Dealing with virtual aggregation -: a new index for analysing heterogeneous point patterns

Point pattern analyses such as the estimation of Ripley's K-function or the pair-correlation function g are commonly used in ecology to characterise ecological patterns in space. However, a major disadvantage of these methods is their missing ability to deal with spatial heterogeneity. A heterogeneous intensity of points causes a systematic bias in estimates of the K- and g-functions, a phenomenon termed virtual aggregation in the recent literature. To address this problem, we derive a new index, called K2-index, as an extension of existing point pattern characteristics. The K2-index has a heuristic interpretation as an approximation to the first derivative of the g-function. We estimate the K-, g- and K2-functions for six different types of simulated point patterns and show that the K2-index may provide important information on point patterns that the other methods fail to detect. The results indicate that particularly the small-scale distributions of points are better represented by the K2-index. This might be important for testing hypotheses on ecological processes, because most of these processes, such as direct neighbour interactions, occur very locally. When applied to empirical patterns of molehill distribution, the results of the K2-analysis show regularity up to distances between 0.1 and 0.4 m in most of the study areas, and aggregation of molehills up to distances between 0.2 and 1.1 m. The type and scale of these deviations from randomness agree with a priori expectations on the hill-building behaviour of moles. In contrast, the estimated g-functions merely indicate aggregation at the full range from 0 to 7 m (or even above). Considering the advantages and disadvantages of the different methods, we suggest that the K2-index should be used as a complement to existing approaches, particularly for point patterns generated by processes that act on more than one scale.