ecespa (version 1.1-17)

K012: 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")

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).

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.

Author

Marcelino de la Cruz

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).

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. tools:::Rd_expr_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")