Statistical inferences and applications of nonparametric regression models based on fourier series

Abstract

This study develops statistical inference methods, including estimation and hypothesis testing procedures, which consist of partial and simultaneous tests for nonparametric regression models based on Fourier series approximations. The Fourier series is effective for data with periodic characteristics, offering flexible nonparametric regression solutions. While previous work primarily focused on estimation techniques, this research introduces a structured hypothesis testing framework using F-test and T-test statistics to evaluate the significance of model parameters. Utilizing real-world expenditure data from 29 districts in Papua, Indonesia, we examine the Gross Regional Domestic Product per capita, poverty rates, and labor force participation rates as predictors of per capita expenditure. The best oscillation parameter is determined through Generalized Cross Validation (GCV), improving the accuracy of the model. The findings demonstrate significant effects of the predictor variables on the model, supported by both simultaneous tests using the F-test and partial tests using T-tests. The highlights of this research are:

• •Introduces hypothesis testing using F-test and T-test for nonparametric regression with Fourier series.
• •Applies model to regional expenditure data with optimal parameter selection.
• •Validates model using real data, showing significant predictor influence.

Graphical abstract

Specifications table

Subject area:	Mathematics and Statistics
More specific subject area:	Nonparametric Regression
Name of your method:	Statistical Inference for Nonparametric Regression Models based on Fourier Series
Name and reference of original method:	Fourier series function developed by Bilodeau (1992), M. Bilodeau, Fourier smoother and additive models, Canad. J. Stat. 20 (1992) 257–269, https://doi.org/10.2307/3315313. F-test and T-test statistic for hypothesis testing of parameters in the book of Rencher and Schaalje (2008) and Hogg and Craig (2019), A.C. Rencher and G. B. Schaalje, Linear Models in Statistics, 2nd ed., John Wiley & Sons, Inc., New Jersey, 2008 R. V. Hogg, J. W. McKean and A. T. Craig, Introduction to Mathematical Statistics, Eighth ed., Pearson Education, Inc, United States, 2019
Resource availability:	Real Expenditure per Capita in Papua and its predictor variables can be accessed through the website of the Central Bureau of Statistics (BPS) as follows: •https://papua.bps.go.id/id/publication/2024/07/02/5fc648ff81310c6eedcd4490/indeks-pembangunan-manusia-provinsi-papua-2023.html •https://www.bps.go.id/id/publication/2024/06/07/f90b4d2293193647cf2faee1/produk-domestik-regional-bruto-kabupaten-kota-di-indonesia-2019–2023.html •https://papua.bps.go.id/id/publication/2024/09/26/6d0fe5bf112b4243f905049c/statistik-daerah-provinsi-papua-2024.html

Background

Regression analysis using nonparametric regression does not necessitate assuming a certain regression curve form.. This method is more flexible, enabling the regression curve to organically fit the data's local patterns. [1]. Nevertheless, determining the regression curve in nonparametric regression requires insights from the data, which can be gained by examining a scatter plot of the data pattern [2,3]. Various estimators are available for nonparametric regression, including Spline functions [[4], [5], [6]], Kernel functions [[7], [8], [9]], Fourier series [[10], [11], [12], [13]], among others. The Fourier series, in particular, has attracted significant interest among researchers because it is well-suited to data with periodic patterns. Bilodeau proposed a variation of the Fourier series that uses only the cosine function and adds a trend component to the regression model [10]. While other researchers describe the Fourier series as a trigonometric polynomial comprising both cosine and sine components [[14], [15], [16]]. This modified Fourier series estimator has gained popularity and has been refined by many researchers to estimate nonparametric regression curves [13,[17], [18], [19], [20]]. In practice, it has been successfully applied to model different types of data [21,22].

Statistical inference in regression analysis is fundamental for achieving optimal model estimation. In this context, statistical inference is generally divided into two primary components, i.e. parameter estimation and hypothesis testing. Parameter estimation pertains to the computation of model parameter values, which subsequently enables the derivation of inferences about the response variable in relation to the predictor variables. On the other hand, hypothesis testing is employed to evaluate if the parameters significantly influence the model. This type of testing is generally categorized into partial and simultaneous hypothesis testing. Simultaneous testing evaluates the overall effect of all parameters on the model. If a significant impact is detected, then partial testing is conducted to determine which individual parameter contribute to the model. Various researchers have thoroughly explored the role of statistical inference in regression analysis [[23], [24], [25]]. Statistical inference in nonparametric regression models using Fourier series approximation has largely focused on the estimation of model parameters [10,12,[17], [18], [19],21,22]. Furthermore, Ramli et al. has explored hypothesis testing with a specific emphasis on simultaneous testing [20]. However, partial hypothesis testing has not yet been explored in this model. Considering the significance of statistical inference in regression analysis, especially in nonparametric regression models utilizing Fourier series approximation, this topic becomes highly relevant for further research. Previous studies have explored topics such as estimation and simultaneous hypothesis testing, both theoretically and practically. However, this research will only briefly address these areas, focusing instead on developing a theoretical framework for hypothesis testing using F-test and T-test, which remains unexplored. Additionally, real world expenditure data from 29 districts in Papua for 2023 will be utilized to implement in this study.

Method details

Nonparametric regression model with fourier series

Assume paired data$(y_i,x_{1i},x_{2i},\dots ,x_{pi});i=1,2,\dots ,n$satisfies the nonparametric regression model as stated in Eq. (1).

$$y_i=m\left(x_{1i},x_{2i},\dots ,x_{pi}\right)+{\epsilon }_i$$ (1)

where$y_i$ is the response variable for observation-$i,x_{ji}$is the predictor variable for observation$i,$for the$j$-th predictor, and${\epsilon }_i$is the error for observation$i$which is normally distributed and independent with a mean of 0 and variance${\sigma }^2$, written as${\epsilon }_i\sim N\left(0,{\sigma }^2\right)$. If the predictor variables are assumed to be uncorrelated, the model for the nonparametric regression can be stated in an additive form as follows:

$$y_i=\sum _{j=1}^{p}m\left(x_{ji}\right)+{\epsilon }_i$$ (2)

Since the regression curve$m\left(x_{ji}\right)$is assumed to be unknown, it can be approximated by one of the estimators in the nonparametric regression model, which is the Fourier series function [10], where the Fourier series function is stated in Eq. (3).

$$m\left(x_{ji}\right)=\frac{{\delta }_j}{2}+{\gamma }_j{x}_{ji}+\sum _{k=1}^{\lambda }{\alpha }_{jk}\cos \left(k{x}_{ji}\right)$$ (3)

Substituting Eq. (3) into Eq. (2) yields the Fourier series nonparametric regression model as stated in Eq. (4).

$$y_i=\sum _{j=1}^p\left[\frac{{\delta }_j}{2}+{\gamma }_j{x}_{ji}+\sum _{k=1}^{\lambda }{\alpha }_{jk}\cos \left(k{x}_{ji}\right)\right]+{\epsilon }_i$$ (4)

In general, for$i=1,2,\dots ,n$, the Fourier series nonparametric regression model in Eq. (4) can be written in matrix form as shown in Eq. (5).

