Package 'ecespa'

Dixon (2002) Nearest-neighbor contingency table analysis

Description

dixon2002 is a wrapper to the functions of Dixon (2002) to test spatial segregation for several species by analyzing the counts of the nearest neighbour contingency table for a marked point pattern.

Usage

dixon2002(datos, nsim = 99)

Arguments

datos	data.frame with three columns: x-coordinate, y-coordinate and sp-name. See swamp.
nsim	number of simulations for the randomization approximation of the p-values.

Details

A measure of segregation describes the tendency of one species to be associated with itself or with other species. Dixon (2002) proposed a measure of the segregation of species i in a multiespecies spatial pattern as:

$S[i] = log{[(N[ii]/(N[i]-N[ii])] / [(N[i]-1)/(N-N[i])]}$

where $N[i]$ is the number of individuals of species i, $N[ii]$ is the frequency of species i as neighbor of especies i and $N$ is the total number of locations. Values of $S[i]$ larger than 0 indicate that species i is segregated; the larger the value of $S[i]$, the more extreme the segregation. Values of $S[i]$ less than 0 indicate that species i is is found as neighbor of itself less than expected under random labelling. Values of $S[i]$ close to 0 are consistent with random labelling of the neighbors of species i.

Dixon (2002) also proposed a pairwise segregation index for the off-diagonal elements of the contingency table:

$S[ij] = log{[(N[ij]/(N[i]-N[ij])] / [(N[i])/(N-N[j])-1]}$

$S[ij]$ is larger than 0 when $N[ij]$, the frequency of neighbors of species j around points of species i, is larger than expected under random labelling and less than 0 when $N[ij]$ is smaller than expected under random labelling.

As a species/neighbor-specific test, Dixon(2002) proposed the statistic

$Z[ij] =(N[ij] -EN[ij])/sqrt(Var N[ij])$

where j may be the same as i and $EN[ij]$ is the expected count in the contingency table. It has an asymptotic normal distribution with mean 0 and variance 1; its asymptotic p-value can be obtained from the numerical evaluation of the cumulative normal distribution; when the sample size is small, a p-value on the observed counts in each cell ($N[ij]$) may be obtained by simulation, i.e, by condicting a randomization test.

An overall test of random labelling (i.e. a test that all counts in the $k$ x $k$ nearest-neighbor contingency table are equal to their expected counts) is based on the quadratic form

$C = (N-EN)' Sigma^- (N - EN)$

where $N$ is the vector of all cell counts in the contingency table, $Sigma$ is the variance-covariance matrix of those counts and $Sigma^-$ is a generalized inverse of $Sigma$. Under the null hypothesis of random labelling of points, $C$ has a asymptotic Chi-square distribution with $k(k-1)$ degrees of freedom (if the sample sizes are small its distribution should be estimated using Monte-Carlo simulation). P-values are computed from the probability of observing equal or larger values of $C$. The overall statistic $C$ can be partitioned into $k$ species-specific test statistics $C[i]$. Each $C[i]$ test if the frequencies of the neighbors of species i are similar to the expected frequencies if the points were randomly labelled. Because the $C[i]$ are not independent Chi-square statistics, they do not sum to the overall $C$.

Value

A list with the following components:

ON	Observed nearest neighbor counts in table format. From row sp to column sp.
EN	Expected nearest neighbor counts in table format.
Z	Z-score for testing whether the observed count equals the expected count.
S	Segregation measure.
pZas	P-values based on the asymptotic normal distribution of the Z statistic.
pNr	If nsim !=0, p-values of the observed counts in each cell based on the randomization distribution.
C	Overall test of random labelling.
Ci	Species-specific test of random labelling.
pCas	P-value of the overall test from the asymptotic chi-square distribution with the appropriate degrees of freedom.
pCias	P-values of the species-specific tests from the asymptotic chi-square distribution with the appropriate degrees of freedom.
pCr	If nsim !=0, p-value of the overall test from the randomization distribution.
pCir	If nsim !=0, p-values of the species-specific tests from the randomization distribution.
tablaZ	table with ON, EN, Z, S, pZas and pNr in pretty format, as in the table II of Dixon (2002).
tablaC	table with C, Ci, pCas,pCias, pCr and pCir in pretty format, as in the table IV of Dixon (2002).

Warning

The $S[i]$ and $S[ij]$ statistics asume that the spatial nearest-neighbor process is stationary, at least to second order, i.e., have the same sign in every part of the entire plot. A biologically heterogeneous process will violate this asumption.

Author(s)

Philip M. Dixon . Marcelino de la Cruz wrote the wrapper code for the ecespa version.

References

Dixon, P.M. 2002. Nearest-neighbor contingency table analysis of spatial segregation for several species. Ecoscience, 9 (2): 142-151. doi:10.1080/11956860.2002.11682700.

See Also

K012 for another segregation test, based in the differences of univariate and bivariate $K$-functions. A faster version of this function, with code implemented in FORTRAN it is available in function dixon in dixon.

Examples

data(swamp)

dixon2002(swamp,nsim=99)

---

Functions for spatial point pattern analysis in ecology

Description

This is a summary of the features of ecespa by way of examples of its main functions.

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. Introducción al análisis de datos mapeados o algunas de las (muchas) cosas que puedo hacer si tengo coordenadas. Ecosistemas 15 (3): 19-39.

De la Cruz, M. 2008. Métodos para analizar datos puntuales. En: Introducción al Análisis Espacial de Datos en Ecología y Ciencias Ambientales: Métodos y Aplicaciones (eds. Maestre, F. T., Escudero, A. y Bonet, A.), pp 76-127. Asociación Española de Ecología Terrestre, Universidad Rey Juan Carlos y Caja de Ahorros del Mediterráneo, Madrid.

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. 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. 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. 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. 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. 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. 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. doi:10.1051/forest:2008074.

Penttinen, A. 2006. Statistics for Marked Point Patterns. In The Yearbook of the Finnish Statistical Society, pp. 70-91.

Rey-Benayas, J.M., de la Montaña, E., Pérez-Camacho, L., de la Cruz, M., Moreno, D., Parejo, J.L. and Suárez-Seoane, S. 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. doi:10.2307/2265656.

Waagepetersen, R. P. 2007. An estimating function approach to inference for inhomogeneous Neymann-Scott processes. Biometrics 63: 252-258. doi:10.1111/j.1541-0420.2006.00667.x.

Examples

## 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)

---

Artificial point data.

Description

The three different point patterns in the figure 3.1 of De la Cruz (2008)

Usage

data(fig1)
data(fig2)
data(fig3)

Format

A data frame with 87 observations on the following 2 variables.

x

x coordinate

y

y coordinate

References

De la Cruz, M. 2008. Métodos para analizar datos puntuales. En: Introducción al Análisis Espacial de Datos en Ecología y Ciencias Ambientales: Métodos y Aplicaciones (eds. Maestre, F. T., Escudero, A. y Bonet, A.), pp 76-127. Asociación Española de Ecología Terrestre, Universidad Rey Juan Carlos y Caja de Ahorros del Mediterráneo, Madrid.

