fractals

[Paolo Cavallini]

Hi all.
Just found: http://nsac.ca/envsci/staff/vnams/Fractal.htm
Does anybody know of a similar implementation with free software?
All the best.
pc
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[Damiano G. Preatoni]

In un messaggio del Friday 30 January 2009, Paolo Cavallini ha scritto:
> Hi all.
> Just found: http://nsac.ca/envsci/staff/vnams/Fractal.htm
> Does anybody know of a similar implementation with free software?
> All the best.
> pc

Nope :(
Vilis Nams produced _lots_ of software (even for the Palm V!) but none of them
free, or based on something different from Windows...

Anyway, here at my research group, we're facing a similar "Nams-originated"
problem. since the most used triangulation software (Locate III) is grom the
same author and suffers the same problem.

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...

...lots of stuff to do!

a nice week to all the list.

--
L'inutilita' del primo Diluvio dissuase Dio dal mandarne un secondo.
                -- Nicolas de Chamfort
-----------------------------------------------------------
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)

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[Clément Calenge-2]

Hi all,

> 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.
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B7W63-4V28T4D-1&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=1a4a3e6e0ad0be0d8e81f8f2bbf2209d

Literature cited:

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,


Clément Calenge.

--
Clément CALENGE
Office national de la chasse et de la faune sauvage
Saint Benoist - 78610 Auffargis
tel. [phone]

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[Paolo Cavallini]

Clément Calenge ha scritto:
> Hi all,
...
> Hope this helps,

Clément: extremely interesting, as always!
Would you mind to add these considerations to the AniMove wiki for
future reference?
All the best.
pc
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[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 [phone] fax [phone]
http://biocenosi.dipbsf.uninsubria.it/
ICQ: 78690321 jabber: prea@... skype: prea.net
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[Clément Calenge-2]

Paolo Cavallini wrote:

> Clément Calenge ha scritto:
>  
>> Hi all,
>>    
> ...
>  
>> Hope this helps,
>>    
>
> Clément: extremely interesting, as always!
> Would you mind to add these considerations to the AniMove wiki for
> future reference?
> All the best.
> pc
>  

No problem! I will wait until this week-end before adding these elements
to the wiki (in addition, this will let some time to potential
contradictors to reply to the mail).
All the best,


Clément

--
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Office national de la chasse et de la faune sauvage
Saint Benoist - 78610 Auffargis
tel. [phone]

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[Anne Ghisla Insubriae-2]

On Wednesday 04 February 2009 10:42:40 Damiano G. Preatoni wrote:

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

I'm going to submit this (re)implementation as Google Summer of Code [0]
application [0,1].

best regards,

Anne Ghisla

[0] http://code.google.com/soc/2008
[1] https://stat.ethz.ch/pipermail/r-devel/2009-February/052020.html
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[Clément Calenge-2]

Hi Damiano,

> 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...)
>  

Many thanks for this answer, you made a very interesting point.
Actually, my point of view is anchored in the field of statistical tools
development for animal movement analysis, and I sometimes overlook the
need of biologists in practical studies. I of course acknowledge that
this approach could be used presently with benefits as a heuristic tool,
to explore scale-related patterns in the trajectory.

However, this leads to an interesting question from a theoretical point
of view: the fractal D has a clear mathematical meaning only when
objects are self-similar (same properties at large and small scale), and
from an "applied studies" point of view, this measure is mainly used on
objects which are not self-similar, precisely to study this deviation
from self-similarity, and the scale at which this deviation occurs (but
IMHO, we should not call it "fractal" D in this context). What I do not
understand is why a parameter measuring a property on one category of
objects would necessarily be a meaningful measure of the deviation of
one object from this category?

The main argument of Nams (2005; Oecologia 143: 179-188), a strong one
actually, is an empirical observation: all the simulations he did seem
to confirm that D changes continuously within domains, and that
discontinuity occur at domain changes (under the hypothesis of spatial
homogeneity of the trajectory). Both your successes in using this
approach and this empirical argument make the approach interesting for
multiscale analysis (together with the present lack of alternative
approaches in the ecological literature).

This is a bit off-topic, so I do not pursue further, but I think that
more theoretical research may be needed here to establish clearly the
theoretical foundations of this approach, which would define the context
on which it relies (hypotheses on which it relies, required properties
for the trajectories, etc.; but I may have missed such work in the
literature)...
Best wishes,


Clément Calenge
--

Clément CALENGE
Office national de la chasse et de la faune sauvage
Saint Benoist - 78610 Auffargis
tel. [phone]

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[Damiano G. Preatoni]

In un messaggio del Thursday 05 February 2009, hai scritto:

> Hi Damiano,
>
> > 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...)
>
> Many thanks for this answer, you made a very interesting point.
> Actually, my point of view is anchored in the field of statistical tools
> development for animal movement analysis, and I sometimes overlook the
> need of biologists in practical studies. I of course acknowledge that
> this approach could be used presently with benefits as a heuristic tool,
> to explore scale-related patterns in the trajectory.

Glad it worked... perhaps this is the result of being the GIS & statistics
person in a field-oriented reseach group... contamination is always friutful,
isnt'it? :) We rely on people that 'just writes software tools', but this
kind of feedback IMHO is necessary to have better and really usable tools!

> However, this leads to an interesting question from a theoretical point
> of view: the fractal D has a clear mathematical meaning only when
> objects are self-similar (same properties at large and small scale), and
> from an "applied studies" point of view, this measure is mainly used on
> objects which are not self-similar, precisely to study this deviation
> from self-similarity, and the scale at which this deviation occurs (but
> IMHO, we should not call it "fractal" D in this context). What I do not
> understand is why a parameter measuring a property on one category of
> objects would necessarily be a meaningful measure of the deviation of
> one object from this category?
>
> The main argument of Nams (2005; Oecologia 143: 179-188), a strong one
> actually, is an empirical observation: all the simulations he did seem
> to confirm that D changes continuously within domains, and that
> discontinuity occur at domain changes (under the hypothesis of spatial
> homogeneity of the trajectory). Both your successes in using this
> approach and this empirical argument make the approach interesting for
> multiscale analysis (together with the present lack of alternative
> approaches in the ecological literature).

Good point... I agree with Nams' conclusions, and the same speculations were
already discused in Millspaugh & Marzluff 2001 (chap 6).
As I wrote above, I'm half a theoretician (my, 25%?) and half an applied
biologist, so I try to adapt the existing theoretical frameworks to our daily
chores...

> This is a bit off-topic, so I do not pursue further, but I think that
> more theoretical research may be needed here to establish clearly the
> theoretical foundations of this approach, which would define the context
> on which it relies (hypotheses on which it relies, required properties
> for the trajectories, etc.; but I may have missed such work in the
> literature)...

Not of topic at all, IMHO... just about off-topics, what about proposing a
Google Summer of Code project either for a vFractal implementation in
adehabitat?

all the best

--
Se i fatti e la teoria non concordano, cambia i fatti.
                -- Albert Einstein
-----------------------------------------------------------
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 [phone] fax [phone]
http://biocenosi.dipbsf.uninsubria.it/
ICQ: 78690321 jabber: prea@... skype: prea.net
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