# Classic exploratory tools and summary statistics for spatial point patterns

Traditional techniques for performing preliminary analysis of spatial point patterns consist essentially of graphical exploratory tools and formal tests for the hypothesis of complete spatial randomness. They can be subdivided into two general classes of methods: (i) quadrat-based methods and (ii) distance-based methods.

Strong differences among quadrat intensities provide evidence against the CSR hypothesis and hence against the assumption that the data generating process is a homogeneous Poisson point process. A Pearson’s chi-squared test based on quadrat counts can be used to assess if the observed differences in intensity are strong enough to reject the (null) CSR hypothesis. Let us consider the partition of A into m quadrats of equal area a, and the let щ,П2,...,п;,...,пт be the observed quadrat counts. If the CSR hypothesis is true then each «, is a realization of an independent Poisson random variable with mean Xa, where A is the (unknown) first-order intensity corresponding to the expected number of points per unitary area. Considering that the natural estimate of A is A = n / (та), where n is the total number of points observed in A and та gives the total area

Figure 6.15 Quadrat-based estimation of intensity for a hypothetical pattern of business units: (a) point pattern; (b) quadrat average intensities (business units per square kilometer); (c) estimated intensity function.

of A, the expected count for quadrat n, under CSR is ej = Xa = n / m. Therefore, the proper chi-square statistic is

Provided that n / m is greater than 5, if the CSR hypothesis is true then yfi follows approximately a Xm-l distribution. Consequently, significantly great or small values of yfi indicate that the observed point pattern in A tends to be, respectively, more aggregated or more regular than a CSR pattern.

For the point pattern of business units depicted in Figure 6.15, yfi =23.97 with a two-sided /(-value = 0.927 which imply that the CSR hypothesis cannot be rejected, and hence that the observed differences amongst the quadrat average intensities are likely due to chance.

The results of quadrat-based methods depend strongly on the size of the quadrats and hence on the partition of A. Unfortunately, the choice of the partitioning scheme is usually arbitrary and an optimal criterion to guide this choice is not available. In light of this, Grieg-Smith (1952) proposed an approach to verifying the robustness of results with respect to the size of quadrats. The approach is based on the use of different alternative partitions of A characterized by differing quadrats’ size. In particular, Greig-Smith (1952) suggests starting with a given grid of quadrats, such as a 32-by-32 one, and then obtain a series of other less granular grids by progressively aggregating the adjacent quadrats into 2-by-2, 4-by-4, 8-by-8 and so on, blocks. For each grid, it is convenient to compute the “index of dispersion” of the quadrat counts, which corresponds to the ratio between the sample variance and the sample mean of quadrat counts. If we let n = n / m indicating the sample mean of quadrat counts, the sample variance can be computed as:

and hence the index of dispersion is given by:

As already discussed, under CSR the quadrat counts are independent realizations of the same Poisson random variable. Since the mean and variance of a Poisson random variable are the same, the index of dispersion of a CSR pattern should be approximately equal to 1. Therefore, plotting the values of the index of dispersion for the different grids against the corresponding block size allows us to assess how the results of the quadrat-based methods are affected by the spatial scale, that is by the way the study area is partitioned. Figure 6.16 shows a plot of the index of dispersion versus block size (k-by-k) for the point pattern of

Figure 6.16 Behavior of the index of dispersion with respect to the block size (k-by-k) for the artificial data depicted in Figure 6.15.

business units depicted in Figure 6.15. The values of the index observed for each block size fluctuate around 1 thus providing evidence that the observed pattern is consistent with the CSR hypothesis at any spatial scale.