Examples

## Not run: 
data(fig1)

data(fig2)

data(fig3)

# transform to ppp format of spatstat with function haz.ppp:

fig1.ppp <- haz.ppp(fig1)

fig2.ppp <- haz.ppp(fig2)

fig3.ppp <- haz.ppp(fig3)

#Analyses as in Fig.3.2 of De la Cruz (2008). First, compute function K:

cosa1 <- Kest(fig1.ppp)

# Plot different estimators. 
# Fig. 3.2a: 

par("mar"=par("mar")+c(0,1,0,0))

plot(cosa1, col=c(1,0,0,1), lwd=c(2,2,2,2), lty=c(1,1,1,2), 
	 main="")

# Fig. 3.2b:

plot(cosa1, sqrt(./pi)-r~r, col=c(1,0,0,1), lwd=c(2,2,2,2), 
         lty=c(1,1,1,2), main="", ylab="L(r)")

# Fig. 3.2c:

plot(cosa1, .-(pi*r^2)~r, col=c(1,0,0,1), lwd=c(2,2,2,2), 
         lty=c(1,1,1,2), main="", ylab=expression(K(r)-pi*r^2))

# Fig. 3.2d:

plot(cosa1,(./(pi*r^2))-1~r, col=c(1,0,0,1), lwd=c(2,2,2,2), 
	 lty=c(1,1,1,2), main="",
         ylab=expression((K(r)/pi*r^2)-1))

## Analyses as in fig. 3.7 of De la Cruz (2008).
## First, compute function K and pointwise envelopes:

cosa1.env <- envelope(fig1.ppp, Kest)

cosa2.env <- envelope(fig2.ppp, Kest)

cosa3.env <- envelope(fig3.ppp, Kest)

## Plot function L with pointwise envelopes:

plot(cosa1.env,sqrt(./pi)-r~r, lwd=c(1,1,2,2), 
	 lty=c(1,1,3,3), col=c(1,1,1,1), xlab="r", 
         ylab="L(r)", main="", ylim=c(-2,2))

## Add simultaneous envelopes of Ripley (+-1.68 *sqrt(A)/N):

abline(h=1.68*sqrt(area.owin(fig1.ppp$window))/fig1.ppp$n, 
         lty=2, lwd=2)

abline(h=-1.68*sqrt(area.owin(fig1.ppp$window))/fig1.ppp$n, 
         lty=2, lwd=2)

## Plot function L with pointwise envelopes:

plot(cosa2.env,sqrt(./pi)-r~r, lwd=c(1,1,2,2), 
         lty=c(1,1,3,3), col=c(1,1,1,1), xlab="r", 
         ylab="L(r)", main="")

## Add simultaneous envelopes of Ripley:

abline(h=1.68*sqrt(area.owin(fig2.ppp$window))/fig2.ppp$n, 
         lty=2, lwd=2)

abline(h=-1.68*sqrt(area.owin(fig2.ppp$window))/fig2.ppp$n, 
         lty=2, lwd=2)

## Plot function L with pointwise envelopes:

plot(cosa3.env,sqrt(./pi)-r~r, lwd=c(1,1,2,2), 
         lty=c(1,1,3,3), col=c(1,1,1,1), xlab="r", 
         ylab="L(r)", main="")

## Add simultaneous envelopes of Ripley:

abline(h=1.68*sqrt(area.owin(fig3.ppp$window))/fig3.ppp$n, 
         lty=2, lwd=2)

abline(h=-1.68*sqrt(area.owin(fig3.ppp$window))/fig3.ppp$n, 
         lty=2, lwd=2)

## End(Not run)

---

Neighbourhood density function

Description

Computes and plots the neighbourhood density function, a local version of the $K$-function defined by Getis and Franklin (1987).

Usage

getis(mippp, nx = 30, ny = 30, R = 10)

## S3 method for class 'ecespa.getis'
plot(x, type="k", dimyx=NULL, xy=NULL, eps=NULL,  color=NULL,
         contour=TRUE, points=TRUE,...)

Arguments

mippp	A point pattern. An object with the ppp format of spatstat.
nx	Grid dimensions (for estimation) in the x-side.
ny	Grid dimensions (for estimation) in the y-side.
R	Radius. The distance argument r at which the function $K$ should be computed.
x	Result of applying getis to a point pattern.
type	Type of local statistics to be ploted. One of k (local-$K$), l (local-$L$), n (local-$n$) or d (deviations from CSR).
color	A list of colors such as that generated by rainbow, heat.colors, topo.colors, terrain.colors or similar functions.
dimyx	pixel array dimensions, will be passed to (i.e. see details in) as.mask.
xy	pixel coordinates, will be passed to (i.e. see details in) as.mask.
eps	width and height of pixels, will be passed to (i.e. see details in) as.mask.
contour	Logical; if TRUE, add a contour to current plot.
points	Logical; if TRUE, add the point pattern to current plot.
...	Additional graphical parameters passed to link{plot}.

Details

Getis and Franklin (1987) proposed the neigbourhood density function, a local version of Ripley's $L$- function. Given a spatial point pattern $X$, the neigbourhood density function associated with the ith point in $X$ is computed by

$L[i](r) = sqrt((a/((n-1))*pi))*sum[j]e[i,j])$

where the sum is over all points j != i that lie within a distance r of the ith point, a is the area of the observation window, n is the number of points in $X$, and e[i,j] is the isotropic edge correction term (as described in Kest). The value of L[i](r) can also be interpreted as one of the summands that contributes to the global estimate of the $L$-function.

The command getis actually computes the local $K$-function using Kcross. As the main objective of getis is to map the local density function, as sugested by Gestis and Franklin (1987: 476) a grid of points (whose density is controled by nx and ny), is used to accurately estimate the functions in empty or sparse areas. The S3 method plot.ecespa.getis plots the spatial distribution of the local $K$ or $L$ function or other related local statistics, such as $n[i](r)$, the number of neighbor points [=$lambda*K[i](r)$] or the deviations from the expected value of local $L$ under CSR [= $L[i](r) -r$]. It some of the arguments dimyx, xy or eps is provided it will use the function interp.im in spatstat package to interpolate the results;otherwise it will plot the estimated values at the origial grid points.

Value

getis gives an object of class ecespa.getis, bassically a list with the following elements:

x	x coordinates of pattern points (ahead) and grid points.
y	y coordinates of pattern points (ahead) and grid points.
klocal	Estimate of local $K[i](r)$ at the point pattern points.
klocalgrid	Estimate of local $K[i](r)$ at the grid points.
R	Distance $r$ at which the estimation is made.
nx	Density of the estimating grid in the x-side.
ny	Density of the estimating grid in the x-side.
dataname	Name of the ppp object analysed.
ppp	Original point pattern.

plot.ecespa.getis plots an interpolated map of the selected local statistics

Note

As plot.ecespa.getis interpolates over rectangular grid of points, it is not apropriate to map irregular windows. In those cases, Smooth.ppp of spatstat can be used to interpolate the local statistics (see examples).

Author(s)

Marcelino de la Cruz Rot

References

Getis, A. and Franklin, J. 1987. Second-order neighbourhood analysis of mapped point patterns. Ecology 68: 473-477. doi:10.2307/1938452.

See Also

localK, a different approach in spatstat.

Examples

## Compare with fig. 5b of Getis & Franklin (1987: 476):
  
  data(ponderosa)
  
  #ponderosa12 <- getis(ponderosa, nx = 30, ny = 30, R = 12)
  ponderosa12 <- getis(ponderosa, nx = 20, ny = 20, R = 12)
  
  plot(ponderosa12, type = "l", dimyx=256)
  
## Not run: 
  ## Plot the same, using Smooth.ppp in spatstat
  
  ponderosa.12 <- setmarks(ponderosa, ponderosa12$klocal)
  
  Z <- Smooth(ponderosa.12, sigma=5, dimyx=256)
  
  plot(Z, col=topo.colors(128), main="smoothed neighbourhood density")
  
  contour(Z, add=TRUE)
  
  points(ponderosa, pch=16, cex=0.5) 
  

  ## Example with irregular window:
  
  data(letterR)
  
  X <- rpoispp(50, win=letterR)
  
  X.g <- getis(X, R=0.2)
  
  plot(X.g,dimyx=c(200,100))
  
   ## Plot the same, using Smooth.ppp in spatstat
    X2 <- setmarks(X, X.g$klocal)
  
    Z <- Smooth(X2, sigma=0.05, dimxy=256)
  
    plot(Z, col=topo.colors(128), main="smoothed neighbourhood density")
  
    contour(Z, add=TRUE)
  
    points(X, pch=16, cex=0.5)
  
    
    
## End(Not run)

---

Spatial point pattern of a plant community

Description

Locations of plants in a gypsophylous plant community in Central Spain. These are part of the data collected by Romao (2003) that have been analyzed several times (Escudero et al. 2005, De la Cruz 2006). The coordinates of the plans are given in cm.