$$\mathit{y}=\mathbf{Z}\beta +\mathit{\varepsilon }$$ (5)

where$\mathit{y}={\left(y_1,y_2,\dots ,y_n\right)}^{{}^{\prime }},\mathbf{Z}=\left({\mathbf{X}}_1\left(\lambda \right),{\mathbf{X}}_2\left(\lambda \right),\dots ,{\mathbf{X}}_{\mathit{p}}\left(\lambda \right)\right),\mathit{\beta }=\left({\mathit{\beta }}_1^{{}^{\prime }},{\mathit{\beta }}_2^{{}^{\prime }},\dots ,{\mathit{\beta }}_p^{{}^{\prime }}\right){}^{\prime }$, with

$${\mathbf{X}}_{\mathrm{j}}\left(\lambda \right)=\left(\begin{array}{ccccc}1/2& x_{j1}& \cos x_{j1}& \cdots & \cos \lambda x_{j1}\\ 1/2& x_{j2}& \cos x_{j2}& \cdots & \cos \lambda x_{j2}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1/2& x_{jn}& \cos x_{jn}& \cdots & \cos \lambda x_{jn}\end{array}\right),{\mathit{\beta }}_{\mathrm{j}}=\left({\delta }_j,{\gamma }_j,{\alpha }_{j1},\cdots ,{\alpha }_{j\lambda }\right);j=1,2,\cdots ,p$$

error vector is assumed$\mathit{\varepsilon }={\left({\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_n\right)}^{\prime }\sim N_n\left(0,{\sigma }^2\mathbf{I}\right)$.

Parameter estimation

To estimate the nonparametric regression model with a Fourier series in Eq. (5), it is equivalent to estimating the parameter vector. There are several methods for estimating the parameter vector, such as the Penalized Least Squares (PLS) method if the regression curve$m(•)$is assumed to be smooth [10,12,21]. If the regression curve$m(•)$is assumed to be unknown and represented in the linear model form as in Eq. (5), it can be estimated using the Ordinary Least Squares (OLS) method by minimizing the sum of squared errors. The estimation of the parameter vector$\mathit{\beta }$using the OLS method is obtained as follows [17,18,22]:

$$\widehat{\beta }=\underset{\mathit{\beta }\in {\mathrm{R}}^{\mathit{p}\left(\mathit{\lambda }+2\right)}}{\arg \min }{\left(\mathit{y}-\mathbf{Z}\beta \right)}^{\prime }\left(\mathit{y}-\mathbf{Z}\beta \right)={\left(\mathbf{Z}{}^{\prime }\mathbf{Z}\right)}^{-1}{\mathbf{Z}}^{\prime }\mathit{y}$$ (6)

where

$$\widehat{\beta }={\left({\widehat{\beta }}_1^{{}^{\prime }},{\widehat{\beta }}_2^{{}^{\prime }},\dots ,{\widehat{\beta }}_p^{{}^{\prime }}\right)}^{{}^{\prime }}={\left({\widehat{\delta }}_1,{\widehat{\gamma }}_1,{\widehat{\alpha }}_{11},\dots ,{\widehat{\alpha }}_{1\lambda },\dots ,{\widehat{\delta }}_p,{\widehat{\gamma }}_p,{\widehat{\alpha }}_{p1},\dots ,{\widehat{\alpha }}_{p\lambda }\right)}^{{}^{\prime }}\text{and}\mathit{y}={\left(y_1,y_2,\dots ,y_n\right)}^{{}^{\prime }}.$$

Substituting Eq. (6) into Eq. (5), the estimated nonparametric regression model with a Fourier series is obtained as stated in Eq. (7):

$$\widehat{\mathit{y}}=\mathbf{Z}\widehat{\beta }=\mathbf{H}\left(\lambda \right)\mathit{y}$$ (7)

where$\mathbf{H}\left(\lambda \right)=\mathbf{Z}{\left(\mathbf{Z}{}^{\prime }\mathbf{Z}\right)}^{-1}{\mathbf{Z}}^{\prime }$.

The vector parameter estimator$\widehat{\beta }$, obtained using the OLS method in Eq. (6), includes an unknown oscillation parameter$\lambda$. In nonparametric regression with the OLS approach, there is typically an unknown parameter, such as the knot location in Spline functions, the bandwidth in Kernel functions, or the oscillation parameter in Fourier series functions. Thus, to achieve the best parameter estimates for modeling nonparametric Fourier series regression, it is crucial to determine the optimal number of oscillation parameters$\lambda$. Methods such as Cross Validation (CV) or Generalized Cross Validation (GCV) can be employed to determine this optimal$\lambda$. GCV, in particular, has been extensively developed by researchers for applications in Spline functions [6] and Fourier series functions [10]. The GCV method selects the optimal oscillation parameter$\lambda$based on the minimum GCV value.

$$\text{GCV}\left(\lambda \right)=\frac{MSE\left(\lambda \right)}{{\left(\frac{1}{n}tr\left[\mathbf{I}-\mathbf{H}\left(\lambda \right)\right]\right)}^2}=\frac{\frac{1}{n}{\mathit{y}}^{\prime }\left(\mathbf{I}-\mathbf{H}\left(\lambda \right)\right)\mathit{y}}{{\left(\frac{1}{n}tr\left[\mathbf{I}-\mathbf{H}\left(\lambda \right)\right]\right)}^2}$$ (8)

Hypothesis testing of parameters using the F-Test statistic and T-Test statistic

Parameter hypothesis testing is essential in modeling and forms a key component of inferential statistics within regression analysis. This testing method determines whether parameter estimates significantly impact the regression model. In the context of nonparametric Fourier series regression models, an F-test for hypothesis testing of parameters has not been previously conducted. According to prior studies, this model has been employed by researchers for modeling or predicting various fields or datasets [17,18,22]. Previous research primarily focused on parameter estimation in nonparametric regression modeling. Recently, parameter hypothesis testing for nonparametric regression models with Fourier series functions has been developed using the Likelihood Ratio Test (LRT) approach [20]. Several researchers have constructed the F-test statistic for hypothesis testing of parameters in parametric regression models [26]. Fundamentally, the F-test statistic is used to compare a reduced model to a full model. This study advances the application of parameter hypothesis testing in nonparametric Fourier series regression models using the F-test statistic.

According to Casella and Berger, the F-test statistic for testing parameter hypotheses in parametric regression models is associated with the distribution of quadratic forms [27].

Lemma 1

Given an identity matrix$\mathbf{J}$of size$n\times n$and a matrix$\mathbf{H}\left(\lambda \right)$as defined in Eq. (7), it follows that

• (i)$\mathbf{H}\left(\lambda \right)$is symmetric and idempotent

• (ii)$\mathbf{H}\left(\lambda \right)\mathbf{J}=\mathbf{J}$and$\mathbf{J}\mathbf{H}\left(\lambda \right)=\mathbf{J}$

Proof

• (i) Symmetry and Idempotence of$\mathbf{H}\left(\lambda \right)$

To show that$\mathbf{H}\left(\lambda \right)$is symmetric, we compute the transpose of$\mathbf{H}\left(\lambda \right):$

$${\left[\mathbf{H}\left(\lambda \right)\right]}^{\prime }={\left[\mathbf{Z}{\left(\mathbf{Z}\prime \mathbf{Z}\right)}^{-1}\mathbf{Z}\prime \right]}^{\prime }=\mathbf{Z}{\left(\mathbf{Z}\prime \mathbf{Z}\right)}^{-1}\mathbf{Z}\prime =\mathbf{H}\left(\lambda \right)$$

Thus,$\mathbf{H}\left(\lambda \right)$is symmetric.

