# A New Method of Hypothesis Test for Truncated Spline Nonparametric Regression Influenced by Spatial Heterogeneity and Application

Authors: Sifriyani; I. N. Budiantara; S. H. Kartiko; Gunardi
Journal: Abstract and Applied Analysis
 (2018)

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## Abstract

This study developed a new method of hypothesis testing of model conformity between truncated spline nonparametric regression influenced by spatial heterogeneity and truncated spline nonparametric regression. This hypothesis test aims to determine the most appropriate model used in the analysis of spatial data. The test statistic for model conformity hypothesis testing was constructed based on the likelihood ratio of the parameter set under H0 whose components consisted of parameters that were not influenced by the geographical factor and the set under the population parameter whose components consisted of parameters influenced by the geographical factor. We have proven the distribution of test statistics V and verified that each of the numerators and denominators in the statistic test V followed a distribution of χ2. Since there was a symmetric and idempotent matrix S, it could be proved that Y~TSY~/σ2~χn-lm-12. Matrix Dui,vi was positive semidefinite and contained weighting matrix Wui,vi which had different values in every location; therefore matrix Dui,vi was not idempotent. If Y~TDui,viY~≥0 and Dui,vi was not idempotent and also Y~ was a N0,I distributed random vector, then there were constants k and r; hence Y~TDui,viY~~kχr2; therefore it was concluded that test statistic V followed an F distribution. The modeling is implemented to find factors that influence the unemployment rate in 38 areas in Java in Indonesia.


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## Body

## 1. Introduction

This study examines theoretically the multivariate nonparametric regression influenced by spatial heterogeneity with truncated spline approach. The model is the development of truncated spline nonparametric regression that takes into account geographic or spatial factors. Truncated spline is a function constructed on the basis of polynomial components and truncated components; i.e., polynomial pieces that have knot points, which can overcome the pattern of changes in data behavior. Truncated spline approach is used as a solution to solve the problem of spatial data analysis modeling; that is, the relationship between the response variable and the predictor variable does not follow a certain pattern and there is a changing pattern in certain subintervals. The response variable in the model contains the predictor variables whose respective regression coefficients depend on the location where the data is observed, due to differences in environmental and geographic characteristics between the observation sites; therefore each observation has different variations (spatial heterogeneity). Spatial is one type of dependent data, where data at a location is influenced by the measurement of data at another location (spatial dependency).This study determines the model conformity hypothesis test between multivariable nonparametric regression that is influenced by spatial heterogeneity with truncated spline approach and multivariable nonparametric regression in general. This hypothesis test aims to determine the model that is most suitable for spatial data analysis. The test statistic was derived using the maximum likelihood ratio test (MLRT) method. The first step in this study was formulating the hypothesis to be tested and then defining the set of parameters under H0 whose components consist of parameters that are not influenced by geographical factors and the set under population parameters whose components consist of parameters influenced by geographical factors. Likelihood ratio Λ was constructed based on the maximum ratio of the likelihood function under H0 as the numerator and set under the population as a denominator. Based on the likelihood ratio test statistic V was obtained. Furthermore, the distribution of test statistic V was determined. To prove the distribution of test statistic V, we first proved that each numerator and denominator are chi square distributed.The purpose of this study is to obtain a new method for the determination of hypothesis test of model conformity between multivariate nonparametric truncated spline regression influenced by spatial heterogeneity versus multivariate nonparametric truncated spline regression in general. This hypothesis test aims to determine what model is most suitable for spatial data analysis.