Usage

data(gypsophylous)

Format

An object of class "ppp" of spatstat representing the point pattern of plants locations. See ppp.object for details of the format.

Source

Romao, R.L. 2003. Estructura espacial de comunidades de gipsófitos: interacciones bióticas y constricciones abióticas. Tesis Doctoral. Universidad Politécnica de Madrid.

References

De la Cruz, M. 2006. Introducción al análisis de datos mapeados o algunas de las (muchas) cosas que puedo hacer si tengo coordenadas. Ecosistemas 15 (3): 19-39.

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. doi:10.1111/j.1654-1103.2005.tb02377.x.

Examples

## Not run: 

data(gypsophylous)

plot(gypsophylous)


## End(Not run)

---

Easily convert xy data to ppp format

Description

ppp maker for the impatient layman

Usage

haz.ppp(W)

Arguments

W	a data frame or matrix with two or three columns (coordinate x, coordinate y, and mark of the point)

Details

This naive function easily transform your xy data to the format required by spatstat (version <2.0). It establishes the window of observation as the rectangle defined by the xy range. It asumes that the first two columns are coordinates x and y, and the third (if any) gives the marks of the points.

Value

A point pattern with the format of spatstat v. < 2.0

Note

This window is an underestimate of the real sampling window of the observed points. For a less biassed estimate, use function ripras in spatstat.

Author(s)

Marcelino de la Cruz

See Also

ppp, as.ppp

Examples

data(fig1)

plot(fig1) #typical xyplot

fig1.ppp <- haz.ppp (fig1)

fig1.ppp

plot(fig1.ppp) # point pattern plot of spatstat

---

Spatial point pattern of Helianthemum squamatum adult plants and seedlings

Description

Locations of H. squamatum adult plants and seedlings in a 6 m x 7 m plot over gypsum soil in Chinchón (near Madrid, Spain). These are part of the data collected by Romao (2003) that have been analyzed several times (Escudero et al.2005, De la Cruz 2006, De la Cruz et al. in press. ). The coordinates of the plans are given in cm.

Usage

data(Helianthemum)

Format

An object of class "ppp" of spatstat representing the point pattern of plants locations marked by their type. See ppp.object for details of the format.The dataset has 866 points with the following levels:

adultHS

adult H. squamatum plants

deadp

dying H. squamatum seedlings

survpl

surviving H. squamatum seedlings

Source

Romao, R.L. 2003. Estructura espacial de comunidades de gipsófitos: interacciones bióticas y constricciones abióticas. Tesis Doctoral. Universidad Politécnica de Madrid.

References

De la Cruz, M. 2006. Introducción al análisis de datos mapeados o algunas de las (muchas) cosas que puedo hacer si tengo coordenadas. Ecosistemas 15 (3): 19-39.

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. doi:10.1111/j.0906-7590.2008.05299.x.

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. doi:10.1111/j.1654-1103.2005.tb02377.x.

Examples

data(Helianthemum)
plot(Helianthemum)

---

Fit the (In)homogeneous Poisson Cluster Point Process by Minimum Contrast

Description

Fits the (In)homogeneous Poisson Cluster point process to a point pattern dataset by the Method of Minimum Contrast.

Usage

ipc.estK(mippp, lambda = NULL, correction = "iso", r = NULL, sigma2 = NULL,
            rho = NULL, q = 1/4, p = 2)


## S3 method for class 'ecespa.minconfit'
plot(x, type="L", add=FALSE, xlim=NULL, ylim=NULL, lwd=c(1,1),
           lty=c(1,2), col=c(1,2), main=NULL, ...)

Arguments

mippp	Point pattern to which the (I)PCP will be fitted. A point pattern with the ppp format of spatstat.
lambda	Optional. Values of the estimated intensity function as a pixel image (object of class "im" of spatstat) giving the intensity values at all locations of mippp.
correction	A character item selecting any of the options "border", "bord.modif", "isotropic", "Ripley" or "translate". It specifies the edge correction(s) to be applied in the computation of the $K(r)$ function.
r	Numeric vector. The values of the argument $r$ at which the $K(r)$ functions should be evaluated.
sigma2	Optional. Starting value for the parameter $sigma2$ of the Poisson Cluster process.
rho	Optional. Starting value for the parameter $rho$ of the Poisson Cluster process.
q	$q$ exponent of the contrast criterion (see mincontrast).
p	$p$ exponent of the contrast criterion (see mincontrast).
x	An object of class 'ecespa.minconfit', resulting of applying ipc.estK to fit a Poisson Cluster Process.
type	Type of function to be ploted. If type="L", function $L(r)$ [= $sqrt(K(r)/pi)-r$] is ploted. Otherwise, function $K(r)$ is ploted.
add	Logical. Should the curves be added to another plot?
xlim	Vector setting the limits of the x-axis.
ylim	Vector setting the limits of the y-axis.
lwd	Vector (length=2) setting the line width for ploting the two functions.
lty	Vector (length=2) setting the line type for ploting the two functions.
col	Vector (length=2) setting the line color for ploting the two functions.
main	Optional. Text to appear as a title of the plot.
...	Additional graphical parameters passed to link{plot}.

Details

The algorithm fits the (inhomogeneous) Poisson cluster point process (PCP) to a point pattern, by finding the parameters of the (inhomogeneous) Poisson cluster model which give the closest match between the theoretical K function of the Poisson cluster process and the observed K function. For a concise explanation of the PCP see pc.estK. For a more detailed explanation of the Method of Minimum Contrast, see mincontrast in spatstat or Diggle (2003: 86).

The inhomogeneous PCP can be thought of as a thinned process of an homogeneous PCP, where the spatially varying thinning probability $f(s)$ is related to the spatially varying intensity function $lambda(s)$ as $f(s) = lambda(s)/max{lambda(s)}$ (Waagepetersen, 2007). As the inhomogeneous K function for the IPCP coincides with the (homogeneous) K function for the corresponding homogeneous PCP, the parameters of the underlying homomgeneous PCP can be estimated as those that give the closest match between the theoretical K function for the homogeneous PCP and the empirical inhomogeneous K function for the observed IPCP.

This Poisson cluster process can be simulated with rIPCP.

Value

ipc.estK gives an object of class 'ecespa.minconfit', basically a list with the following components:

sigma2	Parameter $sigma^2$.
rho	Parameter $rho$.
d.theta	Minimized value of the contrast criterion $D(theta)$.
Kobs	Values of the observed K-function.
Kfit	Values of the fitted K-function.
r	Sequence of distances at which Kobs and Kfit have been estimated.
data	Original point pattern.
lambda	Original intensity function.
dataname	Name of the original point pattern.
lambdaname	Name of the original intensity function image.
q	$q$ exponent of the contrast criterion.
p	$p$ exponent of the contrast criterion.

Author(s)

Marcelino de la Cruz Rot , inspired by some code of Philip M. Dixon

References

Diggle, P. J. 2003. Statistical analysis of spatial point patterns. Arnold, London.

Waagepetersen, R. P. 2007. An estimating function approach to inference for inhomogeneous Neymann-Scott processes. Biometrics 63: 252-258. doi:10.1111/j.1541-0420.2006.00667.x.

See Also

some functions in spatstat: mincontrast for a more general implementation of the method of mimimum contrast; matclust.estK and lgcp.estK fit other appropriate processes for clustered patterns.

