MRW and Ecology – Part VI: The Statistical Property of Return Events

Animals that combine scale-free space use with targeted returns to previous locations generate a self-organized kind of home range. In short, the home range becomes an emergent property from such self-reinforcing revisits. Obviously, any space use pattern from complex processes outside the domain of Markov (mechanistic) theory needs to be analyzed using methods that are coherent with this kind of behaviour. Below I exemplify further the versatility of the MRW approach to adjust for serial auto-correlation (see Part III). I also show the quite surprising model property that the sub-set of inter-fix displacement lengths for return events seems to have a similar statistical distribution as the over-all pattern of exploratory step lengths. This additional emergent property of space use may lead to methods to test a wide range of behaviour-ecological hypotheses, for example to which extent an animal calculates on an energy cost with respect to distance to potential target locations for returns..

In ecological research it is traditionally considered logical that an animal considers a return to a distant familiar location to be less preferred than revisiting closer locations. On the other hand, by default (a priori) the MRW model does not include such a distance penalty on long-distance returns. Recently, the realism of this model premise has gained empirical support from studies on bison and toads (Merkle et al. 2014,2017; Marchand et al. 2017). In the MRW model’s standard version, a given return step is targeting any previous locations with equal probability except for the additive effect of number of previous visits to a given site, which increases the statistical probability for future revisits (self-reinforcing site fidelity). The implicit assumption is that the added energetic cost from long distance returns either is negligible relative to other parts of the energy budget, or the fitness value from keeping in touch with familiar locations regardless of current distance far exceeds the energy consideration. While this property regards a homogeneous environment it is trivial to adjust to a heterogeneous scenario without loss of the general principle. In this post I present more details on the return step property of MRW from a theoretical angle, as a starting point to test the model’s default condition on real data.

First, consider that the robustness of the MRW-based method to estimate an individual’s characteristic scale of space use (CSSU) within a given time and space extent is key to understand the energy aspect of return events as outlined above. The property of return events imposes a characteristic scale; i.e., CSSU, on space use, despite the scale-free nature of exploratory steps. For a given period, CSSU is a combined function of average movement speed and average return frequency. In a previous post I proposed how CSSU may be estimated even in auto-correlated (“over-sampled”) data series of location fixes. In this post I present a pilot analysis which strengthens this approach.

Consider the two simulated Home range ghost results to the right; incidence, I, as a function of number of fixes, N. The first set of fixes (circles) regards weakly auto-correlated series of fixes from return rate 1:10 and fix sampling at 1:100, while the second series (squares) resulted from a strongly autocorrelated path sampling (return rate 1:100 and sampling at 1:10). As was shown in Part III of this set of blog posts, by performing the “averaging trick” on log(I,N) from frequency and continuous sampling (open symbols for respective sets) the average log-log slope remains close to z=0.5 (area expanding proportionally with square root of sample size) even for the strongly auto-correlated series. The slight deviance from z=0.5 in the two series should be considered normal variability to be expected from one simulated series to the next (averaging over large sets of series would bring z closer to 0.5).

Critically, the present result also shows compliance with the expected change of the characteristic scale of space use (CSSU, represented by the parameter c in the Home range ghost formula I=cN0.5) as a function of the ratio between frequency of return events relative to exploratory moves (assuming constant average movement speed). In other words, observation frequency, which represents a sub-set of all displacements along a path (sampling of fixes) should not influence the CCSU estimate despite influencing the degree of auto-correlation. According to MRW theory, fewer returns during a constant average movement speed lead to larger CSSU*. In the analysis of the present two series, ten times smaller return rate led to an optimized unit pixel size (I≡1) of magnitude √10 = 3.3 times larger than for the weakly autocorrelated series with higher return frequency.

In the Figure above, the two CSSU scales have both been rescaled to c=1 ([log(c)=0], but respective series’ unit scale (I=1) is de facto correctly found to be very different in absolute terms.In the present examples, CSSU was estimated to c1 = 1252 area units for the high frequency return scenario and c2 = 4002 area units for the second series with fewer returns (and stronger degree of autocorrelation).

To conclude, after optimizing pixel size in respective series by analyzing I(N) over a range of pixel resolutions as previously described in my book and other posts, this preliminary analysis verifies a strong coherence between return step frequency and the magnitude of CSSU in accordance to the theoretical parameter prediction, despite strong difference in degree of serial autocorrelation in the sample of relocations. On other words, the CSSU estimate is quite resilient to the researcher’s choice of fix sampling scheme.

However, another aspect of the return step component of may turn out to be valuable to test the opposing energy hypotheses with respect to distance penalty, as outlined above.

Quite surprisingly I must admit, even considering the implicit “no distance penalty” model design, the tail part of the step length distribution of returns is quite similar to the tail of observed step lengths (fixes) that are sampled from the total series of steps!

The example series above with the weakest degree of auto-correlation (circles in the top Figure) shows similar functional form between the over-all distribution of binned step lengths [log(L); red circles below] and return distances (open symbols)**.

