Detecting spatial aggregation from distance sampling: a probability distribution model of nearest neighbor distance

Meng Gao

doi:10.1007/s11284-013-1029-x

Ecological Research

May 2013, Volume 28, Issue 3, pp 397–405 | Cite as

Detecting spatial aggregation from distance sampling: a probability distribution model of nearest neighbor distance

Authors
Authors and affiliations

Meng GaoEmail author

Original Article

First Online: 16 February 2013

Received: 04 September 2012

Accepted: 08 January 2013

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2 Citations

Abstract

Spatial point pattern is an important tool for describing the spatial distribution of species in ecology. Negative binomial distribution (NBD) is widely used to model spatial aggregation. In this paper, we derive the probability distribution model of event-to-event nearest neighbor distance (distance from a focal individual to its n-th nearest individual). Compared with the probability distribution model of point-to-event nearest neighbor distance (distance from a randomly distributed sampling point to the n-th nearest individual), the new probability distribution model is more flexible. We propose that spatial aggregation can be detected by fitting this probability distribution model to event-to-event nearest neighbor distances. The performance is evaluated using both simulated and empirical spatial point patterns.

Keywords

Spatial point pattern Negative binomial Distance sampling Barro Colorado Island, Panama

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Notes

Acknowledgments

The author is grateful to the Center for Tropical Forest Science for providing the BCI data. Help from Prof. Fangliang in reading the original manuscript and Dr. G.C. Shen in preparing the data are also acknowledged. The authors also wish to thank the editors and two anonymous reviewers for their useful comments and suggestions related to this manuscript. This work was supported by NNSF of China (No.31000197), as well as Knowledge Innovation Project of CAS (No. KZCX2-EW-QN209).

Supplementary material

11284_2013_1029_MOESM1_ESM.docx (153 kb)

Supplementary material 1 (DOCX 152 kb)

Appendix

To derive the probability distribution model of event-to-event NND, we also assume that the spatial distribution pattern satisfies stationarity and isotropy. Then the pdf of n-th NND (event-to-event) can be derived simply by applying Bayes’ rule and finding the limit of a conditional probability. Here, we only take the NBD case as an example to illustrate the derivation. In Fig. 1, there are two concentric circles of radii r and Δr, and the center is a randomly selected individual. First, the number of individuals in a circle of radius r follows a negative binomial distribution with mean λπr ². Define a random variable r _n as the n-th NND (event-to-event), then probability that r _n is smaller than r is given by P(r _n < r), which is also the probability distribution function of r _n (denoted by F _n(r)). Now, it is in position to derive the analytical expression of P(r _n < r). We have

$$ \begin{aligned} F_{n} (r) & = P(r_{n} < r) \\ & = P({\text{more than }}n + 1{\text{ individuals in the circle}}, {\text{one at the center}}) \\ & = \frac{{P({\text{more than }}n + 1{\text{ individuals in the circle}},{\text{one\,at\,the\,center}})}}{{P({\text{one at the center}})}} \\ & = \mathop {\lim }\limits_{\Updelta r \to 0} \frac{{P(n{\text{ individuals in the annullus}}, {\text{one in the inner circle}})}}{{P({\text{one in the inner circle}})}} \\ \end{aligned} $$

(A1)

The numerator in Eq. (A1) can be approximated by

$$ \mathop \sum \limits_{s = n + 1}^{\infty } \left( {\begin{array}{*{20}c} s \\ 1 \\ \end{array} } \right)q(1 - q)^{s - 1} \frac{\Upgamma (k + s)}{s!\Upgamma (k)}\left( {1 + \frac{k}{{\lambda \pi r^{2} }}} \right)^{ - s} \left( {1 + \frac{{\lambda \pi r^{2} }}{k}} \right)^{ - k} $$