Examples

#####################
## Same example as in pc.estK

data(gypsophylous)

# set the number of simulations (nsim=199 or larger for real analyses)
# and resolution for lambda (ngrid=200  or larger for real analyses)
nsim<- 19
ngrid <- 50

## Estimate K function ("Kobs").

gyps.env <- envelope(gypsophylous, Kest, correction="iso", nsim=nsim)

plot(gyps.env, sqrt(./pi)-r~r, legend=FALSE)

## Fit Poisson Cluster Process. The limits of integration 
## rmin and rmax are setup to 0 and 60, respectively. 

cosa.pc2 <- ipc.estK(gypsophylous, r = gyps.env$r[gyps.env$r<=60])

## Add fitted Kclust function to the plot.


plot(cosa.pc2, add=TRUE, lwd=c(3,3))


## A kind of pointwise test of the gypsophylous pattern been a realisation
## of the fitted model, simulating with rIPCP and using function J (Jest).


gyps.env.sim2 <- envelope(gypsophylous, Jest, nsim=nsim,
                    simulate=expression(rIPCP(cosa.pc2)))


plot(gyps.env.sim2,  main="",legendpos="bottomleft")

## Not run: 

#####################
## Inhomogeneous example

data(urkiola)
   
   #####################
   ## univariate case

   # 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=ngrid)

   # Compute and plot envelopes to Kinhom, simulating from an Inhomogeneous
   #  Poisson Process:
   
   I2.env <- envelope( I.ppp,Kinhom, lambda=I.lam, correction="trans", 
                              nsim=nsim, 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=nsim, ngrid=ngrid)

   plot (Ipc.env, xlab="r (metres)", ylab= "L (r)")
 
   #####################
   ## bivariate case: 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=nsim)
   
   plot(IJ.env, type=12)
   
   plot(IJ.env, type=21)

## End(Not run)

---

Tests against 'independent labelling'

Description

Given a "fixed" point pattern and some process that asign labels (I,J) to another "variable" point pattern, K012 estimates the combined bivariate K function between the fixed pattern and every type of the variable pattern, and test that they are independent (i.e. that the labels are randomly assigned, irrespectively of the fixed pattern).

Usage

K012(X, fijo, i, j, nsim = 99, nrank = 1, r = NULL,
	 correction = "isotropic")

Arguments

X	Multitype marked point pattern. An object with the ppp format of spatstat.
fijo	Number or character string identifying the mark value of the "fixed" pattern in X
i	Number or character string identifying the mark value of the I pattern in X
j	Number or character string identifying the mark value of the J pattern in X
nsim	Number of simulated point patterns to be generated when computing the envelopes.
nrank	Integer. Rank of the envelope value amongst the nsim simulated values. A rank of 1 means that the minimum and maximum simulated values will be used.
r	Numeric vector. The values of the argument r at which the K functions should be evaluated.
correction	A character item selecting any of the options "border", "bord.modif", "isotropic", "Ripley" or "translate". It specifies the edge correction(s) to be applied.

Details

This test was developped to answer some questions about the spatial pattern of survival and mortality of seedlings and its relationships with adult plants in a plant community (De la Cruz et al. 2008). In order to evaluate the spatial structures of seedlings fates (survive or die), the null hypothesis of random labelling (Cuzick & Edwards 1990, Dixon 2002) would be the appropriate one. This kind of pattern is the result of two hierarchical processes: a first one that generates the pattern of points (seedlings) and other that assign "labels" (i.e. "die", "survive") to the points. On the other hand, to analyze the relationships between the spatial pattern of emerging seedlings and the pattern of adult plants (two patterns that have been generated independently), independence would be the appropriate null hypothesis (Goreaud & Pellisier 2003). However, testing the relationship between the pattern of seedling fates and the pattern of adult plants does not completely fit any of the mentioned hypotheses because, although the pattern of adult plants and the pattern of, e.g., dead seedlings are generated independently, their relationship is conditioned by the dependence of the fate "dead" on the locations of emerging seedlings. This implies that one can not apply the usual technique of toroidal shifting one pattern over the other to test the independence hypothesis. Instead one must permute the label of the focal fate (i.e. survive, die) over the global pattern of seedlings points, keeping the locations and labels of adults fixed. This is the method that K012 uses to build the envelopes. The bivariate K functions are computed with the Lotwick's and Silverman's (1982) combined estimator (Kmulti.ls).

Value

A list with two elements.

k01	Bivariate K function of the fixed point pattern and the I variable type, with simulation envelopes
k02	Bivariate K function of the fixed point pattern and the J variable type, with simulation envelopes

Each of the above elements is a fv.object, essentially a data.frame with the following items:

r	the values of the argument r at which the functions kave been estimated
hi	upper envelope of simulations
lo	lower envelope of simulations

together with the observed corrected estimate of the combined bivariate K function ( iso, trans, border, etc).

Author(s)

Marcelino de la Cruz

References

Cuzick, J. and Edwards, R. 1990. Spatial clustering for inhomogeneous populations (with discussion). Journal of the Royal Statistical Society B 52: 73-104.

De la Cruz, M. 2006. Introducción al análisis de datos mapeados o algunas de las (muchas) cosas que puedo hacer si tengo coordenadas. Ecosistemas 15 (3): 19-39.

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. doi:10.1111/j.0906-7590.2008.05299.x.

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.

Goreaud, F. and Pelissier, R. 2003. Avoiding misinterpretation of biotic interactions with the intertype K12-function: population independence vs. random labelling hypotheses. J. Veg. Sci. 14: 681-692.

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.

See Also

dixon2002 for another segregation test, based in the contingency table of counts of nearest neigbors in a marked point pattern.

Examples

data(Helianthemum)


# To get 2.5% and 97.5% envelopes of 999 random labelings.
## as in fig. 10 of De la Cruz (2006), change nsim= 999 and nrank=25
nsim <-19
nrank <-1



## Test asociation/repulsion between the fixed pattern of adult
## H. squamatum plants and the "variable" pattern of surviving and 
## dead seedlings,


cosa <- K012(Helianthemum, fijo="adultHS", i="deadpl", j="survpl",
             r=seq(0,200,le=201), nsim=nsim, nrank=nrank, 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")

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")

---

Differences between univariate and bivariate K-functions

Description

Given two point patterns I and J, K1K2computes the differences between both univariate $K$-functions (i.e. $Ki(r)-Kj(r)$) as well as the differences between the univariate and the bivariate $K$-function (i.e. $Ki(r)-Kij(r)$ and $Kj(r)-Kij(r)$). It also computes simulation envelopes to test that that the observed differences are within the range expected asuming the random labelling hypothesis.

Usage

K1K2(X, i, j, nsim = 99, nrank = 1, r = NULL,
	 correction = "isotropic")

Arguments

X	Multitype marked point pattern. An object with the ppp format of spatstat.
i	Number or character string identifying the mark value of the I pattern in X.
j	Number or character string identifying the mark value of the J pattern in X.
nsim	Number of simulated point patterns to be generated when computing the envelopes.
nrank	Integer. Rank of the envelope value amongst the nsim simulated values. A rank of 1 means that the minimum and maximum simulated values will be used.
r	Numeric vector. The values of the argument $r$ at which the $K(r)$ functions should be evaluated.
correction	A character item selecting any of the options "border", "bord.modif", "isotropic", "Ripley" or "translate". It specifies the edge correction(s) to be applied.

Details

The indiscriminate use of the raw bivariate functions (mainly the $K$ or the $L$-bivariate functions) in ecological studies for testing the association/ repulsion between different point patterns waste some of the most interesting properties of the $K$-function. One of them is that under the random labelling hypothesis every individual pattern would be a random thinning of the corresponding bivariate pattern and therefore $Ki(r)=Kj(r)= Kij(r)=pi*r^2$ (Diggle 2003). Dixon (2002) sugested that some differences of these functions could provide provide interesting ecological information. For example, $D(r)= Ki(r)-Kj(r)$, has an expected value of 0 for all $r$ distances under random labelling and evaluates the differences in the intensity of aggregation of the two point patterns (e.g., in the example bellow, the pattern of drought and herbivory deaths). Other relevant function is $D(r) = Ki(r)-Kij(r$) and the complementary $D(r)= Kj(r)-Kij(r$) which evaluate the degree of segregation of every individual pattern, i.e. if every point of the pattern is more -or less- surrounded by other points of the same type than would be expected under the random labelling hypothesis. K1K2 uses $K^*ij(r)$, the combined estimator of Lotwick and Silverman (a weigthed mean of $Kij(r)$ and $Kji(r)$) as computed by Kmulti.ls.

