In my book I devote several chapters to explain, illustrate, simulate and mathematically formulate the parallel processing (PP) principle. PP represents the foundation of the extended statistical-mechanical framework that I propose as necessary to model complex space use more realistically. In this Part II of the PP description I elaborate further on this unfamiliar approach, which represents an alternative to the standard “Markov process” (mechanistic) methods in movement ecological theory.
Hopefully the reader of my book and these supplementary blog posts will manage the difficult task to be both critical and open-minded. After all, I have formulated several testable hypotheses, where the null hypothesis is the standard framework and the alternative hypothesis is the PP-compliant process. Several pilot tests on real animal space use are presented in my book and in our papers, lending support to a PP kind of space use; spatio-temporal memory utilization in combination with a multi-scaled perception of the environment (I refer to the Scaling cube to illustrate how PP locates itself relative to the standard models).
Now, let us continue the road towards PP illustrations at the simplistic, conceptual level. In the illustration below I seek to show the spatial aspect of multi-scaled space use, from the PP perspective. Some kind of habitat heterogeneity is added as coloured blobs. Each dotted circle has an individual location at its origin (marked by a “star”), representing a given individual’s position at a given point in time. The set of circles/stars represent either a series of relocations of a given individual over time or a simultaneous observation of many individuals in the same landscape at a given point in time (statistical-mechanically, the latter is called an ensemble). The size of the circle is meant to illustrate – in probability terms – at which spatial level the given animal in the ensemble at the given point in time is relating to its environment.
Observe that there are 1/g times as many circles with area g. As described in several posts already, this inverse relationship represents an animal that is postulated to put “equal weight” (in statistical-mechanical terms) into optimizing space use at respective spatial resolutions. In other words, the sum of the areas of small circles of a given size equals the total area of g times larger circles. From a single-individual perspective, most of the time an individual relates to its environment at a fine scale, but occasionally it broadens its perspective. From an ensemble perspective, at any point in time most individuals are in the process of executing short term goals/responses.
Now comes the more tricky part. As observed from a given small increment in time (you may call it the “mechanistic” time scale) the stars with the smallest circles are apparently executing movement behaviour in close compliance with a traditional Markov process. Simply stated, the small circles in the image above are close to the extent of the individual’s perceptual field. However, larger circles implies not only spatial memory-influenced movement events but also execution of these events at a coarser temporal scale! The illustration to the right seeks to show this relationship, where the red square points to the spatio-temporal scale range where a Markov/mechanistic model apparently could be expected to capture the process quite realistically (spatial scale in this image is proportional to “grain size”, or circle area).
However, if the process is PP-like (and not just a superposition of scale-specific dynamics at different time scales; see composite Brownian motion in the scaling cube), the behaviour within the red box will occasionally be “disturbed” by dynamics that take place at coarser temporal resolutions in a non-trivial manner. Since we are observing the process at the temporal resolution of the red box, these coarser-scale influences will happen stochastically. Thus, a given “strategic” behavioural event will by necessity appear as a surprise event from the temporally fine-grained perspective of the red box, not explicable or predictable in deterministic terms from this “tactical” time scale.
The two-axis space-time representation above could (in statistical-mechanical terms) illustrate a composite Brownian motion system that is including spatial memory within this framework (the MemCompRW corner of the Scaling cube), where the stochastic influence on fine-grained behaviour regards the superposition of probabilities from coarser-scale influences at a smaller scale. Such superposition implies compliance with principles from standard statistical mechanics.
In contrast to this familiar framework, to understand and analyze the deterministic property of the coarser-scale behaviour of a PP-compliant multi-scaled process, one has to coarse-grain the temporal resolution of the sequence of observations. Such coarse-graining to make fine-grained stochastic dynamics appear more deterministic from the coarser perspective is indicated by the dotted arrows.
Addendum, August 27: In order to conceptualize the PP’s property of such cross-scale influence, consider that the thin space-time line is standing orthogonal on the two-dimensional space-time plane. Thus, consider the illustration to be 3-dimensional, with scale range as an independent system descriptor and its origin at the mechanistic (Markov) plane. In this manner, each lower left corner of the rectangles are all located at the “here and now” origin of the space-time plane if the scale range axis is compressed to zero. An event that is executed deterministically at the scale of a given rectangle (i.e., one or another strategic goal, somewhere along the scale axis) may then by chance happen to take place inside the smaller “here and now” box. The more strategic the goal the less probable it will take place within the red box, creating a surprise event from this space-time Markov-plane perspective. A 3-dimensional system representation (space-time-scale) shows that an extended kind of statistical mechanics is required to capture a PP-compliant space use process. Compressing this axis to zero invokes dynamic paradoxes from the perspective of executing multi-scaled behaviour on the space-time plane alone!
In my book I show how analysis of a set of GPS fixes may distinguish between a mechanistic and a PP-compliant kind of dynamics, in statistical-mechanical terms.
Still confused? More posts are in the coming. In parallel (sic!) you may shortcut the waiting time till new posts arrive by reading chapters 5-7 of my book.