(A2)

where q = ∆r ²/r ². The formula after the summation in Eq. (A2) represents the probability that only one individual is located at the inner circle while other s − 1 individuals are at the annulus, given there are s individuals falling in the circle. The basic assumption here is that individuals are randomly distributed in the outer circle and follow a binomial distribution. The denominator is relatively simple, which is just the probability of an individual falling in the inner circle,

$$ p(x = 1) = \lambda \pi \Updelta r^{2} \left( {1 + \frac{{\lambda \pi \Updelta r^{2} }}{k}} \right)^{ - k - 1} $$

(A3)

Substituting Eqs. (A2–A3) into Eq. (A1) and evaluating the limit, we can obtain the probability distribution function:

$$ \begin{aligned} F_{n} (r) & = \frac{2\Upgamma (n + k + 1)}{rk\Upgamma (k + 1)\Upgamma (n + 2)}\left( {1 + \frac{k}{{\lambda \pi r^{2} }}} \right)^{ - n} \left( {1 + \frac{{\lambda \pi r^{2} }}{k}} \right)^{ - k - 1} \\ & \quad \times \left[ {\lambda \pi r^{2} k\left( {k + n + 1} \right)F_{1}^{2} \left( {2,2 + k + n,2 + n,\frac{{\lambda \pi r^{2} }}{{k + \lambda \pi r^{2} }}} \right)} \right. \\ \quad - (n + 1)(k + \lambda \pi r^{2} )(\lambda \pi r^{2} \left( {k + 1} \right) - nk)F_{1}^{2} \left( {1,1 + k + n,1 + n,\frac{{\lambda \pi r^{2} }}{{k + \lambda \pi r^{2} }}} \right) \\ \end{aligned} $$

(A4)

where $ F_{1}^{2} $ is Gauss hypergeometric function. The expression of F _n(r) is very complex, but it differential about distance r is relative simple, which is exactly the probability density function of the n-th NND (event-to-event),

$$ f_{n} (r) = \frac{{2(\lambda \pi )^{n} r^{2n - 1} }}{{k^{n} }}\frac{\Upgamma (n + k + 1)}{\Upgamma (n)\Upgamma (k + 1)}\left( {1 + \frac{{\lambda \pi r^{2} }}{k}} \right)^{ - n - k - 1} $$

(A5)

It can be easily verified that Eq. (A5) meets all the conditions of general probability density functions. Following the same procedure, the probability density function of n-th NND (event-to-event) for the Poisson case can also be derived. The result is exactly p _n(r) shown in Eq. (3).

With the analytical expressions of p _n(r), g _n(r), and f _n(r), the moments can be simply derived. For Poisson case, Thompson (1956) presented that the expectation of n-th NND, either point-to-event or event-to-event, was

$$ E\left[ {r_{n} } \right] = \frac{1}{\lambda }\frac{n(2n)!}{{(2^{n} n!)}} \approx \frac{1}{{\sqrt {\lambda \pi } }}n^{0.5} $$

(A6)

The approximation is based on Sterling’s formula, which holds very well even for small n. For negative binomial case, the expectations of point-to-event NND and event-to-event NND are

$$ {\text{Point-to-event}}{:}\,E\left[ {r_{n} } \right] = \frac{1}{{\sqrt {\lambda \pi } }}\frac{\Upgamma (n + 0.5)}{\Upgamma (n)}\frac{\Upgamma (k - 0.5)\sqrt k }{\Upgamma (k)}\quad (k\;>\;0.5), $$

(A7)

$$ {\text{Event-to-event}}{:}\,E[r_{n} ] = \frac{1}{{\sqrt {\lambda \pi } }}\frac{\Upgamma (n + 0.5)}{\Upgamma (n)}\frac{\Upgamma (k + 0.5)\sqrt k }{\Upgamma (k + 1)} $$

(A8)

Other higher-order of moments can also be obtained, and we do not show all of them here.

