A Statistical-Mechanical Perspective on Site Fidelity – Part IV

In Part I of this set of posts I described animal space use from the perspective of ergodicity. This is a key concept of standard statistical mechanics, with similar importance to analysis of individual paths and home range data under the extended theoretical framework (the parallel processing conjecture, expressed by the MRW model). Below I elaborate on this theme. I indicate by simulation examples the transition from a fully ergodic state on the home range scale to a narrowing of this scale as the series of fixes become increasingly autocorrelated (higher-frequency sampling).

From a biophysical perspective, the following system description may point towards my most important theoretical development since the Scaling cube. This will become clearer as I gradually turn my upcoming posts from description of statistical-mechanical properties towards how to explore these properties in novel methods for ecological inference.

First consider the general principle that the transition from non-autocorrelated data towards increased time-dependency (stronger serial autocorrelation) narrows the spatio-temporal scale range for which the statistical-mechanical system description is applicable. In the limit of very high frequency sampling we reach the biological resolution, where the animal’s behaviour is directly observable. In this limit “the hidden layer” has vanished. In my book I introduced the hidden layer concept to explain the ergodic principle in the context of animal GPS series (and allowing for a statistical-mechanical system description), and related ergodicity to the degree of serial autocorrelation.

Thus, by varying sampling frequency of an animal’s path one passes a transitional state between two kinds of system representations of animal space use; direct observation of a systems’ dynamic development from successive causal events to an indirect observation of space use as the hidden layer is increased. The former description requires a high frequency observational approach, while the latter requires lower frequency of path sampling.

However, here comes the novel and interesting part. In several blog posts I have described another important theoretical concept – the individual’s characteristic scale of space use (CSSU). In previous posts I have underscored that its estimation requires serially non-autocorrelated fixes. In the following simulations of MRW I show for the first time (a) how the CSSU concept may be extended in a theoretically consistent manner to auto-correlated series, given that the hidden layer is still sufficiently deep to allow for a statistical-mechanical system description, and (b) how this extension towards a time-dependent process description has implications for the system’s CSSU characteristics.

Autocorr and CSSU

In the illustration above the standard result (serially non-autocorrrelated fixes) is shown by black/white circle symbols. After a “zooming excercise” to find the proper grid resolution for I(N) from the home ghost formula I(N) = cNz , CSSU is found when the y-intercept is close to the optimal, c ≈ 1 [log(c) ≈ 0]. At this grid resolution the power exponent is expected to be close to the default MRW condition, z ≈ 0.5. Filled symbols show the result from a large series, N=100,000. Open symbols show a separate analysis using the last 20% of the series; N=20,000. As expected, z and c are of similar magnitude, due to the series’ non-autocorrelation property. In this domain, z and c are expected to be independent both on sampling frequency and series length. Thus, CSSU estimation is resilient under this condition.

Then turn your focus to the coloured symbols, showing the average result from four replicates. The orange and red circles show log[I(N)] based on the same grid resolution as for the black/white circles, but now from strongly autocorrelated series as a result of higher sampling frequency. This is a project under development, but here follow some initial comments for this scenario.

  • Autocorrelated series require quite large N. The present example – based on averaging over 4 replicate series – shows instability of I(N) for small samples (N <≈ 250. The orange circles show the result for N = 100,000 and the red circles regard a subset of the last 20% of the series.
  • Observe an N-dependency expressed by the difference between the orange and the red circles. I(N) for a given N is somewhat larger (larger intercept) when using the last part of the series and the data are autocorrelated. The difference is indicated by the { character. I will return with an explanation of this effect in a later post. Quite trivial, in fact.
  • In addition to this N-dependency, incidence for a given N is also smaller when the series are autocorrelated and fixes are sampled at an even higher frequency than the coloured series above (not shown here). The stronger the autocorrelation, the smaller the y-intercept.
  • Given that the sampling frequency is still within the domain of providing sufficiently deep hidden layer relative to the individual’s true path, for large series the regression tends to confirm power exponent z ≈ 0.5, as for the non-autocorrelated series.
  • The smaller intercept, log(c), for autocorrelated series implies that a smaller grid scale is required to optimize towards log(c) ≈ 0. Apparently this implies that CSSU is non-stationary under varying degree of autocorrelation.
  • However, consider the red triangles, using N=20% of the total series (data similar to the red circles). The difference between the red circles and the red triangles is due to a change of grid scale to a 1:4 finer resolution, following the procedure in an upcoming post in May to optimize grid scale towards achieving log2(c) ≈ 0 and using z=0.5 as an assumption; i.e. interpolating to log(N) = 0 from Nmax. After this re-scaling of the grid, CSSU is similar in magnitude to the estimate for the non-autocorrelated sampling scheme, but a model extension is needed to account for the fact that the process is now observed in the time-dependent domain (I will return to this model variant later). In other words, incidence (non-empty grid cells) for a given N is of similar magnitude, but the cells are smaller. The model extension thus regards the rescaling factor to account for strength of time-dependency for the sake of estimating CSSU under this condition.
  • The relative change in spatial resolution to maintain log(c) = 0 is reflecting a similar narrowing of the upper limit of the spatial scale where system ergodicity is maintained. Thus, higher sampling frequency in the auto-correlated domain implies a narrowing of the scale range over which ergodicity is satisfied. If this range is further narrowed, we reach the transition from the statistical-mechanical to the biological system representation (see above).
  • Autocorrelation implies time dependent system description, and CSSU becomes a function of both space and time scales. In non-autocorrelated series, CSSU is independent of the time dimension.
  • In this paricular example, CSSU is 1/4 smaller for the given degree of autocorrelation, which is a function of average return frequency relative to path sampling frequency; see my book.

