## Multiscale local density of a spatial point pattern

### Description

Computes local density estimates of a spatial point pattern, i.e. the number of points per unit area, within sample circles of regularly increasing radii r, centred at the nodes of a grid covering a simple (rectangular or circular) or complex sampling window (see Details).

### Usage

```dval(p, upto, by, nx, ny)
```

### Arguments

 `p ` a `"spp"` object defining a 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). `nx,ny ` number of sample circles regularly spaced out in x and y directions.

### Details

The local density is estimated for a regular sequence of sample circles radii given by `seq(by,upto,by)` (see `seq`). The sample circles are centred at the nodes of a regular grid with size nx by ny. Ripley's edge effect correction is applied when the sample circles overlap boundary of the sampling window (see Ripley (1977) or Goreaud & P?Pelissier (1999) for an extension to circular and complex sampling windows). Due to edge effect correction, `upto`, the maximum radius of the sample circles, is half the longer side for a rectangle sampling window (i.e. 0.5*max((xmax-xmin),(ymax-ymin))) and the radius r0 for a circular sampling window (see `swin`).

### Value

A list of class `c("vads","dval")` with essentially the following components:

 `r ` a vector of regularly spaced out distances (`seq(by,upto,by)`). `xy ` a data frame of (nx*ny) observations giving (x,y) coordinates of the centres of the sample circles (the grid nodes). `cval ` a matrix of size (nx*ny,length(r)) giving the estimated number of points of the pattern per sample circle with radius r. `dval ` a matrix of size (nx*ny,length(r)) giving the estimated number of points of the pattern per unit area per sample circle with radius r.

### Warning

In its current version, function `dval` ignores the marks of multivariate and marked point patterns (they are all considered to be univariate patterns).

### Note

There are printing, summary and plotting methods for `"vads"` objects.

### Author(s)

Raphael.Pelissier@ird.fr

### References

Goreaud, F. and P?Pelissier, R. 1999. On explicit formula of edge effect correction for Ripley's K-function. Journal of Vegetation Science, 10:433-438.

P?Pelissier, R. and Goreaud, F. 2001. A practical approach to the study of spatial structure in simple cases of heterogeneous vegetation. Journal of Vegetation Science, 12:99-108.

Ripley, B.D. 1977. Modelling spatial patterns. Journal of the Royal Statistical Society B, 39:172-212.

`plot.vads`, `spp`.

### Examples

```  data(BPoirier)
BP <- BPoirier
## Not run: spatial point pattern in a rectangle sampling window of size [0,110] x [0,90]
swr <- spp(BP\$trees, win=BP\$rect)
dswr <- dval(swr,25,1,11,9)
summary(dswr)
plot(dswr)

## Not run: spatial point pattern in a circle with radius 50 centred on (55,45)
swc <- spp(BP\$trees, win=c(55,45,45))
dswc <- dval(swc,25,1,9,9)
summary(dswc)
plot(dswc)

## Not run: spatial point pattern in a complex sampling window
swrt <- spp(BP\$trees, win=BP\$rect, tri=BP\$tri1)
dswrt <- dval(swrt,25,1,11,9)
summary(dswrt)
plot(dswrt)
```