# Functions for spatial point pattern analysis in ecology This is a summary of the features of `ecespa` by way of examples of its main functions. latin1 package ## Author(s) Marcelino de la Cruz Rot, with contributions of Philip M. Dixon and Jose M. Blanco-Moreno and heavily borrowing Baddeley's & Turner's `spatstat` code. ## References De la Cruz, M. 2006. . **Ecosistemas** 15 (3): 19-39. De la Cruz, M. 2008. De la Cruz, M., Romao, R.L., Escudero, A. and Maestre, F.T. 2008. Where do seedlings go? A spatio-temporal analysis of early mortality in a semiarid specialist. **Ecography**, 31(6): 720-730. tools:::Rd_expr_doi("10.1111/j.0906-7590.2008.05299.x") . Diggle, P. J. 2003. **Statistical analysis of spatial point patterns**. Arnold, London. Dixon, P.M. 2002. Nearest-neighbor contingency table analysis of spatial segregation for several species. **Ecoscience**, 9 (2): 142-151. tools:::Rd_expr_doi("10.1080/11956860.2002.11682700") . Dixon, P. M. 2002. Ripley's K function. In **The encyclopedia of environmetrics** (eds. El-Shaarawi, A.H. & Piergorsch, W.W.), pp. 1976-1803. John Wiley & Sons Ltd, NY. Escudero, A., Romao, R.L., De la Cruz, M. & Maestre, F. 2005. Spatial pattern and neighbour effects on **Helianthemum squamatum** seedlings in a Mediterranean gypsum community. **Journal of Vegetation Science**, 16: 383-390. tools:::Rd_expr_doi("10.1111/j.1654-1103.2005.tb02377.x") . Getis, A. and Franklin, J. 1987. Second-order neighbourhood analysis of mapped point patterns. **Ecology** 68: 473-477. tools:::Rd_expr_doi("10.2307/1938452") . Loosmore, N.B. and Ford, E.D. (2006) Statistical inference using the G or K point pattern spatial statistics. **Ecology** 87, 1925-1931. tools:::Rd_expr_doi("10.1890/0012-9658(2006)87[1925:SIUTGO]2.0.CO;2") . Lotwick,H.W. & Silverman, B. W. 1982. Methods for analysing spatial processes of several types of points. **Journal of the Royal Statistical Society** B, 44: 406-413. tools:::Rd_expr_doi("10.1111/j.2517-6161.1982.tb01221.x") . Olano, J.M., Laskurain, N.A., Escudero, A. and De la Cruz, M. 2009. Why and where adult trees die in a secondary temperate forest? The role of neighbourhood. **Annals of Forest Science**, 66: 105. tools:::Rd_expr_doi("10.1051/forest:2008074") . Penttinen, A. 2006. Statistics for Marked Point Patterns. In **The Yearbook of the Finnish Statistical Society**, pp. 70-91. 2010. Inter-annual dynamics and spatial congruence of a nocturnal bird assemblage inhabiting a Mediterranean agricultural mosaic. **Ardeola**,57(2): 303-320. Syrjala, S. E. 1996. A statistical test for a difference between the spatial distributions of two populations. **Ecology** 77: 75-80. tools:::Rd_expr_doi("10.2307/2265656") . Waagepetersen, R. P. 2007. An estimating function approach to inference for inhomogeneous Neymann-Scott processes. **Biometrics** 63: 252-258. tools:::Rd_expr_doi("10.1111/j.1541-0420.2006.00667.x") . ## Examples ```r ## Not run: ############################################# ### Transfom easily data from a data.frame into the ppp format ### of spatstat: data(fig1) plot(fig1) #typical xyplot fig1.ppp <- haz.ppp (fig1) fig1.ppp plot(fig1.ppp) # point pattern plot of spatstat ############################################# ### Summarize the joint pattern of points and marks at different scales ### with the normalized mark-weighted K-function (Penttinen, 2006). ### Compare this function in two consecutive cohorts of Helianthemum ### squamatum seedlings: ## Figure 3.10 of De la Cruz (2008): data(seedlings1) data(seedlings2) s1km <- Kmm(seedlings1, r=1:100) s2km <- Kmm(seedlings2, r=1:100) plot(s1km, ylime=c(0.6,1.2), lwd=2, maine="", xlabe="r(cm)") plot(s2km, lwd=2, lty=2, add=T ) abline(h=1, lwd=2, lty=3) legend(x=60, y=1.2, legend=c("Hs_C1", "Hs_C2", "H0"), lty=c(1, 