Value

A list with three elements.

k1k2	Difference between $Ki(r)$ and $Kj(r)$, with simulation envelopes.
k1k12	Difference between $Ki(r)$ and $Kij(r)$, with simulation envelopes.
k2k12	Difference between $Kj(r)$ and $Kij(r)$, with simulation envelopes.

Each of the above elements is a fv.object, essentially a data.frame with the following items:

r	The values of the argument r at which the functions have been estimated.
hi	Upper envelope of simulations.
D	The respective difference function $D(r)$, i.e., respectively, $Ki(r)-Kj(r)$, $Ki(r)-K^*ij(r)$ or $Kj(r)-K^*ij(r)$.
lo	Lower envelope of simulations.

Author(s)

Marcelino de la Cruz

References

De la Cruz, M. 2006. Introducción al análisis de datos mapeados o algunas de las (muchas) cosas que puedo hacer si tengo coordenadas. Ecosistemas 15 (3): 19-39.

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. 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. 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.

Examples

data(Helianthemum)

# set the number of simulations (nsim=199 or larger for real analyses)
nsim<- 19

cosa12 <- K1K2(Helianthemum, j="deadpl", i="survpl", r=seq(0,200,le=201),
		 nsim=nsim, correction="isotropic")

## plots of figure 9 in De la Cruz (2006) (they where made with nsim=999)
plot(cosa12$k1k2, lty=c(2, 1, 2), col=c(2, 1, 2), xlim=c(0, 200),
         main= "survival- death")

plot(cosa12$k1k12, lty=c(2, 1, 2), col=c(2, 1, 2), xlim=c(0, 200),
	 main="segregation of surviving seedlings")

plot(cosa12$k2k12, lty=c(2, 1, 2), col=c(2, 1, 2), xlim=c(0, 200),
         main= "segregation of dying seedlings", legend=FALSE)

---

Test against non-Poisson (in-)homogeneous models

Description

Functions to automate testing of 'univariate' and 'bivariate' point pattern hypothesis against non-Poisson (in-)homogeneous models.

Usage

Kci(mod1, mod2, correction="trans", nsim=99, ngrid=200, nrep=1e+05,
     r=NULL, simu="both", spctype=1)

Ki(mod1, correction="trans", nsim=99, ngrid=200, nrep=1e+05, r=NULL,
    spctype=1)

## S3 ploth method for objects of class 'ecespa.kci':
## S3 method for class 'ecespa.kci'
plot(x, type=1, q=0.025, kmean=TRUE, add=FALSE, maine=NULL, 
       xlabe=NULL, ylabe=NULL, xlime=NULL, ylime=NULL, 
       lty=c(1,2,3), col=c(1,2,3), lwd=c(1,1,1), ...)

Arguments

mod1	Fitted model. An object of class ppm or ecespa.minconfit.
mod2	Fitted model. An object of class ppm or ecespa.minconfit.
correction	A character item selecting any of the options "border", "bord.modif", or "translate". It specifies the edge correction to be applied when computing K-functions.
nsim	Number of simulated point patterns to be generated when computing the envelopes.
ngrid	Dimensions (ngrid by ngrid) of a rectangular grid of locations where predict.ppm would evaluate the spatial trend of the fitted models.
nrep	Total number of steps (proposals) of Metropolis-Hastings algorithm that should be run by rmh to simulate models of class ppm.
r	Numeric vector. The values of the argument $r$ at which the $K(r)$ functions should be evaluated.
simu	A character item indicating if both models will be simulated for the computation of the envelopes (simu = "both") or just the second model (simu != "both").
spctype	Type of 'pre-thinning' method employed by rIPCP in the simulation of ecespa.minconfit models.
x	An object of class 'ecespa.kci'. The result of runing Kci or Ki.
type	What to plot. One of 1 (K1), 2 (K2), 12 (K12), 21 (K21), 112 (K1-K12) or 221 (K2-K21).
q	Quantile for selecting the simulation envelopes.
kmean	Logical. Should the mean of the simulated envelopes be ploted?
add	Logical. Should the kci.plot be added to a previous plot?
maine	Title to add to the plot.
xlabe	Text or expression to label the x-axis.
ylabe	Text or expression to label the y-axis.
xlime	Max and min coordinates for the x-axis.
ylime	Max and min coordinates for the y-axis.
lty	Vector with the line type for the estimated Kmm function, the simulated envelopes and the mean of the simulated envelopes.
col	Vector with the color for the estimated K-function, the simulated envelopes and the mean of the simulated envelopes.
lwd	Vector with the line width for the estimated K-function, the simulated envelopes and the mean of the simulated envelopes.
...	Additional graphical parameters passed to plot.

Details

These functions are designed to automate the testing of 'univariate' and(/or) 'bivariate' point pattern hypotheses (based on K-functions) against non-Poisson (in-)homogeneous models. These non-Poisson (in-)homogeneous models should have been fitted with pseudolikelihood tools (spatstat ppm function) or with minimum contrast methods (ecespa.minconfit).

Function Ki is designed to test 'univariate' hypotheses. It will compute the (in-)homogeneous K-function (using spatstat Kinhom function) of the point pattern to which the ppm or ecespa.minconfit model has beeen fitted and will compute 'envelopes' simulating from the fitted model. The computed envelopes can be considered as a pointwise test of the point pattern been a realisation of the fitted model.

Function Kci is designed to test 'bivariate' hypotheses. It will compute the (in-)homogeneous cross K-function (using spatstat Kcross.inhom function) and will compute envelopes simulating from the fitted models. As, when dealing with inhomogeneos patterns $K12$ != $K21$, Kci will compute both functions. If simu = "both" (default option), Kci will simulate from mod2 to test K12 and from mod1 to test K21. If simu != "both", only mod2 will be simulated. This option may be useful when only K12 is of interest. Function Kci will also compute univariate (in-) homogeneous K-functions and envelopes for each individual point pattern.

The S3 ploth method will plot the selected K-function and envelopes (actually, it will plot the most usual L-function = $sqrt[K(r)/pi]-r$). The appropriate K function can be selected with the argument type. If type = 1 (default option), it will plot the univariate K function (of the analized model in Ki or of the first model [mod1] in Kci). If type = 2, it will plot the univariate K function of the second model (mod2 in Kci). When type = 12 or type = 21, it will plot respectively K12 or K21. Options type = 112 or type = 221 will graph a kind of 'segregation test' (see K1K2), and will represent de differences K1-K12, K2-K21 and their envelopes.

Value

Both Kci and Ki return an object of class 'ecespa.kci', basically a list with the following items:

r	Numeric vector. The values of the argument $r$ at which the $K(r)$ functions have been evaluated.
kia	Numeric vector. Observed (in-)homogeneous K function for mod1 point pattern.
kib	Numeric vector. Observed (in-)homogeneous K function for mod2 point pattern.
kci.ab.o	Numeric vector. Observed (in-) homogeneous cross K-function (K12) for mod1 and mod2 point patterns.
kci.ba.o	Numeric vector. Observed (in-) homogeneous cross K-function (K21) for mod2 and mod1 point patterns.
kci.ab.s	Matrix of simulated (in-) homogeneous cross K-function (K12) for mod1 and mod2 point patterns.
kci.ba.s	Matrix of simulated (in-) homogeneous cross K-function (K21) for mod2 and mod1 point patterns.
kib.s	Matrix of simulated (in-)homogeneous K function for mod2 point pattern.
kia.s	Matrix of simulated (in-)homogeneous K function for mod1 point pattern.
datanamea	Name of mod1 point pattern.
datanameb	Name of mod2 point pattern.
modnamea	Name of model mod1.
modnameb	Name of model mod2.
type	Type of analysis. "Kci" or "Ki".

Warning

As this implementation involves the use of images as the means of evaluation of the (inhomogeneous) spatial trend, and a mask based on those images will be used as the point pattern window, the "Ripley's" or "isotropic" edge correction can not be employed.

