where,
d
ij
ST
is the spatio-temporal distance between point
i
and
j
,
h
ST
is the non-negative bandwidth which produces a
decay of influence with distance. A too small
h
ST
will make
the weights decay rapidly with distance and increase the
standard error of estimation while unsuitably large
h
ST
might
cause smooth change of weights and lead to bias rising (Foth-
eringham
et al.
2002). Therefore, it is important to choose a
suitable bandwidth during the weight calculation.
The identification of optimal bandwidth
h
ST
is realized
using cross validation (
CV
)(Huang
et al.,
2010). Supposing
ˆ
y
i
(
h
ST
) is the predicted
PM2.5
value, and
y
i
is the observed
PM2.5
measurement,
h
ST
is determined by minimizing the sum of the
squared error CVRSS(
h
ST
) as expressed in Equation 4.
CVRSS
h
y y h
ST
i
i
ST
(
)
=
−
∑
≠
(
(
))
1
2
ˆ
(4)
The
d
ST
of two points in a three-dimensional space-time
coordinate system can be achieved through a combination of
spatial distance
d
S
ij
and temporal distance
d
S
ij
,
(
)
( )
( )
d
d
d
ij
ST
ij
S
ij
T
2
2
2
=
+
λ
µ
(5)
with
( )
( )
d t t
d
ij
T
i
j
ij
S
i
j
i
j
2
2
2
2
2
= −
(
)
= −
(
)
+ −
(
)
µ µ
ν ν
(6)
where
d
S
ij
and
d
S
ij
are the Euclidean distances between point
i
and
j
in space and time, respectively. Considering the differ-
ent units of space and time (meter for space and day for time
in this case), they may have different scale effects. Thus,
λ
and
μ
are the corresponding scale factors to balance the different
effects of spatial and temporal metric systems. The scale fac-
tors (
λ
and
μ
) are determined by cross validation in terms of
R
2
. Once the spatio-temporal distance and bandwidth are cal-
culated, the spatiotemporal weight matrix in Equation 2 can
be constructed by using the Gaussian
distance decay-based function (Equation
3) and, consequently, the parameters
β
can be derived(Huang
et al.,
2010; Wu
et
al.,
2014).
Extending GTWR with Seasonal Variability
Although the traditional
GTWR
has been
successfully applied in environmental
and social disciplines to address the
spatial and temporal heterogeneity and
generally yields better model-fitting
than
GWR
, the seasonal variability
in temporal dimension has not been
considered in such models especially
for natural variables (i.e.,
PM2.5
in this
study) which exhibit strong seasonal
variation due to meteorological fac-
tors (Chu
et al.,
2015; Li
et al.,
2017).
PM2.5
concentrations in same seasons
are more similar than those in different
seasons. Given the seasonal variation
of
PM2.5
concentrations, the seasonal
distance
d
ij
SE
is added to improve the
traditional
GTWR
model. Since the
PM2.5
concentrations are high in winter and
spring and low in autumn and summer
in this study, showing a similar trend
with the cosine curve, the seasonal dis-
tance
d
ij
SE
is defined as:
(
)
cos
cos
d
t t
T
t t
T
ij
SE
i
SE
j
SE
2
0
0
2
2
2
=
+
−
+
π
π
.
.
(7)
where
T
SE
is denoted as the period of
PM2.5
seasonal variation
(365 days in this study),
t
0
is the offset of the cosine function,
which can be determined by
t
cos
t t
T
PM
t
SE
0
1 2 365
0
2 5
0
2
=
−
−
∈
arg min
{ , , ,
}
.
π
.
(8)
where
t
is the vector of time locations for all observations,
–
PM2.5
is the normalized vector of
PM2.5
concentrations, which
ranges from -1 to 1. Since the samples closer to the estimated
point
i
in space, time and season have more impacts on pre-
diction, the weight in Equation 3 can be deduced to:
+
+
ω
λ
µ
κ
ij
ij
S
ij
T
ij
SE
ST
d
d
d
h
= −
exp
( )
( )
(
)
2
2
2
2
(9)
where
λ
,
μ
and
κ
are the adjustment scale factors for the
balance of different units of spatial, temporal and seasonal
distances, respectively,
h
ST
represents the spatiotemporal
bandwidth in the improved
GTWR
model. In order to simplify
the calculation,
λ
is set to be 1 and
μ
and
κ
are determined by
CV
in terms of R
2
(Huang
et al.,
2010). Once the three param-
eters are appropriately adjusted, the weight
ω
ij
can be derived
from the Gaussian distance decay-based function.
Results and Discussion
Descriptive Statistics
In order to demonstrate the seasonal variations of
PM2.5
in the
study area, the boxplot of the monthly
PM2.5
concentrations
from 2012 to 2014 is plotted in Figure 2. It can be seen that
Figure 2. Boxplot of the average monthly of
PM2.5
concentrations from 2012 to 2014.
764
December 2018
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