For idempotence, we calculate${\left[\mathbf{H}\left(\lambda \right)\right]}^2:$

$${\left[\mathbf{H}\left(\lambda \right)\right]}^2=\mathbf{H}\left(\lambda \right)\mathbf{H}\left(\lambda \right)=\mathbf{Z}{\left({\mathbf{Z}}^{\prime }\mathbf{Z}\right)}^{-1}{\mathbf{Z}}^{\prime }\mathbf{Z}{\left({\mathbf{Z}}^{\prime }\mathbf{Z}\right)}^{-1}{\mathbf{Z}}^{\prime }=\mathbf{Z}{\left({\mathbf{Z}}^{\prime }\mathbf{Z}\right)}^{-1}{\mathbf{Z}}^{\prime }=\mathbf{H}\left(\lambda \right)$$

Therefore, $\mathbf{H}\left(\lambda \right)$is idempotent. ◻

• (ii) Properties$\mathbf{H}\left(\mathit{\lambda }\right)\mathbf{J}=\mathbf{J}$and$\mathbf{J}\mathbf{H}\left(\mathit{\lambda }\right)=\mathbf{J}$

To verify these properties, note that$\mathbf{H}\left(\lambda \right)\mathbf{Z}=\mathbf{Z}$, which implies that$\mathbf{H}\left(\lambda \right)$acts as a projection matrix, preserving the columns of$\mathbf{Z}$. Specifically:

$$\mathbf{H}\left(\lambda \right)\left({\mathbf{X}}_1\left(\lambda \right),{\mathbf{X}}_2\left(\lambda \right),\dots ,{\mathbf{X}}_{\mathit{p}}\left(\lambda \right)\right)=\left({\mathbf{X}}_1\left(\lambda \right),{\mathbf{X}}_2\left(\lambda \right),\dots ,{\mathbf{X}}_{\mathit{p}}\left(\lambda \right)\right)$$

Thus, for any column${\mathbf{X}}_{\mathrm{j}}\left(\lambda \right)$with$j=1,2,\dots ,p$, we have

$$\mathbf{H}\left(\lambda \right){\mathbf{X}}_{\mathrm{j}}\left(\lambda \right)={\mathbf{X}}_{\mathrm{j}}\left(\lambda \right)$$

For$j=1$, let us choose one particular structure:

$$\mathbf{H}\left(\lambda \right)\left[\frac{1}{2}\mathbf{j},{\tilde{\mathbf{X}}}_1\left(\lambda \right)\right]=\left[\frac{1}{2}\mathbf{j},{\tilde{\mathbf{X}}}_1\left(\lambda \right)\right]$$

where$\mathbf{j}$is a unit vector of dimension$n\times 1,{\tilde{\mathbf{X}}}_1\left(\lambda \right)$represents the matrix of${\mathbf{X}}_1\left(\lambda \right)$with the first column removed. We also find

$$\left[\frac{1}{2}\mathbf{H}\left(\lambda \right)\mathbf{j},\mathbf{H}\left(\lambda \right){\tilde{\mathbf{X}}}_1\left(\lambda \right)\right]=\left[\frac{1}{2}\mathbf{j},{\tilde{\mathbf{X}}}_1\left(\lambda \right)\right]$$

This hold because$\mathbf{H}\left(\lambda \right)\mathbf{j}=\mathbf{j}$and$\mathbf{H}\left(\lambda \right){\tilde{\mathbf{X}}}_1\left(\lambda \right)={\tilde{\mathbf{X}}}_1\left(\lambda \right)$

Finally, by definition of$\mathbf{H}\left(\lambda \right)\mathbf{j}{\mathbf{j}}^{\prime }=\mathbf{j}\mathbf{j}{}^{\prime }$we conclude that

$$\mathbf{H}\left(\lambda \right)\mathbf{J}=\mathbf{J}\text{and}{\mathbf{J}}^{\prime }={\left[\mathbf{H}\left(\lambda \right)\mathbf{J}\right]}^{\prime }=\mathbf{J}{\left[\mathbf{H}\left(\lambda \right)\right]}^{\prime }=\mathbf{J}\mathbf{H}\left(\lambda \right)=\mathbf{J}$$

Lemma 2

The matrices $\mathbf{I}-\mathbf{H}\left(\lambda \right)\text{and}\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}$, where$\mathbf{J}$is an$n\times n$identity matrix, have the following properties:

• (i)$\mathbf{I}-\mathbf{H}\left(\lambda \right)$is symmetric and idempotent

• (ii)$\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}$is symmetric and idempotent

Proof

• (i) Symmetry of$\mathbf{I}-\mathbf{H}\left(\lambda \right)$:

Since$\mathbf{H}\left(\lambda \right)$is symmetric, we have

$${\left[\mathbf{I}-\mathbf{H}\left(\lambda \right)\right]}^{\prime }=\mathbf{I}-{\left[\mathbf{H}\left(\lambda \right)\right]}^{\prime }=\mathbf{I}-\mathbf{H}\left(\lambda \right)\left(\text{from}\text{Lemma}1\left(\mathrm{i}\right)\right)$$

thus,$\mathbf{I}-\mathbf{H}\left(\lambda \right)$is symmetric.

Idempotence of$\mathbf{I}-\mathbf{H}\left(\lambda \right)$:

To show idempotence, we compute

$${\left[\mathbf{I}-\mathbf{H}\left(\lambda \right)\right]}^2=\left[\mathbf{I}-\mathbf{H}\left(\lambda \right)\right]\left[\mathbf{I}-\mathbf{H}\left(\lambda \right)\right]=\mathbf{I}-\mathbf{H}\left(\lambda \right)-\mathbf{H}\left(\lambda \right)+{\left[\mathbf{H}\left(\lambda \right)\right]}^2$$

Using Lemma 1(i), where${\left[\mathbf{H}\left(\lambda \right)\right]}^2=\mathbf{H}\left(\lambda \right)$, we obtain:

${\left[\mathbf{I}-\mathbf{H}\left(\lambda \right)\right]}^2=\mathbf{I}-\mathbf{H}\left(\lambda \right)$(idempotence is confirmed) ◻

• (ii) Symmetry of$\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}$:

Since both$\mathbf{H}\left(\lambda \right)$and$\frac{1}{n}\mathbf{J}$are symmetric matrices, their difference is also symmetric:

$${\left[\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right]}^{\prime }={\left[\mathbf{H}\left(\lambda \right)\right]}^{\prime }-{\left[\frac{1}{n}\mathbf{J}\right]}^{\prime }=\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\left(\text{from}\text{Lemma}1\left(\mathrm{i}\right)\right)$$

Thus,$\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}$is symmetric.

