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.
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.
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.
Regarding the second question raised above regarding Markov compliance, see the RW criterion in the Figure below [as was also exemplified by “Markov the robot” in Gautestad (2013)].
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.
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.
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.
- What are the conditions to treat the movement as random walk when analysing the data?
- What are the basic prerequisites for applying the classical random walk theory for the analysis?
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. 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 below [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.