ds.explanation.md

Distribution smoothing in Zonation

The alpha-value

The $\alpha$-value of biodiversity feature-specific scale of landscape use (parameter of negative exponential, $\alpha_j$). This parameter is only used when distribution smoothing is enabled. However, you always need to specify this parameter, and any numerical value will do fine if distribution smoothing is not used. The value indicates the range of connectivity of biodiversity features. For example, it may refer to how a species uses the surrounding landscape.

How to define the degree of smoothing for individual features. The $\alpha$-value indicates how species use the surrounding landscape and can be calculated based on, for example, the dispersal capability or the home range sizes of the species. It can be calculated as:

$\alpha=\frac{2}{Use\ of\ landscape\ (in\ same\ units\ as\ cell\ size)}$

It is extremely important to make sure that the distance units used in this calculation are consistent with the cell size units used in the feature distribution files. The above equation assumes that 'use of landscape' is given in the same units that are use for the cell size value in the feature distribution files (if these are .asc files, the cell size value is given in the 'cellsize' field in the first rows). For example, if the cell size in the distribution files is 1 km and the known/guesstimated mean dispersal capability of a species is 3 km, then the value of alpha for this feature is:

$\alpha=\frac{2}{3}\approx0.667$

Let the cell size of the distribution files is 100 meters. Let's calculate the corresponding $\alpha$-values for mean dispersal distances of 500, 1000 and 3000 meters:

(alpha.500 <- 2/500)

## [1] 0.004

(alpha.1000 <- 2/1000)

## [1] 0.002

(alpha.3000 <- 2/3000)

## [1] 0.0006667

Distribution smoothing

Distribution smoothing transforms the values of each pixel in the distribution file by the values of all the other pixel values according to a negative exponential kernel, where the above calulated alpha value is used as a multiplier.

When using smoothing, the value for species $j$ in a focal cell $i$ is

$O_ij'=\sum\limits_{x}\sum\limits_{y}exp(-\alpha d(x-u,y-r))O_ij$

where $O_ij$ is the original occurrence level of species $j$ at cell $i$. Cell $i$ is located in $(u,r)$ and $d(x-u, y-r)$ is the distance between locations $(x,y)$ and $(u,r)$. The summation is over the landscape grid and $\alpha_j$ is the parameter of the dispersal kernel for species $j$. This is a two-dimension kernel smoothing using a radially symmetric negative exponential (dispersal) kernel.

For the above calculated $\alpha$-values, the kernel takes the following values:

Note that the mean dispersal distances (500, 1000, and 3000 meters) on the x-axis correspond to value

$value=exp(-1)\approx 0.368$

on the y-axis.