**All the new and exciting developments in movement ecology demonstrate that a new era for the science of animal space use may be in the making. However, not all ecologists are that enthusiastic. On one hand we are confronted with new theoretical ideas, criticism of old dogma and off-piste hypotheses that are still in the process of being empirically scrutinized. On the other hand these developments also bring up the expected dose of controversy, confusion, and plain ignorance of what’s on the table. Taking the latter attitude would imply a risk of becoming a conformist, which should not be confounded with sound scientific skepticism. A conformist typically observe a new theoretical direction and its empirical support, but will tend to reject its implications “anyway” – even prior to testing the new approach and methods on one’s own data (or prior to waiting for others to do the job on their data). **

In a recent post, Stepping Away From the Markov Lamppost, I focused on one particular aspect from the heterogeneous zoo of new theoretical proposals. In this post I will be more concrete in my criticism of conformatory attitudes in wildlife ecology. Consider the kernel density estimate (KDE), which has a central position in studies on animal space use by estimating the utilization distribution (UD).

… we consider that the animals’ use of space can be described by a bivariate probability density function, the UD, which gives the probability density to relocate the animal at any place according to the coordinates (x, y) of this place. Calenge (2015), p14.

Many kinds of kernel functions exist, but there is a general agreement that the so-called “smoothing parameter”, *h*, is more crucial to the UD estimate than the qualitative choice of kernel function. Thus, the bivariate normal kernel is a common choice (and is the default choice in – for example – the adehabitatHR package for R).

KDE produces isopleths for a given *h*; contour lines representing expected equal-sized intensity of space use along the specified demarcation, given the actual sample of relocations (fixes), N. So – what is the problem?

**If the dynamics behind the pattern do not comply with standard statistical assumptions, the demarcated area from KDE (from a given h) will overcompensate for the effect from change in sample size of fixes, N. This surprising tendency for shrinking area with increasing N when using the KDE has repeatedly been documented for many species, like brown bear (Belant and Follmann 2002), cerulean warbler (**

**Barg et al. 2005), white-tailed deer (Fieberg and Börger 2012), mule deer (Schuler et al. 2014) and coyotes (Schuler et al. 2014).**

Despite this range of empirical studies showing non-trivial N-dependence on the KDE contour lines (*op. cit*.), the majority of ecologists seem to be quite ignorant. In other words, they show a conformist attitude and continue applying the standard method as if the documented shortcomings did not exist – or at best they are acceptable. 10-15 years of wake-up calls should perhaps be enough? Obviously not. As a result, we continue to see home range analyses with subjective and quasi-objective choices of KDE smoothing and other adjustments. This may block progress towards alternative space use models with potentially stronger predictive power (for example, with improved value for studies on habitat selection).

**What does the empirically observed “overcompensation for the effect of change of N” imply? For example, it means that the home-range ghost paradox (non-asymptotic space-use demarcation with increasing N) under the KDE protocol may be camouflaged by a statistical artifact, due to a specific statistical assumption inherent in the KDE. Instead of observing area increasing with N as under the repeatedly verified home range ghost expectation when using a range of other non-parametric demarcation methods (incidence, MCP, R/SD, …), the KDE shows a tendency to “buffer” this pattern and even tends to make home range area appear shrinking with increasing N.**

The dubious KDE assumption contains two facets;

- animal movement and space use, even under site fidelity conditions, is assumed to be mechanistic and thus Markov compliant (see several previous posts). This follows statistical-mechanically from the choice of (for example) the bivariate normal kernel whether
*h*is small or large. The Markov assumption is apparent also from the choice of model for simulations, for example the bivariate Ornstein-Uhlenbeck design (see below). - N-dependency is assumed to be statistically trivial, implying that given a sufficiently large N, a home range area asymptote is expected. The KDE is in fact ideally designed to “calibrate” for this small-sample artifact (Fieberg and Börger 2012).

My book is in some chapters devoted to arguing and documenting – both theoretically and empirically – that both assumptions may be wrong. The home-range generating process may not be Markovian, and a home range area asymptote may not be expected.

What is at stake, if one allows oneself to consider these nuisances to be more than a question of generators of some statistical noise? Stepping away from the Markov framework means that one has to consider non-mechanistic theory and accompanying statistical methods when studying home range data. Under this alternative framework, core concepts like “home range area asymptote”, “outlier fixes relative to core area fixes”, “home range area”, “home range overlap” and many others need to be critically evaluated and adjusted – possibly replaced – by similar concepts under the alternative theory. The standard framework should definitely not just be accepted *a priori*, because there exists a 50-100 year tradition to do so. Given the piling indications of a shaky theory and dubious methods, just moving on as usual will cement a conformist attitude.

