clement.calenge at oncfs.gouv.fr
Wed Feb 4 10:16:28 CET 2009
> I use Fractal, and the algorithms are rather well clear, both in the
> documentation and in the related papers... since Clément _has_ several
> movement ecology tools in adehabitat (the traj class and all
> trajectory-rerlated stuff IMHO are a good base to start with), a further
> AniMov project could be to add fractal movement analysis estimators...
When we added tools to analyse animal trajectories in adehabitat, we
considered whether we had to include estimators of fractal dimension.
Indeed, fractal dimension seems at first sight to be an interesting
measure of the tortuosity of animal movements. In addition, the papers
of Nams are very interesting in that respect, in that they provide
examples of how this measure could possibly be used. However, it is not
that clear what fractal dimension represents biologically, and a clear
theoretical framework underlying its use in practical analysis is
lacking (research is still needed here, IMHO).
The fractal dimension corresponds to the ability of fractal objects to
fill Euclidean spaces in which they are embedded (Halley et al., 2004).
In our case, it measures the ability of fractal trajectories to fill the
plane. Thus, a trajectory with a fractal dimension equal to 2 should
fill the plane. However, fractal dimension only have a clear meaning
when calculated on fractal objects, i.e. self-similar objects (objects
that "look the same" whatever the scale at which we look at them).
Benhamou (2004) notes that measuring the fractal dimension of a
non-fractal object has no meaning and is therefore only liable to
generate artifactual results.
However, Nams (2005) proposes a use of the fractal D which does not
assume that the studied object is fractal; in other words, D is no
longer the fractal dimension (since D is a fractal dimension only when
objects are fractal) , but simply a measure of the tortuosity of animal
movements. However, as noted by Benhamou (2004):
"a strong mathematical argument against the use of the apparent fractal
dimension F (as computed from the local slope of the log–log relation)
to measure the path tortuosity was provided paradoxally by Nams (1996)
in a paper advocating the opposite point of view. (...) F is no more
than a monotonously decreasing function of the mean cosine of turning
angles c (...) The decrease of the local slope (equal to 1 - F ; from 0
to -1) is eventually the simple reflect of the decrease (from 1 to 0) of
the mean cosine of turning angles".
The fractal dimension eventually turns out to be a measure of the
tortuosity related to the mean cosine of turning angles, which is easier
to compute (especially given that the turning angles are automatically
computed when objects of class "ltraj" are created), and easier to
interpret (see Benhamou 2004). In addition, there is a much larger
literature on the mathematical properties of the mean cosine and related
measures (all the literature on circular statistics, e.g. Batschelet
1981 or Jammalamadaka and SenGupta 2001), which render the mean cosine
more practical to use in all days analysis.
Because of (i) all the drawbacks described by Halley et al. (2004) and
Turchin (1996) when we suppose that the studied trajectories are
fractal, and (ii) the more abundant literature documenting the
properties of closely related measures or tortuosity with a clearer
biological meaning, we decided not to include the fractal dimension as a
measure of tortuosity in adehabitat.
Of course, that is not to say that I think that the fractal dimension is
useless, but rather that I do not clearly see how it can be used
presently (I would of course appreciate pointers). However, if you think
that fractal dimension can bring more than "classical" measures of
tortuosity, it should be quite easily computed in R: the function
redisltraj in adehabitat can be used to rediscretize the trajectory with
different step sizes (returning objects of class "ltraj", see
?redisltraj), and because objects of class "ltraj" store the lengths of
the steps, fractal dimension could be easily computed...
For additional details on the class "ltraj" and its use, see:
Calenge, C., Dray, S. and Royer-Carenzi, M. 2009. The concept of
animals' trajectories from a data analysis perspective. Ecological
informatics, in press.
Batschelet, E. 1981. Circular statistics in biology. Academic Press, London.
Benhamou, S. 2004. How to reliably estimate the tortuosity of an
animal's path: straightness, sinuosity, or fractal dimension? Journal of
Theoretical Biology, 229, 209-220.
Halley, J.M. and Hartley, S. and Kallimanis, A.S. and Kunin, W.E. and
Lennon, J.J. and Sgardelis, S.P. 2004. Uses and abuses of fractal
methodology in ecology. Ecology Letters, 7, 254-271.
Jammalamadaka, S.R. and SenGupta, A. 2001. Topics in circular
statistics. Series on Multivariate analysis. World scientific, London,
Nams, V.O. 2005. Using animal movements paths to measure response to
spatial scale. Oecologia,143, 179-188.
Nams, V.O. 1996. The VFractal: a new estimator for fractal dimension of
animal movement path, Landscape Ecology, 11, 289-297.
Turchin, P. 1996. Fractal analyses of animal movement: a critique.
Ecology, 77, 2086-2090.
Hope this helps,
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