## 2. Truncated Spline Nonparametric Regression Influenced by Heterogeneity Spatial

Truncated spline nonparametric regression influenced by spatial heterogeneity is the development of nonparametric regression for spatial data with parameter estimators local to each location of observation. Truncated spline approach is used to solve spatial analysis problems whose regression curve is unknown [1]. The assumption of the regression model used is the normal distributed error with mean zero and variance σ2ui,vi at each location ui,vi. Location coordinates ui,vi are an important factor in determining the weights used to estimate the parameters of the model. Given data x1i,x2i,…,xli,yi and relationship between x1i,x2i,…,xli and yi, it is assumed to follow multivariate nonparametric regression model as follows:(1)yi=fx1i,x2i,…,xli+εi,i=1,2,…,ny i is response variable and fx1i,x2i,…,xli is unknown regression curve and assumed to be additive. If fx1i,x2i,…,xli is approached with a truncated spline function. Mathematically, the relation between response variable yi and the predictor variable x1i,x2i,…,xli ati-th location for the multivariate nonparametric truncated spline regression model can be expressed as follows [2]:(2)yi=β0ui,vi+∑p=1l∑k=1mβpkui,vixpik+∑p=1l∑h=1rδp,m+hui,vixpi-Kph+m+εiwith truncated function:(3)xpi-Kph+m=xpi-Kph,xpi≥Kph0,xpi<KphEquation (2) is a multivariate nonparametric truncated spline regression model of degreem withn area. The components in (2) are described as follows:y i is a response variable ati-th location, where i=1,2,…,n.x p i is ap-th predictor variable ati-th location with p=1,2,…,l.K p h is anh-th knot point inp-th predictor variable component with h=1,2,…,r.β p k u i , v i is a polynomial component parameter of a multivariate nonparametric truncated spline regression. βpkui,vi is ak-th parameter fromp-th predictor variable ati-th location. δp,m+hui,vi is a truncated component from multivariate nonparametric truncated spline regression. δp,m+hui,vi is an l+h-th parameter inh-th knot point andp-th predictor variable ati-th location.Multivariate nonparametric truncated spline regression in (2) is described as follows:(4)yi=β0ui,vi+β11ui,vix1i+β12ui,vix1i2+⋯+β1mui,vix1im+β21ui,vix2i+β22ui,vix2i2+β23ui,vix2i3+⋯+β2mui,vix2im+⋯+βl1ui,vixli+βl2ui,vixli2+βl3ui,vixli3+⋯+βlmui,vixlim+δ1,m+1ui,vix1i-K11+m+⋯+δ1,m+rui,vix1i-K1r+m+δ2,m+1ui,vix2i-K21+m+⋯+δ2,m+rui,vix2i-K2r+m+⋯+δl,m+1ui,vixli-Kl1+m+⋯+δl,m+rui,vixli-Klr+m+εi,Equation (4) can also be expressed as follows:(5)y1y2⋮yn=β0u1,v1+∑p=1l∑k=1mβpku1,v1xpnkβ0u2,v2+∑p=1l∑k=1mβpku2,v2xpnk⋮β0un,vn+∑p=1l∑k=1mβpkun,vnxpnk+∑p=1l∑h=1rδp,m+hu1,v1xpn-Kph+m∑p=1l∑h=1rδp,m+hu2,v2xpn-Kph+m⋮∑p=1l∑h=1rδp,m+hun,vnxpn-Kph+m+ε1ε2⋮εnThus (5) can be expressed by(6)Y~=f~+ε~=Xβ~ui,vi+Pδ~ui,vi+ε~whose vector f~ contains of truncated spline function with geographical weighting sized n×1; response variable and error, respectively, are given by vectors as follows:(7)Y~=y1y2⋮yn,ε~=ε1ε2⋮εn.Vectors Y~ and ε~ are, respectively, sized n×1.Meanwhile matricesX and P are, respectively, given by(8)X=1x11x112x113⋯x11mx21x212⋯x21m⋯xl1xl12⋯xl1m1x12x122x123⋯x12mx22x222⋯x22m⋯xl2xl22⋯xl2m⋮⋮⋮⋮⋱⋮⋮⋮⋱⋮⋱⋮⋮⋱⋮1x1nx1n2x1n3⋯x1nmx2nx2n2⋯x22m⋯xlnxln2⋯xlnm,P=x11-K11+m⋯x11-K1r+mx21-K21+m⋯xl1-Kl1+m⋯xl1-Klr+mx12-K11+m⋯x12-K1r+mx22-K21+m⋯xl2-Kl1+m⋯xl2-Klr+m⋮⋱⋮⋮⋱⋮⋱⋮x1n-K11+m⋯x1n-K1r+mx2n-K21+m⋯xln-Kl1+m⋯xln-Klr+mVectors β~ui,vi and δ~ui,vi are, respectively, given by(9)β~ui,vi=β0ui,viβ11ui,viβ12ui,viβ13ui,vi⋮β1mui,viβ21ui,viβ22ui,vi⋮β2mui,vi⋮βl1ui,viβl2ui,vi⋮βlmui,vi,δ~ui,vi=δ1,m+1ui,viδ1,m+2ui,vi⋮δ1,m+rui,viδ2,m+1ui,viδ2,m+2ui,vi⋮δ2,m+rui,vi⋮δl,m+1ui,viδl,m+2ui,vi⋮δl,m+rui,viMatrix X is sized n×1+lm; matrix P contains predictor variable of truncated function sized n×lr. Vector β~ui,vi is a parameter vector sized 1+lm×1. Vector δ~ui,vi is a parameter vector containing truncated function sized lr×1. The estimator forms β~^ui,vi, δ~^ui,vi, and f~^ are complete in Theorem 1 and Corollary 2 [2].Theorem 1.
If the regression model (2) with an error εi normally distributed with zero mean and variance σ2ui,vi was given Maximum Likelihood Estimator (MLE), it is used to obtain estimator β~^ui,vi and δ~^ui,vi as follows.(10)β~^ui,vi=AKY~δ~^ui,vi=BKY~where(11)AK=SXTWui,viX-1XT-XTWui,viPPTWui,viP-1PTWui,viBK=RPTWui,viP-1PT-PTWui,viXXTWui,viX-1XTWui,viCorollary 2.
Ifβ~^ui,vi and δ~^ui,vi are given by Theorem 1, then the estimator for the regression curve f~^ is given by(12)f~^=Xβ~^ui,vi+Pδ~^ui,vi=CKY~,where(13)CK=XAK+PBKEstimator of regression curvef~^ contains the polynomial components represented by matrix X and truncated components represented by matrix P [3]. If the matrix P=0, then the estimator multivariable of nonparametric regression curve in the Geographically Weighted Regression (GWR) models with truncated spline approach, f~^, will change to estimator polynomial parametric regression curve in the GWR model. Furthermore, if P=0 and matrix X contains a linear function, the estimator of the multivariable spline nonparametric regression curve in the GWR model, f~^, will change to estimator of linear parametric regression curves in the GWR model or multiple linear regression in the GWR model developed by many researchers such as Brusdon and Fotheringham [4], Fotheringham, Brunsdon, and Charlton (2003), Demsar, Fotheringham, and Charlton [5], Yan Li, Yan Jiao, and Joan A. Browder [6], Shan-shan Wu, Hao Yang, Fei Guo, and Rui -Ming Han [7], and Benassi and Naccarato [8].This study continued the previous research [2]; in this study the test statistics that will be used in the truncated spline nonparametric regression influenced by spatial heterogeneity modeling will be found; further research continued the distribution of test statistics and rejection areas.


