## Multiscale second-order neighbourhood analysis of a multivariate spatial point pattern

### Description

(Formerly `kijfun`) Computes a set of K- and K12-functions for all possible pairs of marks (p,q) in a multivariate spatial point pattern defined in a simple (rectangular or circular) or complex sampling window (see Details).

### Usage

``` kpqfun(p, upto, by)
```

### Arguments

 `p ` a `"spp"` object defining a multivariate spatial point pattern in a given sampling window (see `spp`). `upto ` maximum radius of the sample circles (see Details). `by ` interval length between successive sample circles radii (see Details).

### Details

Function `kpqfun` is simply a wrapper to `kfun` and `k12fun`, which computes either K(r) for points of mark p when p=q or K12(r) between the marks p and q otherwise.

### Value

A list of class `"fads"` with essentially the following components:

 `r ` a vector of regularly spaced distances (`seq(by,upto,by)`). `labpq ` a vector containing the (p,q) paired levels of `p\$marks`. `gpq ` a data frame containing values of the pair density functions g(r) and g12(r). `npq ` a data frame containing values of the local neighbour density functions n(r) and n12(r). `kpq ` a data frame containing values of the K(r) and K12(r) functions. `lpq ` a data frame containing values of the modified L(r) and L12(r) functions. ` ` Each component except `r` is a data frame with the following variables: `obs ` a vector of estimated values for the observed point pattern. `theo ` a vector of theoretical values expected under the null hypotheses of spatial randomness (see `kfun`) and population independence (see `k12fun`).

### Note

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

### Author(s)

Raphael.Pelissier@ird.fr

`plot.fads`, `spp`, `kfun`, `k12fun`, `kp.fun`.

### Examples

```  data(BPoirier)
BP <- BPoirier
## Not run: multivariate spatial point pattern in a rectangle sampling window
swrm <- spp(BP\$trees, win=BP\$rect, marks=BP\$species)
kpqswrm <- kpqfun(swrm, 25, 1)
plot(kpqswrm)

## Not run: multivariate spatial point pattern in a circle with radius 50 centred on (55,45)
swcm <- spp(BP\$trees, win=c(55,45,45), marks=BP\$species)
kpqswcm <- kpqfun(swcm, 25, 1)
plot(kpqswcm)

## Not run: multivariate spatial point pattern in a complex sampling window
swrtm <- spp(BP\$trees, win=BP\$rect, tri=BP\$tri2, marks=BP\$species)
kpqswrtm <- kpqfun(swrtm, 25, 1)
plot(kpqswrtm)

```