It is usual that during the simulation process some warnings are produced. They are related to some simulated points being rejected as lying outside the specified window.

Note

Even when one of the two point patterns is assumed to be homogeneous Poisson (and, apparently not worth of fitting any model), an homogeneous Poisson model can be easily fitted and passed to Kci with ppm. See the examples.

Author(s)

Marcelino de la Cruz Rot

References

De la Cruz, M. and Escudero, A. 2008. Null models and tools for multivariate heterogeneous point patterns. Submitted.

De Soto, L., Olano, J.M., Rozas, V. and De la Cruz, M. 2009. Release of Juniperus thurifera woodlands from herbivore-mediated arrested succession in Spain. Applied Vegetation Science, 13: 1525. doi:10.1111/j.1654-109X.2009.01045.x

Examples

data(urkiola)
    
    # set the number of simulations (nsim=199 or larger for real analyses)
    nsim<- 5

  #####################
   ## univariate case

   # 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=nsim, 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=nsim, ngrid=200)

   plot (Ipc.env, xlab="r (metres)", ylab= "L (r)")
 
  ## Not run: 
   #####################
   ## bivariate case: 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=nsim)
   
   plot(IJ.env, type=12)
   
   plot(IJ.env, type=21)
 
## End(Not run)

---

Simulation envelopes from the fitted values of a logistic model

Description

Computes simulation envelopes for (in-)homogeneous K-function simulating from a vector of probabilitiesn.

Usage

Kinhom.log (A, lambda=NULL, mod=NULL, lifemark="0", prob=NULL,
			r=NULL, nsim=99, correction="trans", ngrid=200)

Arguments

A	A marked point pattern with the ppp format of spatstat.
lambda	Optional. Values of the estimated intensity function as a pixel image (object of class "im" of spatstat) giving the intensity values at all locations of A.
mod	A fitted model. An object of class ppm.
lifemark	Level of the marks of A which represents the "live" or "succes" cases.
prob	Numeric vector, with length equal to the number of points of A, represeting the fitted values of a logistic model fitted to A marks.
r	Numeric vector. The values of the argument $r$ at which the $K(r)$ functions should be evaluated.
nsim	Number of simulated point patterns to be generated when computing the envelopes.
correction	A character item selecting any of the options "border", "bord.modif", or "translate". It specifies the edge correction to be applied when computing K-functions.
ngrid	Dimensions (ngrid by ngrid) of a rectangular grid of locations where predict.ppm would evaluate the spatial trend of the fitted models.

Details

This function is a wrapper to compute the critical envelopes for Monte Carlo test of goodness-of-fit of (in-)homogeneous K functions, simulating from the fittted values of a logistic model (i.e. a binomial GLM with logit link) fitted to the marks ("failure", "success") of a "binomially"-marked point pattern. This is particularly interesting in plant ecology when considering alternatives to the random mortality hypothesis (Kenkel 1988). This hypothesis is usually tested building Monte Carlo envelopes from the "succesful" patterns resulting from a random labelling of a "binomially"-marked point pattern (this is equivalent to a random thinning of the whole pattern irrespective of the marks). As tree mortality is rarely random but instead can be modelled as a function of a certain number of covariates, the most natural alternative to the random mortality hypothesis is the logistic mortality hypothesis, that can be tested thinning the original pattern of trees with retention probabilities defined by the fitted values of a logistic model (Batista and Maguire 1998, Olano et al. 2008).

Kinhom.log will compute the envelopes by thinning the unmarked point pattern A with retention probabilities prob. If no prob vector is provided, all points will be thinned with the same probability ( number of "live" points / number of points ), i.e. Kinhom.log will compute random thinning envelopes.

Kinhom.log will compute envelopes both to homogeneous and inhomogeneous K functions. If no lambda or mode arguments are provided, Kinhom.log assumes that the original pattern is homogeneous and will use a constant lambda to compute the inhomogeneous K (i.e. it will compute the homogeneous K). The most convenient use with inhomogeneous point patterns is to provide the argument mod with an inhomogeneous Poisson model fitted to the original pattern of 'live' points (with spatstat function ppm; see the examples). This model will be used to compute (and to update in the simulations) the inhomogeneous trend (i.e. the "lambda") of the patterns. If the argument lambda is provided but not mod, these lambda will be used as a covariate to fit an inhomogeneous Poisson model that will be used to compute (and to update in the simulations) the inhomogeneous spatial trend.

Kinhom.log will produce an object of class 'ecespa.kci' that can be easily ploted (see the examples). This is accomplished by the S3 ploth method plot.ecespa.kci; it will plot the K-function and its envelopes (actually, it will plot the most usual L-function = $sqrt[K(r)/pi]-r$).

Value

Kinhom.log returns an object of class ecespa.kci, basically a list with the following items:

r	Numeric vector. The values of the argument $r$ at which the $K(r)$ functions have been evaluated.
kia	Numeric vector. Observed (in-)homogeneous K function.
kia.s	Matrix of simulated (in-)homogeneous K functions.
datanamea	Name of point pattern A.
modnamea	Name of model mod.
type	Type of analysis. Always "Kinhom.log".
probname	Name of the vector of fitted retention probabilities prob.
modtrend	Spatial trend (formula) of the model mod.
nsim	Number of simulations.

Warning

As this implementation involves the use of images as the means of evaluation of the (inhomogeneous) spatial trend, and a mask based on those images will be used as the point pattern window, the "Ripley's" or "isotropic" edge correction can not be employed.

Author(s)

Marcelino de la Cruz Rot

References

Batista, J.L.F. and Maguire, D.A. 1998. Modelling the spatial structure of tropical forests. For. Ecol. Manag., 110: 293-314.

Kenkel, N.C. 1988. Pattern of self-thinning in Jack Pine: testing the random mortality hypothesis. Ecology, 69: 1017-1024.

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. doi:10.1051/forest:2008074.

Examples

data(quercusvm)
   
   # set the number of simulations (nsim=199 or larger for real analyses)
   nsim<- 19

   # 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",  prob=probquercus, nsim=nsim)

   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=nsim, mod=mod2)

   
   ################################ 
   ## An example of homogeneous random thinning:
      
   cosa <- Kinhom.log(A=quercusvm, lifemark="0", nsim=nsim)
   
   plot(cosa)

---

Mark-weighted K-function

Description

This is a functional data summary for marked point patterns that measures the joint pattern of points and marks at different scales determined by $r$.

Usage

Kmm(mippp, r = 1:10, nsim=NULL)

## S3 method for ploting objects of class 'ecespa.kmm':
## S3 method for class 'ecespa.kmm'
plot(x, type="Kmm.n", q=0.025, 
            xlime=NULL, ylime=NULL,  maine=NULL, add=FALSE, kmean=TRUE,
            ylabe=NULL, xlabe=NULL, lty=c(1,2,3), col=c(1,2,3), lwd=c(1,1,1),
             ...)

Arguments

mippp	A marked point pattern. An object with the ppp format of spatstat.
r	Sequence of distances at which Kmm is estimated.
nsim	Number of simulated point patterns to be generated when computing the envelopes.
x	An object of class 'ecespa.kmm'. The result of applying Kmm to a marked point pattern.
type	Type of mark-weighted K-function to plot. One of "Kmm" ("plain" mark-weighted K-function) or "Kmm.n" (normalized mark-weighted K-function).
q	Quantile for selecting the simulation envelopes.
xlime	Max and min coordinates for the x-axis.
ylime	Max and min coordinates for the y-axis.
maine	Title to add to the plot.
add	Logical. Should the kmm.object be added to a previous plot?
kmean	Logical. Should the mean of the simulated Kmm envelopes be ploted?
ylabe	Text or expression to label the y-axis.
xlabe	Text or expression to label the x-axis.
lty	Vector with the line type for the estimated Kmm function, the simulated envelopes and the mean of the simulated envelopes.
col	Vector with the color for the estimated Kmm function, the simulated envelopes and the mean of the simulated envelopes.
lwd	Vector with the line width for the estimated Kmm function, the simulated envelopes and the mean of the simulated envelopes.
...	Additional graphical parameters passed to plot.

