{cc}-0.55+4.55\times {10}^{-4}{x}_{7,3}\hfill & x_{7,3}<457,740\hfill \\ -217.98-0.2\times {10}^{-4}{x}_{7,3}\hfill & x_{7,3}\ge 457,740\hfill \end{array}\hfill \end{array}$$ (29)

  The interpretation of this model is, when expenditure on food is less than 457,740 rupiah, then if spending on food increases by 1 rupiah, then the Food Security Index in Indonesia will increase by 4.55 $\times$ 10^–4.

• 8.
  GWSNR model interpretation for the variable $x_8$ the percentage of poor people and other variables is considered constant. The effect of the percentage of poor people on the Food Security Index can be interpreted in the Equation Model (30).

  $$\begin{array}{c}{\widehat{y}}_3=-0.55-2.43x_{8,3}+1.52\left(x_{8,3}+4.67\right)\hfill \\ {\widehat{y}}_3=\{\begin{array}{cc}-0.55-2.43x_{8,3}\hfill & x_{8,3}<4.67\hfill \\ 6.55-0.91x_{8,3}\hfill & x_{8,3}\ge 4.67\hfill \end{array}\hfill \end{array}$$ (30)

  The interpretation of this model is, when the percentage of poor people is less than 4.67 percent, then if the percentage of poor people increases by 1 percent, then the Food Security Index in Indonesia will decrease by 2.43.

• 9.
  GWSNR model interpretation for variable $x_9$ is the percentage of the population according to insufficient consumption status and other variables are considered constant. The effect of the percentage of the population according to the status of insufficient consumption on the Food Security Index can be interpreted in the Equation Model (31).

  $$\begin{array}{c}{\widehat{y}}_3=-0.55+1.58x_{9,3}-2.10\left(x_{9,3}+2.14\right)\hfill \\ {\widehat{y}}_3=\{\begin{array}{cc}-0.55+1.58x_{9,3}\hfill & x_{9,3}<2.14\hfill \\ -5.04-0.52x_{9,3}\hfill & x_{9,3}\ge 2.14\hfill \end{array}\hfill \end{array}$$ (31)

  The interpretation of this model is, when the percentage of the population according to the consumption insufficiency status is less than 2.14 percent, then if the percentage of the population according to the consumption insufficiency status increases by 1 percent, then the Food Security Index in Indonesia will increase by 1.58.

• 10.
  GWSNR model interpretation for the variable $x_{10}$ the percentage of the population with food insecurity and other variables is considered constant. The effect of the percentage of population with food insecurity on the Food Security Index can be interpreted in the Equation Model (32).

  $$\begin{array}{c}{\widehat{y}}_3=-0.55-33.46x_{10,3}-32.88\left(x_{10,3}+2.99\right)\hfill \\ {\widehat{y}}_3=\{\begin{array}{cc}-0.55-33.46x_{10,3}\hfill & x_{10,3}<2.99\hfill \\ 97.76-0.58x_{10,3}\hfill & x_{10,3}\ge 2.99\hfill \end{array}\hfill \end{array}$$ (32)

  The interpretation of this model is, when the percentage of the population with food insecurity is less than 2.99 percent, then if the percentage of the population with food insecurity increases by 1 percent, the Food Security Index in Indonesia will decrease by 33.46.

Conclusion

• 1.Food security index data and predictor variables $x_1$, $x_2$, $x_3$, $x_4$, $x_5$, $x_6$, $x_7$, $x_8,x_9$and $x_{10}$ provinces in Indonesia Year 2022 has a spatial effect.
• 2.The best GWSNR model is using the Gaussian kernel weighting function, while selecting the optimal knot point is one knot point based on the lowest CV and GCV values.
• 3.The results of simultaneous and partial parameter tests are significant. Based on the results of the partial significance test, there are 10 area classifications based on significant predictor variables with different effects on each grouping classification results.
• 4.The results of the model fit test stated that the GWSNR model was most appropriate for modeling food security index data.
• 5.The GWSNR model goodness-of-fit is 95.16 % which indicates that the GWSNR model for variables $x_1$, $x_2$, $x_3$, $x_4$, $x_5$, $x_6$, $x_7$, $x_8,x_9$and $x_{10}$ can explain the diversity of the Food Security Index of 34 provinces in Indonesia of 92.78 % with an RMSE value of 3.41.
• 6.Factors influencing the national food security index are rice production, red chili production, shallot production, palm oil production, beef production, laying hen meat production, average per capita food expenditure per month, percentage of poor people, percentage of population according to inadequate food consumption status, and percentage of population with food insecurity.

CRediT authorship contribution statement

Sifriyani: Software, Visualization, Data curation. I Nyoman Budiantara: Validation, Writing – review & editing, Supervision. Krishna Purnawan Candra: Software, Visualization, Data curation. Marisa Putri: Investigation, Resources, Writing – original draft, Project administration.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

• Data will be made available on request.