Spatial Patterning
4.2 Geographically weighted statistics
Any summary statistic can be computed locally using a geographical weighting scheme. Many weighting schemes, both binary and continuous, have been developed (see Section 2.4). A widely used weighting scheme is the Gaussian function (129). With this function, a weight at the observation i is obtained with:
wij = exp[−0.5(d/τ )2] (4.1)
where d is the Euclidean distance between the location of observation i, and the location j and τ is the bandwidth of the kernel. The Gaussian function (for a bandwidth of (i) 2500 and (ii) 5000 units) is illustrated in Figure 4.1.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 5000 10000 15000 20000
Distance
Weight
τ = 2500 τ = 5000
FIGURE 4.1: Gaussian curve for a bandwidth (i) 2500 and (ii) 5000 units.
The remainder of this section details geographically weighted variants of standard statistical summaries. The geographically weighted mean can be computed with (129):
¯ zi=
Pn
j=1zjwij
Pn
j=1wij
(4.2)
Spatial Patterning 75 Following Fotheringham et al. (129), if the weights are rescaled to sum to one, it is given by:
Table 4.1 shows the computation of the geographically weighted mean, with the weights given in unstandardised form, for a subset of the log-ratio data described in Section 1.8.6 for bandwidths of 1000, 2000, and 3000 m.
TABLE 4.1
Geographically weighted mean (GWmean) for log-ratio data. BW is bandwidth;
dij is distance; wij is the weight.
ln(CathCB/
NonCathCB) dij(m) wij zjwij wij zjwij wij zjwij
−2.483 0.000 1.000 −2.483 1.000 −2.483 1.000 −2.483
−1.450 1000.000 0.607 −0.879 0.882 −1.280 0.946 −1.372
−1.413 1000.000 0.607 −0.857 0.882 −1.247 0.946 −1.337
−1.376 1000.000 0.607 −0.835 0.882 −1.214 0.946 −1.302
−1.673 1414.214 0.368 −0.615 0.779 −1.303 0.895 −1.497
−1.444 1414.214 0.368 −0.531 0.779 −1.125 0.895 −1.292
−1.474 2000.000 0.135 −0.199 0.607 −0.894 0.801 −1.180
−1.700 2000.000 0.135 −0.230 0.607 −1.031 0.801 −1.361
−1.728 2000.000 0.135 −0.234 0.607 −1.048 0.801 −1.384
−0.964 2236.068 0.082 −0.079 0.535 −0.516 0.757 −0.730
−1.187 2236.068 0.082 −0.097 0.535 −0.635 0.757 −0.899
−2.766 2236.068 0.082 −0.227 0.535 −1.481 0.757 −2.095
−1.923 2236.068 0.082 −0.158 0.535 −1.029 0.757 −1.457
−0.679 2236.068 0.082 −0.056 0.535 −0.363 0.757 −0.514
−2.572 2828.427 0.018 −0.047 0.368 −0.946 0.641 −1.649
−1.562 2828.427 0.018 −0.029 0.368 −0.575 0.641 −1.002
−1.195 3000.000 0.011 −0.013 0.325 −0.388 0.607 −0.725
−1.175 3000.000 0.011 −0.013 0.325 −0.381 0.607 −0.713
−1.259 3162.278 0.007 −0.008 0.287 −0.361 0.574 −0.722
−1.649 3162.278 0.007 −0.011 0.287 −0.472 0.574 −0.946
−1.035 3162.278 0.007 −0.007 0.287 −0.297 0.574 −0.594
−1.563 3605.551 0.002 −0.002 0.197 −0.308 0.486 −0.759
−1.439 3605.551 0.002 −0.002 0.197 −0.283 0.486 −0.699
−1.478 3605.551 0.002 −0.002 0.197 −0.291 0.486 −0.718
−1.624 4000.000 0.000 −0.001 0.135 −0.220 0.411 −0.668
−1.108 4123.106 0.000 0.000 0.119 −0.132 0.389 −0.431
−2.626 4123.106 0.000 −0.001 0.119 −0.314 0.389 −1.021
−1.499 4123.106 0.000 0.000 0.119 −0.179 0.389 −0.583
−1.673 4242.641 0.000 0.000 0.105 −0.176 0.368 −0.615
−1.355 4472.136 0.000 0.000 0.082 −0.111 0.329 −0.446 Sum 4.456 −7.619 13.217 −21.084 19.766 −31.193
GWmean −1.710 −1.595 −1.578
BW (m) 1000 2000 3000
76 Local Models for Spatial Analysis The unweighted mean of the observations in the example is −1.569. Note how the geographical weights at larger distances increase in proportion to the geographical weights for smaller distances as the bandwidth is increased.
Lloyd (245) presents a further worked example of the geographically weighted mean. The geographically weighted mean was computed using the Gaussian function with a 2km bandwidth, given the log-ratio data described in Section 1.8.6, and it is illustrated in Figure 4.2. Comparison of Figure 1.10 with Figure 4.2 demonstrates the smoothing effect of the local mean. Figure 4.3 shows the geographically weighted mean for a 5km bandwidth. Varying the bandwidth in this way provides a means of assessing how far local patterns persist across different spatial scales. The maps suggest that there is marked variation in the local mean.