## 3. Method

The hypothesis test for model conformity between multivariate nonparametric truncated spline regression influenced by spatial heterogeneity with nonparametric truncated spline regression is derived.Step 1.
Formulating hypothetical model:H0: βpkui,vi=βpk and δp,m+hui,vi=δp,m+h,
p=1,2,…,l; k=1,2,…,m; h=1,2,…,r; i=1,2,…,nH1: at least, there is one of βpkui,vi≠βpk or δpm+hui,vi≠δp,m+h,
p=1,2,…,l; k=1,2,…,m; h=1,2,…,r; i=1,2,…,n.Step 2.
Defining the set of parameters under populationΩ.Step 3.
Determining estimatorsη~^Ωui,vi and σ^Ω2ui,vi which are parameters in the space under population Ω.Step 4.
Obtaining maximum likelihood function under populationΩ.Step 5.
Defining parameter space under H0, i.e., ω.Step 6.
Determining estimatorsη~ω and σω2 which are parameters under H0.Step 7.
Obtaining maximum likelihood function under space H0.Step 8.
Obtaining likelihood ratioΛ.Step 9.
Obtaining test statisticV from model conformity testing.Step 10.
Specifying the distribution of numeratorτ from test statistic V.Step 11.
Specifying the distribution of denominatorτ∗ from test statistic V.Step 12.
Specifying the distribution of test statisticV.Step 13.
Deciding the rejection area ofH0 and writing the conclusion.


