Exploring spatial patterning in data values - Spatial Data Analysis: An Introduction for GIS Users - page 123

precipitation amount with an increase in elevation). Parts of the far west of the region
have large slope and large r 2 values. h e GWR slopes and r 2 values tend to be larger in
areas with larger elevation values (see Figure 10.5 for a map of elevation in Northern
Ireland) and this suggests that the elevation-precipitation relationship tends to be less
strong (and therefore elevation is a less useful predictor) in areas with small elevation
values. Trends in precipitation amounts are directional and are a function of many
factors not considered here but, as the example demonstrates, GWR is a powerful
means of exploring spatially variable relationships such as those between elevation
and precipitation amount.
Relationships between many variables of interest in the physical and social sciences
are a function of geography. h ese include the previous example of altitude and pre-
cipitation as well as other variables such as employment status and religion. Where
such geographical variations are suspected, standard global regression analyses may
be inadequate and an analysis based on the application of GWR may reveal a far richer
picture than would be obtained through conventional regression analysis. GWR
allows assessment of how far the nature of relationships (e.g. are variables related
positively or negatively) vary spatially and how strongly variables are related in dif er-
ent regions. GWR has been used in many other contexts. Brunsdon et al. (2001)
used GWR to explore the average elevation-precipitation relationship across Britain,
while Lloyd and Shuttleworth (2005) used GWR to explore spatial variation in the
relationship between commuting distance and a range of other variables in Northern
Ireland. Some authors have commented on potential problems associated with
GWR, particularly in terms of multicollinearity (i.e. strong correlations between inde-
pendent variables). h is may have an impact on the interpretation of the local regres-
sion coei cients (see Wheeler and Tiefelsdorf, 2005). In short, where there are multiple
independent variables in the GWR analysis, and these independent variables are
strongly related, the values and signs of coei cients for individual variables may be
highly misleading. Methods for diagnosing collinearity and a solution to this problem
are detailed by Wheeler (2007). Another issue which should be considered is the avail-
ability of enough observations with signii cantly non-negative weights for the pur-
poses of GWR parameter estimation. One way around this problem is to use an
adaptive bandwidth whereby the size of the bandwidth varies as a function of the density
of observations in a given area (see Fotheringham et al. , 2002).

Other approaches

8.6

h is chapter of ers only a brief summary of some widely used approaches for the anal-
ysis of spatial structure in single and multiple variables. h e selection is biased and
many other approaches could have been included. For example, in terms of regression
approaches, a body of models called multilevel models is used widely by geographers
(and others) to explore relationships between variables at dif erent spatial scales (see
Fotheringham et al. (2002) and Lloyd (2006) for summaries of some other approaches).