As expected from the weakly autocorrelated series, the fit to the power law function with Levy exponent β=2 of the exploratory steps is showing a clear “hump” in the extreme part of the Log(L) distribution of fixes, due to influence from intermediate return events.

For the more strongly autocorrelated series with N=10,000 fixes from a total series of 100,000 steps and a lower return frequency 1:100 (Figure below) we see – as theoretically expected – a more subdued hump for the fixes, due to less influence from return events***. The hump would be even less pronounced if the fix sampling frequency had been even larger (Gautestad and Mysterud 2013; in particular Figure A2 in Supplementary material).


Again the tail distribution of return lengths – where the total set of 1,000 events is shown as triangles – is similar to the the over-all distribution of fixes (1,000 first and 1,000 last of the N=10,000 fixes, shown as red and green circles). The median length for return steps is larger under this scenario (740 length units, versus 262) due to a ten times lower return frequency in relative terms. On the other hand, the median length for the actual set of fixes is strongly reduced as a consequence of the ten times larger fix sampling frequency.

To summarize, while the estimate of CSSU is quite resilient to fix sampling frequency, the (observed) median step length of fixes and (unobserved) length of return steps are influenced by fix sampling rate and return rate, respectively. Despite independence between the median length for observed series and hidden return lengths, both aspects of movement show a similar distribution of lengths.

Finally, what if the return step targets had not been set a priori to be independent on distance; i.e., by invoking distance penalty on return events? I have not tested this aspect yet in a modified MRW simulation model, but intuitively I predict the distribution of return steps to morph towards a negative exponential function rather than a power law, as in the exploratory kind of moves. As aconsequence, the “hump” effect in the distribution of fixes should also be more subdued. Hence, by testing the difference in functional form of return steps and step lengths of observed fixes, one may have a method to test empirically the energy hypothesis that was outlined above.

The challenge, of course, is to develop a method to distinguish between exploratory moves and return events in empirical data. In simulation data it is simple to filter out the returns; in true space use data it is necessary to distinguish returns from path crossing by chance. More on this methodology in an upcoming post.


*) Thus, the ratio returns/exploratory moves have a similar influence on CSSU as a change in average movement speed where the speed is expressed as the average staying time in a given grid cell. In Gautestad and Mysterud 2010, Eq. 4, we defined the expected length of step x, Lx, as a function of a scaling parameter for movement speed δ and fractal dimension of the path, d:

Lx = (δ[1 − Rnd])−1/d         (Eq. 4)

where Rnd is  a random number 0 ≤Rnd < 1 and δ is a scaling parameter. In some sense δ may be interpreted as a parameter for expected staying time in a given patch, since larger δ implies smaller Lx and thus increased local fix contagion.
Gautestad and Mysterud 2010, p2744

Thus, by defining the space use’s fractal dimension D as D≡d, we have the relationship with CSSU’s Home range ghost parameter, c, and movement speed:

c ∝ 1/√δ  |  D = 1          (Eq 5).

**) Due to a return step frequency of 1:10 and actual fix sampling frequency of 1:100, the total set of return events exceeds the fix sample by a factor of 10. Thus, I have compared the distribution of of return lengths from the early part of the simulated path (open squares) with return lengths towards the end of the path (open triangles), keeping both samples at same size as the set of observed fixes. Red circles in the Figure above represent 10,000 fixes from a total series of 1 million steps. When studying the first and the last part of the 100,000 hidden return steps specifically, their distribution looks indistinguishable from the series of “observed” fixes. Triangles show the result for the first 10,000 return events, and the squares show the result from the last 10,000 returns during the total 1 of million steps.

***) In this example where observation frequency exceeds the intrinsic return frequency by a factor of 10, the first and last part of the set of fixes (red and green circles, respectively) was used for comparison with the total set of return steps (open triangles).


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.

Gautestad, A. O., and A. Mysterud. 2013. The Lévy flight foraging hypothesis: forgetting about memory may lead to false verification of Brownian motion. Movement Ecology 1:1-18.

Marchand, P, M. Boenke and D. M. Green. 2017. A stochastic movement model reproduces patterns of site fidelity and long-distance dispersal in a population of Fowler’s toads (Anaxyrus fowleri). Ecological Modelling 360:63–69.

Merkle, J. A., D. Fortin and J. M. Morales. 2014. A memory-based foraging tactic reveals an adaptive mechanism for restricted space use. Ecology Letters 17:924–931.