Population Dispersion under Scale-Free Memory Influence

Over the years the term “density” has provided a convenient variable for ecological inference, since it has a clear definition both at the individual level and the population level.  For example, GPS fix density for a given individual and sample size of fixes is proportional with the inverse of the demarcated home range size when this is the area unit. By assuming the standard asymptote model for home range area, fix density N/area is higher when area is smaller. The MRW model, which contradicts the asymptote assumption and is incompatible with the density surface (UD) as a proper proxy for local space use intensity, still presents a proxy variable for intensity of habitat use.

In the MRW model the traditional density=N/area is replaced by H=(√N)/c‘ where c‘ is the individual’s characteristic scale of space use, CSSU. Thus, H is proportional with the inverse of CSSU. Contrary to density, which assumes area is intrinsically constant (the asymptote), H adjusts for N-sensitive area demarcation by depending on c’ rater than on the area asymptote. How is H expected to perform as a proxy for ecological inference – intensity of habitat use – in comparison to the traditional N/area, if selection is estimated at the level of the local population? Simulations provide predictions.


A smaller home range for a given individual is generally found in superior habitats. The Figure above is copied from Gautestad and Mysterud (2010), and shows the simultaneous space use by 100 individuals (superposition of 100 sets of fixes). The pooled spatial utilization pattern is generated from simulation of multi-scale random walk (MRW) at the individual level. A virtual grid is then superimposed, and local N is then counted at the cell level. The result illustrates how an initially random start location for the respective individuals lead to a heterogeneous dispersion of 100*N fixes among cells in a virtual grid (the inset image illustrates the local fix abundance).The two-column band of cells with a different colour in the 3D representation above regards localities with higher resource level, simulated by applying a five times higher parameter than over the rest of the arena. A superabundance of resources is assumed for all patches over the entire simulation period in order to disregard crowding effects in this basic scenario (the result from this variant is described in Gautestad and Mysterud 2010).

Implication_EcolMod_BDoes the density of fixes – the pooled utilization distribution (UD) – reflect this heterogeneous resource map in a clear and consistent manner? How does the UD compare to the locally estimated H index?

First, since we in this case are studying a scenario where the animals have utilized its habitat in a multi-scaled (basically scale-free) manner and under influence of spatial memory, we do not expect the UD to perform well. Such compliance would require that the animals moved classically – i.e., mechanistically and with Markov compliance – as described repeatedly in other posts. The illustration above indicates that the “superior” columns 3 and 4 does indeed have the highest density of fixes. However, the difference to the other columns is not substantial, and the intra-column standard deviation (vertical bars) is high. In short, the UD performs badly as a proxy variable for the true space use intensity in this scenario.

Implication_EcolMod_CThen consider the MRW-derived utilization distribution – the H index – applied on the same data. The results speak for itself. As expected, the intensity of space use stands out clearly in column 3 and 4, and the standard deviation is small.

The crucial question is then – do individuals in real populations comply with the standard framework? If so, one should expect a similarly “sharp” proxy variable result from using the standard density of fixes, and the H index should be expected to perform poorly. On the other hand, if the MRW framework is more proper, a performance like the illustration above should be expected.

Of course, individuals’ response to a heterogeneous landscape will in reality be much more complicated than in the present illustrative example. However, this example illustrates the importance of applying analytical models from a process framework that complies best with the actual animal behaviour in statistical-mechanical terms.

  • If a Markov compliant behaviour is the most suitable framework for the space use process, use the classic density variable as a proxy for intensity of space use.
  • If a non-Markovian behaviour (like MRW) is more suitable, use H.

My book summarizes several approaches to pilot test which choice is the proper one for a given data set. Best of all, the methods are statistically simple to perform.

After this basic choice between two qualitatively different frameworks has been done, move forward with a more realistic model design; i.e., add terms for various habitat attributes that are considered important for local space use intensity and thus for habitat selection. In other words, move into the domain of ecological inference.


Gautestad, A. O., and I. Mysterud. 2010. Spatial memory, habitat auto-facilitation and the emergence of fractal home range patterns. Ecological Modelling 221:2741-2750.

