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<x-charset iso-8859-1>Bonjour chers amis,

Leonardo A. Saravia nous pose un probleme intéressant concernant les points

doubles dans l'analyse de la structure spatiale :

*>
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*>Some time ago you sent me the C source for the computation of Ripley's K
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*>functions, I discovered a small problem, your code do not check for
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*>duplicated points. Duplicated points occur when somebody makes a mistake
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*>entering field data and when you generate confidence intervals. In the
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first

*>case spurious significant values at the first distance interval would be
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*>found and in the second one the confidence band will be wider for the
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*>smaller distances. I confirmed this problem with ADE-4 too.
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*>
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*>best regards,
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*>
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*>Leonardo
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*>
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*>Leonardo A. Saravia - Programa de Invest. en Ecologia Matematica
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*>Universidad Nacional De Lujan - C.C. 221 - (6700) - Lujan - Argentina
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*>Te: 54 2323 423171/421030 Fax: 1 (801) 409-3991
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*>e-mail:lsaravia@mail.retina.ar
*

Voici quelques elements de reponse, en anglais. Le debat reste bien sur

ouvert.

1) As mentioned by Leonardo, duplicated points (i.e. two points having the

same co-ordinates in a data set) can occur when somebody makes a mistake

entering field data, and the automatic detection of duplicated points could

help correcting the data set.

BUT duplicated points can also occur because two (or more) individuals are

very close to each other, at a smaller distance than the precision of the

co-ordinates measurement. It may be a very rare event when trees are

concerned (but I have seen some such cases with coppice), but not that rare

with smaller individuals.

IN THIS CASE, there is no reason why we should not compute the K(r) function

with such duplicated points, or delete some of this points. The obtained

values DO characterise the particular structure of the pattern (for instance

low distance aggregation).

2) Moreover, the K(r) function or other similar function are defined for all

(homogeneous and isotropic) point processes, with or without duplicated

points, and no problem occur either in the definition or in the computation

of the functions in case of duplicated points. There are no theoretical

reason why we should not compute the K(r) function in this case.

EVENTUALLY, we decided with Raphael Pelissier not to change the computation,

so that it is still possible to compute K(r) with duplicated points, but to

inform the user of the eventual presence of such points. He will then be

free to correct or not his data set.

3) As far as the computation of the confidence interval is concerned, the

problem is similar : there is no reason why we should delete duplicated

points. In a real Poisson pattern duplicated points can occur. Moreover,

because we truncate the co-ordinates to a fixed precision in our Poisson

pattern simulation, virtual points that are very close to each other (but

still different) will have the same co-ordinates. As we want to simulate a

confidence interval for the Poisson hypothesis, we must keep this

possibility. In fact, if you delete duplicated points in the Monte Carlo

simulation, you create a bias and thus obtain a confidence interval that

does no more correspond to a Poisson pattern, but to a hard-core Poisson

pattern. That's why the result is different.

THEREFORE, we do not change the computation of the confidence interval.

bien amicalement,

-------------------------------------------------------

François Goreaud

CEMAGREF, LISC

24 avenue des Landais - BP 50085

63172 AUBIERE CEDEX 1 - France

Tel. : 33 (0)4.73.44.06.80

Fax : 33 (0)4.73.44.06.96

email : francois.goreaud@clermont.cemagref.fr

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