Criticizing a broadly applied method in wildlife ecology should of course be supplemented with a feasible alternative. With respect to the two dubious KDE assumptions above, in my book I propose these alternatives:

**Exploring the bivariate Cauchy distribution**as an alternative candidate model for the kernel function, or – more simply – apply incidence (virtual grid cells containing at least one fix at the optimized grid resolution, CSSU). These approaches are better fit to cope with multi-scaled (even scale-free, to some extent) space use under influence of spatial memory.**Acknowledging that memory-influenced space use is expected to generate a statistical fractal**with respect to spatial fix dispersion. Consequently, one should explore 1/*c*from the Home range ghost formula as an expression for “space use intensity”, rather than auto-applying the traditional density of fixes approach (the UD), which is inherently non-fractal. As a first-level approach the density based UD provides a good representation of space use in coarse terms, but may not be a proper tool for more detailed analyses of habitat utilization and other aspects of ecological inference. I will be more specific in an upcoming post.

Finally, what does this post’s illustrations show? Using a commonly applied method in R (KDE under the adehabitatHR Package), the respective 50-99% isopleths for small and large samples of serially non-autocorrelated fixes tend to show a contraction of home range area for larger N. For example, while the 99% isopleth for N=1,000 demarcated ca 15% smaller area than the 99% isopleth for N=100, the difference was 49% smaller area for the 90% isopleths! According to the theory for the actual simulation, the MRW model, this contraction is in fact *predicted* from applying a demarcation method which implies a “smooth” and differentiable function (represented by the density surface of the UD) upon a “rugged” and *non-differentiable* kind of density surface (fractal dispersion of fixes, emerging from a tendency for self-reinforcing re-visits to previous locations).

Observe that “rugged” in the context of multi-scaled space use is a different concept than a multi-modal UD. The latter is inherently smooth despite revealing more peaks in the UD at fine resolutions, regardless of the actual magnitude of the KDE’s smoothing parameter.

Interestingly, the KDE’s N-dependency issue has been documented also in other simulation studies, for example based on the bivariate Ornstein-Uhlenbeck model; a mechanistic convection model for the home range (Fieberg 2007). The bivariate Ornstein-Uhlenbeck model, due to its combination of Markov compliance and scale-specificity, simulates data that is statistical-mechanically compliant with the concept of a “smooth” UD and also with the classical area asymptote. However, the N-dependency may be due to serially autocorrelated samples (see table 1 in Fieberg 2007, and an upcoming post *in prep.*).

**The MRW example above is based on serially non-autocorrelated samples. Thus, MRW is the first theoretical model that is able to demonstrate coherence between simulations and real data with respect to the KDE’s N-dependency issue in addition to showing coherence with the empirically observed fractal structure of the UD and the non-asymptotic HR area property (the home range ghost).**

So here you have it again. Are you ignoring the empirically verified weaknesses (or paradoxes) inherent to the KDE and similar statistical models like Brownian bridge? Are you ignoring a statistical-mechanical theory that claims to offer a solution to these paradoxes, under a broader statistical-mechanical approach? In other words, are you a conformist, or are you ready to engage in a sound critical evaluation of the proposed alternative methods?

If you are reading these questions you are *not* a conformist. They have already jumped off further up.

REFERENCES

Barg, J. J., J. Jones, and R. J. Robertson. 2005. Describing breeding territories of migratory passerines: suggestions for sampling, choice of estimator, and delineation of core areas. Journal of Animal Ecology 74:139-149

Belant, J. L., and E. H. Follmann. 2002. Sampling considerations for American black and brown bear home range and habitat use. Ursus 13:299-315.

Calenge, C. 2015. Home Range Estimation in R: the adehabitatHR Package.

Fieberg, J. 2007. Kernel density estimators of home range: smoothing and the autocorrelation red herring. Ecology 88:1059-1066.

Fieberg, J., and L. Börger. 2012. Could you please phrase “home range” a question? Journal of Mammalogy 93:890-902.

Schuler, K. L., G. M. Schroeder, J. A. jenks, and J. G. Kie. 2014. Ad hoc smoothing parameter performance in kernel estimates of GPS-derived home ranges. Wildlife Biology 20:259-266.