Application of Change-Point Problem to the Detection of Plant Patches
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Application of Change-Point Problem to the Detection of Plant Patches I. Lo´pez • M. Ga´mez • J. Garay • T. Standova´r Z. Varga
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Received: 24 February 2009 / Accepted: 6 October 2009 / Published online: 6 November 2009 Springer Science+Business Media B.V. 2009
Abstract In ecology, if the considered area or space is large, the spatial distribution of individuals of a given plant species is never homogeneous; plants form different patches. The homogeneity change in space or in time (in particular, the related change-point problem) is an important research subject in mathematical statistics. In the paper, for a given data system along a straight line, two areas are considered, where the data of each area come from different discrete distributions, with unknown parameters. In the paper a method is presented for the estimation of the distribution change-point between both areas and an estimate is given for the distributions separated by the obtained change-point. The solution of this problem will be based on the maximum likelihood method. Furthermore, based on an adaptation of the well-known bootstrap resampling, a method for the estimation of I. Lo´pez (&) M. Ga´mez Department of Statistics and Applied Mathematics, University of Almerı´a, La Can˜ada de San Urbano, s/n, 04120 Almerı´a, Spain e-mail: [email protected] M. Ga´mez e-mail: [email protected] J. Garay Research Group of Theoretical Biology and Ecology of Hungarian Academy of Sciences, Budapest, Hungary e-mail: [email protected] J. Garay T. Standova´r Department of Plant Taxonomy and Ecology, Institute of Biology, L. Eo¨tvo¨s University, Pazmany Peter Setany 1/c., 1117 Budapest, Hungary e-mail: [email protected] Z. Varga Institute of Mathematics and Informatics, Szent Istva´n University, Pa´ter K. u. 1., 2103 Go¨do¨ll}o, Hungary e-mail: [email protected]
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the so-called change-interval is also given. The latter approach is very general, since it not only applies in the case of the maximum-likelihood estimation of the changepoint, but it can be also used starting from any other change-point estimation known in the ecological literature. The proposed model is validated against typical ecological situations, providing at the same time a verification of the applied algorithms. Keywords Bootstrap Change-interval Change-point Ecological boundary Edge detection Plant patches
1 Introduction In most field studies ecologists encounter some sort of patchiness in the vegetation. This patchiness can be observed at several spatial scales, starting from fine scale patterns caused by the growth characteristics and clonal behaviour of individual species, to global patterns in the distribution of biomes along climatic gradients. Since the works of Watt (1947), studying vegetation patterns has had an utmost importance in ecology, as the quantitative description of the spatial and temporal aspects of these patterns is the key to reveal underlying ecological processes. Moreover, the patterns of plants have profound effects on int
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