# Multiscale second-order neighbourhood analysis of a multivariate spatial point pattern using Rao quadratic entropy Computes distance-dependent estimates of Rao's quadratic entropy 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 either a random labelling or a species equivalence (see Details). latin1 ```r krfun(p, upto, by, nsim=0, dis = NULL, H0 = c("rl", "se"), 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. - `dis`: (optional) a `"dist"` object defining Euclidean distances between species. By default $dis = NULL$ so that species are considered equidistant. - `H0`: one of c("rl","se") to select either the null hypothesis of random labelling (H0 = "rl") or species equivalence (H0 = "se") (see Details). By default, the null hypothesis is random labelling. - `alpha`: if `nsim>0`, significant level of the confidence limits. By default $\alpha = 0.01$. ## Details Function `krfun` computes distance-dependent functions of Rao (1982) quadratic entropy (see `divc` in package `ade4`). For a multivariate point pattern consisting of $S$ species with intensity $\lambda$p, such functions can be estimated from the bivariate $Kpq$-functions between each pair of different species $p$ and $q$. Function `krfun` is thus a simple wrapper function of `k12fun` and `kfun`, standardized by Rao diversity coefficient (Pelissier & Goreaud 2014): $Kr(r) = sum(\lambda p * \lambda q * Kpq(r)*dpq) / (\lambda * \lambda * K(r) * HD)$. $gr(r) = sum(\lambda p * \lambda q * gpq(r)*dpq) / (\lambda * \lambda * g(r) * HD)$. where $dpq$ is the distance between species $p$ and $q$ defined by matrix `dis`, typically a taxonomic, phylogenetic or functional distance, and $HD=sum(Np*Nq*dpq/(N(N - 1)))$ is the unbiased version of Rao diversity coefficient (see Shimatani 2001). When `dis = NULL`, species are considered each other equidistant and `krfun` returns the same results than `ksfun`. 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 & Pelissier (1999). Theoretical values under the null hypothesis of either random labelling or species equivalence as well as local Monte Carlo confidence limits and p-values of departure from the null hypothesis (Besag & Diggle 1977) are estimated at each distance $r$. The random labelling hypothesis (H0 = "rl") is tested by reallocating species labels at random among all points of the pattern, keeping the point locations unchanged, so that expectations of $gr(r)$ and $Kr(r)$ are 1 for all $r$. The species equivalence hypothesis (H0 = "se") is tested by randomizing the between-species distances, keeping the point locations and distribution of species labels unchanged. The theoretical expectations of $gr(r)$ and $Kr(r)$ are thus $gs(r)$ and $Ks(r)$, respectively (see `ksfun`). ## Returns A list of class `"fads"` with essentially the following components: - **r**: a vector of regularly spaced out distances (`seq(by,upto,by)`). - **gr**: a data frame containing values of the function $gr(r)$. - **kr**: a data frame containing values of the function $Kr(r)$. 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 selected null hypothesis. - **sup**: (optional) if `nsim>0` a vector of the upper local confidence limits of a random distribution of the selected null hypothesis at a significant level $\alpha$. - **inf**: (optional) if `nsim>0` a vector of the lower local confidence limits of a random distribution of the selected null hypothesis at a significant level $\alpha$. - **pval**: (optional) if `nsim>0` a vector of local p-values of departure from the selected null hypothesis. ## References Rao, C.R. 1982. Diversity and dissimilarity coefficient: a unified approach. **Theoretical Population Biology**, 21:24-43. Shimatani, K. 2001. On the measurement of species diversity incorporating species differences. **Oikos**, 93, 135-147. Goreaud F. & Pelissier 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. Pelissier, R. & Goreaud, F. 2014. ads package for R: A fast unbiased implementation of the k-function family for studying spatial point patterns in irregular-shaped sampling windows. **Journal of Statistical Software**, in press. ## Author(s) Raphael.Pelissier@ird.fr ## Note There are printing and plotting methods for `"fads"` objects. ## See Also `plot.fads`, `spp`, `ksfun`, `kdfun`, `divc`. ## Examples ```r data(Paracou15) P15<-Paracou15 ## Not run: spatial point pattern in a rectangle sampling window of size 125 x 125 swmr <- spp(P15$trees, win = c(175, 175, 250, 250), marks = P15$species) ## Not run: testing the random labeling hypothesis krwmr.rl <- krfun(swmr, dis = P15$spdist, H0 = "rl", 25, 2, 50) ## Not run: running more simulations is slow krwmr.rl <- krfun(swmr, dis = P15$spdist, H0 = "rl", 25, 2, 500) plot(krwmr.rl) ## Not run: testing the species equivalence hypothesis krwmr.se <- krfun(swmr, dis = P15$spdist, H0 = "se", 25, 2, 50) ## Not run: running more simulations is slow krwmr.se <- krfun(swmr, dis = P15$spdist, H0 = "se", 25, 2, 500) plot(krwmr.se) ## Not run: 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) krwmc <- krfun(swmc, dis = P15$spdist, H0 = "rl", 25, 2, 100) ## Not run: running more simulations is slow krwmc <- krfun(swmc, dis = P15$spdist, H0 = "rl", 25, 2, 500) plot(krwmc) ## Not run: spatial point pattern in a complex sampling window swrt <- spp(P15$trees, win = c(125,125,250,250), tri = P15$tri, marks = P15$species) krwrt <- krfun(swrt, dis = P15$spdist, H0 = "rl", 25, 2) ## Not run: running simulations is slow krwrt <- krfun(swrt, dis = P15$spdist, H0 = "rl", 25, 2, 500) plot(krwrt) ```

krfun function