## 4. Parameter Estimation under Space H0 and Space Population in the Model

A hypothesis testing of model conformity for nonparametric spline regression with spatial heterogeneity was designed by using hypothesis formulation:H0: βpkui,vi=βpk and δp,m+hui,vi=δp,m+h,H1: at least, there is one ofβpkui,vi≠βpk or δpm+hui,vi≠δp,m+h.This hypothesis test was derived using maximum likelihood ratio test method by defining the parameter spaces under H0ω and under population Ω. The parameter space under H0ω is given by(14)ω=β11,β21,…,βlm,δ1,m+1,δ1,m+2,…,δ1,m+r,…,δl,m+1,…,δl,m+r,σω2=β~,δ~,σω2=η~ω,σω2where η~ω=β~δ~.(15)β~=β11β12…β1mβ21…β2m…βlmTδ~=δ1,m+1δ1,m+2…δ1,m+rδ2,m+1δ2,m+2…δ2,m+r…δl,m+rT.While the parameter space under the population Ω is given by(16)Ω=β0ui,vi,β11ui,vi,β21ui,vi,⋯,βlmui,vi,δ1,m+1ui,vi,⋯,δl,m+rui,vi,σΩ2ui,vi=β~ui,vi,δ~ui,vi;i=1,2,…,n,σΩ2ui,vi=η~Ωui,vi;i=1,2,…,n,σΩ2ui,viObtaining the test statistic of hypothesis above required some lemmas as follows.Lemma 3.
Ifη~Ωui,vi is a parameter under population Ω from nonparametric spline regression with spatial heterogeneity (2), then estimator η~^Ωui,vi is given by(17)η~^Ωui,vi=QTWui,viQ-1QTWui,viY~Proof.
To obtain estimatorη~^Ωui,vi we form likelihood function under population parameter space LΩ. Therefore yi has normal distribution with mean(18)β0ui,vi+∑p=1l∑k=1mβpkui,vixpik+∑p=1l∑h=1rδp,m+hui,vixpi-Kph+mand variance σΩ2ui,vi; then probability functions y1,y2,…,yn are given by(19)fy1,y2,…,yn=∏j=1n12πσΩ2ui,viexp⁡-12σΩ2ui,viwijyj-β0ui,vi+∑p=1l∑k=1mβpkui,vixpjk+∑p=1l∑h=1rδp,m+hui,vixpj-Kph+m2Obtained likelihood function is as follows:(20)LΩ=2π-n/2σΩ2ui,vi-n/2exp⁡-12σΩ2ui,vi∑j=1nwjiyj-β0ui,vi+∑p=1l∑k=1mβpkui,vixpik+∑p=1l∑h=1rδp,m+hui,vixpi-Kph+m2Equation (20) in matrix form is(21)LΩ=2π-n/2σΩ2ui,vi-n/2exp⁡-12σΩ2ui,viY~-Qη~Ωui,viTWui,viY~-Qη~Ωui,vi.Estimator η~^Ωui,vi is obtained on the basis of the following derivative results:(22)∂ln⁡LΩ∂η~Ωui,vi=0Then the following is obtained:(23)∂ln⁡LΩ∂η~Ωui,vi=∂-2η~Ωui,viTQTWui,viY~∂η~Ωui,vi+∂η~Ωui,viTQTWui,viQη~Ωui,vi∂η~Ωui,vi=-2QTWui,viY~+2QTWui,viQη~^Ωui,viTherefore,(24)η~^Ωui,vi=QTWui,viQ-1QTWui,viY~Furthermore, estimatorσ^Ω2ui,vi is shown in Lemma 4.Lemma 4.
IfσΩ2ui,vi is a parameter in space under population Ω from nonparametric spline regression with spatial heterogeneity spatial (2), then estimator σ^Ω2ui,vi which is obtained from likelihood function:(25)LΩ=2π-n/2σΩ2ui,vi-n/2exp⁡-12σΩ2ui,vi∑j=1nyj-β0ui,vi+∑p=1l∑k=1mβpkui,vixpik+∑p=1l∑h=1rδp,m+hui,vixpi-Kph+m2is given by(26)σ^Ω2ui,vi=Y~-Qη~^Ωui,viTY~-Qη~^Ωui,vinProof.
Estimatorσ^Ω2ui,vi is obtained using likelihood function:(27)LΩ=2π-n/2σΩ2ui,vi-n/2exp⁡-12σΩ2ui,vi∑j=1nyj-β0ui,vi+∑p=1l∑k=1mβpkui,vixpjk+∑p=1l∑h=1rδp,m+hui,vixpj-Kph+m2The ln likelihood function is given by(28)ln⁡LΩ=-n2ln⁡2π-n2σΩ2ui,vi+-12σΩ2ui,viY~-Qη~Ωui,viTY~-Qη~Ωui,viFurthermore, estimator σ^Ω2ui,vi is obtained on the basis of the following derivative results:(29)∂ln⁡LΩ∂σΩ2ui,vi=0Then the following was obtained:(30)∂ln⁡LΩ∂σΩ2ui,vi=∂-n/2ln⁡2π-n/2σΩ2ui,vi-1/2σΩ2ui,viY~-Qη~Ωui,viTY~-Qη~Ωui,vi∂σΩ2ui,vi=-n2σ^Ω2ui,vi+12σ^Ω2ui,vi2Y~-Qη~^Ωui,viTY~-Qη~^Ωui,viTherefore,(31)σ^Ω2ui,vi=Y~-Qη~^Ωui,viTY~-Qη~^Ωui,vinBased on estimatorsη~^Ωui,vi and σ^Ω2ui,vi which are given by Lemmas 3 and 4 the following is obtained:(32)LΩ^=Lη~^Ωui,vi,σ^Ω2ui,vi=2π-n/2σ^Ω2ui,vi-n/2exp⁡-12σ^Ω2ui,viY~-Qη~^Ωui,viTY~-Qη~^Ωui,vi=2π-n/2σ^Ω2ui,vi-n/2exp⁡-12σ^Ω2ui,viY~-QQTWui,viQ-1QTWui,viY~TY~-QQTWui,viQ-1QTWui,viY~Lemma 5.
Ifη~ω and σω2 are parameters under H0 from multivariate nonparametric truncated spline influenced by spatial heterogeneity model (2), then estimator for η~^ω is given by(33)η~^ω=QTQ-1QTY~and estimator for σω2 is given by(34)σ^ω2=Y~-Qη~ωTY~-Qη~ωnProof.
To obtain estimatorsη~^ω and σ^ω2 we form likelihood function under parameter space H0Lω. Therefore yi has normal distribution with mean(35)β0+∑p=1l∑k=1mβpkxpik+∑p=1l∑h=1rδp,m+hxpi-Kphand variance σω2; then probability functions y1,y2,…,yn are given by(36)fy1,y2,…,yn=∏j=1n12πσω2exp⁡-12σω2yj-β0+∑p=1l∑k=1mβpkxpik+∑p=1l∑h=1rδp,m+hxpi-Kph+m2The following likelihood functions were obtained:(37)Lω=2π-n/2σω2-n/2exp-12σω2∑j=1nyj-β0+∑p=1l∑k=1mβpkxpik+∑p=1l∑h=1rδp,m+hxpi-Kph+m2The equation is in the form of a matrix(38)Lω=2π-n/2σω2-n/2exp⁡-12σω2Y~-Qη~ωTY~-Qη~ω.Estimators η~^ω and σ^ω2 are obtained:(39)η~^ω=QTQ-1QTY~σ^ω2=Y~-Qη~^ωTY~-Qη~^ωnBased on Lemma5, maximum likelihood function is obtained as follows:(40)Lω^=Lη~^ω,σ^ω2=2π-n/2σ^ω2-n/2exp⁡-12σ^ω2Y~-Qη~^ωTY~-Qη~^ωη~^ω and σ^ω2 are parameters estimator of under H0 from multivariate nonparametric regression with truncated spline approach.