Details

Penttinnen (2006) defines $Kmm(r)$, the mark-weighted $K$-function of a stationary marked point process $X$, so that

$lambda*Kmm(r) = Eo[sum(mo*mn)]/mu^2$

where $lambda$ is the intensity of the process, i.e. the expected number of points of $X$ per unit area, $Eo[ ]$ denotes expectation (given that there is a point at the origin); $m0$ and $mn$ are the marks attached to every two points of the process separated by a distance $<= r$ and $mu$ is the mean mark. It measures the joint pattern of marks and points at the scales determmined by $r$. If all the marks are set to 1, then $lambda*Kmm(r)$ equals the expected number of additional random points within a distance $r$ of a typical random point of $X$, i.e. $Kmm$ becomes the conventional Ripley's $K$-function for unmarked point processes. As the $K$-function measures clustering or regularity among the points regardless of the marks, one can separate clustering of marks with the normalized weighted K-function

$Kmm.normalized(r) = Kmm(r)/K(r)$

If the process is independently marked, $Kmm(r)$ equals $K(r)$ so the normalized mark-weighted $K$-function will equal 1 for all distances $r$.

If nsim != NULL, Kmm computes 'simulation envelopes' from the simulated point patterns. These are simulated from nsim random permutations of the marks over the points coordinates. This is a kind of pointwise test of $Kmm(r) == 1$ or $normalized Kmm(r) == 1$ for a given $r$.

Value

Kmm returns an object of class 'ecespa.kmm', basically a list with the following items:

dataname	Name of the analyzed point pattern.
r	Sequence of distances at which Kmm is estimated.
nsim	Number of simulations for computing the envelopes, or NULL if none.
kmm	Mark-weighted $K$-function.
kmm.n	Normalized mark-weighted $K$-function.
kmmsim	Matrix of simulated mark-weighted $K$-functions, or or NULL if none.
kmmsim.n	Matrix of simulated normalized mark-weighted $K$-functions, or or NULL if none.

Note

This implementation estimates $Kmm(r)$ without any correction of border effects, so it must be used with caution. However, as $K(r)$ is also estimed without correction it migth compensate the border effects on the normalized $Kmm$-function.

Author(s)

Marcelino de la Cruz Rot

References

De la Cruz, M. 2008. Métodos para analizar datos puntuales. En: Introducción al Análisis Espacial de Datos en Ecología y Ciencias Ambientales: Métodos y Aplicaciones (eds. Maestre, F. T., Escudero, A. y Bonet, A.), pp 76-127. Asociación Española de Ecología Terrestre, Universidad Rey Juan Carlos y Caja de Ahorros del Mediterráneo, Madrid.

Penttinen, A. 2006. Statistics for Marked Point Patterns. In The Yearbook of the Finnish Statistical Society, pp. 70-91.

See Also

markcorr

Examples

## Figure 3.10 of De la Cruz (2008):
  # change r to r=1:100
  
   r = seq(1,100, by=5)
  
  data(seedlings1)
  
  data(seedlings2)
  
  s1km <- Kmm(seedlings1, r=r)
  
  s2km <- Kmm(seedlings2, r=r)
  
  plot(s1km, ylime=c(0.6,1.2), lwd=2, maine="", xlabe="r(cm)")

  plot(s2km,  lwd=2, lty=2, add=TRUE )

  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")
## Not run: 
## A pointwise test of normalized Kmm == 1 for seedlings1:

   s1km.test <- Kmm(seedlings1, r=1:100, nsim=99)

   plot(s1km.test,  xlabe="r(cm)")

  
## End(Not run)

---

Lotwick's and Silverman's combined estimator of the marked K-function

Description

For a multitype point pattern, calculates the combined estimator of the bivariate $Kij(r)$ and $Kji(r)$ functions.

Usage

Kmulti.ls(X, I, J, r = NULL, corre = "isotropic")

Arguments

X	Multitype marked point pattern. An object with the ppp format of spatstat.
I	Subset index specifying the points of the first pattern.
J	Subset index specifying the points of the second pattern.
r	Numeric vector. The values of the argument r at which the multitype K function $K^*ij(r)$ should be evaluated.
corre	A character item selecting any of the options "border", "bord.modif", "isotropic", "Ripley" or "translate", as described in Kest. It specifies the edge correction(s) to be applied.

Details

As a consequence of edge effects, the estimators $Kij(r)$ and $Kji(r)$ of the same bivariate pattern could differ. $K^*ij(r)$ is the combined estimator defined by Lotwick and Silverman (1982) as

$nj*Kij(r)+ ni*Kji(r) / (ni + nj) ,$

$ni$ and $nj$ being respectively the number of points in $I$ and $J$.

Value

An object of class "fv" (see fv.object). Essentially a data frame containing numeric columns

r	The values of the argument r at which the function $K^*ij(r)$ has been estimated

.

theo	The theoretical value of $K*ij(r)$ for a marked Poisson process, namely $pi * r^2$

.

together with a column or columns named "border", "bord.modif", "iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function $K^*ij(r)$ obtained by the edge corrections named.

Note

Kmulti.ls is a wrapper for a convenient use of the Kmulti function of spatstat. Please refer to its help page for additional documentation.

Author(s)

Marcelino de la Cruz

References

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. doi:10.1111/j.2517-6161.1982.tb01221.x.

Examples

data(amacrine)

plot(Kmulti.ls(amacrine, I=amacrine$marks=="on", J=amacrine$marks=="off", 
	 corre="isotropic"), sqrt(./pi)-r~r, main="")

# compare with Kmulti

plot(Kmulti(amacrine, I=amacrine$marks=="on", J=amacrine$marks=="off"),
         sqrt(iso/pi)-r~r, add=TRUE, col=3)

plot(Kmulti(amacrine, J=amacrine$marks=="on", I=amacrine$marks=="off"),
      sqrt(iso/pi)-r~r, add=TRUE, col=4)

---

Loosmore and Ford Goodness of Fit Test

Description

Performs the Loosmore and Ford (2006) test or the Maximum Absolute Deviation test for a spatial point pattern.

Usage

LF.gof(X, rmin=NULL, rmax=NULL, na.rm=TRUE)

Arguments

X	An object resulting from the function envelope, i.e., with an attribute "simfuns" (obtained using the argument savefuns=TRUE in envelope) ,which is an object of class "fv" containing the summary functions computed for each of the simulated patterns.
rmin	Minimum value of the function argument r over which the maximum absolute deviation, or the integral, will be computed for the test.
rmax	Maximum value of the function argument r over which the maximum absolute deviation, or the integral, will be computed for the test.
na.rm	Should NA's be removed to compute the integral?

Details

These function perform a tests for goodness-of-fit of a point pattern dataset to a point process model, based on Monte Carlo simulation from the model. The simulations should have been previously computed with the function envelope, applied with the argument savefuns=TRUE in order to save all the simulated functions, required for the computation of the test.

The test, popularized in the ecological field by Loosmore and Ford (2006) is also described in Diggle (2003, page 14), and according to Baddeley and Turner (2005) also in Diggle (1986) and Cressie (1991, page 667, equation (8.5.42)). If the arguments rmin and rmax are set to NULL, the integral of the GoF statistics will be computed over the complete range of r values.

Value

A list with the following components:

u

The GoF statistic, i.e., the value of the integral over the range of r's

p

The p-value of the test

na.count.by.r

Number of NA values for each r. It helps to evaluate the reliability of the computed u's, specially for small r's

Author(s)

Marcelino de la Cruz

References

Cressie, N.A.C. (1991) Statistics for spatial data. John Wiley and Sons, 1991.

Diggle, P. J. (1986). Displaced amacrine cells in the retina of a rabbit : analysis of a bivariate spatial point pattern. J. Neuroscience Methods 18, 115-125.

Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.

Loosmore, N.B. and Ford, E.D. (2006) Statistical inference using the G or K point pattern spatial statistics. Ecology 87, 1925-1931. doi:10.1890/0012-9658(2006)87[1925:SIUTGO]2.0.CO;2.

