Geography Reference
In-Depth Information
7.3.2 Kernel estimation
Kernel estimation (KE) is a more sophisticated means of exploring spatial variations
in event intensity. Such approaches are ot en used in the identii cation of clustering
'hot spots. If we want to estimate the intensity of points over an area we can simply
calculate the number of events within a radius around the nodes of a grid and divide
this amount by the area concerned. In other words, we in ef ect overlay onto the point
pattern a set of points on a regular grid and compute the event intensity within the
neighbourhood of each point on the grid. h is is called the naïve estimator. h e naïve
intensity estimate is given by:
#(
Cd
d
(
x
,
))
ˆ ()
(7.5)
l
x
=
p
2
where #( C ( x , d )) indicates the number (#) of events in the circle C ( x , d ) that has as its
centre the location x , and has the radius d , with its area given by pd 2 . h e bold lower
case x is matrix vector notation as introduced in Section 3.3. In this case, x refers to a
location with coordinates x,y .
Figure 7.7 shows intensity estimates for PP1 using radii of 25 and 50 units. h e dif-
ferences in minimum and maximum intensities and in the spatial patterning are
immediately apparent. h e maximum intensity is greater for a radius of 25 units as this
smaller radius 'picks out' the small clusters. h eir ef ect is diminished when the larger
radius (50 units) is used. h e two groups of clusters notable when the 25 unit radius
is used become, in ef ect, merged into one as the size of the radius is increased to
50 units.
KE can be expanded to make use of a geographical weighting scheme (a kernel
function) whereby the inl uence of the points varies depending on how far they
are from the centre of the window. h e general idea of geographical weighting was
Cell = 5 units square
Radius = 50 units
Cell = 5 units square
Radius = 25 units
Val u e
High : 0.01324
Value
High : 0.00700
Low : 0.00267
Low : 0.00255
A
B
Figure 7.7 Intensity estimates for PP1 using radii of 25 units (A) and 50 units (B).
The values are events per square unit.
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