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First published online March 2, 2007
Journal of Experimental Biology 210, 935-945 (2007)
Published by The Company of Biologists 2007
doi: 10.1242/jeb.02710
Fractal landscape method: an alternative approach to measuring area-restricted searching behavior
1 University of California, Santa Cruz, Long Marine Laboratory, Center for
Ocean Health, 100 Shaffer Road, Santa Cruz, CA 95060, USA
2 Department of Mathematics and Computing, University of Southern
Queensland, Toowoomba 4352, Australia
* Author for correspondence (e-mail: tremblay{at}biology.ucsc.edu)
Accepted 8 January 2007
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Summary |
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 TOP Summary IntroductionMaterials and methodsResults and discussionReferences |
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Key words: fractal dimension, fractal landscape, elephant seal, albatross, foraging, prey-searching strategy, tracking, Argos, top predator, area-restricted search
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TOPSummaryIntroductionMaterials and methodsResults and discussionReferences |
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). Ultimately,
efficient searching behavior results in higher energy gains relative to the
time and/or energy expended (Bell,
1991; Pyke et al.,
1977). Therefore, greater understandings of the factors that shape
foraging behavior are crucial to evaluate ecological specialization and
adaptation.
The use of electronic devices to track animal movement has provided
detailed information on the movement patterns and prey-searching behavior of a
wide variety of organisms (Austin et al.,
2004; Block et al.,
2002; Costa, 1993;
Hays et al., 2004). As top
predators forage in a patchy, hierarchical environment
(Fauchald, 1999), they display
alterations in traveling speed and direction between different behavioral
phases in the track. These phases are scale dependent
(Fauchald et al., 2000), and
result in hierarchical area-restricted searching (ARS) pattern of movement.
This pattern has been shown to be a response to patchy resource distribution
(Benhamou, 1992;
Fauchald and Tveraa, 2003;
Lode, 2000). To investigate
foraging behavior in a patchy environment it is necessary to independently and
accurately describe each ARS zone along a track.
Several techniques have been developed to measure the scales at which ARS
occurs (Robinson et al.,
2007). One of the simplest methods is to measure changes in
transit speed along a track (LeBoeuf et al., 2000). The assumption here is
that intensive searching is associated with reduced speed; other approaches
measure the changes in sinuosity and/or angularity of the track
(Erlandsson and Kostylev, 1995;
Laidre et al., 2004). However,
none of these methods provide an index of the scale or domain of the region of
ARS. Several approaches have been developed to measure the region or spatial
scale of the ARS. First-passage time (FPT) provides a measure of the time an
animal takes to cross a circle of a certain radius that is moved along the
track (Fauchald and Tveraa,
2003). The circle radius corresponding to the highest variance in
FPT is used to define the animal's operational spatial scale. One type of
fractal analysis uses a similar approach, but uses fractal dimension (D) (or
fractal dimension estimator) instead of time
(Nams, 1996). A segment of a
given length is moved along the track, and D is calculated for each segment.
As D generally increases with track convolutions, the segment lengths
corresponding to highest average D, and/or highest variance in D, are used as
a cut-off to identify the operational spatial scales
(Nams, 1996;
Nams, 2005). These two methods
are similar in principle, but produce different results. FPT analysis
identifies a radius (Pinaud and
Weimerskirch, 2005) whereas the above fractal analysis identifies
a segment length (Fritz et al.,
2003; Nams, 2005).
Intense searching behavior is often a combination of longer time spent and
higher track convolutions in an ARS zone, which in theory should be identified
by both methods. However, these methods have fundamental differences. One
focuses on time whereas the other focuses on space coverage, both in an
exclusive way. Therefore, the two methods are not really measuring the same
thing (Robinson et al., 2007)
and the use of one or the other has to be driven by underlying questions
and/or theory about ARS behavior. Both methods may accurately detect the major
ARS zones but they may differ in describing them, and they have a major flaw:
they both identify scales that are roughly constant throughout the track
(Fauchald and Tveraa, 2003;
Fritz et al., 2003;
Pinaud and Weimerskirch,
2005). In order to finely describe ARS behavior, it is important
to identify the scale and position of each ARS zone individually.
The present study describes and tests a new method for quantifying ARS behavior in tracking data, and tests the effect of the track quality on the efficiency of the method.
