[AniMov] fractals

[Damiano G. Preatoni]

In un messaggio del Wednesday 04 February 2009, Clément Calenge ha scritto:

First, thank you Clément for the concise review!
Some comments follow...

[snip]

> 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".

[snip]

> 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).

This is a first point: I've used Nams' FRACTAL on several species up to now, 
that is Brown bear, the Alpine Ibex, Leisler's noctule and a putativa 
invasive bird species (Paradoxornis webbianus).
I found the vFractal estimator rather useful to have an idea not just of 
the 'path tortuosity', but of the scale level at which 'search patterns' 
occur. The results achieved are quite interesting, i.e. Brown bears released 
in NE Italy showed significant differences in D calculated in the release 
year and in subsequent years, Ibexes showed a 'random walk peak' in the 
D-vs-scale diagram  ad about 1 km (which does make perfect sense for an Ibex) 
and P. webbianus, already known as a short distamce mover, showed a 300 to 
500 m search range.
I found this very useful to have an idea of the scale of habitat perceprtion 
by a species, and since I run all my analyses in R (what else? :) I'm annoyed 
to go back and forth from FRACTAL to adehabitat.

This could be an example of fractal dimension use (...hope so...)

> 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.

This is definitely true. I have some (yet) unpublished data on NE Italy brovn 
bears where the correlation between mean cosines showed an interesting 
pattern when comparing autochtonous Slovenian bears (n=49) with the 9 
reintroduced in Trentino: in this case D wasn't enough, since the 'radius pr 
habitat perception' was not significantly different... but from a certain 
scale level (something like 20 km, I'd have to recheck my data), in the 
Slovenian population the correlation coefficients started to be negative, 
whereas in Trentino they stayed > 0, i.e. over a certain distance, slovenian 
bears "did turn back", and reintroduced italian bears still showed an 
exploratory pattern.

IMHO, both D and r(cosine) are useful tools, and maybe worth to be offered as 
a tool (volounteers needed? :) in adehabitat...

I missed Turchin paper and I'll give it a read... I love "countering" 
papers :)



-- 
It is almost impossible to make something "idiot-proof".
Idiots are so resourceful...
-----------------------------------------------------------
Damiano G. Preatoni, PhD

Unità di Analisi e Gestione delle Risorse Ambientali
Dipartimento Ambiente-Salute-Sicurezza
Università degli Studi dell'Insubria
Via J.H. Dunant, 3 - 21100 Varese (ITALY)

tel +39 0332421538 fax +39 0332421446
http://biocenosi.dipbsf.uninsubria.it/
ICQ: 78690321 jabber: prea at jabber.org skype: prea.net
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