0 10 20 Kilometres GW mean
-2.55 - -1.26 -1.25 - -0.01 0.00 - 1.25 1.26 - 2.76 Inland water
FIGURE 4.2: Geographically weighted mean for a Gaussian function with a 2km bandwidth: log-ratio of Catholics/Non-Catholics in Northern Ireland in 2001 by 1km grid squares. Northern Ireland Census of Population data — c° Crown Copyright.
For standardised weights, the geographically-weighted standard deviation is given by:
si=hPn
j=1(zj− ¯zi)2· wij
i1/2
(4.4)
Spatial Patterning 77
0 10 20 Kilometres GW mean
-2.26 - -1.26 -1.25 - -0.01 0.00 - 1.25 1.26 - 2.54 Inland water
FIGURE 4.3: Geographically weighted mean for a Gaussian function with a 5km bandwidth: log-ratio of Catholics/Non-Catholics in Northern Ireland in 2001 by 1km grid squares. Northern Ireland Census of Population data — c° Crown Copyright.
The geographically weighted standard deviation (for a 2km bandwidth) is given in Figure 4.4, and this highlights areas with mixed population characteristics by community background. For example, the large values in the Belfast region (i.e., the central eastern part of Northern Ireland), indicate neighbouring areas which are dominated by Catholics or Protestants. In contrast, areas with small standard deviations are more homogeneous in terms of the community background of their residents.
The geographically weighted standardised differences of the global and local means (termed the geographically weighted standard score by Fotheringham et al. (129)) provides a means of assessing locally marked deviations from the global mean:
zsi= z¯i− µ σqPn
j=1w2ij (4.5)
where µ is the global mean, and σ is the global standard deviation. As before, the weights are standardised. An example of the geographically weighted standard score computed using the log-ratio data is given in Figure 4.5. The map indicates the high degree of variation in the geographically weighted
78 Local Models for Spatial Analysis
0 10 20 Kilometres GW std. dev.
0.25 - 0.75 0.76 - 1.00 1.01 - 1.50 1.51 - 2.26 Inland water
FIGURE 4.4: Geographically weighted standard deviation for a Gaussian function with a 2km bandwidth: log-ratio of Catholics/Non-Catholics in Northern Ireland in 2001 by 1km grid squares. Northern Ireland Census of Population data — c° Crown Copyright.
mean and the distinction between the west and the east of Northern Ireland is clear.
Following the previous definitions, the geographically weighted coefficient of variation is given by:
CVi= si/¯zi (4.6)
When the mean is close to zero, as is the case with some of the local means (see Figure 4.2), the coefficient of variation is unstable. Therefore, the upper and lower ranges of the geographically weighted coefficient of variation in Figure 4.6 are simply given as being larger (for positive values) or smaller (for negative values) than the specified thresholds. There are several areas in the map with large (positive or negative) values. As one example, there is a group of contrasting values around the central Belfast area, illustrating very different characteristics of the population within parts of Belfast, and between the city and the suburbs.
Spatial Patterning 79
0 10 20 Kilometres GW std. score
-8.85 - -4.00 -3.99 - -0.01 0.00 - 4.00 4.01 - 9.40 Inland water
FIGURE 4.5: Geographically weighted standard score for a Gaussian function with a 2km bandwidth: log-ratio of Catholics/Non-Catholics in Northern Ireland in 2001 by 1km grid squares. Northern Ireland Census of Population data — c° Crown Copyright.
0 10 20 Kilometres GW coeff. var.
< -5.00 -5.00 - -1.01 -1.00 - 1.00 1.01 - 5.00
> 5.00 Inland water
FIGURE 4.6: Geographically weighted coefficient of variation for a Gaussian function with a 2km bandwidth: log-ratio of Catholics/Non-Catholics in Northern Ireland in 2001 by 1km grid squares. Northern Ireland Census of Population data — c° Crown Copyright.
80 Local Models for Spatial Analysis Geographically weighted skewness can be given by (67):
bi= 3 s Pn
j=1(zj− ¯zi)3wij
s3i (4.7)
Figure 4.7 shows the geographically weighted skewness for the log-ratio data. The map suggests that there is marked variation in local skewness.
With any geographically weighted statistic, altering bandwidth size is impor-tant. Exploration of maps generated using alternate bandwidths may aid interpretation of complex patterns such as those shown in Figure 4.7.
Brunsdon et al. (67) and Fotheringham et al. (129) present descriptions and example applications of geographically weighted versions of a range of summary statistics. Harris and Brunsdon (177) present an analysis of freshwater acidification critical load data, and they find evidence for spatial variation in the local mean, variance, coefficient of variation, and skewness.
Brunsdon and Charlton (66) discuss the locally-based analysis of directional data such as wind direction data. Fotheringham et al. (129) utilise a randomisation approach to identify ‘interesting’ locations — that is, those local statistics with values which are deemed significant. Section 4.3.1 discusses a similar approach with respect to spatial autocorrelation. A range of geographically weighted statistics can be computed using the R package spgwr∗.