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ksfun: Multiscale second-order neigbourhood analysis of a multivariate spatial point pattern using Simpson diversity

Description

Computes estimates of Shimatani alpha and beta functions of Simpson diversity from a multivariate spatial point pattern in a simple (rectangular or circular) or complex sampling window. Computes optionally local confidence limits of the functions under the null hypothesis of a random allocation of species labels (see Details).

Usage

ksfun(p, upto, by, nsim=0, alpha=0.01)

Arguments

p

a "spp" object defining a spatial point pattern in a given sampling window (see spp).

upto

maximum radius of the sample circles (see Details).

by

interval length between successive sample circles radii (see Details).

nsim

number of Monte Carlo simulations to estimate local confidence limits of the null hypothesis of a random allocation of species labels (see Details). By default nsim=0, so that no confidence limits are computed.

alpha

if nsim>0, significant level of the confidence limits. By default $\alpha=0.01$.

Value

• A list of class "fads" with essentially the following components:
• r
  a vector of regularly spaced out distances (seq(by,upto,by)).
• gs
  a data frame containing values of the function $gs(r)$.
• ks
  a data frame containing values of the function $Ks(r)$.
• Each component except r is a data frame with the following variables:
• obs
  a vector of estimated values for the observed point pattern.
• theo
  a vector of theoretical values expected under the null hypothesis of random labelling, i.e. 1 for all $r$.
• sup
  (optional) if nsim>0 a vector of the upper local confidence limits of a random distribution of species labels at a significant level $\alpha$.
• inf
  (optional) if nsim>0 a vector of the lower local confidence limits of a Prandom distribution of species labels at a significant level $\alpha$.
• pval
  (optional) if nsim>0 a vector of local p-values of departure from a random distribution of species labels.

encoding

latin1

Details

Function ksfun computes Shimatani $\alpha(r)$ and $\beta(r)$ functions of Simpson diversity, called here $Ks(r)$ and $gs(r)$, respectively. For a multivariate point pattern consisting of $S$ species with intensity $\lambda$p, Shimatani (2001) showed that a distance-dependent measure of Simpson (1949) diversity can be estimated from Ripley (1977) $K$-function computed for each species separately and for all the points grouped toghether (see also Eckel et al. 2008). Function ksfun is thus a simple wrapper function of kfun, standardized by Simpson diversity coefficient: $Ks(r) = 1 - sum(\lambda p * \lambda p * Kp(r)) / (\lambda * \lambda * K(r) * D)$ which is a standardized estimator of $\alpha(r)$ in Shimatani (2001). $gs(r) = 1 - sum(\lambda p * \lambda p * gp(r)) / (\lambda * \lambda * g(r) * D)$ is the derivative of $Ks(r)$ corresponding to a standardized version of $\beta(r)$ in Shimatani (2001). $Kp(r)$ and $K(r)$ (resp. $gp(r)$ and $g(r)$) are univariate K-functions computed for species $p$ and for all species toghether; $D = 1 - sum(Np * (Np - 1) / (N*(N - 1)))$ is the unbiased version of Simpson diversity, with $Np$ the number of individuals of species $p$ in the sample and $N = sum(Np)$.

The program introduces an edge effect correction term according to the method proposed by Ripley (1977) and extended to circular and complex sampling windows by Goreaud & P�lissier (1999). The theoretical values of $gr(r)$ and $Kr(r)$ under the null hypothesis of random labelling is 1 for all $r$. Local Monte Carlo confidence limits and p-values of departure from this hypothesis are estimated at each distance $r$ by reallocating at random the species labels among points of the pattern, keeping the point locations unchanged.

References

Shimatani K. 2001. Multivariate point processes and spatial variation in species diversity. Forest Ecology and Managaement, 142:215-229.

Eckel, S., Fleisher, F., Grabarnik, P. and Schmidt V. 2008. An investigation of the spatial correlations for relative purchasing power in Baden-W�rttemberg. AstA - Advances in Statistical Analysis, 92:135-152. Simpson, E.H. 1949. Measurement of diversity. Nature, 688:163. Goreaud F. & P�lissier R. 1999. On explicit formulas of edge effect correction for Ripley's K-function. Journal of Vegetation Science, 10:433-438.

Ripley B.D. 1977. Modelling spatial patterns. Journal of the Royal Statistical Society B, 39:172-192.

See Also

plot.fads, spp, kfun, kpqfun, kp.fun, krfun.

Examples

data(Paracou15)
  P15<-Paracou15
  # spatial point pattern in a rectangle sampling window of size 125 x 125
  swmr <- spp(P15$trees, win = c(125, 125, 250, 250), marks = P15$species)
  kswmr <- ksfun(swmr, 50, 5, 500)
  plot(kswmr)

  # spatial point pattern in a circle with radius 50 centred on (125,125)
  swmc <- spp(P15$trees, win = c(125, 125, 50), marks = P15$species)
  kswmc <- ksfun(swmc, 50, 5, 500)
  plot(kswmc)
  
  # spatial point pattern in a complex sampling window
  swrt <- spp(P15$trees, win = c(125, 125, 250, 250), tri=P15$tri, marks=P15$species)
  kswrt <- ksfun(swrt, 50, 5, 500)
  plot(kswrt)