See Also

dclf.test for an alternative implementation of the test in spatstat.

---

Mark-sum measure

Description

An exploratory data analysis technique for marked point patterns. The marked point pattern is mapped to a random field for visual inspection.

Usage

marksum(mippp, R = 10, nx = 30, ny = 30)

## S3 method for ploting objects of class 'ecespa.marksum':
## S3 method for class 'ecespa.marksum'
plot(x, what="normalized",  contour=FALSE, grid=FALSE,
	ribbon=TRUE,col=NULL ,main=NULL,xlab="",ylab="",...)

Arguments

mippp	A marked point pattern. An object with the ppp format of spatstat.
R	Radius. The distance argument r at which the mark-sum measure should be computed
nx	Grid density (for estimation) in the x-side.
ny	Grid density (for estimation) in the y-side.
x	An object of class 'ecespa.marksum'. Usually, the result of applying marksum to a point pattern.
what	What to plot. One of "marksum" (raw mark sum measure), "point" (point sum measure) or "normalized" (normalized sum measure).
contour	Logical; if "TRUE" add contour to map.
grid	Logical; if "TRUE" add marked grid to map.
ribbon	Logical; if "TRUE" add legend to map.
col	Color table to use for the map ( see help file on image for details).
main	Text or expression to add as a title to the plot.
xlab	Text or expression to add as a label to axis x.
ylab	Text or expression to add as a label to axis y.
...	Additional parameters to Smooth.ppp, density.ppp or as.mask, to control the parameters of the smoothing kernel, pixel resolution, etc.

Details

Penttinen (2006) defines the mark-sum measure as a smoothed summary measuring locally the contribution of points and marks. For any fixed location $x$ within the observational window and a distance $R$, the mark-sum measure $S[R](x)$ equals the sum of the marks of the points within the circle of radius $R$ with centre in $x$. The point-sum measure $I[R](x)$ is defined by him as the sum of points within the circle of radius $R$ with centre in $x$, and describes the contribution of points locally near $x$. The normalized mark-sum measure describes the contribution of marks near $x$ and is defined (Penttinen, 2006) as

$S.normalized[R](x) = S[R](x)/I[R](x)$

This implementation of marksum estimates the mark-sum and the point-sum measures in a grid of points whose density is defined by nx and ny.

Value

marksum gives an object of class 'ecespa.marksum'; basically a list with the following elements:

normalized	Normalized mark-sum measure estimated in the grid points.
marksum	Raw mark-sum measure estimated in the grid points.
pointsum	Point-sum measure estimated in the grid points.
minus	Point-sum of the grid points. For advanced use only.
grid	Grid of points.
nx	Density of the estimating grid in the x-side.
ny	Density of the estimating grid in the x-side.
dataname	Name of the ppp object analysed.
R	Radius. The distance argument r at which the mark-sum measure has been computed.
window	Window of the point pattern.

plot.ecespa.marksum plots the selected mark-sum measure.

Author(s)

Marcelino de la Cruz Rot

References

Penttinen, A. 2006. Statistics for Marked Point Patterns. In The Yearbook of the Finnish Statistical Society, pp. 70-91.

See Also

getis, related to the point-sum measure, and markstat for designing different implementations.

Examples

data(seedlings1)
   
 seed.m <- marksum(seedlings1, R=25)

 # raw mark-sum measure; sigma is bandwith for smoothing
 plot(seed.m, what="marksum", sigma = 5)  

 # point sum measure
 plot(seed.m, what="pointsum", sigma = 5) 
   
 # normalized  mark-sum measure
 plot(seed.m,  what="normalized", dimyx=200, contour=TRUE, sigma = 5) 

# the same with added grid and normalized  mark-sum measure
plot(seed.m,  what="normalized", dimyx=200,
      contour=TRUE, sigma = 5, grid=TRUE)

---

P-value for a discrete distribution on small sample data

Description

Computes the p-value for a two-sided hypothesis test following Dixon's (2002:145) description of the method of Agresti & Min (2001).

Usage

p2colasr(Z)

Arguments

Z	vector with the observed Z-score in the first position and all the simulated values behind.

Value

P-value of the two-sided hypothesis test

Note

This function is usually not to be called by the user. It is internally used by dixon2002.

Author(s)

Marcelino de la Cruz Rot

References

Agresti, A. & Min, Y. 2001. On small-sample confidence intervals for parameters in discrete distributions. Biometrics, 57: 963-971.

Dixon, P.M. 2002. Nearest-neighbor contingency table analysis of spatial segregation for several species. Ecoscience, 9(2): 142-151.

---

Fit the Poisson Cluster Point Process by Minimum Contrast

Description

Fits the Poisson Cluster point process to a point pattern dataset by the Method of Minimum Contrast.

Usage

pc.estK(Kobs, r, sigma2 = NULL, rho = NULL)
Kclust(r, sigma2, rho)

Arguments

Kobs	Empirical $K$-function.
r	Sequence of distances at which function $K$ has been estimated.
sigma2	Optional. Starting value for the parameter $sigma2$ of the Poisson Cluster process.
rho	Optional. Starting value for the parameter $rho$ of the Poisson Cluster process.

Details

The algorithm fits the Poisson cluster point process to a point pattern, by finding the parameters of the Poisson cluster model which give the closest match between the theoretical K function of the Poisson cluster process and the observed K function. For a more detailed explanation of the Method of Minimum Contrast, see mincontrast in spatstat or Diggle (2003: 86).

The Poisson cluster processes are defined by the following postulates (Diggle 2003):

PCP1	Parent events form a Poisson process with intensity $rho$.
PCP2	Each parent produces a random number of offspring, according to a probability distribution
	$p[s]: s = 0, 1, 2, ...$
PCP3	The positions of the offspring relative to their parents are distributed according to a bivariate pdf $h$.

This implementation asumes that the probability distribution $p[s]$ of offspring per parent is a Poisson distribution and that the position of each offspring relative to its parent follows a radially symetric Gaussian distribution with pdf

$h(x, y) = [1/(2*pi*sigma^2)]* exp[-(x^2+y^2)/(2*sigma^2)]$

The theoretical $K$-function of this Poisson cluster process is (Diggle, 2003):

$pi*r^2 + [1- exp(-r^2/4*sigma^2)]/rho$

The command Kclust computes the theoretical $K$-function of this Poisson cluster process and can be used to find some initial estimates of $rho$ and $sigma^2$. In any case, the optimization usually finds the correct parameters even without starting values for these parameters.

This Poisson cluster process can be simulated with sim.poissonc.

Value

sigma2	Parameter $sigma^2$.
rho	Parameter $rho$.

Note

The exponents $p$ and $q$ of the contrast criterion (see mincontrast) are fixed respectively to $p = 2$ and $q = 1/4$. The $rmin$ and $rmax$ limits of integration of the contrast criterion are set up by the sequence of values of $r$ and $Kobs$ passed to pc.estK.

Author(s)

Marcelino de la Cruz Rot, inspired by some code of Philip M. Dixon

References

Diggle, P. J. 2003. Statistical analysis of spatial point patterns. Arnold, London.

See Also

ipc.estK for fitting the inhomogeneous Poisson cluster process; some functions in spatstat ( matclust.estK and lgcp.estK) fit other appropriate processes for clustered patterns; mincontrast performs a more general implementation of the method of mimimum contrast.

Examples

data(gypsophylous)


# set the number of simulations (nsim=199 or larger for real analyses)
nsim<- 19

## Estimate K function ("Kobs").

gyps.env <- envelope(gypsophylous, Kest, correction="iso", nsim=nsim)

plot(gyps.env, sqrt(./pi)-r~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=nsim,
                    simulate=expression(sim.poissonc(gypsophylous,
		    sigma=sqrt(cosa.pc$sigma2), rho=cosa.pc$rho)))

plot(gyps.env.sim,  main="",legendpos="bottomleft")

---