ecespa (version 1.1-17)

K1K2: 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")

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.

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.

Author

Marcelino de la Cruz

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.

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. 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. 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)