Stosowanie funkcji Ripleya do badania anizotropicznych rozmieszczeń drzew

The use of Ripley's function for investigations of anisotropic spatial patterns of trees


  • Leszek Bolibok Szkoła Główna Gospodarstwa Wiejskiego w Warszawie, Wydział Leśny, Katedra Hodowli Lasu, ul. Nowoursynowska 159/34, 02-776 Warszawa;
    Tel. +48 22 5938106, e-mail:


One of the most frequently used methods of spatial analysis – Ripley’s K(t) function has limitations which constrain the usefulness of this method. The classical estimator of Ripley’s K(t) function can only be used in case when examined point pattern formed by trees is homogenous and isotropic. In the paper, the idea of spatial stochastic point process was described in rather illustrative manner. Special attention was paid to anisotropy as a feature of the point process. Detection of anisotropy in the horizontal distribution of trees can direct further investigations of agents shaping tree stand structure. Classical estimator of Ripley’s K(t) function is not able to provide any information about the anisotropy of the spatial pattern. The review of the current methods of anisotropy detection in the point pattern is presented with special attention for directional estimators of Ripley’s K(t) function. Unfortunately, suitable software for such analyses is hardly available (e.g. SPPA). There are graphical methods implemented in noncommercial statistical package R which can help the investigator to examine the point pattern. The library ecespa contains procedure getis which can calculate local values Li(t) of Ripley’s function in the way described by Getis and Franklin. The map of Li(t) values can show the shape of potential point clusters and encourage a researcher to make formal test for anisotropy. The library spatstat contains the procedure Kmeasure which can calculate a reduced second measure of the point process. The graphical output of this procedure shows the probability of finding another tree in the neighborhood of a ‘typical tree’ in the investigated point pattern and can show any directional difference in such probability.

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