Bivariate Ripley's K

From: Martin Béland (
Date: Wed Apr 05 2000 - 22:43:28 MET DST

<x-charset iso-8859-1>Dear all,

I found an answer to my question (pasted below). Andersen(1992) uses an
estimator of K1,2(d) that combines the estimators K1,2 and K2,1 into a
single estimator. This estimator, mentionaed by Lotwick and Silverman
(1983), is the linear combination:

(n2K12(d) + n1K21(d))/(n1 + n2)

where n1 and n2 are the number of type 1 and type 2 events.

I hope this can help others with the same problem.

Martin Béland wrote:
> Dear netters,
> I recently posted a summary of questions and responses about analyses of
> point patterns to study competition in mixed jack pine stands. Here is
> one more question that arose from running the software called
> "Potempkin" to compute intertype Ripley's K(d) analysis :
> I thought that bivariate analysis of interaction between species 1 and 2
> should be the same as that of interaction between species 2 and 1. The
> output given by potemkin for the bivariate analysis includes K1,1 K1,2
> K2,1 and K2,2. The values for K1,2 and K2,1 for short values of d are
> the same but as d increases, K2,1 becomes larger that K1,2. The
> confidance intervals are different from the beginning. How do you
> explain this?
> To this, John Brzustowski, the author of the programm replied to me
> > Good question. I don't seem to have a paper describing the bivariate
> > K here, but what I think is happening is this:
> > The obvious definition for bivariate K(t) would be the proportion of
> > pairs of individuals, the first of species 1, the second of species 2,
> > that lie within a distance t or less of each other. That would give a
> > symmetric definition. But I think Ripley does this a bit differently:
> > K1,2 (t) is the average, over all individuals in species 1, of the
> > proportion of species 2 neighbours that are within distance t of the
> > individual in species 1. This is a bit tricky, but it just amounts to
> > weighting the pairs differently in each case. I suppose an advantage
> > of this approach is it should allow one to detect attraction of
> > species 1 by species 2, as opposed to the other way around. Maybe.
> >
> Does any of you know:
> 1- if other software compute the bivariate K(t) in a different way that
> use a symmetric definition of the interaction? and
> 2- if there is no other way to compute the bivariate K(t), how do you
> suggest the result must be interpreted, that is, which of K1,2 or K2,1
> must be used in which situation?
> Any help would be appreciated,
Martin Béland, biologiste, Ph.D. Env.
Unité de recherche et de développement forestiers de
Université du Québec en Abitibi-Témiscamingue
445, boulevard Université
Rouyn-Noranda (Québec) J9X 5E4
Téléphone : (819) 762-0971 #2458
Fax : (819) 797-4727
Courriel :


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