## 5. Statistics Test for Truncated Spline Nonparametric Regression with Spatial Heterogeneity

The test statistic for the model conformity hypothesis test can be obtained by using Lemmas3, 4, and 5. In the next step, we show the likelihood ratio for test statistic presented in Lemma 6.Lemma 6.
IfLΩ^ and Lω^, respectively, are given by (32) and (40), then the likelihood ratio Λ is given by(41)Λ=Y~TSY~Y~TDui,viY~where(42)S=I-QQTQ-1QTDui,vi=I-Wui,viQQTWui,viQ-1QTI-QQTWui,viQ-1QTWui,viProof.
Based on Lemmas3, 4, and 5, and also (32) and (40), the likelihood ratio is obtained:(43)Λ=Lω^LΩ^=2π-n/2σ^ω2-n/2exp⁡-n/22π-n/2σ^Ω2ui,vi-n/2exp⁡-n/2=σ^ω2σ^Ω2ui,vi-n/2=Y~-Qη~^ωTY~-Qη~^ωY~-Qη~^Ωui,viTY~-Qη~^Ωui,vi-n/2Based on (3) and (5), the likelihood ratio (44)=Y~-QQTQ-1QTY~TY~-QQTQ-1QTY~Y~-QQTWui,viQ-1QTWui,viY~TY~-QQTWui,viQ-1QTWui,viY~-n/2=Y~TI-QQTQ-1QTTI-QQTQ-1QTY~Y~TI-QQTWui,viQ-1QTWui,viTI-QQTWui,viQ-1QTWui,viY~-n/2=Y~TI-QQTQ-1QTY~Y~TI-Wui,viQQTWui,viQ-1QTI-QQTWui,viQ-1QTWui,viY~-n/2=Y~TSY~Y~TDui,viY~-n/2Given test statistic for model conformity hypothesis is presented by Theorem7.Theorem 7.
If likelihood ratioΛ is given by Lemma 6, then test statistic for H0 versus H1 in (2) is given by(45)V=Y~TSY~/n-lm-1Y~TDui,viY~/trI-ξTI-ξProof.
Based on Lemma6, the likelihood ratio is as follows:(46)Λ=Y~TSY~Y~TDui,viY~-n/2Based on MLRT method, H0 is rejected if(47)Λ=Y~TSY~Y~TDui,viY~-n/2<cFor a constant c, (43) is equivalent to(48)Y~TSY~Y~TDui,viY~>c-2/nIn the two sections of the inequality above, each numerator is divided by n-lm-1 and each denominator is divided by trI-ξTI-ξ; then the following inequality is obtained:(49)Y~TSY~/n-lm-1Y~TDui,viY~/trI-ξTI-ξ>c-2/ntrI-ξTI-ξn-lm-1=c∗Based on (44), the test statistic for H0 versus H1 is given by(50)V=Y~TSY~/n-lm-1Y~TDui,viY~/trI-ξTI-ξFurthermore, the distribution of statistics testV will be found.The statistics test given in Theorem7 is test statistics developed from the spline truncated approach in the GWR model, different from the one developed by Leung, Mei, and Zhang [9], Leung, Mei, and Zhang [10], and Mennis and Jordan [11] using GWR without using the Truncated Spine approach.