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).

KDE paradox vs incidence

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.


Follow-up on the KDE’s Sample Size Paradox

In a previous post I showed how the combination of scale-free movement and spatial memory utilization generates space use that is incompatible with statistical descriptors under the KDE approach. In short, I advocated that the home range – as observed from the scatter of relocation data (fixes) – should be described by methods that are in better coherence with the underlying process in statistical-mechanical terms. The KDE does not comply with scale-free movement, since the emerging fix scatter is a statistical fractal with dimension D≈1, which makes space use incompatible with a density “surface” (utilization distribution, UD). In the present post I elaborate on this issue, and provide further simulation explorations.

KDE from MemRW (Levy1)The illustration to the right shows a home range with a “well-behaved” space use – in close compliance with standard KDE assumptions. The model animal moves scale-specifically (Markov compliant, here represented by correlated random walk), in combination with occasional return events to previously visited locations. In the Scaling cube, this model satisfies the MemRW universality class. The simulated movement path is then sampled at sufficiently large intervals to ensure non-autocorrelated fixes (I will return to autocorrelated series in later posts, where I discuss the Brownian bridge variant of KDE).

The habitat in the present example is homogeneous in order to disentangle the space use effect from intrinsic behaviour from the added effect from habitat heterogeneity (the general argument for homogeneous environment was given in a previous post).

According to MemRW theory, one should expect ca (1000/100)0.25 = 3.2% smaller area demarcation for N=100 relative to N=1,000, since N-dependency is weak; Area ∝ N0.25 (area measured as incidence, see Gautestad et al., 2013). Two sample sizes are superimposed in the illustration above; N=100 (red crosses) and N=1,000. The respective 99% isopleths are quite similar but area from N=100 is ca 10% smaller. It indicates KDE’s ability to almost compensate for the trivial sample size artifact that small samples of fixes tend to show smaller home range areas. The net  discrepancy seen here in this “quick and dirty” test is ca (10-100.25)% = 6.8%. Proper small-N compensation from KDE depends on the assumption that the process is scale-specific, as it is in this example.

KDE from MemRW w 10 seeds (Levy7)What about habitat heterogeneity? The home range to the right shows an example. Habitat heterogeneity is simulated in a simplistic manner, by randomly placing 10 initial start locations for the animal to choose from when accessing its memory map of previously visited locations during return events. The seed locations may be interpreted as a priori preferred patches, for example due to expectation of a particularly high foraging success at these localities, or patch attributes that satisfy the given animal’s habitat preference better than the surrounding area. Due to the MemRW condition (“denser” UD due to a larger fractal dimension, D), this initial set of “preferred patches” is not sufficiently influential to produce strong multi-modality. Other specifications of habitat heterogeneity could have produced stronger modality, though. The 95% isopleth for N=1,000 is also in this example ca 10% larger than the N=100 isopleth. In other words, again a statistically trivial discrepancy relative to expectation, (10-100.25)% = 6.8%.

KDE from MRW_multi-patch (Levy3)Using the same “preferred patches” method to impose habitat heterogeneity under scale-free condition (MRW), the initial set of 10 attraction points has a profound effect on modality of the UD. The image to the right shows clear influence from environmental heterogeneity. Thus, the UD from a MRW pattern is more “sensitive” to even subtle levels of environmental heterogeneity, relative to a scale-specific process like MemRW.

For a MRW process, theory predicts Area ∝ N0.5 (area measured as incidence, see Gautestad and Mysterud, 2010, or this post). Contrary to the “well-behaved” MemRW space use above, the MRW model with habitat heterogeneity confirms the KDE paradox: the 99% isoplets N=100 demarcates ca 15% larger area in comparison to N=1,000. The discrepancy is even larger for narrower isopleths, e.g. 90% (N=100 produce ca 30% larger area than N=1,000; not shown). Hence, the total discrepancy under the 99% isopleth is (15+100.5)%=18.2%. Despite the strong multi-modal effect on the UD from adding heterogeneity the result on home range demarcation is quite similar to the same model in a homogeneous environment, as was shown in the previous post.

In other words, the KDE sample size paradox – as found both in empirical studies and in the present theoretical pilot tests on MRW – seems to be quite resilient to environmental conditions. On the other hand, the classic space use condition for home range simulations (here represented by MemRW) behaves in better compliance with statistical expectations, in this case KDE assumptions.

To conclude, also these preliminary tests seem to support scale-free MRW over scale-specific MemRW as model representation for memory-influenced movement, due to their closer compliance with real data’s statistical properties (the KDE paradox). What appears as a statistical paradox under Markov conditions for space use in fact represents a model prediction under the parallel processing conditions (MRW).


Gautestad, A. O., L. E. Loe, et al. (2013). Inferring spatial memory and spatiotemporal scaling from GPS data: comparing red deer Cervus elaphus movements with simulation models. Journal of Animal Ecology 82(3): 572-586.