A Statistical-Mechanical Perspective on Site Fidelity – Part IIX
The power law expansion of observed space use in the Multi-scaled random walk model (MRW) shows a non-trivial relationship with sample size N of spatial relocations (fixes). In this post I introduce imaginary numbers to resolve more precisely what I have called the apparent paradox of the time-independent inwards/outwards expansion with increasing N, as it emerges both from the spatial and the temporal aspect of the process. This novel development hopefully contributes to the continued bridge-building between biophysics and space use aspects of behavioural ecology.
The inwards/outwards expansion in the MRW model is popping out of the Home range ghost formula I(N) ≈ c√N (Gautestad and Mysterud 2005), where incidence, I, is the total area of fix-containing virtual grid cells. As verified repeatedly both theoretically and empirically, I(N) is not simply a statistical small-sample size artifact, but an emerging property of the combination of spatio-temporal memory and scale-free habitat utilization. From the perspective of classic (Markovian-based) dynamic and statistical modelling of space use this non-trivial statistical pattern is paradoxical, but it is resolved in the MRW formulation.
Below I propose at a deeper mathematical level to expand the model's real number formulation with an imaginary part; i.e., invoking vectors in the complex plane of real and imaginary numbers, for the sake of improved biophysical coherence with respect to the inherent inwards/outwards paradox. In particular, I will formulate both the spatial and the temporal complements of observation intensity.
First a summary of the spatial aspect. The spatial scale c ≈ CSSU in the Home range ghost formula represents the spatial resolution (grid scale) of the "balancing level" of memory-influenced site fidelity and scale-free movement*, as was conceptualized in this post: "A Statistical-Mechanical Perspective on Site Fidelity – Part VII".
I(N) should be considered being a product of "observed" (outward) and "hidden" (inward) structures of the spatial pattern of fixes at the given CSSU resolution. To ensure compliance with entropy principles, in Part VII I argued that
Iinwards * Ioutwards = Itotal
where Ioutwards regards "observed" space use at the CSSU scale, and Iinwards is "hidden" at finer resolutions than the CSSU observation scale. This model implies that Iinwards and Ioutwards are of equal strength; i.e., Ioutwards = √Itotal, as outlined by the Home range ghost formula.
Next, the temporal aspect of the MRW process. Below I expand this statistical-mechanical description of complex space use by focusing on observation intensity, N, per se rather than observed space use I(N). Also from this perspective, Ninwards and Noutwards are of equal strength (Gautestad and Mysterud 2010). This implies that Noutwards = √Ntotal.
In this temporal N-complement of the spatial I(N) described above, consider conjugate vectors in the complex plane of real and imaginary numbers; describing inward and outward expansion of observed space use at the Characteristic scale of space use (CSSU).
The process from this complementary temporal scale perspective can be formulated as
Ninwards * Noutwards = N | c = 1 at the CSSU scale, Ninwards = Noutwards ∝ √Ntotal, Ntotal = N
Thus, Elements for a preliminary approach to this space-time complementary scaling relationship was presented in Gautestad and Mysterud (2010). However, the expression outlined above is imprecise, considering that the product is expanding equally strong when N ∝ 1/t and N ∝ T; i.e., expanding observation intensity T/t from increasing observation frequency 1/t within a given time extent T, or from increasing T by keeping t constant (see Part VII). This duality of observation intensity raises a deep challenge with respect to inwards expansion √(Ninwards). Why?
In the temporal domain of process scaling we get a similarly paradoxical expression for space use intensity as explained in Part VII, but now given by the integer N, since square root of N due to the logic of this framework is "hidden" at a lower value of the "Elacs" axis** relative to the given temporal observation scale ratio N ∝ T/t. The undenominated ratio T/t, as expressed by the Elacs axis (larger N gives larger Elacs), implies a negatively shifted lower level of the Elacs-interval for Ninwards as N is increased. Similarly, Noutwards is increasing the upper level of the Elacs-interval to the same extent as N is increased. The "negative N" component is expressing the process of "resetting" space use expansion during return events to known locations.
This resetting of a movement path at a given average frequency is in statistical-mechanical terms represented by Ninwards. A return event functions like restarting the movement path from a previously visited location. However, since the base of potential return targets grows over time, a new resetting will on average both be distributed among a larger spatially dispersion of fixes, in addition to being distributed among a larger historic time extent since the animal started utilizing this particular area.
Loosely formulated, the dynamics is in operational terms partly running backwards in time (the "negative N" concept). However, since the average return interval since last visit to a location depends on the average time that has lapsed since all historic locations the animal currently relates to, a return does not bring the process back to start.
However, a mathematical problem needs to be resolved:
√(-N) * √(N) = N ??? | c = 1, N >= 1
The question marks express that real numbers do not allow for this operation of taking the square root of a negative number. Negative numbers don't have real square roots since a square is either positive or 0. However, introducing complex numbers (the combination of real an imaginary numbers) do the trick:
√[(N)(-i)] * √[(N)(i)] = N,
expressing the product of "inward" and "outward" temporal component of space use expansion.
Similarly, we have the formulation of the spatial aspect I(N):
√[(I(N)(-i)] * √[(I(N)(i)] = I(N).
By invoking a complex plane with imaginary numbers in both the temporal and spatial domain of the MRW process we have hereby achieved a product of inwards and outwards expansion of space use in a mathematically consistent manner.
*) This property of "observer-dependent" and non-trivial (paradoxical) N-dependence was described in Part I-VI. In particular, recall from Part VI:
**) In my book and here i my blog you may search for the concept of Elacs, expressing complex dynamics over a temporal scale range t/T, where t regards observation interval and T regards the temporal observation window.
Gautestad, A. O., and I. Mysterud. 2005. Intrinsic scaling complexity in animal dispersion and abundance. The American Naturalist 165:44-55.
Gautestad, A. O. and I. Mysterud. 2010. The home range fractal: from random walk to memory dependent space use. Ecological Complexity 7:458-470.