Key concepts 3 Spatial data analysis - Spatial Data Analysis: An Introduction for GIS Users - page 50

illustrated in Figure 4.5, where the mean average of neighbouring pixels is computed.
Note that when such procedures are employed, the output image ot en has fewer rows
and columns than the input image. In the example below, the window is 3 ¥ 3 pixels in
size and the mean is only computed where there are neighbours on all sides of a pixel.
When the window is centred on pixels at the edge of the image, there are fewer than
3 ¥ 3 pixels and so no value is computed. h e moving window statistic could still be
calculated from the smaller number of pixels, but in many cases the procedure
employed in this example is followed.
In the case of position 1, the value in the centre of the window is 42 and its neigh-
bouring values are 45, 44, 44, 43, 39, 38, 32, and 34. Adding these values together and
dividing the sum by nine gives a value (the mean average) of 40.11, as shown in the
top-let cell of the output grid.
h e next section extends the moving window idea by treating each of the observa-
tions in the window dif erently according to where they are located.

Geographical weights

4.7

h e tendency for observations close together in space to be more similar than obser-
vations that are separated by larger distances (see Section 4.1) is ot en accounted for in
spatial analyses. For example, a summary statistic computed in a moving window of a
particular size may be based equally on all of the data in the window at a certain posi-
tion. Alternatively, observations close to the centre of the window may be given more
weight (or inl uence). Logically this is sensible: if the summary statistic is allocated to
a point in the centre of the window, it is sensible to allow close-by observations to have
most inl uence on the estimated statistic at that location since these close-by values are
most likely to be similar. h e objective is to obtain a more reliable statistic as distance
to neighbours is taken into account.
Weights can be based on adjacency or they can, for example, be a function of dis-
tance. h e example application of the Moran's I statistic in the following section uses
adjacency: neighbouring cells are given a weight of one while all other cells are given
a weight of zero, i.e. they are not included in the calculations. Alternatively, all cells (or
points/areas) or some subset could be used in the calculations but with larger weights
given to cells closer to the cell of interest. h ere is a large variety of weighting functions
that determine how much weight should be given to observations as a function of dis-
tance. A simple linear weighting function could be used whereby an observation twice
as far away receives half as much weight, e.g. an observation at 10 km receives twice as
much weight as an observation 20 km away. In practice, more sophisticated schemes
are used for most applications. One well-known weighting function is based on taking
the inverse of the squared distance from the location of interest (following the inverse
square law). In other words, the weight is a function of (is dependent on) the inverse
squared distance, d -2 (this can be obtained with 1 / d 2 , as detailed below, and see Section 9.5
for an application of this weighting function). Whatever distance decay weighting