## 6. Distribution of Test Statistic and Critical Area of Hypothesis

To prove the distribution of test statisticV, we first prove τ=Y~TSY~/σ2~χn-lm-12 and Y~TDui,viY~/σ2∑i=1nλi2/trI-ξTI-ξ~χtrI-ξTI-ξ2/∑i=1nλi22. The proofs are presented in Theorems 8 and 9 as follows.Theorem 8.
IfS is a matrix given by Lemma 6 then statistic is(51)τ=Y~TSY~σ2~χn-lm-12.Proof.
To prove this Lemma, the following steps are taken.
MatrixS is shown which is a symmetric and idempotent matrix as follows:(52)ST=I-QQTQ-1QTT=I-QQTQ-1QTBased on equation above, it is proved that matrix S is symmetric.(53)STS=I-QQTQ-1QTTI-QQTQ-1QT=I-2QQTQ-1QT+QQTQ-1QTQQTQ-1QT=I-2QQTQ-1QT+QQTQ-1QT=I-QQTQ-1QTIt is proved that matrix S is idempotent. Furthermore, trS is calculated as follows:(54)trS=trI-QQTQ-1QT=n-lm-1Therefore, it is proved that (55)τ=Y~TSY~σ2~χn-lm-12Theorem 9.
IfDui,vi is a matrix given by Lemma 6 then statistic is(56)τ∗=Y~TDui,viY~σ2∑i=1nλi2/trI-ξTI-ξ~χtrI-ξTI-ξ2/∑i=1nλi22Proof.
Based on (24), we obtain(57)η~^Ωui,vi=QTWui,viQ-1QTWui,viY~And estimator is obtained:(58)Y~^=Qη~^Ωui,vi=QQTWui,viQ-1QTWui,viY~=ξY~in which(59)ξ=QQTWui,viQ-1QTWui,vi.=q1TQTWu1,v1Q-1QTWu1,v1q2TQTWu2,v2Q-1QTWu2,v2⋮qnTQTWun,vnQ-1QTWun,vnqiT=1x1ix1i2⋯x1imx2i⋯xli-Klr+mFurther, error vector is given as follows:(60)ε~^=Y~-Y~^=I-ξY~Sum Square of Error (SSE) of model is obtained by squaring the following error vectors:(61)SSE=Y~TI-ξTI-ξY~=Y~-ξY~TY~-ξY~=I-ξY~-EY~TI-ξY~-EY~=Y~-EY~TI-ξTI-ξY~-EY~=ε~TI-ξTI-ξε~Furthermore,(62)ESSE=Eε~TI-ξTI-ξε~=σ2trI-ξTI-ξSince SSE is a quadratic form from random variable:(63)Y~TI-ξTI-ξY~≥0,hence, matrix I-ξTI-ξ is positive semidefinite but not idempotent. Next, we obtain(64)SSEσ2=ε~∗TI-ξTI-ξε~∗,ε~∗=ε~σSince ε~~N0~,σ2I, hence ε~∗~N0~,I, and since matrix I-ξTI-ξ is not idempotent, the distribution of statistic is(65)SSEσ2~kχr2For constantsk andr, based on (55), we obtain(66)ESSE=σ2trI-ξTI-ξESSEσ2=trI-ξTI-ξSince I-ξTI-ξ is symmetric and positive semidefinite, hence there is an orthogonal matrix Θ; therefore,(67)ΘI-ξTI-ξΘT=Λ=diagλ1,λ2,…,λnΛ is a diagonal matrix in which λ1,λ2,…,λn are eigenvalues from matrix I-ξTI-ξ. Hence(68)SSEσ2=ε~∗TI-ξTI-ξε~∗=ε~∗TΘTI-ξTI-ξΘε~∗=ε~∗TΛε~∗=∑i=1nλiεi∗2in which ε~∗=ε1∗ε2∗⋯εn∗T. Random variables ε1∗,ε2∗,…,εn∗ are independent, identical, and normal distributed; therefore(69)εi∗2~χ12with mean 1 and variance 2; therefore(70)varSSEσ2=var∑i=1nλiεi∗2=∑i=1nλi2varεi∗2=2∑i=1nλi2Since SSE/σ2~kχr2,(71)ESSEσ2=trI-ξTI-ξ=krvarSSEσ2=2∑i=1nλi2=2k2rHence the values ofk andr are as follows: r=trI-ξTI-ξ2/∑i=1nλi2 and k=(∑i=1nλi2)/trI-ξTI-ξ; as a result,(72)Y~TDui,viY~σ2~∑i=1nλi2trI-ξTI-ξχtrI-ξTI-ξ2/∑i=1nλi22Hence(73)τ∗=Y~TDui,viY~σ2∑i=1nλi2/trI-ξTI-ξ~χtrI-ξTI-ξ2/∑i=1nλi22Corollary 10.
If statisticV is given by Theorem 7, then (74)V=Y~TSY~/n-lm-1Y~TDui,viY~/trI-ξTI-ξ~Fn-lm-1,trI-ξTI-ξ2/∑i=1nλi2Proof.
Based on Theorem8, statistic is obtained:(75)τ=Y~TSY~σ2~χn-lm-12Based on Theorem 9, statistic is obtained:(76)τ∗=Y~TDui,viY~σ2∑i=1nλi2/trI-ξTI-ξ~χtrI-ξTI-ξ2/∑i=1nλi22Hence(77)V=Y~TSY~/n-lm-1Y~TDui,viY~/∑i=1nλi2/trI-ξTI-ξ/trI-ξTI-ξ2/∑i=1nλi2=Y~TSY~/n-lm-1Y~TDui,viY~/trI-ξTI-ξ=Y~TSY~/σ2/n-lm-1Y~TDui,viY~/σ2∑i=1nλi2/trI-ξTI-ξ/trI-ξTI-ξ2/∑i=1nλi2~Fn-lm-1,trI-ξTI-ξ2/∑i=1nλi2The critical area for the model conformity hypothesis is derived which is given by Lemma11.Lemma 11.
If given test statisticV is as in Theorem 7, then the critical area for H0 is given by(78)Cy,x1,x2,…,xl=y,x1,x2,…,xl;V>cA constantc is obtained according to(79)PV>c=α,0<α<1in which α is a determined level of significance and (80)V~Fn-lm-1,trI-ξTI-ξ2/∑i=1nλi2.Proof.
Based on Theorem7, the following relationship is obtained:(81)Y~TSY~/n-lm-1Y~TDui,viY~/trI-ξTI-ξ2/∑i=1nλi2>c-2/nn-lm-1trI-ξTI-ξ2/∑i=1nλi2=c∗for a constant c∗. According to Corollary 10, statistic is obtained:(82)V=Y~TSY~/n-lm-1Y~TDui,viY~/trI-ξTI-ξ2/∑i=1nλi2~Fn-lm-1,trI-ξTI-ξ2/∑i=1nλi2for a level of significance α given by H0 which is rejected if (83)V>Fαn-lm-1,trI-ξTI-ξ2/∑i=1nλi2After finding the hypothesis test formulation, the suitability of the model between the truncated spline nonparametric regression model which is influenced by spatial heterogeneity and nonparametric regression (global) will then be implemented on unemployment rate data in 38 regions in Java Indonesia.