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Materials and methods |
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TOPSummaryIntroductionMaterials and methodsResults and discussionReferences |
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;
Laidre et al., 2004), but none
focused strictly on space coverage. The fractal dimension of a set of
two-dimensional points can be seen as a measure of its propensity to cover the
plane, being a value of one for no plane coverage (a straight line or a
circle, for example) and two for full coverage of some area in the plane (a
Hilbert curve, for example). The fractal dimension does not measure track
sinuosity per se, and, if sinuosity is the question of interest, some
other sinuosity indexes should be used instead
(Benhamou, 2004). When
analyzing searching behavior it makes sense to quantify how well an animal
covers the region or plane of interest both spatially and temporally: for
example, the greatest chance of recovering your watch requires that you
maximize the amount of area searched within the plane and this will require
more time. If your watch is not precious, you will give up earlier and perhaps
not cover the plane as well. Since plane coverage and time are two fundamental
ingredients for searching behavior, we argue that methods of track analysis
should quantify both components of searching behavior: time and space
coverage. Based on these considerations, we develop a new method for detecting and quantifying searching behavior focusing on space coverage and time. The underlying idea is that animals should increase their local plane coverage (i.e. fractal dimension of the track) when they exhibit ARS behavior. After testing the accuracy of the method on simulated tracks of known characteristics, we tested the effect of track inaccuracy on the process by altering the perfect simulated tracks in a way that resembles Argos track qualities, and reanalyzed these tracks to determine the effect of track quality.
Argos track qualities vary considerably with the type of animal tracked
(Tremblay et al., 2006).
Consequently, differences in track qualities can affect apparent plane
coverage (and sinuosity) and/or spatial distribution of presence (i.e. time),
and it is therefore important to quantify these effects. We therefore
simulated albatross-like and elephant seal-like tracks. These two species were
chosen because they represent two extremes in animal speed and in Argos track
qualities (Tremblay et al.,
2006).
Note that we focused here on the description of small-scale ARS, where
animals are presumably searching for prey. At a larger scale, the animal is
probably searching for suitable areas that include small-scale ARS
(Pinaud and Weimerskirch,
2005).
Track simulations
Tracks were simulated by concatenating portions of tracks determined by
correlated random walks (CRW) with different degrees of correlation between
successive steps. The degree of the correlation between successive angles was
controlled by the standard deviation (s.d.) used to generate the circular
normal distribution from which the turning angles were randomly selected. When
s.d. is small, the path is straighter, and when s.d. is large, the track
approaches an uncorrelated random walk (i.e. Brownian motion). The process
essentially follows the simulation process described by Fauchald and Tveraa,
with some adjustments (Fauchald and
Tveraa, 2003).
Adjustments include the following:
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The parameters were chosen for simulating each type of track and for each portion according to Table 1. An example of a simulated track is given in Fig. 1A. The time between each step was 5 min.
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The number of locations per day used to sub-sample each track was randomly selected from a truncated normal distribution centered on 17.770 and 5.125 with a s.d. of 4.450 and 2.826 in albatross and elephant seal tracks, respectively. Minima were set to six and three locations per day and maxima were set to 26 and 16 locations per day, for albatross and elephant seals, respectively. These values were based on data collected for the TOPP program during 2002–2005 (http://www.toppcensus.org/).
Similarly, we introduced spatial error to each selected point
(Fig. 1B). For this, a quality
class was randomly attributed to each selected point, in order to match mean
proportions of the various quality classes in real deployments
(Table 2). An error was then
randomly selected from a normal distribution centered on the mean error and
with the same s.d. as the errors for the corresponding class
(Table 2), as obtained during
static tests at the laboratory [similar results to those found in tests by
Vincent et al. (Vincent et al.,
2002)].
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The resulting tracks were then considered to be similar to Argos tracks,
and were therefore filtered and further interpolated along a Bezier curve
(µ=0.2), following Tremblay et al.
(Tremblay et al., 2006)
(Fig. 1C).
The filtering process involved a speed filter set at 80 and 10 km h–1 for albatross and elephant seal tracks, respectively. An angle filter set at 170° was also used in both types of track, to reject small-scale spikes not removed by the speed filter. In addition, when locations were closer than 10 min, the location with the least quality was removed irrespective of its quality class. This allows for the removal of induced sinuosity, created by location proximity, and avoids taking into account locations that are calculated using common hits from the transmitters (in real data). All quality classes were allowed to be removed.
Calculating fractal dimension
A fractal dimension (D) can be calculated from a set of points using
several distinct methods that differ in accuracy, in the sensitivity to the
number of points used and computing time required
(Esteller et al., 1999;
Kallimanis et al., 2002;
Nams, 2006). It is therefore
essential to carefully select the appropriate algorithm for a given
application (Esteller et al.,
1999; Jelinek et al.,
1998). In this study, fractal D was calculated using the
information dimension (Halley et al.,
2004) as it is less prone to errors in finite datasets (A. J.
