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,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 location ui,vi can be stated as follows:(4)e∼h=Y∼h-Y∼^h=I-SY∼h,
where I 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 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 e∼Tuj,vje∼uj,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)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.
Since ESSEhuj,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
∼
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, 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,
where I−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.
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)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 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 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 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^hh∗uj,vj=SSEhEh∗uj,vj/n−2trS+trSTS.
By using the characteristics of the matrix I−STI−S, Eσ^h2uj,vj, and Eσ^hh∗uj,vj can be determined to satisfy the unbiased.
Theorem 3.
If σ^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
−
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^hh∗uj,vj=ESSEhEh∗uj,vjtrI−STI−S=σhh∗uj,vj,
where σ^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 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,vj⋯σ^ 1quj,vjσ^ 22uj,vj⋯σ^ 2quj,vjsimetris⋱⋮σ^ 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,vi⋯σ^ 1qui,viσ^ 22ui,vi⋯σ^ 2qui,vi simetris⋱⋮σ^ 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.