Idempotence of$\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}$:

To confirm idempotence, we calculate

$${\left[\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right]}^2=\left[\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right]\left[\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right]$$

Expanding this expression:

$${\left[\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right]}^2={\left[\mathbf{H}\left(\lambda \right)\right]}^2-\frac{1}{n}\mathbf{H}\left(\lambda \right)\mathbf{J}-\frac{1}{n}\mathbf{J}\mathbf{H}\left(\lambda \right)+\frac{1}{n^2}\mathbf{JJ}$$

Based on Lemma 1(i) where${\left[\mathbf{H}\left(\lambda \right)\right]}^2=\mathbf{H}\left(\lambda \right)$and Lemma 2(i) that$\mathbf{H}\left(\lambda \right)\mathbf{J}=\mathbf{J}$and noting that$\mathbf{JJ}=n\mathbf{J}$, we obtain:

$${\left[\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right]}^2=\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}-\frac{1}{n}\mathbf{J}+\frac{1}{n^2}n\mathbf{J}=\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\left(\text{idempotence}\text{is}\text{confirmed}\right)\square$$

Based on the nonparametric Fourier series regression model, we can define the Total Sum of Squares (SST), Regression Sum of Squares (SSR), and Error Sum of Squares (SSE) as follows:

$$\text{SST}=\sum _{i=1}^n{(y_i-\overline{y})}^2,\text{SSR}=\sum _{i=1}^n{({\widehat{y}}_i-\overline{y})}^2,\text{and}\text{SSE}=\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2.$$

The formulas for SST, SSR, and SSE can be represented in matrix notation, as stated in the following theorem.

Theorem 1

The Total Sum of Squares (SST), Regression Sum of Squares (SSR), and Error Sum of Squares (SSE) can be expressed as follows:

$$\text{SST}={\mathit{y}}^{\prime }\left(\mathbf{I}-\frac{1}{n}\mathbf{J}\right)\mathit{y},\text{SSR}=\mathit{y}{}^{\prime }\left(\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right)\mathit{y},\text{and}\text{SSE}=\mathit{y}{}^{\prime }\left(\mathbf{I}-\mathbf{H}\left(\lambda \right)\right)\mathit{y}$$

Proof

$$SST=\sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2=\sum _{i=1}^n{y}_i^2-n{\overline{y}}^2=y^{\prime }y-y^{\prime }\frac{1}{n}Jy=y^{\prime }\left(I-\frac{1}{n}J\right)y$$

$$SSR=\sum _{i=1}^n{({\widehat{y}}_i-\overline{y})}^2={\left(\widehat{y}-\overline{y}j\right)}^{\prime }\left(\widehat{y}-\overline{y}j\right)={\widehat{y}}^{\prime }\widehat{y}-2\overline{y}{j}^{\prime }\widehat{y}+{\overline{y}}^2{j}^{\prime }j$$

$$={\left(H\left(\lambda \right)y\right)}^{\prime }\left(H\left(\lambda \right)y\right)-2\overline{y}{j}^{\prime }H\left(\lambda \right)y+n{\overline{y}}^2=y^{\prime }H{\left(\lambda \right)}^{2}y-2\overline{y}{j}^{\prime }y+n{\overline{y}}^2$$

$$=y^{\prime }H\left(\lambda \right)y-2n{\overline{y}}^2+n{\overline{y}}^2=y^{\prime }H\left(\lambda \right)y-y^{\prime }\frac{1}{n}Jy$$

$$=y^{\prime }\left(H\left(\lambda \right)-\frac{1}{n}J\right)y$$

$$SSE=\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2={\left(y-\widehat{y}\right)}^{\prime }\left(y-\widehat{y}\right)=y^{\prime }y-2y^{\prime }\widehat{y}+{\widehat{y}}^{\prime }\widehat{y}$$

$=y^{\prime }y-y^{\prime }H\left(\lambda \right)y=y^{\prime }\left(I-H\left(\lambda \right)\right)y$ ◻

Based on Theorem 1, the relationship between SST, SSR, and SSE can be expressed as:

$$\text{SST}=\text{SSR}+\text{SSE}$$ (9)

According to Rencher and Schaalje in Theorem 5.5A, let$\mathit{y}\sim N_p\left(\mathit{\mu },\mathbf{\Sigma }\right)$, where$\mathbf{A}$is a symmetric$p\times p$matrix of constants with rank$r$, and let$\nu =\frac{1}{2}{\mathbf{\mu }}^{\prime }\mathbf{A}\mathit{\mu }$. Then${\mathit{y}}^{\prime }\mathbf{A}\mathit{y}\sim {\chi }^2\left(r,\nu \right)$if and only if$\mathbf{A}\Sigma$is idempotent. Additionally, Rencher and Schaalje states that if A and B are two symmetric matrices of constants and$\mathit{y}\sim N_p\left(\mathit{\mu },\mathbf{\Sigma }\right)$, then${\mathit{y}}^{\prime }\mathbf{A}\mathit{y}$and${\mathit{y}}^{\prime }\mathbf{B}\mathit{y}$are independent if and only$\mathbf{A}\Sigma \mathbf{B}=0$[26].

Theorem 2

If$\mathit{y}\sim N_n\left(\mathbf{Z}\beta ,{\sigma }^2\mathbf{I}\right)$, then:

• (i)$\frac{\text{SSR}}{{\sigma }^2}\sim {\chi }^2\left(p\left(\lambda +2\right)-1,{\nu }_1\right)$, with${\nu }_1=\mathit{\beta }{}^{\prime }\mathbf{Z}{}^{\prime }\left(\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right)\mathbf{Z}\beta /2{\sigma }^2$
• (ii)$\text{SSE}/{\sigma }^2\sim {\chi }^2\left(n-p\left(\lambda +2\right)\right)$
• (iii)SSR and SSE are independent

Proof

• (i) Distribution of$\text{SSR}/{\sigma }^2:$

From Theorem 1, we have

$$\text{SSR}/{\sigma }^2=\mathit{y}{}^{\prime }\left(\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right)\mathit{y}/{\sigma }^2$$

Since$\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}$is symmetric and idempotent (from Lemma 2(ii)), we get

$$\text{rank}\left[\left(\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right)/{\sigma }^2\right]=\text{rank}\left(\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right)=\text{tr}\left(\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right)=p\left(\lambda +2\right)-1$$

Now, considering that

$$\mathbf{A}\Sigma =\left[\left(\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right)/{\sigma }^2\right]{\sigma }^2\mathbf{I}=\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}$$

is idempotent (by Lemma 2(ii)), we can apply the result of Rencher and Schaalje in Theorem 5.5A [26]. The noncentrality parameter is given by

$$\nu =\frac{1}{2}{\mathbf{\mu }}^{\prime }\mathbf{A}\mathit{\mu }=\frac{1}{2}\left(\mathbf{Z}\beta \right){}^{\prime }\left[\left(\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right)/{\sigma }^2\right]\left(\mathbf{Z}\beta \right)=\mathit{\beta }{}^{\prime }\mathbf{Z}{}^{\prime }\left(\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right)\mathbf{Z}\beta /2{\sigma }^2$$

