AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi 10.1155/2020/4657151 4657151 Research Article Estimation of Error Variance-Covariance Parameters Using Multivariate Geographically Weighted Regression Model https://orcid.org/0000-0001-9664-027X Harini Sri 1 Soliman Abdel-Maksoud A. 1 Mathematics Department Faculty of Science and Technology Maulana Malik Ibrahim State Islamic University Malang East Java Indonesia uin-malang.ac.id 2020 122020 2020 18 10 2019 29 11 2019 122020 2020 Copyright © 2020 Sri Harini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The Multivariate Geographically Weighted Regression (MGWR) model is a development of the Geographically Weighted Regression (GWR) model that takes into account spatial heterogeneity and autocorrelation error factors that are localized at each observation location. The MGWR model is assumed to be an error vector ε that distributed as a multivariate normally with zero vector mean and variance-covariance matrix Σ at each location ui,vi, which Σ is sized qxq for samples at the i-location. In this study, the estimated error variance-covariance parameters is obtained from the MGWR model using Maximum Likelihood Estimation (MLE) and Weighted Least Square (WLS) methods. The selection of the WLS method is based on the weighting function measured from the standard deviation of the distance vector between one observation location and another observation location. This test uses a statistical inference procedure by reducing the MGWR model equation so that the estimated error variance-covariance parameters meet the characteristics of unbiased. This study also provides researchers with an understanding of statistical inference procedures.

Research and Community Service Institutions Directorate General of Islamic Higher Education
1. Introduction

In statistical inference, estimation of spatial data parameters using the GWR approach has been carried out by many researchers. According to , the GWR method is selected due to the weaknesses of the ordinary least square (OLS) parameter estimation results, where the variance error in the OLS model is still assumed to be fixed (homoscedasticity) and there is no dependency between errors (spatial effects) at each observation location. Spatial problems, specifically in parameter estimation has been studied by Cressie . The author discussed spatial analysis in detail by using OLS and estimator of spatial regression models with the maximum likelihood estimation (MLE) methods. Yasin  proposed the GWR stepwise method in order to choose a significant variable. The selection of the stepwise GWR method reduces several predictor variables that are not significant to the response variable. A Mixed Geographically Weighted Regression model (MGWR) is a combination of linear regression and the GWR. A statistical test of MGWR models with the maximum likelihood ratio test (MLRT) method have been carried out by , Cressie  and Harini et al. . By inference, the MLRT method can maximize the probability value of the resulting parameters. Furthermore, to complete the MGWR model, which in inference analysis, the first derivative analytical solution of the log-likelihood function is unavailable in closed form.

Harini and Purhadi  used the Matrix Laboratory algorithm approach, a high-level programming language based on numerical computational techniques to solve problems involving mathematical operations with database arrays and vector formulations. The advantage of this approach is the absence of variable dimension constraints. Referring to , Triyanto et al.  discussed the parameter estimation of the Geographically Weighted Multivariate Poisson Regression (GWMPR) model using the Maximum Likelihood Estimation (MLE) methods. The GWMPR is used to model the spatial data with response variables that are distributed Poisson.

Another problem that often arises in the GWR model is to validate hypothesis testing using statistical inference analysis because invalidating hypothesis test requires several stages of parameter estimation that cannot be done globally . Therefore, the R and GWR4 programs can be used to check the validity level of hypothesis testing. The advantages of the program parameter estimation results are both global and local and can be done together. Soemartojo et al.  analyzed the spatial heterogeneity problem of the GWR model using Weighted Least Squares (WLS) method with Gaussian kernel weight function. Spatial heterogeneity occurs because there is a strong dependence between one observation with other observations that are nearby (nearest neighboring) to cause spatial effects. The process of non-stationarity by applying an extended hyper-local GWR is examined by Comber et al. . This model optimizes the covariates of each local regression simultaneously, to determine the local bandwidth specifications based on lots of data at each location and evaluates different bandwidths in each location to choose the right local regression model.

In this research, we focus on the form and properties of the estimated error variance-covariance parameters of the MGWR model using the MLE and WLS methods. This test uses statistical inference procedures to obtain the estimated error variance-covariance parameters that meet the unbiased nature.

2. Theoretical GWR

Supporting theories for completing this research refer to the Geographically Weighted Regression (GWR)  and Statistics for Spatial Data .

3. Methods of MGWR

The MGWR method refers to  and .

4. Results

The MGWR is the development of a multivariate linear model with known location information. In the multivariate spatial linear model, the relationship between the response variable Y1,Y2,,Yq and the predictor variable X1,X2,,Xp at the-ith location is given by(1)Yhi=βh0ui,vi+βh1ui,viX1i+βh2ui,viX2i++βhpui,viXpi+εhi,h=1,2,,qandi=1,2,,n.

The assumptions used in the MGWR model are error vector ε with multivariate normal distributions with zero vectors mean and variance-covariance matrix Σ at each location ui,vi, which the size of Σ is qxq for the samples at the-ithlocation.(2)Σui,vi=σ12ui,viσ12ui,viσ1qui,viσ22ui,viσ2qui,viσq2ui,vi.