Merkle, J. A., J. R. Potts and D. Fortin. 2017. Energy benefits and emergent space use patterns of an empirically parameterized model of memory-based patch selection. Oikos 126:185–195

MRW and Ecology – Part V: Black Bear Home Ranges Revisited

Back in 1994 I enjoyed an unforgettable and extremely inspiring 2-month stay at University of Tennessee, visiting professor Stuart L. Pimm (Department of Ecology and Evolutionary Biology)  and Professor Mike L. Pelton (Department of Forestry, Wildlife and Fishery). During some hectic weeks I worked on transforming the mathematical formulation of the Zoomer model for complex population dynamics into a spatially explicit simulation model (Stuart’s lab) in parallel with interaction with many dedicated students of the biology and ecology of black bear Ursus americanus (Mike’s lab).The Zoomer model is published in my book and already commented on this blog. Regarding the stay at Mike’s lab we published a test on the bears’ general space use, where we found close compliance with the Multi-scaled random walk model, MRW (Gautestad et al. 1998). In this post I revisit the black bear data and find this model’s additional potential to cast light on behavioural ecology in a wildlife management context.

In the 1998 paper we applied the so-called re-scaled range analysis, R/SD (Mandelbrot 1983:pp247-25), to study home range area as a function of fix sample size. Details on the method and the raw data can be found in Gautestad et al. 1998. Below I revisit these bear data and re-test the Home range ghost function with the presently preferred method. Following this exercise I show a new result, which indicates that a collared bear’s characteristic scale of space use may have been inflated during the initial period of fix sampling of its path. Implicitly, the result rises the question if the experience of being collared and getting used to bearing the collar is influencing the bear’s space use towards a more coarse-grained habitat utilization on average; i.e., a larger CSSU, during the first months following the capture/release. The home range ghost regards the “paradoxical” pattern whereby home range area, A, apparently expands non-asymptotically with sample size of fixes, N, in compliance with the power law A = cNz. and z≈0.5. The parameter c is the Characteristic scale of space use, CSSU, emerging from memory-dependent tendency to return to familiar locations.

The inset exemplifies the Home range ghost for one of the black bear individuals. The apparent area asymptote [inset, showing arithmetic axes for A(N)] disappears under log-transformation of the axes; i.e., the expansion with N is non-asymptotic and power law compliant with exponent 0.5. Thus, the area expands with the square root of N.

In the present results to the right and below the A(N) pattern was analyzed with the most recent MRW method, using incidence (number of virtual grid cells of size, I, embedding at least one location) at the estimated grid resolution of CSSU as a representative for home range utilization intensity. The respective (N,I) plots were calculated as the average from two sampling schemes; frequency and time-continuous (see this post). The advantage over R/SD is the I(N) method’s ability to estimate the parameters c (representing CSSU) and power exponent (z) even for autocorrelated series of fixes.

In the present pilot test using a subset of 15 individual data sets (the first part of the total database of 77 series) the result shows strong coherence for estimates of z between the previous R/SD method and the present I(N) method with respect to how home range area responded to sample size, N. Once more I stress the advantage of using a model which accounts for the non-trivial N-dependency in home range size estimates, making CSSU – where the N-dependency is filtered out – a superior ecological proxy relative to the traditional method, direct area estimate.

As expected from scale-free space use, the distribution of CSSU is approximately log-normal (observe the log-scaling x-axis).

On this background it’s time to move in the direction of ecological analysis of respective individual’s space use.The histogram to the right shows a strongly variable CSSU between the 15 individuals. This raises interesting ecological questions. For example, why has the leftmost individual (female F170) 1:240 magnitude of CSSU – ca 1:15 in terms of linear rather than area scale – relative to the rightmost score (female F060)? In other words, according to the present analysis F170 utilized its habitat with substantially higher intensity than F060 (intensity ∝ 1/CSSU). F170 showed a relatively small I(N) for a given N, meaning that F170 utilized its habitat in a more area-restricted manner. Unfortunately I don’t have access to individual behaviour details nor environmental GIS data for the respective space use patterns.

However, consider the relationship between sample size, N, and the CSSU (right). F060 and F170 are found as the largest and smallest CSSU in the scatterplot. The average CSSU for all 15 individuals is ca 866,000 m2 (0.866 km2). Under constant conditions one should expect CSSU to be stationary under larger sample size (larger N) and non-autocorrelated series. However, the plot indicates a negative relationship between N and CSSU in these data, which is unexpected from the standard MRW model a priori (as shown by simulated data in this post)*.

Why? Many hypotheses may be invoked and tested – here I indicate one in particular. The capturing, radio collaring and subsequent tracking might have stressed the bears for some time after release. To study this, I re-analyzed the three individuals with the largest fix sampling period, including the first 100 fixes only. As shown by open symbols, all individuals (F182, F201 and F243) showed larger CSSU during the initial period for 100 fixes following release – a period of ca 1-2 years (!) – relative to the total fix sampling period of almost four years**), and thereby became similar to the other bears with respect to average spatial scale.

Is this an indication of a more restless space use behaviour in the first period following capture, release with radio collar and subsequent telemetry tracking 1-2 times/week (distance observer-animal is unknown)? To explore this aspect, several alternative hypotheses should also be considered (the three individuals were collared in August, June and June, and were 7, 3 and 3 years of age) – but the present pilot test does at least rise an interesting hypothesis. It also illustrates how the MRW model’s CSSU parameter may be applied to cast light on a potentially concerning aspect of data collection and its interpretation in the context of wildlife management***).