Thus, we have

$$\text{SSR}/{\sigma }^2\sim {\chi }^2\left(p\left(\lambda +2\right)-1,\nu \right)\text{with}\nu ={\mathit{\beta }}^{\prime }{\mathbf{Z}}^{\prime }\left(\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right)\mathbf{Z}\beta /2{\sigma }^2\square$$

• (ii) Distribution of$\text{SSE}/{\sigma }^2:$

From Theorem 1, we have

$$\text{SSE}/{\sigma }^2={\mathit{y}}^{\prime }\left(\mathbf{I}-\mathbf{H}\left(\lambda \right)\right)\mathit{y}/{\sigma }^2$$

Since$\mathbf{I}-\mathbf{H}\left(\lambda \right)$is symmetric and idempotent (from Lemma 2(i)), we have

$$\text{rank}\left[\left(\mathbf{I}-\mathbf{H}\left(\lambda \right)\right)/{\sigma }^2\right]=\text{rank}\left(\mathbf{I}-\mathbf{H}\left(\lambda \right)\right)=\text{tr}\left(\mathbf{I}-\mathbf{H}\left(\lambda \right)\right)=n-p\left(\lambda +2\right)$$

Similarly, for$\mathbf{A}\Sigma =\left[\left(\mathbf{I}-\mathbf{H}\left(\lambda \right)\right)/{\sigma }^2\right]{\sigma }^2\mathbf{I}=\mathbf{I}-\mathbf{H}\left(\lambda \right)$, which is idempotent (by Lemma 2(i)), the noncentrality parameter is

$$\nu =\frac{1}{2}{\mathbf{\mu }}^{\prime }\mathbf{A}\mathit{\mu }=\frac{1}{2}\left(\mathbf{Z}\beta \right){}^{\prime }\left[\left(\mathbf{I}-\mathbf{H}\left(\lambda \right)\right)/{\sigma }^2\right]\left(\mathbf{Z}\beta \right)$$

Simplifying, we get

$$\nu =\frac{{\beta }^{\prime }{{\mathbf{Z}}^{\prime }}^{\left(\mathbf{I}-\mathbf{H}\left(\lambda \right)\right)}\mathbf{Z}\beta }{\mathbf{2}{\sigma }^{\mathbf{2}}}=\frac{{\beta }^{\prime }\left[{\mathbf{Z}}^{\prime }\mathbf{Z}-{\mathbf{Z}}^{\prime }\mathbf{H}\left(\lambda \right)\mathbf{Z}\beta \right]}{\mathbf{2}{\sigma }^{\mathbf{2}}}=\frac{{\beta }^{\prime }\left[{\mathbf{Z}}^{\prime }\mathbf{Z}-{\mathbf{Z}}^{\prime }\mathbf{Z}\right]\beta }{\mathbf{2}{\sigma }^{\mathbf{2}}}=\mathbf{0}$$

Thus, we conclude

$$\text{SSE}/{\sigma }^2\sim {\chi }^2\left(n-p\left(\lambda +2\right)\right).$$

• (iii) Independence of$\text{SSR}$and$\text{SSE}:$

From Theorem 1, we have$\text{SSR}={\mathit{y}}^{\prime }\left(\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right)\mathit{y}$and$\text{SSE}={\mathit{y}}^{\prime }\left(\mathbf{I}-\mathbf{H}\left(\lambda \right)\right)\mathit{y}$. Note that

$$\mathbf{A}\Sigma \mathbf{B}=\left[\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}\right]{\sigma }^2\mathbf{I}\left[\mathbf{I}-\mathbf{H}\left(\lambda \right)\right]$$

Expanding this, we obtain

$$\mathbf{A}\Sigma \mathbf{B}={\sigma }^2\left(\mathbf{H}\left(\lambda \right)-{\left[\mathbf{H}\left(\lambda \right)\right]}^2-\frac{1}{n}\mathbf{J}+\frac{1}{n}\mathbf{JH}\left(\lambda \right)\right)$$

Using the properties from Lemma 1(i) and (ii), we get

$$\mathbf{A}\Sigma \mathbf{B}={\sigma }^2\left(\mathbf{H}\left(\lambda \right)-\mathbf{H}\left(\lambda \right)-\frac{1}{n}\mathbf{J}+\frac{1}{n}\mathbf{J}\right)=0\square$$

Thus, according to Rencher (2008),$\text{SSR}$and$\text{SSE}$are independent.

According to Hog et al., if the random variables $U$and$V$ are independent and each follows a chi-square distribution with degrees of freedom$r_1$ and$r_2$, respectively, then the random variable defined by [27]:

$$F=\frac{U/r_1}{V/r_2}\sim F\left(r_1,r_2\right)$$

Theorem 3

Let$\mathit{y}\sim N_n\left(\mathbf{Z}\beta ,{\sigma }^2\mathbf{I}\right)$and define the F-statistic as follows:

$$F=\frac{\text{SSR}/\left(p\left(\lambda +2\right)-1\right)}{\text{SSE}/\left(n-p\left(\lambda +2\right)\right)}$$

If the null hypothesis$H_0:\mathit{\beta }=0$is true, then$\nu =0$and$F\sim F\left(p\left(\lambda +2\right)-1,n-p\left(\lambda +2\right)\right)$.

Proof

According to Hog et al. [27] and by applying Theorem 2(i) and (ii), we have

$F=\frac{\text{SSR}/\left(p\left(\lambda +2\right)-1\right)}{\text{SSE}/\left(n-p\left(\lambda +2\right)\right)}=\frac{\text{SSR}/{\sigma }^2\left(p\left(\lambda +2\right)-1\right)}{\text{SSE}/{\sigma }^2\left(n-p\left(\lambda +2\right)\right)}\sim F\left(p\left(\lambda +2\right)-1,n-p\left(\lambda +2\right)\right)$ ◻

If the Fourier series coefficients are found to be jointly significant in testing, we can proceed to test individual parameters of the nonparametric Fourier regression model [28]. According to Hog et al. [27], if the random variables$W$and$V$are independent with$W\sim N\left(0,1\right)$and$V\sim {\chi }^2\left(r\right)$, then the random variable

$$T=\frac{W}{\sqrt{V/r}}\sim t\left(r\right)$$

The individual parameter tests in the nonparametric Fourier regression model include testing the intercept parameter${\delta }_j,$the trend parameter${\gamma }_j$, and the cosine function parameter${\alpha }_{ji}$for$j=1,2,\dots ,p;i=1,2,\dots ,n$. To test these parameters individually, the first step is to compute the covariance of the estimator$\widehat{\beta }$ in Eq. (6) as follows:

$$\text{cov}\left(\widehat{\beta }\right)={\sigma }^2\mathbf{G}$$ (10)

where$\mathbf{G}={\left({\mathbf{Z}}^{\prime }\mathbf{Z}\right)}^{-1}$. Since the error variance${\sigma }^2$in Eq. (10) is unknown, it can be estimated by

$$s^2=\frac{{\mathit{y}}^{\prime }\left(\mathbf{I}-\mathbf{H}\left(\lambda \right)\right)\mathit{y}}{n-p\left(\lambda +2\right)}$$ (11)