2, 3), lwd=c(3, 2, 2), bty="n") ## A pointwise test of normalized Kmm == 1 for seedlings1: s1km.test <- Kmm(seedlings1, r=1:100, nsim=99) plot(s1km.test, xlabe="r(cm)") ############################################# ### Explore the local relationships between marks and locations (e.g. size ### of one cohort of H. squamatum seedlings). Map the marked point pattern ### to a random field for visual inspection, with the normalized mark-sum ### measure (Penttinen, 2006). data(seedlings1) seed.m <- marksum(seedlings1, R=25) plot(seed.m, what="marksum", sigma = 5) # raw mark-sum measure; sigma is bandwith for smoothing plot(seed.m, what="pointsum", sigma = 5) # point sum measure plot(seed.m, what="normalized", dimyx=200, contour=TRUE, sigma = 5) # normalized mark-sum measure # the same with added grid plot(seed.m, what="normalized", dimyx=200, contour=TRUE, sigma = 5, grid=TRUE) ############################################# ### Test against the null model of "independent labelling", ### i.e. test asociation/repulsion between a "fixed" pattern (e.g. adult ### H. squamatum plants) and a "variable" pattern (e.g. of surviving and ### dead seedlings), with 2.5% and 97.5% envelopes of 999 random ### labellings (De la Cruz & al. 2008). data(Helianthemum) cosa <- K012(Helianthemum, fijo="adultHS", i="deadpl", j="survpl", r=seq(0,200,le=201), nsim=999, nrank=25, correction="isotropic") plot(cosa$k01, sqrt(./pi)-r~r, col=c(3, 1, 3), lty=c(3, 1, 3), las=1, ylab=expression(L[12]), xlim=c(0, 200), main="adult HS vs. dead seedlings", legend=FALSE) plot(cosa$k02, sqrt(./pi)-r~r, col=c(3, 1, 3), lty=c(3, 1, 3), las=1, ylab=expression(L[12]), xlim=c(0, 200), main="adult HS vs. surviving seedlings", legend=FALSE) ############################################# ### Test differences of agregation and segregation between two patterns, ### e.g. surviving and dying H. squamatum seedlings (De la Cruz & al. 2008). data(Helianthemum) cosa12 <- K1K2(Helianthemum, j="deadpl", i="survpl", r=seq(0,200,le=201), nsim=999, nrank=1, correction="isotropic") plot(cosa12$k1k2, lty=c(2, 1, 2), col=c(2, 1, 2), xlim=c(0, 200), main= "survival- death",ylab=expression(K[1]-K[2]), legend=FALSE) plot(cosa12$k1k12, lty=c(2, 1, 2), col=c(2, 1, 2), xlim=c(0, 200), main="segregation of surviving seedlings",ylab=expression(K[1]-K[12]), legend=FALSE) plot(cosa12$k2k12, lty=c(2, 1, 2), col=c(2, 1, 2), xlim=c(0, 200), main= "segregation of dying seedlings",ylab=expression(K[2]-K[12]), legend=FALSE) ############################################# ### Test 'univariate' and 'bivariate' point patterns ### against non-Poisson (in-)homogeneous models ### (De la Cruz and Escudero, submited). data(urkiola) ##################### ## univariate example # get univariate pp I.ppp <- split.ppp(urkiola)$birch # estimate inhomogeneous intensity function I.lam <- predict (ppm(I.ppp, ~polynom(x,y,2)), type="trend", ngrid=200) # Compute and plot envelopes to Kinhom, simulating from an Inhomogeneous # Poisson Process: I2.env <- envelope( I.ppp,Kinhom, lambda=I.lam, correction="trans", nsim=99, simulate=expression(rpoispp(I.lam))) plot(I2.env, sqrt(./pi)-r~r, xlab="r (metres)", ylab= "L (r)", col=c(1,3,2,2),legend=FALSE) # It seems that there is short scale clustering; let's fit an Inhomogeneous # Poisson Cluster Process: I.ki <- ipc.estK(mippp=I.ppp, lambda=I.lam, correction="trans") # Compute and plot envelopes to Kinhom, simulating from the fitted IPCP: Ipc.env <- Ki(I.ki, correction="trans", nsim=99, ngrid=200) plot (Ipc.env, xlab="r (metres)", ylab= "L (r)") ##################### ## bivariate example: test independence between birch and quercus in Urkiola J.ppp <- split.ppp(urkiola)$oak # We want to simulate oak from a