## 7. Empirical Study on Unemployment Rate in Java Indonesia

### 7.1. Description of Research Data

In this study, the nonparametric truncated spline regression model influenced by spatial heterogeneity was applied to Open Unemployment Rate (OUR) data in province of Java, Indonesia, and some predictor variables that were suspected to affect it, i.e., population density (X1), percentage of the poor (X2), percentage of population with low education (X3), percentage of population working in agriculture sector (X4), area of agricultural land (X5), economic growth rate (X6), regional minimum wage (X7), and ratio number of large industries being number of labor force (X8). The amount of data used is 382 from 38 provinces and 8 predictor variables. Table 1 shows the description of our research data and the predictor variables.Table 1
Description of our research data and the predictor variables.
Variable Data Minimum Maximum Rata-rata Standard Deviation Y 38 0,61 8,59 4,3939 1,81385 X1 38 4,59 25,80 12,0963 4,99263 X2 38 0,2105 7,6445 2,631579 1,6160512 X3 38 0,07 7,19 5,5747 1,22772 X4 38 35,10 33.548,70 3.378,7658 6.497,23435 X5 38 851582 2507632 1516816.21 420415.074 X6 38 0,0074281 0,1640892 0,052024430 0,0474488599 X7 38 0,0405726 12,4704588 2,631578947 2,8535682531 X8 38 474 85122 28730,32 22372,865 Source: BPS (2017a, 2017b, and 2017c).The spread of Open Unemployment Rate in East Java is shown by Figure1. It shows the percentage of East Java unemployment rate in 2015.Figure 1
Open Unemployment Mapping in province of Java, Indonesia.


### 7.2. Spatial Heterogeneity Test

Each region has different characteristics and different parameters, as well as different functional forms; this is what proves spatial aspect. Breusch-Pagan testing is used to see the spatial heterogeneity of each location. Table2 shows the Breusch-Pagan test.Table 2
Breusch-Pagan test.
Test Significance Value Decision Breusch-Pagan 0.002414 Reject H0Since spatial effect testing is fulfilled, i.e., there are effects of spatial heterogeneity, then the case can be solved by using the point approach. Furthermore, an analysis was performed using nonparametric truncated spline regression model influenced by spatial heterogeneity.


### 7.3. Model Conformity Test

Hypotheses for model conformity test between multivariate nonparametric truncated spline regression model influenced by heterogeneity spatial and multivariate nonparametric truncated spline regression model (global) are as follows:H0: βpkui,vi=βpk and δp,m+hui,vi=δp,m+h,
p=1,2,…,8; k=1; h=1,2,3; i=1,2,…,38H1: at least, there is one of βpkui,vi≠βpk or δpm+hui,vi≠δp,m+h,
p=1,2,…,8; k=1; h=1,2,3; i=1,2,…,38The test statistic is given by Theorem7 as follows:(84)V=Y~TSY~/n-lm-1Y~TDui,viY~/trI-ξTI-ξMatrixS was constructed by multivariate nonparametric truncated spline. Matrix Dui,vi was constructed by multivariate nonparametric truncated spline regression. Hence, the numerator is obtained:(85)Y~TSY~n-lm-1=3,7814with degree of freedom n-lm-1=29. Meanwhile the denominator is obtained:(86)Y~TDui,viY~trI-ξTI-ξ=1,8356with degree of freedom trI-ξTI-ξ= 27,0762. Test statistic V= 2,06 with level of significance α= 0,05 was obtained and concluded to reject H0 since V>F(0,05;29,27)= 1,88. Therefore, there is a significant difference between multivariate nonparametric truncated spline regression influenced by spatial heterogeneity and nonparametric truncated spline regression. Due to the influence of geographical factors on the model, the appropriate model used is multivariate nonparametric truncated spline regression influenced by spatial heterogeneity.The modeling application used Open Unemployment Rate (OUR) data in 38 districts/cities in East Java. The results of the empirical study showed that the OUR data has a geographical influence, namely, spatial heterogeneity, and based on the results of the model conformity hypothesis test, the appropriate model used is a multivariable nonparametric truncated spline regression model influenced by spatial heterogeneity with the weighted Gaussian kernel function. The modeling produced a coefficient of determination of 80.42%.


