Improving Accuracy and Precision in Estimating Fractal Dimension of Animal movement paths
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ABSTRACT It is difficult to watch wild animals while they move, so often biologists analyse characteristics of animal movement paths. One common path characteristic used is tortuousity, measured using the fractal dimension (D). The typical method for estimating fractal D, the divider method, is biased and imprecise. The bias occurs because the path length is truncated. I present a method for minimising the truncation error. The imprecision occurs because sometimes the divider steps land inside the bends of curves, and sometimes they miss the curves. I present three methods for minimising this variation and test the methods with simulated correlated random walks. The traditional divider method significantly overestimates fractal D when paths are short and the range of spatial scales is narrow. The best method to overcome these problems consists of walking the dividers forwards and backwards along the path, and then estimating the path length remaining at the end of the last divider step.
Key Words: tortuousity, bias, efficiency, divider
1. INTRODUCTION The behaviour of animals while travelling affects aspects of their ecology at different spatial scales: for example, foraging behaviour (M˚arell et al., 2002) and animal orientation (Benhamou, 1989) at small scales, dispersal patterns at intermediate scales (Zollner and Lima, 1999), and population dynamics at large scales (With et al., 2002). However it is difficult to study behavioural aspects of these directly because it is hard to watch wild animals while they are moving, especially at large distances and over long time intervals. It is much easier to record animal locations. Thus as a proxy to watching moving animals we often analyse various characteristics of their movement paths. One common movement path characteristic used is tortuousity, measured using the fractal dimension (D). Fractal dimension is the continuous analogue of discrete geometric dimensions (Mandelbrot, 1967; Milne, 1991); for example, lines have a geometric dimension of 1 and planes a dimension of 2. The fractal dimension for movement paths lies between 1 and 2 - i.e. D = 1 when the path is straight and D = 2 when the path is so tortuous that over time it will completely cover a plane. Fractal analysis has been used in various types of studies of animal movement – ranging from the landscape perceptions of grasshoppers (With, 1994), habitat selection at different spatial scales of marten (Nams and Bourgeois, 2004), to scale-dependent movements of seabirds (Fritz et al., Acta Biotheoretica (2006) 54: 1–11 DOI: 10.1007/s10441-006-5954-8
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Springer 2006
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V.O. NAMS
Figure 1. Example of path truncation. Gross distance is estimated for this path with dividers of various step sizes. At divider sizes 5 and 15 the gross distance is underestimated by up to 5 and 15 units. Since D is based on the slope of the plot, this truncation could have major effects
2003). Fractal analysis is especially useful when studying how animals change in their response to their environment with changes in spatial
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