We consider a response variable measured at n different sites denoted Yi (i stands for a site), L local variables which can be continuous or discrete and are denoted as xil (l stands for a local variable and i for a site) and K landscape variables denoted as zrk (k stands for a landscape variable and r for a polygon in the landscape). In the Bsiland method, the effect of landscape variables is modelled using buffers with (p_{{i},delta_{k}}^{k}), the percentage of the landscape variable k in a buffer of radius δk and centered on site i. Since the Bsiland model is based on the generalized linear models framework, the expected value of the response variable Yi is modelled as follows:
$$ mu_{i} = mu + sumlimits_{l in L} {alpha_{l} x_{i}^{l} } + sumlimits_{k in K} {beta_{k} p_{{i},delta_{k}}^{k} } $$
(1)
where µ is the intercept, αl and βk are the effects of local and landscape variables, respectively.
The Fsiland method is based on Spatial Influence Functions (SIFs) in a similar framework to Chandler & Hepinstall-Cymerman 9. To simplify computations, the entire study area is not considered as continuous but rasterized, i.e. pixelated on a regular grid, named R. The value of each landscape variable k at a pixel r is described in zrk. For instance, if the landscape variable k is a presence/absence variable, zrk is equal to one or zero. The expected value of the response variable Yi is then modelled as follows:
$$ mu_{i} = mu + sumlimits_{l in L} {alpha_{l} x_{i}^{l} } + sumlimits_{k in K} {beta_{k} } sumlimits_{r in R} {f_{{delta}_{k}} (d_{i,r} )z_{r}^{k} } $$
(2)
where fδk(.) is the SIF associated with the landscape variable k and di,r is the distance between the center of pixel r and the observation at site i. The SIF is a density function decreasing with the distance. The scale of effect of a landscape variable k is calibrated through the parameter δk, the mean distance of fδk. Two families of SIF are currently implemented in the siland package, exponential and Gaussian families defined as fδ(d) = 2/(πδ2)exp(–2d/δ) and fδ(d) = 1/(2δ)2exp(-π(d/2δ)2), respectively19. The effect of a landscape variable k is modelled by two parameters: an intensity parameter, βk describing its strength and its direction and a scale parameter, δk, describing how this effect declines with distance. Each pixel potentially has an effect on the response variable at any observation site. No set of scales of effects is initially determined. In Eq. 2, the sum on the regular grid R is an approximation of the integration on the continuous study area. The choice of the grid definition is a tradeoff between computing precision and computing time. The smallest the mesh size of the grid is, the better are the precision but the longer the computing time is (and the larger the required memory size is). The parameters estimation may be very sensitive to this mesh size. To obtain a reliable estimation, we recommend to ensure, after the estimation procedure, that mesh size is at least three times smaller than the smallest estimated SIF (see Supplementary Fig. S2 online for details). If not, it is recommended to proceed with a new estimation with a smaller mesh (by using the wd argument of the Fsiland function, set at 30 by default).
All parameters, µ, {α1,…, αK}, {β1,…, βK} but also {δ1,…, δK} are simultaneously estimated by likelihood maximization for both Bsiland and Fsiland methods. We have developed a sequential algorithm. At the initialization stage, values are arbitrarily defined for the {δ1,..,δK} scales parameters. In step A, the µ, {α1,.., αK}, {β1,.., βK} parameters are estimated using the classical maximization procedures implemented in the lm and glm functions knowing the fixed values of the scale parameters. In step B, the scale parameters are estimated by likelihood maximization knowing the parameters estimated in step A. The values of the scale parameters are then fixed at the new estimated values. Steps A and B are thus repeated until the relative increase in likelihood decreased below a threshold or the maximum number of repetitions is reached. Tests performed (obtained using the summary function) are similar to those given by summary.lm or summary.glm function (see R Core Team16 for details, this implies that tests are given conditionally to the estimated scale parameters.).
Source: Ecology - nature.com