Finally, an estimator for$\text{cov}\left(\widehat{\beta }\right)$in Eq. (10) is given by

$$\widehat{\text{cov}}\left(\widehat{\beta }\right)=s^2\mathbf{G}$$ (12)

Theorem 4

If $s^2$ is as defined in Eq. (11), then

$$\frac{\left[n-p\left(\lambda +2\right)\right]s^2}{{\sigma }^2}\sim {\chi }^2\left(n-p\left(\lambda +2\right)\right)$$

Proof

$\frac{\left[n-p\left(\lambda +2\right)\right]s^2}{{\sigma }^2}=\frac{{\mathit{y}}^{\prime }\left(\mathbf{I}-\mathbf{H}\left(\lambda \right)\right)\mathit{y}}{{\sigma }^2}=\frac{\text{SSE}}{{\sigma }^2}\sim {\chi }^2\left(n-p\left(\lambda +2\right)\right)\left(\text{from}\text{Theorem}2\left(\text{ii}\right)\right)$ ◻

The least squares estimator$\widehat{\beta }$in Eq. (6) is an unbiased estimator for$\mathit{\beta }$. Given$\mathit{y}\sim N_n\left(\mathbf{Z}\beta ,{\sigma }^2\mathbf{I}\right),$from Eq. (10), we have

$$\widehat{\beta }\sim {\mathbf{N}}_{\mathbf{p}\left(\lambda +\mathbf{2}\right)}\left(\beta ,{\sigma }^{\mathbf{2}}{\left({\mathbf{Z}}^{\prime }\mathbf{Z}\right)}^{-\mathbf{1}}\right)$$ (13)

From Eq. (13), the distribution of the individual parameter estimates is

$${\widehat{\delta }}_j\sim N\left({\delta }_j,\text{var}\left({\widehat{\delta }}_j\right)\right),{\widehat{\gamma }}_j\sim N\left({\gamma }_j,\text{var}\left({\widehat{\gamma }}_j\right)\right),and{\widehat{\alpha }}_{ji}\sim N\left({\alpha }_j,\text{var}\left({\widehat{\alpha }}_{ji}\right)\right)$$

where:

• •$\text{var}\left({\widehat{\delta }}_j\right)={\sigma }^2{a}_{jj}$, with$a_{jj}$is the$\left(j-1\right)\left(\lambda +2\right)+1$th main diagonal element of$\mathbf{G}$,
• •$\text{var}\left({\widehat{\gamma }}_j\right)={\sigma }^2{b}_{jj}$, with$b_{jj}$is the$\left(j-1\right)\left(\lambda +2\right)+2$th main diagonal element of$\mathbf{G}$,
• •$\text{var}\left({\widehat{\alpha }}_{ji}\right)={\sigma }^2{c}_{ji}$, with$c_{ji}$is the$\left(j-1\right)\left(\lambda +2\right)+2+i$th main diagonal element of$\mathbf{G}$for$j=1,2,\dots ,p$and$i=1,2,\dots ,\lambda .$

From${\widehat{\delta }}_j\sim N\left({\delta }_j,\text{var}\left({\widehat{\delta }}_j\right)\right)$, we obtain

$$Z_j=\frac{{\widehat{\delta }}_j-{\delta }_j}{\sqrt{\text{var}\left({\widehat{\delta }}_j\right)}}=\frac{{\widehat{\delta }}_j-{\delta }_j}{\sigma \sqrt{a_{jj}}}\sim N\left(0,1\right)$$ (14)

For$j=1,2,\dots ,p$, based on Eq. (14) and Theorem 4, we define the random variable

$$T_{1j}=\frac{\frac{{\widehat{\delta }}_j-{\delta }_j}{\sigma \sqrt{a_{jj}}}}{\sqrt{\frac{\left[n-p\left(\lambda +2\right)\right]s^2}{{\sigma }^2\left[n-p\left(\lambda +2\right)\right]}}}=\frac{{\widehat{\delta }}_j-{\delta }_j}{s\sqrt{a_{jj}}}\sim t\left(n-p\left(\lambda +2\right)\right)$$ (15)

To test the parameter${\delta }_j$ individually, we use the hypothesis

$$H_0:{\delta }_j=0\text{versus}{H}_1:{\delta }_j\ne 0$$ (16)

Under the null hypothesis$H_0:{\delta }_j=0$, the test statistic becomes:

$$T_{1j}=\frac{{\widehat{\delta }}_j}{s\sqrt{a_{jj}}}\sim t\left(n-p\left(\lambda +2\right)\right)$$

with significance level$\alpha$, the critical region for rejecting$H_0$ is$|T_{1j}|>t_{\frac{\alpha }{2}}\left(n-p\left(\lambda +2\right)\right)$.

For$j=1,2,\dots ,p$, based on Eq. (14) and Theorem 4, we define the random variable

$$T_{2j}=\frac{\frac{{\widehat{\gamma }}_j-{\gamma }_j}{\sigma \sqrt{b_{jj}}}}{\sqrt{\frac{\left[n-p\left(\lambda +2\right)\right]s^2}{{\sigma }^2\left[n-p\left(\lambda +2\right)\right]}}}=\frac{{\widehat{\gamma }}_j-{\gamma }_j}{s\sqrt{a_{jj}}}\sim t\left(n-p\left(\lambda +2\right)\right)$$ (17)

To test the parameter${\gamma }_j$ individually, we use the hypothesis

$$H_0:{\gamma }_j=0\text{versus}{H}_1:{\gamma }_j\ne 0$$ (18)

Under the null hypothesis$H_0:{\gamma }_j=0$, the test statistic is

$$T_{2j}=\frac{{\widehat{\gamma }}_j}{s\sqrt{b_{jj}}}\sim t\left(n--p\left(\lambda +2\right)\right)$$

with significance level$\alpha$, the critical region for rejecting$H_0$ is$|T_{2j}|>t_{\frac{\alpha }{2}}\left(n-p\left(\lambda +2\right)\right)$.

