Application of Change-Point Problem to the Detection of Plant Patches

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 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 change-point, 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.

Appendix

1.1 Algorithm 1 (Estimation of the Change-Point K)

• 1. Introduce sample X. N := Size (X).

• 2. FOR K = 1 until N − 1:

  • (a) Calculate: $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{p}_{K,i}$ and $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{q}_{K,i} \;,$ for each i.

  • (b) Calculate: $Log\;P_{K} = \sum\limits_{i = 1}^{K} {Log\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{p}_{{K,x_{i} }} } + \sum\limits_{s = K + 1}^{N} {Log\;\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{q}_{{K,x_{s} }} } .$ (Logarithm is introduced to avoid too small probability values.)

• 3. LogProbSample: = (Log P ₁,…, Log P _N − 1).

• 4. EstimateK := Position K with maximum value among the coordinates of LogProbSample.

• 5. Return Estimate K.

1.2 Algorithm 2 (Estimation of the Change-Interval for K)

• 1. Introduce the sample X := (x ₁, x ₂,…, x _N ). N := Size (X).

• 2. FOR K = 1 until N − 1

  • (a) Calculate a weight for each K:

    $$ W_{K} = \left( {\prod\limits_{i = 1}^{K} {10\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{p}_{{K,x_{i} }} } } \right)\left( {\prod\limits_{s = K + 1}^{N} {10\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{q}_{{K,x_{s} }} } } \right) $$

  • (b) Normalize the weights (and denote them by WN _K ).

  • (c) FOR L = 1 until m (we generate m samples for each K to estimate its change-point):

    • (c1) We draw random samples with replacement from both “homogenous” zones of the original sample separately (using “sample” function of the program “R”):

    • (c2) We apply Algorithm 1 to the sample X _L , to obtain an estimate K _L for the change-point.

  • (d) From the obtained values K ₁,…, K _m , (m large enough) we calculate a distribution d _k of the change-point, for each fixed K.

• 3. We combine all these new distributions to obtain a unique distribution $\sum\limits_{K} {WN_{K} d_{K} }$ for K, for which we calculate the 90%-level change-interval, with percentile 5 and percentile 95 as extremes.