homogeneous Poisson model: J.ppm <- ppm(J.ppp, trend=~1, interaction=Poisson() ) IJ.env <- Kci (mod1=I.ki, mod2=J.ppm, nsim=99) plot(IJ.env, type=12) plot(IJ.env, type=21) ############################################# ### Simulate envelopes from the fitted values of a logistic model, ### as in Olano et al. (2009) data(quercusvm) # read fitted values from logistic model: probquercus <-c(0.99955463, 0.96563477, 0.97577094, 0.97327199, 0.92437309, 0.84023396, 0.94926682, 0.89687281, 0.99377915, 0.74157478, 0.95491518, 0.72366493, 0.66771787, 0.77330148, 0.67569082, 0.9874892, 0.7918891, 0.73246803, 0.81614635, 0.66446411, 0.80077908, 0.98290508, 0.54641754, 0.53546689, 0.73273626, 0.7347013, 0.65559655, 0.89481468, 0.63946334, 0.62101995, 0.78996371, 0.93179582, 0.80160346, 0.82204428, 0.90050059, 0.83810669, 0.92153079, 0.47872421, 0.24697004, 0.50680935, 0.6297911, 0.46374812, 0.65672284, 0.87951682, 0.35818237, 0.50932432, 0.92293014, 0.48580241, 0.49692053, 0.52290553, 0.7317549, 0.32445982, 0.30300865, 0.73599359, 0.6206056, 0.85777043, 0.65758613, 0.50100406, 0.31340849, 0.22289286, 0.40002879, 0.29567678, 0.56917817, 0.56866864, 0.27718552, 0.4910667, 0.47394411, 0.40543788, 0.29571349, 0.30436276, 0.47859015, 0.31754526, 0.42131675, 0.37468782, 0.73271225, 0.26786274, 0.59506388, 0.54801851, 0.38983575, 0.64896835, 0.37282031, 0.67624306, 0.29429766, 0.29197755, 0.2247629, 0.40697843, 0.17022391, 0.26528042, 0.24373722, 0.26936163, 0.13052254, 0.19958585, 0.18659692, 0.36686678, 0.47263005, 0.39557661, 0.68048997, 0.74878567, 0.88352322, 0.93851375) ################################ ## Envelopes for an homogeneous point pattern: cosap <- Kinhom.log(A=quercusvm, lifemark="0", nsim=99, prob=probquercus) plot(cosap) ################################ ## Envelopes for an inhomogeneous point pattern: ## First, fit an inhomogeneous Poisson model to alive trees : quercusalive <- unmark(quercusvm[quercusvm$marks == 0]) mod2 <- ppm(quercusalive, ~polynom(x,y,2)) ## Now use mod2 to estimate lambda for K.inhom: cosapm <- Kinhom.log(A=quercusvm, lifemark="0", prob=probquercus, nsim=99, mod=mod2) plot(cosapm) ############################################# ### Test segregation based on the counts in the contingency table ### of nearest neighbors in a multitype point pattern (Dixon, 2002) data(swamp) dixon2002(swamp,nsim=99) ############################################# ### Fit the Poisson cluster point process to a point pattern with ### the method of minimum contrast (Diggle 2003). data(gypsophylous) # Estimate K function ("Kobs"). gyps.env <- envelope(gypsophylous, Kest, correction="iso", nsim=99) plot(gyps.env, sqrt(./pi)-r~r, ylab="L(r)", legend=FALSE) # Fit Poisson Cluster Process. The limits of integration # rmin and rmax are setup to 0 and 60, respectively. cosa.pc <- pc.estK(Kobs = gyps.env$obs[gyps.env$r<=60], r = gyps.env$r[gyps.env$r<=60]) # Add fitted Kclust function to the plot. lines(gyps.env$r,sqrt(Kclust(gyps.env$r, cosa.pc$sigma2,cosa.pc$rho)/pi)-gyps.env$r, lty=2, lwd=3, col="purple") # A kind of pointwise test of the gypsophylous pattern been a realisation # of the fitted model, simulating with sim.poissonc and using function J (Jest). gyps.env.sim <- envelope(gypsophylous, Jest, nsim=99, simulate=expression(sim.poissonc(gypsophylous, sigma=sqrt(cosa.pc$sigma2), rho=cosa.pc$rho))) plot(gyps.env.sim, main="",legendpos="bottomleft") ############################################# ### Compute Syrjala's test for the difference between the spatial ### distributions of two populations, as in Rey-Benayas et al. ### (submited) data(syr1); data(syr2); data(syr3) plot(syrjala.test(syr1, syr2, nsim=999)) plot(syrjala.test(syr1, syr3, nsim=999)) ## End(Not run) ```

00ecespa function