References

Bailey TC, Gatrell AC (1995) Interactive spatial data analysis. Longman Scientific and Technical, Harlow, UKGoogle Scholar
Bliss CI, Fisher RA (1953) Fitting the negative binomial distribution to biological data. Biometrics 9:176–200CrossRefGoogle Scholar
Boswell MT, Patil GP (1970) Chance mechanisms generating the negative binomial distribution. In: Patil GP (ed) Random counts in scientific work. Pennsylvania State, University Press, University ParkGoogle Scholar
Clark PJ, Evans FC (1954) Distance to nearest neighbor as a measure of spatial relationships in populations. Ecology 35:445–453CrossRefGoogle Scholar
Condit R et al (2000) Spatial patterns in the distribution of tropical tree species. Science 288:1414–1418PubMedCrossRefGoogle Scholar
Cox DR, Lewis PAW (1966) The statistical analysis of series of events. Methuen, London, UKGoogle Scholar
Diggle PJ (2003) Statistical analysis of spatial point patterns. Academic Press, LondonGoogle Scholar
Eberhardt LL (1967) Some developments in distance sampling. Biometrics 23:207–216PubMedCrossRefGoogle Scholar
Green RH (1966) Measurement of non-randomness in spatial distributions. Res Popul Ecol 8:1–7CrossRefGoogle Scholar
Grevstad N (2010) Mapping spatial aggregation from counts data: a penalized likelihood approach. Environmetrics 21:834–848CrossRefGoogle Scholar
He F, Gaston KJ (2000) Estimating species abundance from occurrence. Am Nat 156:553–559CrossRefGoogle Scholar
Hurlbert SH (1997) Spatial distribution of the montane unicorn. Oikos 58:257–271CrossRefGoogle Scholar
John R et al (2007) Soil nutrients influence spatial distributions of tropical tree species. Proc Natl Acad Sci 104:864–869PubMedCrossRefGoogle Scholar
Legendre P, Fortin MJ (1989) Spatial pattern and ecological analysis. Vegetatio 80:107–138CrossRefGoogle Scholar
Magnussen S, Kleinn C, Picard N (2008) Two new density estimator for distance sampling. Eur J For Res 127:213–224CrossRefGoogle Scholar
Pielou EC (1960) A single mechanism to account for regular, random and aggregated populations. J Ecol 48:575–584CrossRefGoogle Scholar
Pielou EC (1961) Segregation and symmetry in two-species population as studied by nearest-neighbor relationships. J Ecol 49:255–269CrossRefGoogle Scholar
Prayag VR, Deshmukh SR (2000) Testing randomness of spatial pattern using Eberhardt’s index. Environmetrics 11:571–582CrossRefGoogle Scholar
Ripley BD (1976) The second-order analysis of stationary point processes. J Appl Probab 13:255–266CrossRefGoogle Scholar
Ripley BD (1977) Modelling spatial patterns (with discussion). J R Statist Soc Ser B 39:172–212Google Scholar
Ripley BD (1988) Statistical inference for spatial processes. Cambridge University Press, Cambridge, UKCrossRefGoogle Scholar
Stoyan D, Penttinen A (2000) Recent applications of point process methods in forestry statistics. Stat Sci 15:61–78CrossRefGoogle Scholar
Stoyan D, Stoyan H (1996) Estimating pair correlation functions of planar cluster processes. Biom J 38:259–271CrossRefGoogle Scholar
Thompson HR (1956) Distribution of distance to n-th nearest neighbor in a population of randomly distributed individuals. Ecology 37:391–394CrossRefGoogle Scholar
Wiegand T, Moloney KA (2004) Rings, circles, and null-models for point pattern analysis in ecology. Oikos 104:209–229CrossRefGoogle Scholar
Zillio T, He F (2010) Modeling spatial aggregation of finite populations. Ecology 91(12):3698–3706PubMedCrossRefGoogle Scholar

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Meng Gao
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1.Key Laboratory of Coastal Zone Environmental Processes, Yantai Institute of Coastal Zone ResearchChinese Academy of SciencesYantaiChina

Detecting spatial aggregation from distance sampling: a probability distribution model of nearest neighbor distance

Abstract

Keywords

Notes

Acknowledgments

Supplementary material

References

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