From Equation (2), the estimation of the variance-covariance error matrix parameters Σ^ui,vi is observed at each study location using the MLE and WLS methods. To get the estimation of the variance-covariance matrix parameter Σ^ui,vi, the parameter estimation is determined at one the-jth location σh2uj,vj as follows:(3)σ^h2uj,vj=i=1nwijuj,vjYhi-β^h0uj,vj+k=1pβ^hkuj,vjXki2n=Yh-Xβ^huj,vjTWuj,vjYh-Xβ^huj,vjn=SSEuj,vjn.

The vector error at the location ui,vi can be stated as follows:(4)eh=Yh-Y^h=I-SYh,

where I is the matrix identity with order n and S is the symmetric matrix sized n×n,(5)Sn×n=X1TXTWu1,v1X1XTWu1,v1X2TXTWu2,v2X1XTWu2,v2XnTXTWun,vnX1XTWun,vn.

About the local character of the MGWR model (3), the sum of square error SSE and the estimated parameters of the error variance-covariance can be determined.

Proposition 1.

If SSE the location of uj,vj the MGWR model is eTuj,vjeuj,vj, then it can be determined SSEh and the expectation value SSEh.

Proof.

To get the SSE from the MGWR model using squaring (4) at the location to uj,vj is:

(6) S S E = e h T u j , v j e h u j , v j = I S Y h T I S Y h = Y h T I S T I S Y h ,

where(7)Eehuj,vj=EYh-Y^h=XTβuj,vj-XTβ^uj,vj=0,

and variance error is(8)Varehuj,vj=Eehuj,vjEehuj,vjehuj,vjEehuj,vjT=Eehuj,vjehTuj,vj=σh2uj,vj.

Based on (8), then (6) can be described as follows:(9)SSEhuj,vj=ehTuj,vjehuj,vj=euj,vjEeuj,vjTeuj,vjEeuj,vj=ehTuj,vjISTISehuj,vj.

From Equation (9), we can find the expected value SSEhuj,vjas follows:(10)ESSEhuj,vj=EehTuj,vjISTISehuj,vj=EtrehTuj,vjISTISehuj,vj=trISTISEehuj,vjehTuj,vj=n2trS+trSTSσh2uj,vj=r1σh2uj,vj.

Since ESSEhuj,vj=r1σh2uj,vj, then we have r1=1/σh2uj,vjESSEhuj,vj with r1=trISTIS.

Proposition 2.

If the errors of estimated parameter variance-covariance MGWR model at the-jth location are σ^hhuj,vj=EehTuj,vjehuj,vj and σ^h2uj,vj=σ^hhuj,vj, then we can determine SSEhEh and the expected value SSEhEh at each location uj,vj mathematically.

Proof.

First, the variance-covariance error at the-ith location is shown as follows:

(11) σ ^ h 2 u j , v j = σ ^ h h u j , v j V a r e h u j , v j , e h u j , v j = E e h T u j , v j e h u j , v j σ ^ h 2 u j , v j = E e h T u j , v j e h u j , v j E e h u j , v j T E e h u j , v j = E e h T u j , v j e h u j , v j = σ ^ h h u j , v j .

Furthermore, SSEhEhuj,vj is searched using (9), we obtain(12)SSEhEhuj,vj=ehuj,vjEehuj,vjTehuj,vjEehuj,vj=ISYhEISYhTISYhEISYh=YhEYhTISTISYhEYh=ehTuj,vjISTISehuj,vj,

where ISTIS is a definite and symmetrical semi-definite matrix n×n with εhuj,vjN0,σhhuj,vj. Then we have(13)ESSEhEhuj,vj=EehTuj,vjISTISehuj,vj=EtrehTuj,vjISTISehuj,vj=trISTISEehuj,vjehTuj,vj=trISTISσhhuj,vj.

Theorem 1.

If SSEh is given by Proposition 1 and the estimation of variance σ^h2uj,vj is given by Proposition 2, the estimated variance-covariance error of the MGWR model is given as follows:

(14) σ ^ h h u j , v j = Y h - X β ^ h u j , v j T W u j , v j Y h - X β ^ h u j , v j n = S S E h E h u j , v j n . Proof.

From Equation (1) of the MGWR model,

(15) Y h i = β h 0 u i , v i + k = 1 p β h k u i , v i X k i + ε h i .

To determine SSEhEh at each location uj,vj, it can be approached using Equation (5),(16)εhWuj,vjεh=Yh-Xβhuj,vjTWuj,vjYh-Xβhuj,vjEεhWuj,vjεh=EYh-Xβhuj,vjTWuj,vjYh-Xβhuj,vjSSEhEhuj,vj=Yh-Xβhuj,vjTWuj,vjYh-Xβhuj,vj,

and(17)σhhuj,vj=SSEhEhuj,vjn.

Based on Propositions 1 and 2, the theorems of estimation parameter variance-covariance error matrix for MGWR model are determined.

Theorem 2.

If ESSEhuj,vj satisfies Proposition 1 and ESSEhEhuj,vj satisfies Proposition 2, the estimated parameter variance-covariance errors matrix of the MGWR model are σ^hhuj,vj=SSEhEhuj,vj/trISTIS and Eσ^hhuj,vj=σhhuj,vj.

