Spatial interpolation - Spatial Data Analysis: An Introduction for GIS Users - page 136

remaining observations to predict the value of the removed value. h e removed value
is returned to the data set and the next observation in order is removed, at er which the
procedure is repeated for all observations. h e prediction errors can then be assessed.
h e accuracy of prediction is commonly assessed using summaries such as the mean
error, the mean absolute error, and the root mean square error (RMSE).
In terms of selecting a data subset for interpolation, several common strategies
exist. h e n (four in the example) nearest neighbours to a prediction location could be
selected. Alternatively, all observations within a specii ed distance of the prediction
location could be used. Another strategy is to divide the search neighbourhood into
quadrants, for example north-east, south-east, north-west, and south-west of the pre-
diction location. h e weights could then be scaled according to the number of obser-
vations in each quadrant and this would help to overcome the ef ect of clustering of
observations in particular areas.
IDW was used to generate a map of precipitation amount in July 2006 in Northern
Ireland using the 16 nearest neighbours to each cell of the prediction grid. h e data
locations are shown in Figure 8.7 and the IDW-derived map is shown in Figure 9.7.
h e map in Figure 9.7 is very smooth in appearance and there are clear clusters of
values around the sample locations—this is a common feature of maps generated
using IDW. With IDW, there tend to be clusters of similar values around data points
(see Lloyd (2005) for another example).
IDW is rapid and easy to implement, although it ot en performs less well than more
sophisticated approaches (e.g. see Lloyd, 2005).

Thin plate splines

9.6

TPS are, like IDW, a very widely used approach to spatial interpolation. TPS functions
are available in ArcGIS™, the GRASS GIS (Neteler and Mitásová, 2007), and in other
sot ware packages. TPS can be viewed as surfaces that are i tted to some local subset of
the data. h e spline can be i tted exactly to the data points or it can be smoothed—that
is, if the spline is not forced to i t to the data points the resulting surface can be made
smoother than if the surface runs through every point. In ef ect, the thin plate smooth-
ing spline generated map is a map of local weighted averages. With TPS, the aim is to
obtain a prediction of the unknown value

ˆ ()

z x with a smooth function g by

0

minimizing (as dei ned below):

n

Â

(9.2)

(( ) z x - g ( )) x

2

+ r J ( ) g

i i m

i = 1

where J m ( g ) is a measure of the roughness of the spline function, m is the degree of the
polynomial used in the model, and r is the smoothing parameter. We seek to i nd the
function g so that it is as close as possible to the observations (indicated by z ( x i )), with
the smoothing function determining if the i t of the function to the observations is