For$j=1,2,\dots ,p$and$i=1,2,\dots ,\lambda ,$based on Eq. (14) and Theorem 4, we define the random variable

$$T_{3ji}=\frac{\frac{{\widehat{\alpha }}_{ji}-{\alpha }_{ji}}{\sigma \sqrt{c_{ji}}}}{\sqrt{\frac{\left[n-p\left(\lambda +2\right)\right]s^2}{{\sigma }^2\left[n-p\left(\lambda +2\right)\right]}}}=\frac{{\widehat{\alpha }}_{ji}-{\alpha }_{ji}}{s\sqrt{c_{ji}}}\sim t\left(n-p\left(\lambda +2\right)\right)$$ (19)

To test the parameter${\alpha }_{ji}$ individually, we use the hypothesis

$$H_0:{\alpha }_{ji}=0\text{versus}{H}_1:{\alpha }_{ji}\ne 0$$ (20)

Under the null hypothesis $H_0:{\alpha }_{ji}=0$, the test statistic is

$$T_{3ji}=\frac{{\widehat{\alpha }}_{ji}}{s\sqrt{c_{jj}}}\sim t\left(n-p\left(\lambda +2\right)\right)$$

with significance level$\alpha$, the critical region for rejecting$H_0$ is$|T_{3ji}|>t_{\frac{\alpha }{2}}\left(n-p\left(\lambda +2\right)\right).$

Method validation

Real data application

Statisical inference on hypothesis testing for nonparametric regression models with fourier series have been theoretically discussed in the previous section. The application of the concept to real data is carried out by analyzing real expenditure data in the province of Papua. The data is obtained from the Central Bureau of Statistics (BPS) in the form of the 2023 Provincial Publication in Figures, the 2023 Human Development Index (HDI) of Papua Province, the 2019–2023 Gross Regional Domestic Product (GRDP) of districts/cities in Indonesia, and the Regional Statistics of Papua Province. In this study, 2023 research data is used with the unit of observation being 29 districts/cities in Papua. The variables used are Real Expenditure per Capita$\left(\mathrm{y}\right)$in billion rupiah, Gross Regional Domestic Product per Capita$\left({\mathrm{x}}_1\right)$, Percentage of Poor Population$\left({\mathrm{x}}_2\right)$, and Labor Force Participation Rate$\left({\mathrm{x}}_3\right)$. Real expenditure per capita in Papua is interesting to study because it ranks the lowest among other provinces in Indonesia, which serves as an indicator in assessing the quality of human resources that can enhance a country's competitiveness. Therefore, the government continues to make efforts and create strategies to improve the quality of human resources in the Province of Papua. Improving Indonesia's human development index is one of the main components in achieving the Sustainable Development Goals (SDGs), particularly points 3, 4, and 8, which are Good Health and Well-being, Quality Education, and Decent Work and Economic Growth.

The data analysis in this study utilizes the R software. The program does not rely on any specific libraries or packages but is developed using basic R syntax based on the following algorithms:

Algorithm for Estimating the Model Parameters

• 1.Input the data containing the values of the response variable$Y$and predictors$X_1,X_2,\dots ,X_p$
• 2.Determine the oscillation parameter values for the$p$predictor variables, denoted as${\lambda }_1,{\lambda }_2,\dots ,\text{and}{\lambda }_p$
• 3.Set the structure of the predictor matrix${\mathit{X}}_1\left({\mathit{\lambda }}_1\right),{\mathit{X}}_2\left({\mathit{\lambda }}_2\right),\dots ,{\mathit{X}}_{\mathit{p}}\left({\mathit{\lambda }}_{\mathit{p}}\right)$
• 4.Set the matrix$\mathit{Z}$
• 5.Calculate the estimator$\widehat{\beta }$.

Algorithm for Determining the Optimal Oscillation Parameters

1. Input the data containing the values of the response variable$Y$ and predictors $X_1,X_2,\dots ,X_p$

2. Determine the maximum trial values for each oscillation parameter, denoted as$max\_{\lambda }_1,max\_{\lambda }_2,\dots ,max\_{\lambda }_p$

3. For${\lambda }_1=1,2,\dots ,$max_${\lambda }_1$;${\lambda }_2=1,2,\dots ,$max_${\lambda }_2$; and${\lambda }_p=1,2,\dots ,$max_${\lambda }_p$, iterate the following steps:

• •Calculate the estimator$\widehat{\beta }$
• •Calculate$\widehat{\mathit{y}}$
• •Calculate$MSE\left({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_p\right)$
• •Calculate$H\left({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_p\right)$
• •Calculate$GCV\left({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_p\right)$

4. Determine the minimum$GCV$

5. Select${\lambda }_1,{\lambda }_2,\dots ,\text{and}{\lambda }_p$corresponding to the minimum$GCV$as the optimal oscillation parameters.

Algorithm for Hypothesis Testing

1. Input the data containing the values of the response variable$Y$and predictors$X_1,X_2,\dots ,X_p$

2. Input the significance level$\alpha$

3. Input the optimal oscillation parameter values, denoted as ${\lambda }_{1opt},{\lambda }_{2opt},\dots ,\text{and}{\lambda }_{popt}$

4. Calculate the estimator$\widehat{\beta }$

5. Calculate The Total Sum of Squares (SST), Regression Sum of Squares (SSR), and Error Sum of Squares (SSE)

6. Perform the simultaneous testing steps:

• a.Calculate the test statistic$F$based on Theorem 3
• b.Compute the$p-value$for the simultaneous test
• c.Reject$H_0$if$p-value<\alpha$

7. Perform the partial testing steps:

• a.Calculate the variance-covariance matrix$\text{cov}\left(\widehat{\beta }\right)$
• b.Compute the standard error of$\widehat{\beta }$as the square root of the main diagonal of the variance-covariance matrix$\text{cov}\left(\widehat{\beta }\right)$
• c.Calculate the test statistic$T$for each regression parameter
• d.Compute the$p-value$for each partial test
• e.Reject$H_0$if$p-value<\alpha$.

As an initial identification, data exploration was conducted to understand the pattern of the relationship between the response and predictor variables using a scatter plot. Based on Fig. 1(a)-(c), it can be seen that the pattern of the relationship between Real Per Capita Expenditure$\left(\mathrm{y}\right)$and the three predictor variables does not follow a specific pattern and tends to change at certain sub-intervals. In districts/cities with a per capita GDP below 100 million, an increase in per capita GDP affects all response variables with sharp movements. Conversely, in the interval of per capita GDP above 100 million, the movements slow down. In the predictor variable of the percentage of poor population, a pattern of relationship that tends to repeat at certain intervals follows a downward trend line. The downward trend line indicates a negative relationship. Although it follows a downward trend line, there is a repeating pattern within it. In the predictor variable of the Labor Force Participation Rate, a relationship pattern that tends to be dispersed and does not form a clear pattern is observed. The plot in Fig. 1(a)-(c) shows no specific relationship pattern and indicates a periodic pattern resembling a distribution, so the data can be analyzed using a Fourier series.

Fig. 1.

Scatter plot of real expenditure per capita (Y) vs predictor variables, (a) X1, (b) X2 and (c) X3.

The model of the relationship between the response variable and the predictor variable is best obtained using the optimal oscillation parameter$\left(\lambda \right)$with the Minimum GCV criterion. The plot of GCV values for each combination of K value trials is presented in the following Fig. 2.

Fig. 2.

GCV plot for each experimental combination.

By using a maximum K value of 4, 64 experimental combinations are obtained.The minimum GCV was obtained at the 17th combination with a GCV value of 2.4679. More detailed information about the optimal oscillation parameter$\left(\lambda \right)$obtained by testing the value of K on each predictor is presented in the following Table 1.

Table 1.

Optimal oscillation parameter values and GCV.

Trial Num.	K1	K2	K3	GCV Value
1	1	1	1	2.6690
2	1	1	2	2.8490
3	1	1	3	3.0675
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
17	2	1	1	2.4679
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
64	4	4	4	4.0991

After obtaining the optimal oscillation parameter$\left(\lambda \right)$, the output results of the Fourier model parameters are obtained in the following Table 2.

