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

Authors: Sri Harini
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

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 vectorε 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.


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## Body

## 1. 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]. The 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. 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 [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. The 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. 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 [7]. 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. [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. The process of non-stationarity by applying an extended hyper-local GWR is examined by Comber et al. [9]. 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) [1] and Statistics for Spatial Data [2].


## 3. Methods of MGWR

The MGWR method refers to [4] and [10].


## 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 variableY1,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 vectorε 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,vi⋯σ1qui,viσ22ui,vi⋯σ2qui,vi⋱⋮σq2ui,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=Y∼h-Xβ∼^huj,vjTWuj,vjY∼h-Xβ∼^huj,vjn=SSEuj,vjn.The vector error at the locationui,vi can be stated as follows:(4)e∼h=Y∼h-Y∼^h=I-SY∼h,whereI is the matrix identity with order n and S is the symmetric matrix sized n×n,(5)⋅Sn×n=X∼1TXTWu1,v1X−1XTWu1,v1X∼2TXTWu2,v2X−1XTWu2,v2⋮X∼nTXTWun,vnX−1XTWun,vn.About the local character of the MGWR model (3), the sum of square errorSSE and the estimated parameters of the error variance-covariance can be determined.Proposition 1.
IfSSE the location of uj,vj the MGWR model is e∼Tuj,vje∼uj,vj, then it can be determined SSEh and the expectation value SSEh.Proof.
To get theSSE 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)Ee∼huj,vj=EY∼h-Y∼^h=XTβ∼uj,vj-XTβ∼^uj,vj=0,and variance error is(8)Vare∼huj,vj=Ee∼huj,vj−Ee∼huj,vje∼huj,vj−Ee∼huj,vjT=Ee∼huj,vje∼hTuj,vj=σh2uj,vj.Based on (8), then (6) can be described as follows:(9)SSEhuj,vj=e∼hTuj,vje∼huj,vj=e∼uj,vj−Ee∼uj,vjTe∼uj,vj−Ee∼uj,vj=e∼hTuj,vjI−STI−Se∼huj,vj.From Equation (9), we can find the expected value SSEhuj,vjas follows:(10)ESSEhuj,vj=Ee∼hTuj,vjI−STI−Se∼huj,vj=Etre∼hTuj,vjI−STI−Se∼huj,vj=trI−STI−SEe∼huj,vje∼hTuj,vj=n−2trS+trSTSσh2uj,vj=r1σh2uj,vj.SinceESSEhuj,vj=r1σh2uj,vj, then we have r1=1/σh2uj,vjESSEhuj,vj with r1=trI−STI−S.Proposition 2.
If the errors of estimated parameter variance-covariance MGWR model at the-jth location are σ^hh∗uj,vj=Ee∼hTuj,vje∼h∗uj,vj and σ^h2uj,vj=σ^hh∗uj,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
∼
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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,SSEhEh∗uj,vj is searched using (9), we obtain(12)SSEhEh∗uj,vj=e∼huj,vj−Ee∼huj,vjTe∼h∗uj,vj−Ee∼h∗uj,vj=I−SY∼h−EI−SY∼hTI−SY∼h∗−EI−SY∼h∗=Y∼h−EY∼hTI−STI−SY∼h∗−EY∼h∗=e∼hTuj,vjI−STI−Se∼h∗uj,vj,whereI−STI−S is a definite and symmetrical semi-definite matrix n×n with ε∼huj,vj∼N0,σhh∗uj,vj. Then we have(13)ESSEhEh∗uj,vj=Ee∼hTuj,vjI−STI−Se∼h∗uj,vj=Etre∼hTuj,vjI−STI−Se∼h∗uj,vj=trI−STI−SEe∼h∗uj,vje∼hTuj,vj=trI−STI−Sσhh∗uj,vj.Theorem 1.
IfSSEh 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 determineSSEhEh∗ at each location uj,vj, it can be approached using Equation (5),(16)ε∼hWuj,vjε∼h=Y∼h-Xβ∼huj,vjTWuj,vjY∼h-Xβ∼huj,vjEε∼hWuj,vjε∼h=EY∼h-Xβ∼huj,vjTWuj,vjY∼h-Xβ∼huj,vjSSEhEh∗uj,vj=Y∼h-Xβ∼huj,vjTWuj,vjY∼h∗-Xβ∼h∗uj,vj,and(17)σhh∗uj,vj=SSEhEh∗uj,vjn.Based on Propositions1 and 2, the theorems of estimation parameter variance-covariance error matrix for MGWR model are determined.Theorem 2.
IfESSEhuj,vj satisfies Proposition 1 and ESSEhEh∗uj,vj satisfies Proposition 2, the estimated parameter variance-covariance errors matrix of the MGWR model are σ^hh∗uj,vj=SSEhEh∗uj,vj/trI−STI−S and Eσ^hh∗uj,vj=σhh∗uj,vj.Proof.
Based on Proposition1 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
=
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e
∼
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T
u
j
,
v
j
e
∼
h
u
j
,
v
j
σ
^
h
2
u
j
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=
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e
∼
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j
,
v
j
e
∼
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∗
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
,andσ^hh∗uj,vj=SSEhEh∗uj,vj/n−2trS+trSTS.By using the characteristics of the matrixI−STI−S, Eσ^h2uj,vj, and Eσ^hh∗uj,vj can be determined to satisfy the unbiased.Theorem 3.
Ifσ^hh∗uj,vj=SSEhEh∗uj,vj/trI−STI−S is an unbiased estimator σhh∗uj,vj, then Eσ^h2uj,vj, and Eσ^hh∗uj,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
−
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T
I
−
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t
r
I
−
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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^hh∗uj,vj=ESSEhEh∗uj,vjtrI−STI−S=σhh∗uj,vj,
whereσ^h2uj,vj and is σ^hh∗uj,vj an estimate of the unbiased error variance-covariance matrix for σh2uj,vj and σhh∗uj,vj.By using Theorem3, 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,vj⋯σ^   1quj,vjσ^       22uj,vj⋯σ^      2quj,vjsimetris⋱⋮σ^   q2uj,vj.Since the variance-covariance error matrixΣuj,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,vi⋯σ^         1qui,viσ^         22ui,vi⋯σ^         2qui,visimetris⋱⋮σ^         q2ui,vi.Thus, it is proven that ifΣ^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 thatΣ^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.



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*Source: 4657151-2020-02-01.xml*