Metoda Monte Carlo w badaniu istotności wyników funkcji Ripleya, czyli jak się ustrzec fałszywego stwierdzenia nielosowości struktury przestrzennej drzewostanu

The use of Monte Carlo method in significance tests of Ripley's function outcome or how to avoid false discovery of nonrandom spatial structure of tree stand


  • 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:


Hypothesis that investigated pattern of tree distribution described by estimator of Ripley’s K(t) is not random is often tested by means of Monte Carlo method. The method involves generation of rather big number of random tree stands with stand area and number of trees identical as in investigated tree stand. For each random stand estimator of Ripley’s function is calculated. The main goal of this procedure is to define extent of estimator variability in the case of random placement of trees in investigated stand. For each spatial scale t the lowest and the highest values of estimator are recorded. Using extreme values of estimator one can draw two lines (lower and upper) determining maximum estimator variability across spatial scales. They are called envelops. Unfortunately sometimes these lines are interpreted as „confidence bands” which is obvious mistake. The case that estimator calculated for investigated tree stand crosses the upper or lower envelop is wrongly interpreted as a proof for non-randomness of investigated pattern. This assumption may be partially justified when only one previously determined spatial scale (eg. 4 m) is considered. In case that many spatial scales are investigated simultaneously (eg. from 0 to 10 m) this assumption can lead very easily to false discovery of non-randomnes of investigated pattern. The interpretation of investigated pattern based only on visual comparison of estimator with envelopes can be used only in explanatory analysis. Instead the formal rank test based on carefully selected statistic should be carried out.

Print ISSN
Online ISSN
Type of article
Original title
Translate »