These are complicated questions, and yes, it does depend on your data but also on your hypothesis. For example you may have an explicit hypothesis regarding boat activity, requiring a set of sigma (bandwidth) values to critically evaluate, whereas the sigma for the turtles could be based simply on a maximum dispersal distance. Thinking through the intent of your analysis is the first step in defining kernel forms and bandwidths.
In ESRI parlance, you really do not need to consider the functional form of the kernel however, this can be a very useful aspect of kernel estimation and should not be ignored due to software limitations. You can change the form of the kernel to approximate different spatial expectations eg., spatial die-off rates thus, moving away from fixed Gaussian assumptions eg., a liner die-off could be represented using an Triangular kernel or percentile forms can be modeled with a Epanechnikov kernel or an explicit die-off distance (process is uniform until d) could be represented using a uniform kernel. Both Python and R have libraries for very robust kernel estimation.
There are some automatic bandwidth selection methods however, the key consideration is what spatial order process are you wanting to represent. A first order approach is going to produce a very smooth surface, with a large sigma, that represents more of a global trend in the data (think first order polynomial). Whereas, a second-order method will emphasis localized density processes, using a small sigma (higher order polynomial).
Here is some general guidance on univariate kernel density sigma selection. Pseudocode for two simple sigma selection approaches are presented. The Scott approach represents first-order spatial process where Stoyan's approach represent very strong second-order (local) spatial process.
Scott's Rule for Bandwidth Selection first-order (Scott 1992)
n = number of points
sdx = sqrt(variance(x.coord))
sdy = sqrt(variance(y.coord))
bw = (sdx, sdy) * n^(-1/6)
Using pair-correlation function strong second-order (Stoyan & Stoyan 1995)
n = number of points
a = total area of points envelope
bw = 0.15 / sqrt(5 * n / a)
Below are some citations for other sigma selection approaches. If your data is multivariate then these approaches fall short and more robust statistics should be implemented (Bowman 1984). Many of these statistics, for deriving sigma, are available in R as well as alternate kernel functions. The
spatstat packages are all very robust for kernel estimation. The
sp.kde function, in
spatialEco, provides a kernel density estimation that is more comparable to ESRI's but with more flexibility is defining kernel functions that can be used in conjugation with functions in
spatstat for modeling sigma.
I will also note that one can take a multi-scale approach and integrate across density estimates representing different sigmas. In evaluating niche overlap using volumes, Hurlbert (1978) give some guidance on the integration across densities/volumes. In application, the Doherty et al., (2016) paper utilizes the Hurlbert (1978) approach in evaluating sage grouse niche volumes.
- Minimize the mean-square error via cross validation, 2nd order
(Berman & Diggle 1989).
- Maximum likelihood cross validation, 2nd order (Loader 1999).
- Least-squares cross-validation matrix selector for multivariate data,
data driven (Bowman 1984).
Berman, M. and Diggle, P. (1989) Estimating weighted integrals of the second-order intensity of a spatial point process. Journal of the Royal Statistical Society, series B 51:81-92.
Bowman, A. (1984) An alternative method of cross-validation for the smoothing of kernel density estimates. Biometrika, 71:353-360.
Doherty K.E., J.S. Evans, P.S. Coates, L.M Juliusson, B.C. Fedy (2016) Importance of regional variation in conservation planning: A range-wide example of the Greater Sage-grouse. Ecosphere 7(10): e01462
Hurlbert, S.H., (1978) The Measure of Niche Overlap and Some Relatives. Ecology 59(1):66-77
Loader, C. (1999) Local Regression and Likelihood. Springer, New York.
Scott, D.W. (1992) Multivariate Density Estimation. Theory, Practice and Visualization. New York, Wiley.
Stoyan, D. and Stoyan, H. (1995) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.