Roberts:
http://www.sci.usq.edu.au/staff/robertsa/soft.html).
We also tested another algorithm implementing the box-counting method to
calculate fractal D (Giorgilli et al.,
1986; Liebovitch and Toth,
1989). The results were noticeably less accurate (data not
shown).
The accuracy of our calculation was assessed using five mathematical curves of known fractal dimension (Fig. 2): a Hilbert curve composed of 4096 points (D=2), a Sierpinski triangle of 2409 points (D=1.58), a Koch's snowflake of 3075 points (D=1.26), a circle of 2000 points (D=1) and a straight line of 2000 points (D=1). For these curves there is no difference between the information dimension and any other fractal dimension. The information dimensions D calculated on these sets were 2.01, 1.56, 1.29, 1.05 and 1.05, respectively, corresponding to a mean error of 3%.
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100 points are used
(Fig. 2). The stability of the
fractal D estimates, with respect to the number of points, reflects the
sensitivity of the shape of the curve to sub-sampling. As an illustration,
consider the decrease in estimated fractal D of the circle in
Fig. 2: the shape of a circle
is indeed affected by the number of points that defines it, being here a
polyhedron of number_of_points –1 number of edges. The higher the
numbers of points, the closer to the real shape of the circle and, therefore,
the more the estimated fractal D tends to 1. The fact that fractal D is <1
for the straight line data set is normal, because, in this case, it is a
measure of line filling (D
1) rather than plane filling (1D2).
A fractal dimension does not characterize a dataset at some specific length
scale; instead it characterizes a self-similar pattern that occurs in the data
across many length scales. Thus, basic fractal dimension estimates come from
fitting a straight line to data on a log-log plot where one axis covers a
range of length scales. As part of a least-squares fit, the algorithm not only
determines the two parameters of the best fractal similarity, but also the two
length scales between which the fractal similarity holds. For the data in this
work, we found that fractal self-similarity holds over an order of magnitude
in scale. Such a decade of length scales is acceptable to claim fractality
(see Feder, 1988). In our
application, it is necessary to note that consistent estimates of fractal
dimension (i.e. higher values for higher plane coverage) are more important
than strictly accurate estimates. Therefore, if the self-similarity in our
tracks is different to the mathematical curves, the accuracy of fractal
dimension may change slightly, but this is not detrimental because consistency
is maintained (see below).
Implementation of the fractal landscape method
The fractal landscape method consisted of four distinct steps. Step 1: we
determined a particular segment length (i.e. a distance) for the track at
which the largest most common ARS occurred. Since speed is reduced in ARS, we
determined the segment length as follows. First, the rank of speed between
successive steps was computed. Second, the portion of the consecutive lowest
third of ranks was determined. Third, the distance traveled during these
portions of relative low speed was calculated. Finally, we used the longest of
these distances as the segment length. This segment length is used as the
window size for the next step.
Step 2: we calculated fractal D along the track. For each point, D was calculated using the locations belonging to the segment centered on each point (the window size). When the number of points was more than 200, we randomly selected 200 points for the given window. This considerably reduced the computing time while preserving reasonable accuracy (Fig. 2), ensuring that sufficient data were used in each D estimate. The plot of D against time describes fractal peaks and valleys, i.e. the fractal landscape (Fig. 3A).
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Step 4: based on this threshold, we then extracted fractal peak characteristics, which were used to visualize and quantify ARS behavior. For each peak we extracted its duration, area above the threshold value, mean fractal D, and the circle position and size where each fractal peak was overlaid on the mapped track (Fig. 3C). The zone corresponding to each fractal peak is determined by a circle centered on its mean coordinates. Standard deviations of the coordinates were calculated for the x and y axes. The radius of the circle was then calculated as three times the smaller of the two s.d. This reduced the effects of outliers, and kept the majority of ARS coordinates within the circle. High-intensity-searching behavior can be seen as the most efficient method of covering the localized circle (high fractal D) over a long time interval and is therefore defined by a high area of fractal peak. This area was thus calculated and used as an index of searching intensity, and color coded for visualization in Fig. 3C.
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Results and discussion |
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TOPSummaryIntroductionMaterials and methodsResults and discussionReferences |
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Analysis of corrected simulated tracks
The introduction of Argos-like errors into the tracks resulted in two
different effects. Some ARS were lost (i.e. existent but not found), and some
ARS were produced (i.e. found but non-existent). As a result, the number of
ARS regions was more variable when Argos tracks were used
(Fig. 4B).