## 7.1. Description of Research Data

In this study, the nonparametric truncated spline regression model influenced by spatial heterogeneity was applied to Open Unemployment Rate (OUR) data in province of Java, Indonesia, and some predictor variables that were suspected to affect it, i.e., population density (X1), percentage of the poor (X2), percentage of population with low education (X3), percentage of population working in agriculture sector (X4), area of agricultural land (X5), economic growth rate (X6), regional minimum wage (X7), and ratio number of large industries being number of labor force (X8). The amount of data used is 382 from 38 provinces and 8 predictor variables. Table 1 shows the description of our research data and the predictor variables.Table 1
Description of our research data and the predictor variables.
Variable Data Minimum Maximum Rata-rata Standard Deviation Y 38 0,61 8,59 4,3939 1,81385 X1 38 4,59 25,80 12,0963 4,99263 X2 38 0,2105 7,6445 2,631579 1,6160512 X3 38 0,07 7,19 5,5747 1,22772 X4 38 35,10 33.548,70 3.378,7658 6.497,23435 X5 38 851582 2507632 1516816.21 420415.074 X6 38 0,0074281 0,1640892 0,052024430 0,0474488599 X7 38 0,0405726 12,4704588 2,631578947 2,8535682531 X8 38 474 85122 28730,32 22372,865 Source: BPS (2017a, 2017b, and 2017c).The spread of Open Unemployment Rate in East Java is shown by Figure1. It shows the percentage of East Java unemployment rate in 2015.Figure 1
Open Unemployment Mapping in province of Java, Indonesia.


## 7.2. Spatial Heterogeneity Test

Each region has different characteristics and different parameters, as well as different functional forms; this is what proves spatial aspect. Breusch-Pagan testing is used to see the spatial heterogeneity of each location. Table2 shows the Breusch-Pagan test.Table 2
Breusch-Pagan test.
Test Significance Value Decision Breusch-Pagan 0.002414 Reject H0Since spatial effect testing is fulfilled, i.e., there are effects of spatial heterogeneity, then the case can be solved by using the point approach. Furthermore, an analysis was performed using nonparametric truncated spline regression model influenced by spatial heterogeneity.


## 7.3. Model Conformity Test

Hypotheses for model conformity test between multivariate nonparametric truncated spline regression model influenced by heterogeneity spatial and multivariate nonparametric truncated spline regression model (global) are as follows:H0: βpkui,vi=βpk and δp,m+hui,vi=δp,m+h,
p=1,2,…,8; k=1; h=1,2,3; i=1,2,…,38H1: at least, there is one of βpkui,vi≠βpk or δpm+hui,vi≠δp,m+h,
p=1,2,…,8; k=1; h=1,2,3; i=1,2,…,38The test statistic is given by Theorem7 as follows:(84)V=Y~TSY~/n-lm-1Y~TDui,viY~/trI-ξTI-ξMatrixS was constructed by multivariate nonparametric truncated spline. Matrix Dui,vi was constructed by multivariate nonparametric truncated spline regression. Hence, the numerator is obtained:(85)Y~TSY~n-lm-1=3,7814with degree of freedom n-lm-1=29. Meanwhile the denominator is obtained:(86)Y~TDui,viY~trI-ξTI-ξ=1,8356with degree of freedom trI-ξTI-ξ= 27,0762. Test statistic V= 2,06 with level of significance α= 0,05 was obtained and concluded to reject H0 since V>F(0,05;29,27)= 1,88. Therefore, there is a significant difference between multivariate nonparametric truncated spline regression influenced by spatial heterogeneity and nonparametric truncated spline regression. Due to the influence of geographical factors on the model, the appropriate model used is multivariate nonparametric truncated spline regression influenced by spatial heterogeneity.The modeling application used Open Unemployment Rate (OUR) data in 38 districts/cities in East Java. The results of the empirical study showed that the OUR data has a geographical influence, namely, spatial heterogeneity, and based on the results of the model conformity hypothesis test, the appropriate model used is a multivariable nonparametric truncated spline regression model influenced by spatial heterogeneity with the weighted Gaussian kernel function. The modeling produced a coefficient of determination of 80.42%.


## 8. Conclusion

Multivariate nonparametric regression with truncated spline approach influenced by spatial heterogeneity is given as follows:(87)yi=β0ui,vi+∑p=1l∑k=1mβpkui,vixpik+∑p=1l∑h=1rδp,m+hui,vixpi-Kph+m+εiBased on the results of the discussion and data analysis, some conclusions can be drawn as follows:(1)
The hypotheses for model conformity between multivariate nonparametric truncated spline regression model influenced by spatial heterogeneity and nonparametric truncated spline regression (global) are as follows:
H0: βpkui,vi=βpk and δp,m+hui,vi=δp,m+h,
p=1,2,…,l; k=1,2,…,m;h=1,2,…,r; i=1,2,…,nH1: at least, there is one of βpkui,vi≠βpk or δpm+hui,vi≠δp,m+h,
p=1,2,…,l; k=1,2,…,m;h=1,2,…,r; i=1,2,…,nTest statistic derived using Maximum Likelihood Ratio Test (MLRT) is obtained as follows:(88)V=Y~TSY~/n-lm-1Y~TDui,viY~/trI-ξTI-ξ(2)
The distribution of multivariate nonparametric truncated spline regression model influenced by spatial heterogeneity is as follows:(89)V~Fn-lm-1,trI-ξTI-ξ2/∑i=1nλi2with level of significanceα; therefore H0 is rejected if (90)V>Fαn-lm-1,trI-ξTI-ξ2/∑i=1nλi2.



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*Source: 9769150-2018-09-12.xml*