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!

 

REFERENCES

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.

Random Walk Should Not Imply Random Walking

Random walk is one of the most sticky concepts of movement ecology. Unfortunately, this versatile theoretical model approach to simplify complex space use under a small set of movement rules often leads to confusion and unnecessary controversy. As pointed out by any field ecologist, unless an individual is passively shuffled around in a stochastic sequence of multi-directional pull and push events, the behavioural response to local events and conditions is deterministic! An animal behaves rationally. It successively interprets and responds to environmental conditions – within limits given by its perceptive and cognitive capacity – rather than ignoring these cues like a drunken walker. Any alternative strategy would lose in the game of natural selection. Still, from a theoretical perspective an animal path may still be realistically represented by random walk – given that the randomness is based on properly specified biophysical premises and the animal adhere to these premises.

Photo: AOG

Outside our house I can study a magpie pica pica moving around, apparently randomly, until something catches its attention. An insect larva? A spider or other foraging rewards? After some activity at this patch it restarts its exploratory movement. As ecologist it is easy to describe the behaviour as ARS (area restricted search). In more general terms, the bird apparently toggles between relatively deterministic behaviour during patch exploration and more random exploratory moves in-between. If I had radio-tagged the magpie with high resolution equipment, I could use a composite random walk model (or more contemporary: a Brownian bridge formulation) derived from ARS to estimate the movement characteristics for intra- and inter-patch steps respectively, and test ecological hypotheses.

However, what if the assumptions behind the random walk equations are not fulfilled by the magpie behaviour? Now and then the magpie flies back in a relatively direct line to a previous spot for further exploration. In other words, the path is self-crossing more frequently than expected by chance. Also, the next day the magpie may be return to our lawn in a manner that indicates stronger site fidelity than expected from chance, considering all the other available gardens in the county. The magpie explores, but also returns in a goal-oriented manner, meaning that the home range concept should be invoked. Looking closer, when exploring the garden the magpie also seems to choose every next step carefully, constantly scanning its immediate surroundings, rather than changing direction and movement speed erratically. Occasional returns to a previous spot, in addition to returning repeatedly to our garden, indicates utilization of a memory map. In short, this magpie example may not fit the premises of an ARS the way it is normally modeled in movement ecology, namely as a toggling between fine- and coarser-scale random walk.

Hence, two challenges have to be addressed.

  1. What are the conditions to treat the movement as random walk when analysing the data?
  2. What are the basic prerequisites for applying the classical random walk theory for the analysis?

Regarding the first question, contemporary ecological modelling of movement typically defines the random parts of an animal’s movement path as truly stochastic (rather than as a model simplification of the multitude of factors that influence true movement), in the meaning of expressing real randomness in behavioural terms. The Lévy flight foraging hypothesis is an example of this specification. The remaining parts of the path are then expressing deterministic rules, like pausing and foraging when a resource patch is encountered, or triggering of a bounce-back response when sufficiently hostile environment is encountered. In my view this stochastic/deterministic framework is counterproductive with respect to model realism, since it tends to cover up the true source of randomness.

To clarify the concept of randomness in movement models one should be explicit about the model’s biophysical assumptions. Different sets of assumptions lead to different classes of random walk. In my book I summarized these classes as eight corners of the Scaling cube. Sloppiness with respect to model premises hinders the theory of animal space use to evolve towards stronger realism.

  • Random walk (RW) in the classical sense; i.e., Brownian motion-like, regards a statistical-mechanical simplification of a series of deterministic responses to a continuous series of particle shuffling. Collision between two particles is one example of such shuffling events. In other words, during a small increment of time a passively responding particle performs a given displacement in compliance with environmental factors (“forces”) and physical laws at the given point in space and time. Until new forces are acting on the particle (e.g., new collisions), it maintains its current speed and direction. In other words, under these physical conditions the process is also Markov-compliant: regardless of which historic events that brought the particle to it current position, its next position is determined by the updated set of conditions during this increment. The next step is independent of its past steps.
  • The average distance between change of movement direction of a RW is captured by the mean free path parameter. This implies that RW is a scale-specific process, and the characteristic scale is given by the mean free path during the defined time extent.
  • Since the RW particle is responding passively, its path is truly stochastic even at the spatio-temporal resolution of the mean free path. When sampling a RW path at coarser temporal resolutions a larger average distance between successive particle locations is observed. Basically, this distance increases proportionally with the square root of the sampling interval. This and other mathematical relationships of a RW (and its complementary diffusion formulation) is predictable and coherent from a well-established statistical-mechanical theory.
  • Stepping from a physical RW particle to a biophysical representation of an individual in the context of movement ecology implies specification and realism of two assumptions: (1) the movement behaviour should be Markov compliant (i.e., scale-specific), and (2) the path should be sampled at coarser intervals than the characteristic time interval that  accompanies the mean free path (formulated in the average “movement speed” at the mean free path scale). At these coarser spatio-temporal resolutions even deterministic movement steps becomes stochastic by nature, due to lumping together the resultant displacement from a series inter-independent finer-grained steps.

    An animal is observed at position A and re-located at position B after t time units. The vector AB may be considered a RW compliant step if – and only if – the intermediate path locations (dotted circles) in totality are sufficiently independent of the respective previous displacement vectors to make the resultant vector AB random. Each of the intermediate steps may be caused by totally deterministic behaviour. Still, the sum of the sequence of more or less inter-independent displacements makes position B unpredictable from the perspective of position A. The criterion for accepting AB as a step in a RW sequence is fulfilled at temporal scale (sampling resolution) t, even it the “hidden layer” steps are more or less deterministic at finer resolutions <<t.

    In my book I refer to such observational coarse-graining as increasing the depth of the hidden layer, from a fine-resolved unit scale – where local causality of respective displacements are revealed – to a coarser resolution where deterministic (and Markov-compliant) behaviour requires a statistical-mechanical description.

Regarding the second question raised above regarding Markov compliance, see the RW criterion in the Figure to the right [as was also exemplified by “Markov the robot” in Gautestad (2013)].

However, what if the animal violates Markov compliance? In other words, what if it is responding in a non-Markovian manner, meaning that path history counts to explain present movement decisions? Is the magpie-kind of non-Markovian movement typical for animal space use, from a parsimonious model perspective, or is multi-scaled site fidelity the exception rather than the rule? These are the core questions any modeller of animal movement should ask him/herself. One should definitely not just accept old assumptions just because several generations of ecologists have done so (many with strong reluctance, though).

Instead of accepting classical RW or its trivial variants correlated RW and biased RW as a proper representation of basic movement by default, albeit while closing your nose, you should explore a broader application of other corners of the Scaling cube, each with respective sets of statistical-mechanical assumptions.

 

REFERENCE

Gautestad, A. O. 2013. Lévy meets Poisson: a statistical artifact may lead to erroneous re-categorization of Lévy walk as Brownian motion. The American Naturalist 181:440-450.