Estimation of Error Variance-Covariance Parameters Using Multivariate Geographically Weighted Regression Model

and Applied Analysis 2 program parameter estimation results are both global and local and can be done together. Soemartojo et al. [8] 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. e process of non-stationarity by applying an extended hyper-local GWR is examined by Comber et al. [9]. is 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. is 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) [1] and Statistics for Spatial Data [2]. 3. Methods of MGWR e MGWR method refers to [4] and [10]. 4. Results e 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 푌1, 푌2, . . . , 푌푞 and the predictor variable 푋1, 푋2, . . . , 푋푝 at theth location is given by e 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 (푢 , v ), which the size of Σ is for the samples at theth location. (1) 푌h푖 =훽h0(푢푖, v푖) + 훽h1(푢푖, v푖)푋1푖 + 훽h2(푢푖, v푖)푋2푖 + . . . + 훽h푝(푢푖, v푖)푋푝푖 + 휀h푖, h =1, 2, . . . , 푞 and 푖 = 1, 2, . . . , 푛. (2) Σ(푢푖, v푖) = [[[ [ 휎2 1(푢푖, v푖) 휎12(푢푖, v푖) ⋅ ⋅ ⋅ 휎1푞(푢푖, v푖) 휎2 2(푢푖, v푖) ⋅ ⋅ ⋅ 휎2푞(푢푖, v푖) . . . . . . 휎2 푞(푢푖, v푖) ]]] ] . From Equation (2), the estimation of the variance-covariance error matrix parameters Σ̂(푢 , v ) is observed at each study location using the MLE and WLS methods. To get the estimation of the variance-covariance matrix parameter Σ̂(푢 , v ), the parameter estimation is determined at one theth location (휎2 h(푢푗, v푗)) as follows: e vector error at the location (푢 , v ) can be stated as follows: where is the matrix identity with order and is the symmetric matrix sized × , About the local character of the MGWR model (3), the sum of square error (푆푆퐸) and the estimated parameters of the error variance-covariance can be determined. Proposition 1. If the location of (푢 , v ) the MGWR model is 푒 ∼ (푢 , v ) 푒 ∼ (푢 , v ), then it can be determined h and the expectation value h. Proof. To get the from the MGWR model using squaring (4) at the location to (푢 , v ) is: where (3) ̂휎2h(푢푗, v푗) = ∑푛푖=1w푖(푗)(푢푗, v푗)(푌h푖 − (훽h0(푢푗, v푗) + ∑푝푘=1훽h푘(푢푗, v푗)푋푘푖)) 2 푛 = (푌 ∼h − X̂훽 ∼h푗 v푗)) 푇 W(푢푗, v푗)(푌 ∼h − X̂훽 ∼h푗 v푗)) 푛 = 푆푆퐸(푢푗, v푗) 푛 . (4) 푒 ∼h = 푌 ∼h − ̂푌 ∼h = (I − S)푌 ∼h, (5) ⋅S(푛×푛) = [[[[[[ [ 푋∼1 (X푇W(푢1, v1)X)X푇W(푢1, v1) 푋∼2 (X푇W(푢2, v2)X)X푇W(푢2, v2) . . . 