Figure 1.
(a) Brownian motion compliant movement from four individuals is marked by successive relocations (‘fixes’ with individual-specific colour) over a total simulation time span T = 100 × 20 000 t time units, and sampled (‘observed’) every 100th time unit. All individuals move with a constant average speed, but different mean free path for the respective individuals leads to difference in number of steps and diffusion rate (net displacement during T). A 10-step sequence for the two series with the largest mean free path (black and blue) is shown with line segments inter-connecting the successive observed fixes. Owing to the time lag of 100 t for sampling, on average 100 times as many direction-influencing events have taken place between successive fixes for each of the four individuals, relative to sampling 20 000 fixes at unit time interval t. Thus, the straight line segments illustrating step vectors hide some finer-grained jaggedness of the respective paths. (b) The fixes from the four individuals in figure 1a are pooled and scrambled, to mimic a four-level single-individual BM. A 100-step representative sequence from the series is enlarged, to visualize the random successive mixture of steps from various scale levels. (c) N = 20 000 fixes from a Lévy walk, collected at lag tobs = 100 (i.e. total time period T = 100 × 20 000, which equals the total series length at unit time scale prior to sampling at lag 100). With reference to the iteration procedure specified in Methods, β = 2 was used. (d) N = 20 000 fixes from multi-scaled random walk under condition β = 2, collected at lag tobs = 100. Owing to return steps taking place at the same time scale (frequency 1 : 100 on average), successive fixes are collected from the transition zone for a spatially auto-correlated and non-auto-correlated series. At a smaller tobs, successive fixes would have tended to appear closer together than more distant fixes in time, as illustrated by the LW in figure 1c.