*) Update: see also this post. which illustrates a transient effect in the early phase of home range establishment, and stationary CSSU for mature home range utilization.  In the present context, is it reasonable to assume mature home ranges even for the initial 1-2 years of fix sampling.

**) Alternative methods to test for inter-sample difference in space use intensity abound, but have drawbacks. For example, the mean displacement length in a set of fix sampling intervals (lags) will be influenced by intermediate return events. Further, due to the very long-tailed (leptocurtic) distribution of step lengths in data from black bear, which has been verified to comply with the MRW space use model, comparing the relative difference in median displacement size in two or more samples is subject to large error terms due to the extreme outlier issue (“occasional sallies”). CSSU resolves these obstacles.

***) It should bee mentioned that  the radio telemetry collars for black bear back in 1978 were substantially heavier than today’s standard. Triangulation also required close stalking to get a fix, in contrast to modern GPS.


Gautestad, A. O., I. Mysterud, and M. R. Pelton. 1998. Complex movement and scale-free habitat use: testing the multi-scaled home range model on black bear telemetry data. Ursus 10:219-234.

Mandelbrot, B. B. 1983, The fractal geometry of nature. New York, W. H. Freeman and Company.


MRW and Ecology – Part IV: Metapopulations?

In light of the recent insight that individuals of a population generally seem to utilize their environment in a multi-scaled and even scale-free manner, the metapopulation concept needs a critical evaluation. Even more so, since many animals under a broad range of ecological conditions are simultaneously mixing scale-free space use with memory map-based site fidelity. In fact, both properties, multi-scaled movement and targeted return events to previous locations, undermine key assumptions of the metapopulation concept.

Levins (1969) model of “populations of populations” – termed metapopulation – rattled many corners of theoretical and applied ecology, despite previous knowledge of the concept from the groundbreaking research by Huffaker (1958) and others (Darwin, Gause, etc.). Since then, Ilkka Hanski (1999) and others have produced broad theoretical and empirical research on the metapopulation concept.

The Levins model describes a metapopulation in a spatially implicit manner, where close and more distant sub-populations are assumed to have same degree of connectivity. Later models (including Hanski’s work) made the dynamics spatially explicit. Hence, distant sub-populations are in this class of design more closely connected dynamically than more distant populations. Sub-populations (or “local” populations) are demarcated by large difference in internal individual mixing during a reproduction cycle relative to the rate of mixing with neighbouring sub-populations at this temporal scale. As a rule-of-thumb, the migration rate between neighbour populations during a reproduction cycle should be smaller than 10-15% to classify the system as a metapopulation. Simultaneously, intrinsic mixing during a cycle in a given sub-population is assumed to approximate 100%; i.e., “full spatial mixing” (spatial homogenization when averaging individual locations over a generation period).


According to the prevailing metapopulation concept, high rate of internal mixing in sub-populations is contrasted by substantially lower mixing rate between sub-populations. The alternative view – advocated here – is a hierarchical superposition of mixing rates if the individual-level movement is scale-free over a broad scale range. The hierarchy is indicated by three levels, with successively reduced intra- and inter-population mixing rate towards higher levels.

The spatially explicit model of a metapopulation is based on three core assumptions:

  1. The individual movement process for the population dynamics should comply with a scale-specific process; i.e., a Brownian motion-like kind of space use in statistical-mechanical terms, both within and between sub-populations. This property allows intrinsic population dynamics of sub-populations to be modelled as “homogeneous” at this temporal scale. This property is also assumed by the theory of differential and difference equations. It also allows the migration between sub-populations to be described as a classical diffusion process.
  2. Following from Point 1, more distant sub-populations are always less dynamically linked (smaller diffusion rate) than neighbour populations. In fact, dispersal between distant sub-populations may be ignored in spatially explicit models.
  3. Emigration from a given sub-population may be stochastic (random) or deterministic (e.g., density dependent emigration rate), while immigration rate is stochastic only. The latter follows logically from compliance with point 1. In other words, emigrating individuals may occasionally return, but only by chance and thus on equal terms with the other immigrants from neighbour populations. Hence, both the intrinsic and inter-population mixing process is assumed to lack spatial memory capacity for targeted returns at the individual level.

In my alternative idea for a spatially (and temporally) structured kind of population dynamics, individual movement is assumed to comply with multi-scaled random walk (MRW). Contrary to a classical Brownian motion and diffusion-like process, MRW defines both a scale-free kind of movement and a degree of targeted returns to previous locations. Thus, both emigration and immigration may be implicitly deterministic. The two perceptions of a structured population are conceptualized in the present illustrations. “Present idea” regards the prevailing metapopulation concept, and the “Alternative idea” regards population dynamics under the MRW assumptions.