푋∼푛(XW(푢푛, v푛)X)X푇W(푢푛, v푛) ]]]]]] ] . (6) 푆푆퐸 = 푒 ∼ 푇h(푢푗, v푗)푒 ∼h(푢푗, v푗) = ((I − S)푌 ∼h) 푇 ((I − S)푌 ∼h) = 푌 ∼ 푇h (I − S)푇(I − S)푌 ∼h, (7) 퐸(푒 ∼h(푢푗, v푗)) = 퐸(푌 ∼h − ̂푌 ∼h = X푇 훽 ∼ (푢푗, v푗) − X푇̂훽 ∼푗 v푗) = 0, 3 Abstract and Applied Analysis and variance error is Based on (8), then (6) can be described as follows: From Equation (9), we can find the expected value 푆푆퐸h(푢푗, v푗) as follows: Since 퐸(푆푆퐸h(푢푗, v푗)) = 푟1 휎2 h(푢푗, v푗), then we have 푟1 = (1/휎2 h (푢푗, v푗))퐸(푆푆퐸h(푢푗, v푗)) with 푟1 = 푡푟((I − S)푇(I − S)). ⬜ Proposition 2. If the errors of estimated parameter variance-covariance MGWR model at theth location are ̂휎hh∗(푢푗, v푗) = 퐸(푒 ∼ 푇h(푢푗, v푗)푒 ∼h∗(푢푗, v푗)) and ̂휎 2h(푢푗, v푗) = ̂휎hh∗(푢푗, v푗), then we can determine h h∗ and the expected value h h∗ at each location (푢 , v ) mathematically. Proof. First, the variance-covariance error at theth location is shown as follows: (8) 푉푎푟(푒 ∼h(푢푗, v푗)) = 퐸[(푒 ∼h(푢푗, v푗) − 퐸(푒 ∼h(푢푗, v푗))) ⋅(푒 ∼h(푢푗, v푗) − 퐸(푒 ∼h(푢푗, v푗))) 푇 ] = 퐸(푒 ∼h(푢푗, v푗)푒 ∼ 푇h(푢푗, v푗)) = 휎2 h(푢푗, v푗). (9) 푆푆퐸h(푢푗, v푗) = 푒 ∼ 푇h(푢푗, v푗)푒 ∼h(푢푗, v푗) = (푒 ∼ (푢푗, v푗) − 퐸(푒 ∼ (푢푗, v푗))) 푇 ⋅ (푒 ∼ (푢푗, v푗) − 퐸(푒 ∼ (푢푗, v푗))) = 푒 ∼ 푇h(푢푗, v푗)(I − S)푇(I − S)푒 ∼h(푢푗, v푗). (10) 퐸(푆푆퐸h(푢푗, v푗)) = 퐸(푒 ∼ 푇h(푢푗, v푗)(I − S)푇(I − S)푒 ∼h(푢푗, v푗)) = 퐸(푡푟(푒 ∼ 푇h(푢푗, v푗)(I − S)푇(I − S)푒 ∼h(푢푗, v푗))) = 푡푟((I − S)푇(I − S))퐸(푒 ∼h(푢푗, v푗)푒 ∼ 푇h(푢푗, v푗)) = (푛 − 2푡푟(S) + 푡푟(S푇S))휎2 h(푢푗, v푗) = 푟1 휎2 h(푢푗, v푗). (11) ̂휎2h(푢푗, v푗) = ̂휎hh∗(푢푗, v푗) 푉푎푟(푒 ∼h(푢푗, v푗), 푒 ∼h(푢푗, v푗)) = 퐸(푒 ∼ 푇h(푢푗, v푗)푒 ∼h∗(푢푗, v푗)) ̂휎2h(푢푗, v푗) = 퐸(푒 ∼ 푇h(푢푗, v푗)푒 ∼h(푢푗, v푗)) − 퐸(푒 ∼h(푢푗, v푗)) 푇 퐸(푒 ∼h(푢푗, v푗)) = 퐸(푒 ∼ 푇h(푢푗, v푗)푒 ∼h(푢푗, v푗)) = ̂휎hh∗(푢푗, v푗). Furthermore, 푆푆퐸h퐸h∗(푢푗, v푗) is searched using (9), we obtain where (I − S) (I − S) is a definite and symmetrical semi-definite matrix × with ε∼h(푢푗, v푗) ∼ 푁(0, 휎hh∗(푢푗, v푗)). en we have