Proof.

Based on Proposition 1 and 2, the estimated error variance-covariance parameters from the MGWR model are:

(18) V a r e h u j , v j , e h u j , v j = E e h T u j , v j e h u j , v j σ ^ h 2 u j , v j = E e h T u j , v j e h u j , v j σ h 2 u j , v j = S S E h u j , v j n 2 t r S + t r S T S ,

and σ^hhuj,vj=SSEhEhuj,vj/n2trS+trSTS.

By using the characteristics of the matrix ISTIS, Eσ^h2uj,vj, and Eσ^hhuj,vj can be determined to satisfy the unbiased.

Theorem 3.

If σ^hhuj,vj=SSEhEhuj,vj/trISTIS is an unbiased estimator σhhuj,vj, then Eσ^h2uj,vj, and Eσ^hhuj,vj can be determined to satisfy the unbiased.

Proof.

(19) E σ ^ h 2 u j , v j = E S S E h u j , v j t r I S T I S = 1 t r I S T I S E S S E h u j , v j = 1 t r I S T I S t r I S T I S σ h 2 u j , v j = σ h 2 u j , v j ,

and in the same way, we obtain(20)Eσ^hhuj,vj=ESSEhEhuj,vjtrISTIS=σhhuj,vj,

where σ^h2uj,vj and is σ^hhuj,vj an estimate of the unbiased error variance-covariance matrix for σh2uj,vj and σhhuj,vj.

By using Theorem 3, an unbiased estimate is obtained from the variance-covariance error matrix Σuj,vj at the-jth location as follows:(21)Σ^uj,vj=σ^ 12uj,vjσ^ 12uj,vjσ^ 1quj,vjσ^ 22uj,vjσ^ 2quj,vjsimetrisσ^ q2uj,vj .

Since the variance-covariance error matrix Σuj,vj satisfies the unbiased nature, then in the same way in other locations, it also meets the unbiased nature. Mathematically, the estimation of the variance-covariance matrix parameters Σ at the location to ui,vi can be stated as follows:(22)Σ^ui,vi=σ^12ui,viσ^12ui,viσ^ 1qui,viσ^ 22ui,viσ^ 2qui,vi simetrisσ^ q2ui,vi .

Thus, it is proven that if Σ^uj,vj as an unbiased estimate of the variance-covariance error matrix Σuj,vj, then Σ^ui,vi is also an unbiased estimate of the variance-covariance error matrix Σui,vi.

5. Conclusion

This research concludes that the MGWR model using MLE and WLS methods is suitable to obtain the estimated error variance-covariance parameters. The results prove that Σ^uj,vj is an unbiased estimate of the variance-covariance error matrix Σuj,vj. Since Σ^uj,vj is an unbiased estimate, then Σ^ui,vi is also an unbiased estimate of the variance-covariance error matrix Σui,vi at all locations.

Data Availability

The authors declare that all of data is original and there is no data from others publication.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgments

We would like to express our sincere gratitude to the Research Sub-Directorate, Community Development and Scientific Publications of the Directorate General of Islamic Higher Education (Dirjen DIKTIS) for providing funds for this research in 2018. Research and Community Service Institutions provide funding support with this publication.

Fotheringham A. Brunsdon C. Charlton M. Geographically Weighted Regression 2002 UK John Wiley and Sons Cressie N. Statistics for Spatial data 2015 2nd NY, USA John Wiley and Sons Yasin H. Selection of variables in the geographically weighted regression model Media Statistika 2011 4 2 63 72 10.14710/medstat.4.2.63-72 Harini S. Purhadi M. M. Sunaryo S. Statistical test for multivariate geographically weighted regression model using the method of maximum likelihood ratio test International Journal of Applied Mathematics & Statistics 2012 29 5 110 115 Harini S. Purhadi Parameter estimation of multivariate geographically weighted regression model using matrix laboratory International Conference on Statistics in Science, Business and Engineering (ICSSBE) 2012 Langkawi, Malaysia IEEE 10.1109/ICSSBE.2012.63966222-s2.0-84872976740 Triyanto P. Bambang W. O. Santi W. P. Parameter estimation of geographically weigthed multivariate poisson regression Applied Mathematical Sciences 2015 9 82 4081 4093 10.12988/ams.2015.543292-s2.0-84936881887 Syerrina N. E. A statistical analysis for geographical weighted regression 169 IOP Conference Series: Earth and Environmental Science 2018 Malaysia IOP 012105 Soemartojo S. Ghaisani R. Siswantining T. Shahab R. M. Parameter estimation of geographically weighted regression (GWR) model using weighted least square and its application AIP Conference Proceedings 2018 Comber A. Wang Y. Y. Xingchang Y. Hyper-local geographically weighted regression: extending GWR through local model selection Journal of Spatial Information Science 2018 17 63 84 Harini S. Purhadi M. M. Sunaryo S. Linear model parameter estimator of spatial multivariate using restricted maximum likelihood Journal of Mathematics and Technology 2010 56 61