Geography Reference
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data. Several approaches exist which take into account the spatial structure in vari-
ables and therefore allow for spatial dependence (e.g. Rogerson, 2006; Ward and
Gleditsch, 2008). With generalized least squares (GLS) regression, information on
spatial dependence can be utilized when estimating the regression coei cients. More
specii cally, the GLS regression coei cients can be estimated given information on the
degree of similarity between variables as a function of the distance by which they are
separated. Bailey and Gatrell (1995) provide an account of GLS. Spatial autoregressive
models provide another means of accounting for spatial structure. Lloyd (2006)
outlines the simultaneous autoregressive model, which includes an interaction param-
eter representing interactions between neighbouring observations. If the interaction
parameter is unknown (the usual case), then such models cannot be i tted using ordi-
nary least squares and specialist sot ware is required. h e simultaneous estimation of
the interaction parameter and the regression ( b ) coei cients can be conducted using
a maximum likelihood procedure, as described by Schabenberger and Gotway (2005).
h e GeoDa sot ware of Anselin et al . (2006) allows for spatial autoregressive model-
ling. Such models help to overcome the problem of analysing relations between spa-
tially referenced variables, but they provide only a single set of coei cients. Increasingly,
in GIS contexts, studies take into account the local context. Local approaches entail
estimating regression coei cients using either local data subsets or a geographical
weighting scheme. Two local regression approaches are outlined next.
8.5.2 Moving window regression
Section 8.4.1 introduced the idea that spatial autocorrelation of variables is ot en
observed to vary spatially. Local measures which take these variations into account
may therefore be worthwhile. Similarly, relationships between variables may dif er
markedly across an area. As an example, many studies have shown that altitude and
precipitation amount are related in some regions. However, while the two variables
may be strongly related in some areas, a global regression of altitude and precipitation
amount may demonstrate only a weak relationship (see Lloyd (2005) for a relevant
case study). Some kind of local regression procedure is therefore needed to enable
exploration of some geographically variable relationships.
One straightforward approach to exploring how relationships vary spatially is sim-
ply to conduct a standard regression in a moving window. In other words, regression
is carried out using only the data in the moving window and the end result is a set of
maps of regression coei cients. Moving window regression (MWR) has been used in
various studies (see Lloyd (2005, 2006) and Lloyd and Shuttleworth (2005) for exam-
ples). MWR is identical to the regression procedure detailed in Section 3.3, the only
dif erence being that regression is conducted for data subsets in a moving window
rather than for all data simultaneously. Building on the geographical weighting
principles outlined previously (see Section 4.7), this approach can be extended such
that the inl uence of observations in the regression is decreased as distance from
the centre of the moving window increases. Such an approach is the subject of the
following section.
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