Introduction
In statistical inference, estimation of spatial data parameters using the GWR approach has been carried out by many researchers. According to [1], 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 [2]. e author discussed spatial analysis in detail by using OLS and estimator of spatial regression models with the maximum likelihood estimation (MLE) methods. Yasin [3] proposed the GWR stepwise method in order to choose a significant variable. e 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 [1], Cressie [2] and Harini et al. [4]. 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 [5] 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. e advantage of this approach is the absence of variable dimension constraints. Referring to [4], Triyanto et al. [6] discussed the parameter estimation of the Geographically Weighted Multivariate Poisson Regression (GWMPR) model using the Maximum Likelihood Estimation (MLE) methods. e GWMPR is used to model the spatial data with response variables that are distributed Poisson.
Another problem that o en 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 [7]. erefore, the R and GWR4 programs can be used to check the validity level of hypothesis testing. e advantages of the program parameter estimation results are both global and local and can be done together. Soemartojo et al. [8] 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. e process of non-stationarity by applying an extended hyper-local GWR is examined by Comber et al. [9].
is 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. is test uses statistical inference procedures to obtain the estimated error variance-covariance parameters that meet the unbiased nature.

Theoretical GWR
Supporting theories for completing this research refer to the Geographically Weighted Regression (GWR) [1] and Statistics for Spatial Data [2].

Results
e 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 푌 1 , 푌 2 , . . . , 푌 푞 and the predictor variable 푋 1 , 푋 2 , . . . , 푋 푝 at theth location is given by e 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 푢 , v , which the size of Σ is for the samples at the-th location. (1) ℎ =1, 2, . . . , 푞 and 푖 = 1, 2, . . . , 푛.
From Equation (2), the estimation of the variance-covariance error matrix parameters Σ 푢 , v is observed at each study location using the MLE and WLS methods. To get the estimation of the variance-covariance matrix parameter Σ 푢 , v , the parameter estimation is determined at one the-th location 휎 2 ℎ 푢 푗 , v 푗 as follows: e vector error at the location 푢 , v can be stated as follows: where is the matrix identity with order and is the symmetric matrix sized × , About the local character of the MGWR model (3), the sum of square error (푆푆퐸) and the estimated parameters of the error variance-covariance can be determined. Proof. To get the from the MGWR model using squaring (4) at the location to 푢 , v is: and variance error is Based on (8), then (6) can be described as follows: From Equation (9), we can find the expected value 푆푆퐸 ℎ 푢 푗 , v 푗 as follows: If the errors of estimated parameter variance-covariance MGWR model at the-th location are then we can determine ℎ ℎ * and the expected value ℎ ℎ * at each location 푢 , v mathematically.
Proof. First, the variance-covariance error at the-th location is shown as follows: (10)

Abstract and Applied Analysis 4
Proof. and in the same way, we obtain where 휎 2 ℎ 푢 푗 , v 푗 and is 휎 ℎℎ * 푢 푗 , v 푗 an estimate of the unbiased error variance-covariance matrix for 휎 2 ℎ 푢 푗 , v 푗 and 휎 ℎℎ * 푢 푗 , v 푗 . By using eorem 1.3, an unbiased estimate is obtained from the variance-covariance error matrix Σ 푢 , v at the-th location as follows: Since the variance-covariance error matrix Σ 푢 , v 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 푢 , v can be stated as follows: us, it is proven that if Σ 푢 , v as an unbiased estimate of the variance-covariance error matrix Σ 푢 , v , then Σ 푢 , v is also an unbiased estimate of the variance-covariance error matrix Σ 푢 , v . ⬜

Conclusion
is research concludes that the MGWR model using MLE and WLS methods is suitable to obtain the estimated error variance-covariance parameters.
e results prove that Σ 푢 , v is an unbiased estimate of the variance-covariance Proof. Based on Proposition 1 and 2, the estimated error variance-covariance parameters from the MGWR model are: By using the characteristics of the matrix (I − S) (I − S), 퐸 휎 2 ℎ 푢 푗 , v 푗 , and 퐸 휎 ℎℎ * 푢 푗 , v 푗 can be determined to satisfy the unbiased. ⬜ is an unbiased estimator 휎 ℎℎ * 푢 푗 , v 푗 , then 퐸 휎 2 ℎ 푢 푗 , v 푗 , and 퐸 휎 ℎℎ * 푢 푗 , v 푗 can be determined to satisfy the unbiased. error matrix Σ 푢 , v . Since Σ 푢 , v is an unbiased estimate, then Σ 푢 , v is also an unbiased estimate of the variance-covariance error matrix Σ 푢 , v at all locations.
Data Availability e authors declare that all of data is original and there is no data from others publication.

Conflicts of Interest
e authors declare that there is no conflict of interests regarding the publication of this article.