Table 2.

Model parameter results.

Parameter Estimation	${\widehat{\delta }}_1$	${\widehat{\gamma }}_1$	${\widehat{\alpha }}_{11}$	${\widehat{\alpha }}_{12}$	${\widehat{\delta }}_2$	${\widehat{\gamma }}_2$	${\widehat{\alpha }}_{21}$	${\widehat{\delta }}_3$	${\widehat{\gamma }}_3$	${\widehat{\alpha }}_{31}$
Coefficient	3.6326	0.0134	−0.3221	−0.7187	3.6326	−0.1685	−0.9947	3.6326	0.0062	−0.1451

Based on the output results in Table 2, a Fourier series model can be formed as follows:

$$=\frac{3.6326}{2}+0.0134{\mathrm{x}}_1-0.3221\cos {\mathrm{x}}_1-0.7187\cos 2{\mathrm{x}}_1+\frac{3.6326}{2}-0.1685{\mathrm{x}}_2-0.9947\cos {\mathrm{x}}_2+\frac{3.6326}{2}+0.0062{\mathrm{x}}_3-0.1451\cos {\mathrm{x}}_3$$

The next step is to simultaneously test the Fourier series coefficients, using the following hypothesis:

${\mathrm{H}}_0:{\delta }_1={\gamma }_1={\alpha }_{11}={\alpha }_{12}={\delta }_2={\gamma }_2={\alpha }_{21}={\delta }_3={\gamma }_3={\alpha }_{31}=0$

${\mathrm{H}}_1:$at least one parameter is not equal to 0

From the test results, an F-value of 8.318087 was obtained with a P-value of 0.000059. Since the P-value is <5 %, the decision is to reject${\mathrm{H}}_0$, thus it can be concluded that the parameters in the nonparametric Fourier series regression model for Real Per Capita Expenditure have a significant simultaneous effect. In addition, individual/partial tests will also be conducted on the Fourier series model, with the following hypothesis in Table 3:

Table 3.

Partial test hypothesis.

H0:	${\delta }_1=0$	${\gamma }_1=0$	${\alpha }_{11}=0$	${\alpha }_{12}=0$	${\delta }_2=0$	${\gamma }_2=0$	${\alpha }_{21}=0$	${\delta }_3=0$	${\gamma }_3=0$	${\alpha }_{31}=0$
H1:	${\delta }_1\ne 0$	${\gamma }_1\ne 0$	${\alpha }_{11}\ne 0$	${\alpha }_{12}\ne 0$	${\delta }_2\ne 0$	${\gamma }_2\ne 0$	${\alpha }_{21}\ne 0$	${\delta }_3\ne 0$	${\gamma }_3\ne 0$	${\alpha }_{31}\ne 0$

Based on the hypothesis in Table 3 and the partial test results in Table 4, with a significance level of 5 %, it was found that all parameters have a significant influence on the model except for the parameters${\alpha }_{11},{\alpha }_{12},{\gamma }_3,{\alpha }_{31}$which are not statistically significant.

Table 4.

Partial test results.

Parameter	Estimated Coefficients	SE Coefficients	T-Test	P-value	Decision
${\delta }_1$	36.326	0,63,140,731	5,7532	0	Reject${\mathrm{H}}_0$
${\gamma }_1$	0,093,055,556	0,00,556,812	2,4132	0,0261	Reject${\mathrm{H}}_0$
${\alpha }_{11}$	−0.3221	0,45,155,398	−0,7133	0,4843	Failed to reject${\mathrm{H}}_0$
${\alpha }_{12}$	−0.7187	0,38,134,101	−1,8846	0,0749	Failed to reject${\mathrm{H}}_0$
${\delta }_2$	36.326	0,63,140,731	5,7532	0	Reject${\mathrm{H}}_0$
${\gamma }_2$	−0.1685	0,03,867,522	−4,3567	0,0003	Reject${\mathrm{H}}_0$
${\alpha }_{21}$	−0.9947	0,415,009	−2,3968	0,027	Reject${\mathrm{H}}_0$
${\delta }_3$	36.326	0,63,140,731	5,7532	0	Reject${\mathrm{H}}_0$
${\gamma }_3$	0.0062	0,02,344,497	0,2647	0,7941	Failed to reject${\mathrm{H}}_0$
${\alpha }_{31}$	−0.1451	0,43,192,972	−0,336	0,7406	Failed to reject${\mathrm{H}}_0$

Model interpretation

Modeling Real Per Capita Expenditure in Papua using a nonparametric Fourier series regression model depends on the selection of the optimal oscillation parameter$\left(\lambda \right)$with the Minimum GCV criterion, resulting in a combination of K (2,1,1) with a minimum GCV value of 2.4679. The complete combination of various experiments can be seen in Table 1. By using the estimated model parameters in Table 2, the regression model equation is obtained as follows

$$\widehat{\mathrm{y}}=\frac{3.6326}{2}+0.0134{\mathrm{x}}_1-0.3221\cos {\mathrm{x}}_1-0.7187\cos 2{\mathrm{x}}_1+\frac{3.6326}{2}-0.1685{\mathrm{x}}_2-0.9947\cos {\mathrm{x}}_2+\frac{3.6326}{2}+0.0062{\mathrm{x}}_3-0.1451\cos {\mathrm{x}}_3$$

Where$\widehat{\mathrm{y}}$is the estimated model of Real Per Capita Expenditure in Papua in 2023. This model produces an${\mathrm{R}}^2$value of 79.76 % and a MAPE of 12.51, which means the predictor variables used, namely Gross Regional Domestic Product per Capita$\left({\mathrm{x}}_1\right)$, Percentage of Poor Population$\left({\mathrm{x}}_2\right)$, and Labor Force Participation Rate$\left({\mathrm{x}}_3\right)$, can explain 79.76 % of the model's variability. This indicates that the model fits the actual data. The results obtained by this model are better when compared to the parametric regression method with an${\mathrm{R}}^2$value of 68.67 % and a MAPE of 14.33. The plot between the actual data and the predictions using a nonparametric Fourier series regression model is presented in Fig. 3 below.

Fig. 3.

Comparison between actual and predicted for the variable Real Expenditure per Capita in Papua.

Limitations

Not applicable.

Ethics statements

The data is obtained from the Central Bureau of Statistics (BPS) in the form of the 2023 Provincial Publication in Figures, the 2023 Human Development Index (HDI) of Papua Province, the 2019–2023 Gross Regional Domestic Product (GRDP) of districts/cities in Indonesia, and the Regional Statistics of Papua Province. The data is available at https://papua.bps.go.id/id/publication and https://www.bps.go.id/id/publication.

CRediT author statement

Suliyanto: Conceptualization, Methodology, Writing – original draft. Toha Saifudin: Conceptualization, Methodology, Software, Visualization. Marisa Rifada: Methodology, Writing – review & editing, Validation. Dita Amelia: Conceptualization, Visualization, Writing – original draft, Validation.

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.