The upper part of the illustration to the right shows the two classical metapopulation assumptions in a simplistic manner. Shades of blue regards strength of inter-population mixing, which basically is reaching neighbour populations only (by a rate of less than 10-15%, to satisfy a metapopulation structure) but not more distant ones. For example, inter-generation dispersal rate between next-closest sub-populations is expected to be less than (10%)*(10%) = 1%, and falls further towards zero at longer distances. The Alternative idea at the lower part describes a more leptocurtic (long-tailed) dispersal kernel – in compliance with a power law (scale-free dispersal) – rather than an exponentially declining kernel (scale-specific dispersal), as in the standard metapopulation representation. Separate arrows for immigration from one sub-population to a neighbour population of the “Present idea” part illustrates the standard diffusion principle, while the single dual-pointing arrow of the “Alternative idea” illustrates that immigration an emigration are not independent processes, due to spatial memory-dependent return events. The emergent property of targeted returns connects even distant sub-populations in a partly deterministic manner.

The Alternative idea design is termed the Zoomer model, which is explored both theoretically and by preliminary simulations in my book. A summary was presented in this post. A long-tailed dispersal kernel may embed and connect subpopulations that are separated by a substantial width of matrix habitat. Since the tail is thin (only a small part of individual displacements are reaching these distances), long-distance moves and directed returns happen with a small rate. 

The Zoomer model as an alternative to the classical metapopulation concept has far-reaching implications for population dynamical modelling and ecological interpretation. For example, the property that two distant sub-populations may sometimes be closer connected than connectivity to intermediate sub-populations due to emergence of a complex network structure at the individual level was illustrated by my interpretation of the Florida snail kite research (see this post). At the individual level, research on Fowler’s toad (see this post) and the Canadian bison (see this post) shows how distant foraging patches may be closer connected than some intermediate patches. Also this is in compliance with the Zoomer concept and in opposition to the classical metapopulation concept. In many posts I’ve shown examples of the leptocurtic distribution of an individual’s histogram of displacement lengths, covering very long distances in the tail part – potentially well into the typical scale regime of a metapopulation (for example, the lesser kestrel).


Hanski, I. Metapopulation Ecology. Oxford University Press. 1999.

Huffaker, C.B. 1958. Experimental Studies on Predation: Dispersion factors and predator–prey oscillations. Hilgardia 27:83-

Levins, R. 1969. Some demographic and genetic consequences of environmental heterogeneity for biological control. Bulletin of the Entomological Society of America 15:237–240

MRW and Ecology – Part III: Autocorrelation

Ideally, when studying ecological aspects of an individual’s whereabouts based on (for example) series of GPS fixes, N should not only be large. The series of fixes should also be non-autocorrelated to ensure statistically independent samples of space use. Since these two goals are difficult to fulfill simultaneously (the latter tend to undermine the former), two workarounds are common. Either the autocorrelation issue is ignored albeit recognized, or space use is analyzed by path analytical methods rather than the more classical use-availability approach. Both workarounds have drawbacks. In this post I show for the first time a surprisingly simple method to compensate for the oversampling effect  that leads to autocorrelated series of fixes.

Again, as in Part II of this series, I focus on how to improve realism and reduce the statistical error term when studying ecological aspects of habitat selection, given that data compliance with the MRW framework has been verified (see, for example, this post regarding red deer) or can be feasibly assumed. Hence, the individual’s characteristic scale of space use (CSSU) is the primary response variable we are looking for. In part II the proper proxy for local intensity of space use was described as the inverse of CSSU (actually, the inverse of the parameter c).

However, by default the basic version of the Home range ghost equation I = c√N, where I is the total area of fix-embedding virtual grid boxes at the CSSU scale, assumes a data set of N serially non-autocorrelated fixes. This is difficult to achieve, due to the simultaneous goal to have a large N available for the analysis. Splitting the data into sub-sets of N from several habitat classes makes the autocorrelation issue even more challenging. Thus, over-sampling of the animal’s movement seems unavoidable. In the following example I illustrate how such an oversampling effect on local and temporal CSSU estimates may be accounted for.

As a reference scenario, consider the default MRW condition of non-autocorrelated fix sampling of an animal moving in a homogeneous environment. Non-autocorrelation is achieved by sampling at larger intervals than the average interval between successive return events. In the illustration above the spatial scatter of 10,000 fixes (grey dots) shows a relatively stationary space use when comparing N=100 fixes from early, middle and late part of the sampling period (blue, red and yellow dots, respectively). However, return events that took place during the last part of the series have a more spread-out set of historic locations to return to, and this explains why the 100 yellow fixes cover a somewhat larger range than a similar sample size from the series’ early part.

