Geography Reference
In-Depth Information
al., 2013a), and dengue fever pattern discovery (Delmelle et al., 2014). The
STKDE value of each voxel (volumetric pixel) in the three-dimensional space-
time cube is estimated as:
1
x
x
y
y
t
t
D
(
x
,
y
,
t
)
K
(
i
,
i
)
K
(
i
)
s
t
2
nh
h
h
h
h
i
s
t
s
s
t
(1)
where
D (x, y, t)
is the density estimation of each voxel based on the data in
neighboring volumetric pixels;
n
is the number of point events, and
h
s
and
h
t
are the spatial and temporal neighboring bandwidths. Each point in the
neighboring pixels is weighted based on the proximity in both space and time
to the voxel using kernel functions (
K
s
and
K
t
). In this study, the Epanechnikov
kernel is used for multivariate probability density estimation within the band-
widths (Epanechnikov, 1969).
Similar to the 2D spatial KDE, larger bandwidths may result in smooth
surface while smaller bandwidths may result in the lack of trending patterns,
so we need to calibrate both spatial and temporal bandwidths of STKDE based
on the experiments with actual datasets.
The results of STKDE are volume data, i.e., 3D-grids. Direct visualization
of such STKDE would require four-dimensional space because of their
volumetric data structure consisting of 2D geographic space, time and another
one for the density estimation scalar. Such volume visualization is not very
common in GIS but very popular in geophysics, geology, medical science, and
in computer graphics (Kaufman, 1990). The three main approaches for volume
visualization were discussed by Demšar and Virrantaus (2010): (1) direct
volume rendering by assigning color and transparency to voxels; (2) isosurface
that is the equivalent of isoline connecting points of equal value on a two-
dimensional map; and (3) volume slicing by planes. We apply the volume
slicing approach with color schema and transparency to the voxels regarding
the consistency of KDE visualization in GIS.
2.3. Spatio-Temporal Autocorrelation Analysis
Analyzing spatio-temporal autocorrelation structures of human activities
would be helpful to understand the urban dynamic patterns in space and time
simultaneously. In statistics, autocorrelation can be taken as the correlation of
a variable with a lagged specification of itself (Box et al., 2008).
