An Introduction to R for Spatial Analysis and Mapping by Brunsdon Chris & Comber Lex

Author:Brunsdon, Chris & Comber, Lex [Brunsdon, Chris]
Language: eng
Format: epub
ISBN: 9781473911192
Publisher: SAGE Publications
Published: 2015-01-21T22:00:00+00:00

where is an estimate of the intensity – given by

with |A| being the area of a study region defined by a polygon A. Also I(.) is an indicator function taking the value 1 if the logical expression in the brackets is true, and 0 otherwise. To consider whether this sample comes from a clustered or dispersed process, it is helpful to compare to KCSR(d).

Statistical inference is important here. Even if the dataset had been generated by a CSR process, an estimate of the K-function would be subject to sampling variation, and could not be expected to match KCSR(d) perfectly. Thus, it is necessary to test whether the sampled is sufficiently unusual with respect to the distribution of estimates one might expect to see under CSR to provide evidence that the generating process for the sample is not CSR. The idea is illustrated in Figure 6.9. Here, 100 K-function estimates (based on equation (6.5)) from random CSR samples of 100 points (the same number of points as in Figure 6.8) are are superimposed, together with the estimate from the point set shown in Figure 6.8. From this it can be seen that the estimate from the clustered sample is quite different from the range of estimates expected from CSR.

Another aspect of sampling inference for K-functions is the dependency of on the shape of the study area. The theoretical form KCSR(d) = λπd2 is based on assumption of points occurring in an infinite two-dimensional plane. The fact that a ‘real-world’ sample will be taken from a finite study area (denoted here by A) will lead to further deviation of sample-based estimates of from the theoretical form. This can also be seen in Figure 6.9 – although for the lower values of d the CSR estimated K-function curves resemble the quadratic shape expected, the curves ‘flatten out’ for higher values of d. This is due to the fact that for larger values of d, points will only be observed in the intersection of a circle of radius d around a random xi and the study area A. This will result in fewer points being observed than the theoretical K-function would predict. This effect continues, and when d is sufficiently large any circle centred on one of the points will encompass the entirety of A. At this point, any further increase in d will result in no change in the number of points contained in the circle – this provides an explanation of the flattening-out effect seen in the figure.

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