k12fun {ads} | R Documentation |

Computes estimates of the intertype *K12*-function and associated neighbourhood functions from a bivariate spatial point pattern
in a simple (rectangular or circular) or complex sampling window. Computes optionally local confidence limits of the functions
under the null hypotheses of population independence or random labelling (see Details).

k12fun(p, upto, by, nsim=0, H0=c("pitor","pimim","rl"), prec=0.01, nsimax=3000, conv=50, rep=10, alpha=0.01, marks)

`p` |
a |

`upto` |
maximum radius of the sample circles (see Details). |

`by` |
interval length between successive sample circles radii (see Details). |

`nsim` |
number of Monte Carlo simulations to estimate local confidence limits of the selected null hypothesis (see Details).
By default |

`H0` |
one of |

`prec` |
if |

`nsimax` |
if |

`conv` |
if |

`rep` |
if |

`alpha` |
if |

`marks` |
by default c(1,2), otherwise a vector of two numbers or character strings identifying the types (the |

Function `k12fun`

computes the intertype *K12(r)* function of second-order neighbourhood analysis and the associated functions *g12(r)*,
*n12(r)* and *L12(r)*.

For a homogeneous isotropic bivariate point process of intensities *λ1* and *λ2*,
the second-order property could be characterized by a function *K12(r)* (Lotwick & Silverman 1982), so that the expected
number of neighbours of type 2 within a distance *r* of an arbitrary point of type 1 is:
*N12(r) = λ2*K12(r)*.

*K12(r)* is an intensity standardization of *N12(r)*: *K12(r) = N12(r)/λ2*.

*n12(r)* is an area standardization of of *N12(r)*: *n12(r) = N12(r)/(π*r^2)*, where *π*r^2* is the area of the disc of radius *r*.

*L12(r)* is a linearized version of *K12(r)*, which has an expectation of 0 under population independence: *L12(r) = √(K12(r)/π)-r*. *L12(r)* becomes positive when the two population show attraction and negative when they show repulsion.
Under the null hypothesis of random labelling, the expectation of *L12(r)* is *L(r)*. It becomes greater than *L(r)* when the types tend to be positively correlated and lower when they tend to be negatively correlated.

*g12(r)* is the derivative of *K12(r)* or bivariate pair density function, so that the expected
number of points of type 2 at a distance *r* of an arbitrary point of type 1 (i.e. within an annuli between two successive circles with radii *r* and *r-by*) is:
*O12(r) = λ2*g12(r)* (Wiegand & Moloney 2004).

The program introduces an edge effect correction term according to the method proposed by Ripley (1977)
and extended to circular and complex sampling windows by Goreaud & Pelissier (1999).

Theoretical values under the null hypothesis of either population independence or random labelling as well as
local Monte Carlo confidence limits and p-values of departure from the null hypothesis (Besag & Diggle 1977) are estimated at each distance *r*.

The population independence hypothesis assumes that the location of points of a given population is independent from the location
of points of the other. It is therefore tested conditionally to the intrinsic spatial pattern of each population. Two different procedures are available:
`H0="pitor"`

just shifts the pattern of type 1 points around a torus following Lotwick & Silverman (1982); `H0="pimim"`

uses a mimetic point process (Goreaud et al. 2004)
to mimic the pattern of type 1 points (see `mimetic`

.

The random labelling hypothesis `"rl"`

assumes that the probability to bear a given mark is the same for all points of the pattern and
doesn't depends on neighbours. It is therefore tested conditionally to the whole spatial pattern, by randomizing the marks over the points'
locations kept unchanged (see Goreaud & Pelissier 2003 for further details).

A list of class `"fads"`

with essentially the following components:

`r ` |
a vector of regularly spaced out distances ( |

`g12 ` |
a data frame containing values of the bivariate pair density function |

`n12 ` |
a data frame containing values of the bivariate local neighbour density function |

`k12 ` |
a data frame containing values of the intertype function |

`l12 ` |
a data frame containing values of the modified intertype function |

` ` |
Each component except |

`obs ` |
a vector of estimated values for the observed point pattern. |

`theo ` |
a vector of theoretical values expected under the selected null hypothesis. |

`sup ` |
(optional) if |

`inf ` |
(optional) if |

`pval ` |
(optional) if |

There are printing and plotting methods for `"fads"`

objects.

Besag J.E. & Diggle P.J. 1977. Simple Monte Carlo tests spatial patterns. *Applied Statistics*, 26:327-333.

Goreaud F. & Pelissier R. 1999. On explicit formulas of edge effect correction for Ripley's K-function. *Journal of Vegetation Science*, 10:433-438.

Goreaud, F. & Pelissier, R. 2003. Avoiding misinterpretation of biotic interactions with the intertype *K12*-function: population independence vs. random labelling hypotheses. *Journal of Vegetation Science*, 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*, 44:403-413.

Ripley B.D. 1977. Modelling spatial patterns. *Journal of the Royal Statistical Society B*, 39:172-192.

Wiegand, T. & Moloney, K.A. 2004. Rings, circles, and null-models for point pattern analysis in ecology. *Oikos*, 104:209-229.
Goreaud F., Loussier, B., Ngo Bieng, M.-A. & Allain R. 2004. Simulating realistic spatial structure for forest stands: a mimetic point process. In *Proceedings of Interdisciplinary Spatial Statistics Workshop*, 2-3 December, 2004. Paris, France.

`plot.fads`

,
`spp`

,
`k12val`

,
`kfun`

,
`kijfun`

,
`ki.fun`

,
`mimetic`

,
`kmfun`

.

data(BPoirier) BP <- BPoirier ## Not run: spatial point pattern in a rectangle sampling window of size [0,110] x [0,90] swrm <- spp(BP$trees, win=BP$rect, marks=BP$species) #testing population independence hypothesis k12swrm.pi <- k12fun(swrm, 25, 1, 500, marks=c("beech","oak")) plot(k12swrm.pi) #testing random labelling hypothesis k12swrm.rl <- k12fun(swrm, 25, 1, 500, H0="rl", marks=c("beech","oak")) plot(k12swrm.rl) ## Not run: spatial point pattern in a circle with radius 50 centred on (55,45) swc <- spp(BP$trees, win=c(55,45,45), marks=BP$species) k12swc.pi <- k12fun(swc, 25, 1, 500, marks=c("beech","oak")) plot(k12swc.pi) ## Not run: spatial point pattern in a complex sampling window swrt.rl <- spp(BP$trees, win=BP$rect, tri=BP$tri2, marks=BP$species) k12swrt.rl <- k12fun(swrt.rl, 25, 1, 500, H0="rl",marks=c("beech","oak")) plot(k12swrt.rl) ## Not run: testing population independence hypothesis requires minimizing the outer polygon xr<-range(BP$tri3$ax,BP$tri3$bx,BP$tri3$cx) yr<-range(BP$tri3$ay,BP$tri3$by,BP$tri3$cy) rect.min<-swin(c(xr[1], yr[1], xr[2], yr[2])) swrt.pi <- spp(BP$trees, window = rect.min, triangles = BP$tri3, marks=BP$species) k12swrt.pi <- k12fun(swrt.pi, 25, 1, nsim = 500, marks = c("beech", "oak")) plot(k12swrt.pi)

[Package *ads* version 1.5-5 Index]