NOTEBOOK FOR SPATIAL DATA ANALYSIS Part I. Spatial Point Pattern Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 I.5-1 Tony E. Smith 5. Comparative Analyses of Point Patterns Up to this point, our analysis of point patterns has focused on single point patterns, such as the locations of redwood seedlings or lung cancer cases. But often the relevant questions of interest involve relationships between more than one pattern. For example if one considers a forest in which redwoods are found, there will invariably be other species competing with redwoods for nourishment and sunlight. Hence this competition between species may be of primary interest. In the case of lung cancers, recall from Section 1.2 that the lung cancer data for Lancashire was primarily of interest as a reference population for studying the smaller pattern of larynx cancers. We shall return to this example in Section 5.8 below. But for the moment we start with a simple forest example involving two species. 5.1 Forest Example The 600 foot square section of forest shown in Figure 5.1 below contains only two types of trees. The large dots represent the locations of oak trees, and the small dots represent locations of maple trees. Although this is a fairly small section of forest, it seems clear that the pattern of oaks is much more clustered than that of maples. This is not surprising, given the very different seed-dispersal patterns of these two types of trees. As shown in Figure 5.2, oaks produce largest acorns that fall directly from the tree, and are only partially dispersed by squirrels. Maples on the other hand produce seeds with individual “wings” that can transport each seed a considerable distance with even the slightest breeze. Hence there are clear biological reasons why the distribution of oaks might be more clustered than that of maples. So how might we test this hypothesis statistically? 0 ---100 200 feet Figure 5.1. Section of Forest Figure 5.2. Patterns of Seed Dispersal OAK MAPLE !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!NOTEBOOK FOR SPATIAL DATA ANALYSIS Part I. Spatial Point Pattern Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 I.5-2 Tony E. Smith 5.2 Cross K-Functions As one approach to this question, observe that if oaks tend to occur in clusters, then one should expect to find that the neighbors of oak trees tend to be other oaks, rather than maples. Alternatively put, one should expect to find fewer maples near oak locations than other locations. While one could in principle test these ideas in terms of nearest neighbor statistics, we have already seen in the Bodmin tors example that this does not allow any analysis of relationships between point patterns at different scales. Hence a more flexible approach is to extend the above K-function analysis for single populations to a similar method for comparing two populations.1 The idea is simple. Rather than looking at the expected number of oak trees within distance h of a given oak, we look at the expected number of maple trees within distance h of the oak. More generally, if we now consider two point populations, 1 and 2, with respective intensities, 1 and 2, and denote the members of these two populations by i and j, respectively, then the cross K-function, 12()Kh, for population 1 with respect to population 2 is given for each distance h by the following extension of expression (4.2.1) above: (5.2.1) 1221( ) (number of - within distance of an arbitrary - )K h E j events h i event Notice that there is an asymmetry in this definition, and that in general, 12 21() ()Kh Kh . Notice also that the word “additional” in (4.2.1) is no longer meaningful, since populations 1 and 2 are assumed to be distinct. This definition can be formalized in a manner paralleling the single population case as follows. First for any realized point patterns, 11( : 1,.., )iSsi n and 22( : 1,.., )iSsi n , from populations 1 and 2 in region R, let (, )ij i jddss denote the distance between member i of population 1 and j of population 2 in R. Then for each distance h the indicator function (5.2.2) 1,() [(,)]0,ijhij h ijijdhId Idssdh now indicates whether or member j of population 2 is within distance h of a given member i of population 1. In terms of this indicator, the cross K-function in (5.2.1) can be formalized [in a manner paralleling (4.3.3)] as (5.2.3) 212121() ( )nhijjKh E Id 1 Note that while our present focus is on two populations, analyses of more than two populations are usually formulated either as (i) pairwise comparisons between these populations (as with correlation analyses), or (ii) comparisons between each population and the aggregate of all other populations. Hence the two-population case is the natural paradigm for both these approaches.NOTEBOOK FOR SPATIAL DATA ANALYSIS Part I. Spatial Point Pattern Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 I.5-3 Tony E. Smith where both the size, 2n , of population 2 and the distances 2( : 1,.., )ijdj n are here regarded as random variables.2 This function plays a fundamental role in our subsequent comparative analyses of populations. 5.3 Estimation of Cross K-Functions Given the definition in (5.2.3) it is immediately apparent that cross K-functions can be estimated in precisely the same way as K-functions. First, since the expectation in (5.2.3) does not depend on which random reference point i is selected from population 1, the same argument as in (4.3.4) now shows that for any given size, 1n , of population 1, (5.3.1) 2212 11( ) ( ) , 1,..,nhijjEId Khin 1212