When sampling a series of fixes from the actual path for a given time period*, two methods may be applied; continuous sampling containing a section of the series, varying in length N; and frequency-based sampling where N fixes are uniformly spread over the entire time interval for the total series (higher sampling frequency implies larger N). With reference to the Home range ghost formula above, I shows compliance with a non-asymptotic power law with exponent z≈0.5 (log-log slope close to 0.5). Grid resolution (pixel size) has been optimized in accordance to previously described method. The well-behaved pattern in this scenario is due to lack of strong auto-correlation under both sampling regimes. In other words, the animal’s path has not been over-sampled. Still, the difference between continuous sampling (open triangles) and frequency-based sampling (open squares) shows that the former is more prone to short term random effects, in this example seen as the “plateau” of I(N) in the range N=23 to 27. The characteristic average scale, log(c), is given by the I(N) intercept with the y-axis, where log2(N)=0.

Observe the set of black circles, which represent the average log[I(N)] from the two sampling methods covering the same sampling period at Ntotal.

Next, consider an example with strongly autocorrelated fixes. The ecological condition will be described below as “semi-punctuated site fidelity”**. Again, the colour codes in the spatial scatter of fixes (above) describe subsets of 100 fixes from the early, middle and late part of the total sampling period.

What is important under this condition is the behaviour of log[I(N)] under the two sampling methods, continuous and frequency. As expected with autocorrelated series, sub-sampling the total series by the frequency method – relative to continuous sampling – will tend to show a larger I for a given N over the middle range of log(N). Similarly, continuous sampling tends to show a smaller area for a given N, relative to expectation from the Home range ghost equation.

However, when averaging the respective log(N,I) points the compliance with I ∝ √N is restored! Thus, the estimate of CSSU may be properly estimated also from over-sampled paths. Despite the substantial under-detection of true space use based on N autocorrelated fixes, the statistical-mechanical theory of MRW in fact predicts the true I(N) – and hence also the CSSU – by performing the averaging trick above.

Why does the average of continuous and frequency-sampled estimates represent the true I(N)? Consider the vertical distance between the respective pairs of log(N,I) points to represent un-observed “ghost area” as a result of over-sampling. The stronger the over-sampling the larger the ghost area. If the sampling regime had regarded non-autocorrelated series, the ghost area would have been small (as in the first example above), due to weak degree of over-sampling. Stronger auto-correlation leads to stronger ghost area. Why is the ghost area splitting the area from frequency sampling and continuous sampling by 50% in log-log terms? This theoretical question requires a deeper statistical-mechanical explanation, which is still in theoretical progress. However, the answer is linked to the 50%/50% inward/outward expansion property of MRW (see this post).


*) If the total sampling period is not kept constant (same time period for Ntotal), CSSU will be influenced by the fact that late return events are targeting a more spread-out scatter of previous locations. Despite this, CSSU will tend to contract somewhat with total observation period (temporal extent). This transient effect will be explored in an upcoming blog post

**) An extreme form of temporal space use heterogeneity is achieved by “punctuated site fidelity”. For example, for every 1/50th part of the total series length the animal erases its affinity to previous locations and begin developing affinity to newer locations only. For example, in the third section of such a path, return events the following return events do not target the initial two parts of the series. The first location in each of the 50 successive parts (time sections) is chosen randomly within the total arena, hence a “punctuated” kind of site fidelity. This scenario could in model-simplistic terms illustrate GPS sampling of an animal that occasionally is changing its space use in accordance to changing food distribution during the season. It could also illustrate an intrinsic predator avoidance strategy, whereby fitness may improve by occasional abrupt changes of patch use, and this may under specific conditions be more advantageous than the cost of occasionally giving up utilization of familiar patches. The scenario could also illustrate patch deterioration with respect to a critical resource; energy profit in utilized patches may deteriorate owing to foraging, and thus trigger a “reset” of over-all patch use in conceptual compliance with a variant of the marginal value theorem.

A less dramatic and more realistic variant of temporal heterogeneity, “partially punctuated site affinity”, is simulated by keeping – for example – the last 10% or 2% of the path locations of the foregoing part of the path as potential return targets on equal footing with the successively emerging locations in the present part. This condition leads to a tendency for a “drifting home range” (Doncaster and Macdonald 1991), with some degree of locking towards previous patch use, similar to the condition that was numerically explored in Gautestad and Mysterud (2006).


Gautestad A. O. and I. Mysterud. 2006 Complex animal distribution and abundance from memory-dependent kinetics. Ecological Complexity 3:44-55.

Doncaster C. P. and D. W. Macdonald. 1991 Drifting territoriality in the red fox Vulpes vulpes. Journal of Animal Ecology 60, 423-39.

MRW and Ecology – Part II: Space Use Intensity

Through the history of ecological methods, local intensity of habitat use has been equalized with local density of relocations. Using relative density as a proxy variable for intensity of habitat use rests on a critical assumption which few seems to be aware of or pay attention to. In this second post on Multi-scaled random walk (MRW) applications for ecological inference I describe a simple method, which rests on an alternative assumption with respect to space use intensity, applicable under quite broad behavioural and ecological conditions. One immediate proposal for application is analysis of habitat selection.

