### KDE – an Obstacle to Ecological Inference?

**In the two foregoing posts I have been quite critical to one of the preferred home range analytical tools among wildlife ecologists, the kernel density estimation (KDE). I have underscored two main points; the well-documented (but mainly ignored) N-paradox and the issue this raises with respect to camouflaging important aspects of the animals’ space use behaviour. In this post I provide an additional illustration of the latter point, and I discuss the serious problem this shortcoming raises for ecological inference based on – for example – GPS fixes (for details and additional examples, see my book).**

First, a more detailed confirmatory illustration of KDE’s N-paradox. Above I have plotted the 99% isopleths based on a range of sample sizes of fixes, N (open symbols) and compared the result to the alternative area demarcation method incidence (number of non-empty virtual grid cells; filled symbols). The data sets are the series from the two foregoing posts.

The “KDE compatible” space use process, MemRW, shows a slight area inflation for small N, but no more than a tolerable nuisance. In short: the KDE provides a good compensation for the statistically trivial small-N-artifact on home range demarcation when the pattern-generating behaviour is mechanistic (Markov-compliant) and scale-specific (correlated random walk in my example).

What is worrying, though, is that KDE shows a similarly “flat” regression, or even an increased tendency for smaller area with increasing N for multi-scaled movement, which I have argued is substantially more commonin the real world (perhaps dominating) in comparison to the standard assumption in animal ecology.

As is well known for readers of my book and my blog, multi-scaled random walk (MRW) generates the home range ghost paradox: area expanding proportionally with square root of N, due to the inherent scale-free nature of the behaviour in statistical-mechanical terms. This intrinsic process property leads to emergence of a statistical fractal for the scatter of fixes, resulting in a power law-like area demarcation with respect to N-dependence, with exponent ca 0.5. Home range ghost compliance is verified in the illustration above, which also shows similarly close compliance with the theoretical expectation for MemRW (exponent ca 0.25).

**In other words, KDE is not able to differentiate area demarcation from scale-specific and a scale-free space use, as shown by the two open-symbol plots above. This crucial difference between the two scenaria is camouflaged by KDE-specific statistical assumptions that are not satisfied by a scale-free process.**

Thus, two paradoxes are arising under the current home range paradigm:

**The home range ghost**: an unexpected positive N-dependency on area demarcation when incidence is applied as the area demarcator (or other methods like minimum convex polygon or R/SD). These alternative methods are not dependent on assumptions of a scale-specific process.**The KDE issue**: the frequently observed zero-to-negative N-dependency on area demarcation when the KDE is applied on real data, with increasing problem for isoclines smaller than 99%. Interestingly, the MRW theory resolves both paradoxes, which in fact represent statistical expectations under this extended framework.

Does the discrepancies matter? In my view, it may be a crucial obstacle for proper ecological inference based on GPS fixes to apply a statistical toolbox (containing, for example KDE) that is not able to differentiate between scale-specific and a multi-scaled space use.

Equally concerning is KDE’s buffering effect on the home range ghost property in scenaria where the animal has in fact behaved in a multi-scaled manner. This inability to differentiate will tend to cement the view that the proper descriptor of space use intensity is local density of relocations; i.e., the utilization distribution (UD), also for a scale-free process. Many of my foregoing posts deal with this issue; explaining and exploring the difference between quantifying local intensity of space use by the classic “density of fixes” proxy versus the 1/c parameter (the inverse of the individual’s characteristic scale of space use, CSSU).

In later posts I intend to explore the KDE issues from the perspective of serially auto-correlated fixes, and in that context also bring in an evaluation of the KDE-related Brownian bridge methods.