The multipoint Morisita index for the analysis of spatial patterns

In many fields, the spatial clustering of sampled data points has significant consequences. Therefore, several indices have been proposed to assess the degree of clustering affecting datasets (e.g. the Morisita index, Ripley's KK-function and Renyi's information). The classical Morisita index measures how many times it is more likely to randomly select two sampled points from the same quadrat (the dataset is covered by a regular grid of changing size) than it would be in the case of a random distribution generated from a Poisson process. The multipoint version takes into account mm points with m greater than or equal to 2m greater than or equal to 2. The present research deals with a new development of the multipoint Morisita index (mm-Morisita) which is directly related to multifractality. This relationship to multifractality is first demonstrated and highlighted on a mathematical multifractal set. Then, the new version of the mm-Morisita index is adapted to the characterization of environmental monitoring network clustering. And, finally, an additional extension, the functional mm-Morisita index, is presented for the detection of structures in monitored phenomena.