First, consider counting number of GPS fixes, N, within respective area segments of a given habitat type h, Ah1, Ah2, …, Ahi, …Ahk,  and calculating the average N pr. area unit of type h. Next, consider comparing this density Dh with another density within a second habitat type j; i.e., Dj, using the same area scale for comparison. If Dh>Dj one traditionally assumes that the intensity of use of habitat h has been stronger than habitat j. Ecological inference about habitat selection then typically follows under this assumption. In the following I describe the critical assumption for applicability of this traditional method, and I conclude with an alternative proxy variable.

The assumption that space use intensity can be represented by density of space use rests on a specific statistical-mechanical property of particle movement. In our context, the particle is an individual.

The concept of space use intensity in ecology may be represented by local particle density of relocations if – and only if – the particle follows the physics of a Markovian process, and the particle has no affinity towards previously visited locations.

Why? Consider the simulation result to the right, where a particle has moved in compliance with a classical random walk (Brownian motion-like; filled circles) and a Lévy walk (open circles). In both instances the number of pixels (“virtual grid squares”, I, at respective unit process scales) that embed at least one relocation of the particle grows proportionally with N. This property of proportionality applies both to the present condition where N represents the entire path of the particle – meaning that N grows in proportion to path length – and an alternative condition where the path is sampled at larger intervals than the given unit time scale, t. In short, at the unit spatial scale and the complementary unit temporal scale t, every new step tends to hit a previously non-visited pixel. Since each displacement at this dual space-time scale is independent of previous steps (the Markov condition) and every path crossing happens by chance (the independence of previous locations condition), dividing N by total area visited, N/I ≡ D, gives a constant density value. If the environment is spatially heterogeneous (“habitat heterogeneity”, as illustrated by the types h and j above), the local density will vary accordingly.

In other words, under this specific and quite restrictive condition intensity of space “use” is equal to density of space use. If the respective unit process scales were changed, D would also change proportionally (recall from above, that respective D-estimates should be compared under a similar spatial scale; i.e., same pixel size). Hence, in an ecological context habitat selection could be inferred by studying difference in density under assumed different characteristic scales for the movement process as  the animal passes through respective types of habitat.

Then consider the widespread condition where an animal’s path is not self-crossing by chance only, due to some degree of affinity to previously visited locations in a manner that is not compatible with a Markovian process. In ecology we call it a home range compliant kind of space use (the home range becomes and emergent property from site fidelity).

Under this condition, density should not be applied as a proxy variable for local intensity of space use. You should be critical to the fact that more than 99% of researchers disregard this piece of advice in situations where the individual has shown home range behaviour in an obvious non-Markovian manner! Misuse of D as a proxy for intensity will clearly inflate the error term substantially in ecological use-availability analyses. I’m happy to see that this fact finally starts percolating into basic models of space use data (Campos et al. 2014; Morellet et al 2013). Downplaying the density-intensity issue by applying kernel density or Brownian bridge representations of local D estimates will not resolve the challenge. These methods also rest on the same classical assumption as described above. Garbage in – garbage out!

As all readers of my book and my blog are aware of, I advocate MRW as a more realistic substitute. The result marked by MRW (filled triangles) in the Figure above shows how I increases proportionally with the square root of N (log-log slope z=0.5); i.e., a non-proportional and very “dampened” expansion of I relative to the classical condition of Brownian motion and Lévy walk.

  • As a consequence, average intensity of space use in respective habitat types should be represented by D’ = (√N)/I, not by D = N/I. According to the MRW theory, D’ = 1/c, where c is the actual MRW process’ characteristic scale.

The exploratory part of MRW is per se scale-free (like Lévy walk), but the site fidelity part of the model behaviour introduces a the particular scale c (I refer the reader to a multitude of previous posts on this theme, or to my book).

The histogram above shows local density (N pr. grid cell = D) of fixes (left pane) and local c of simulated MRW path in a homogeneous environment (Gautestad, in prep.). Since the environment is homogeneous in this scenario, the expectation is c of same magnitude in all grid cells, regardless of local density of fixes. Within each grid cell, c is estimated from the function I(N) = c√N (“The home range ghost” formula), where I is the chosen pixel size within each cell. Contrary to local D estimates, and despite a very crude and simple first-approach method to estimate of local c, the respective columns are quite concentrated around the average c score for the arena as a whole (dotted blue level.

Observe that in this regression of c(N), the superimposed text “Dstep = 1″ refers to the fractal dimension of the movement path.

As shown to the right, local D and c estimates are shown to be independent. While local D in this homogeneous scenario varies tremendously (x-axis shows N pr. grid cell), c varies within a relatively restricted range.

In a previous post I showed by example how the estimate of c can be further improved by fine-tuning the pixel resolution (unit scale for I) for respective grid cell samples of fixes. In the present illustration, pixel size was constant and a priori set somewhat smaller than the given characteristic c = (2 length units)2 = 4 area units. In other words, the pixel scale was set to 1/4 of true c. Still the estimated average c = 21.8 = 3.4 times larger than pixel size, which is close to the true c = 4 area units in over-all terms. Choosing a smaller pixel scale a priori than the true unit scale may explain some deviance from the expected constancy of I in some grid cells (in particular in cells with N ≈ 24).

Additional methodological details: Towards an Alternative Proxy for Space Use Intensity

In a follow-up post I’ll show the influence of serial auto-correlation in the set of fixes, and how it can be accounted for when using 1/c as an alternative – and more realistic – proxy variable for local intensity of space use. A very nagging problem under the present paradigm may be resolved with ease under the MRW statistical-mechanical model assumptions!



Campos, F. A., M. L. Bergstrom, A. Childers, J. D. Hogan, K. M. Jack, A. D. Melin, K. N. Mosdossy et al. 2014. Drivers of home range characteristics across spatiotemporal scales in a Neotropical primate, Cebus capucinus. Animal Behaviour 91:93-109.

Morellet, N., C. Bonenfant, L. Börger, F. Ossi, F. Cagnacci, M. Heurich, P. Kjellander et al. 2013. Seasonality, weather and climate effect home range size in roe deer across a wide latitudinal gradient within Europe. Journal of Animal Ecology 82:1326-1339.

MRW and Ecology – Part I: Introduction

In this “MRW and Ecology” series of posts I plan to present some proposals for simple ecological methods – based on alternative basic assumptions – for analysis of common ecological aspects of animal space use. These proposals are spin-offs of the first direction of research to explicitly break out of the Markovian strait-jacket in the present context. A broadened analytical approach – involving a qualitative shift of direction – is in my view clearly needed, as documented by the rapidly growing line of high quality and deep-level analyses of empirical data  now appearing. For such an alternative direction the parsimonious Multi-scaled random walk model (MRW) may provide a feasible starting point. Over the years the MRW approach has been successfully tested empirically against the prevailing paradigm’s basic assumptions, or indirectly supported by alternative interpretations of respective analyses of space use and movement. Thus, now it’s time to step forward from testing behavioural feasibility of the theory towards ecological hypothesis testing, starting with summaries of some practical methods that are actually rooted in the MRW theory and its parallel processing postulate.

In parallel with high frequency scanning and responses to environmental input from its immediate surroundings, the monk parakeet Myiopsitta monachus probably also relates to its environment at coarser spatial and temporal resolutions; i.e, in statistical-mechanical terms it may comply with the parallel processing kind of space use. “In parallel with…” implies a different process regime than the standard mechanistic kind of dynamics, where the animal is assumed to successively toggle between various states and behavioural response regimes from one moment to the next.  Over the time scale continuum a parallel processing compliant animal is assumed to be mixing frequent short term tactics with more long term strategy in a manner that is incompatible with a Markov process. In other words, a single time resolution is not sufficient to represent the behaviour. Over the spatial scale range the animal is assumed to cognitively be able to “zoom in and out” between fine-grained habitat details to more coarse representations of the previously explored environment (as stored in the individual’s memory map). A Parallel processing-based model provides alternative ecological methods with respect to basic assumptions.  Photo: AOG.

Readers of my papers, my book and this accompanying blog have for a long time been presented with one particular approach towards the qualitative shift of direction, conceptualized as one of the “ceiling” corners of the Scaling cube. In this corner of the cube – the MRW model – the traditional Markovian kind of statistical mechanics is replaced by the parallel processing postulate, covering both spatial and temporal scaling of process memory. In my view the present random walk paradigm (standard random walk and diffusion, composite and or biased random walk,  specific kinds of Lévy walk and truncated Lévy walk, and other statistical-mechanical variants of Markovian formulations) seems to offer a too narrow theory to account for a realistic modelling of spatial memory effects in simultaneous combination with multi-scaled habitat utilization; i.e., covering a broad range of spatial and temporal scales). Fortunately I now register a gradually expanding network of ecologists who feel uncomfortable with the foundation of this general toolbox for dynamic and statistical modelling.

On one hand, my goal is to expand the network of curious peers. On the other hand I want to develop the theory towards applicability in various ecological contexts. After 25 years of theory development and tests of the parallel process postulate per se, as exemplified by the Multi-scaled random walk model as the alternative hypothesis, I feel time is more than ripe for this step in the direction of actual ecology.

Application of an extended statistical-mechanical theory of animal movement and space use is not a walk in the park, in particular when the large majority of theoretical ecologists in this field apparently are happy with the present set of basic model assumptions. Better to add more line segments to the regular convex polygon than to replace this geometry with the mathematics of a circle! In other words, to seek better coherence with the data, just add more terms to the present model. This dampens the hurdle of the peer review process and it makes it easy to find a suitable instrument in the R library of models and methods.

Hopefully, the upcoming series of “MRW and Ecology” posts will contribute to further clarification of the potential of the alternative approach, and spark additional tests and actual model applications on real data.