In albatross-like degraded tracks, 9.6% of the track's ARS were lost whereas 90.4% appropriately identified areas of actual ARS. In elephant seal-like degraded tracks, these values were 16.3 and 83.7%, respectively. In addition, the number of produced ARS regions was 1 out of 105 and 26 out of 108 in albatross- and elephant seal-like tracks, respectively. As a result of the differences in track qualities between albatross- and elephant seal-like tracks, elephant seal tracks had more lost and produced ARS regions (i.e. more errors) than in albatross-like tracks. However, some of these errors can be easily flagged: produced ARS were generally short in time and intensity, with 65.6% of them belonging to the lowest quintile of ARS intensities (area under the peak, see Materials and methods). The deletion of all ARS regions where the search-intensity index falls within the lowest quintile can remove many of the wrong ARS regions, while losing only a few of the good but not intense ARS regions, in elephant seal Argos tracks.
Nevertheless, the proportion of locations (i.e. time) correctly identified
as in or out of an ARS was still reasonably high
(Table 3; 89.5% in albatrosses
and 83.8% in elephant seals). Note that Argos inaccuracies caused a
disproportionate error towards lost ARS rather than produced ARS in both kinds
of tracks (Table 3). Logically,
ARS are missing because of temporal degradation, whereas they are produced
through spatial degradation introducing more spatial coverage. This indicates
that the biggest problem with using Argos data is not the spatial inaccuracy
but the relatively low sampling rate. In this respect, recently developed
methods for filtering Argos data without deleting points
(Jonsen et al., 2005) should
considerably improve the situation, since it is common to delete a third to a
half of the raw Argos data with classical filters
(Austin et al., 2003;
McConnell et al., 1992).
However, differences in spatial accuracies between albatross-like and elephant seal-like corrected tracks had a visible effect: 4.7 times more locations were incorrectly attributed to belonging to an ARS in elephant seal-compared with albatross-like tracks. This was the largest difference in the results between the two kinds of tracks (Table 3).
Spatial accuracy also affected ARS size. Not surprisingly, ARS radii are larger in degraded tracks than in intact tracks, and this difference was higher in elephant seal-than in albatross-like tracks (Fig. 6). We determined that ARS radii in corrected tracks could be corrected by a factor of 1.2 and 2.2 in albatross- and elephant seal-like tracks, respectively. Dividing the radii by these factors allowed us to match the distribution of the ARS radii obtained with intact tracks (Fig. 6). Note that as we chose two extreme cases in terms of Argos track qualities, it is expected that correction factors applied to datasets of intermediate qualities should necessarily be intermediate as well.
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Does the method work on real data?
We applied the method on real non-duty-cycled Argos tracks of 31
albatrosses and 20 elephant seals and the output was consistent with results
from this study. In the Laysan albatross (Phoebastria immutabilis)
track shown in Fig. 7, nine
peaks were clearly identified in the fractal landscape, each peak
corresponding to an ARS on the track. Interestingly, the ARS regions with the
highest intensity index (highest area under the peak, see Materials and
methods) were distributed along a west–east transect whereas the lowest
indexes were found in ARS regions north and south of this line
(Fig. 7). This line is situated
approximately 1700–2000 km north of the breeding colony (Tern island,
Hawaii), and corresponds with the position of the food-rich North Pacific
Transition Zone between sub-Arctic and subtropical waters at this time period
(Hyrenbach et al., 2002;
Polovina et al., 2000). This
result indicates that more intensive ARS behavior occurred when closer to the
front, and shows the utility of this index in identifying biologically
important regions.
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Later, we realized that degrading the track affects the speed calculation along the track, and this sometimes posed problems for the calculation of the window's size (step 1 of the method). This was solved by smoothing the speed data using a classic moving average filter. The window used for the filter was simply chosen by visually interpreting the plot of speed against time, in a way that maintains the major oscillations and removed high-frequency noise. The process was always obvious and straightforward to achieve and therefore we did not automate it.
A similar problem arose with the high-frequency noise in the consecutive fractal D calculations (quick oscillations in the fractal landscape), and in the determination of the threshold. The problem was solved in the same manner, by using a moving average smoothing.
Comments on the method
Concerns have been raised about the use of fractal geometry in biological
sciences (Halley et al., 2004;
Turchin, 1996) and, in
particular, the use of fractal dimension (D) to estimate track sinuosity
(Benhamou, 2004). Some
criticism results from the confusion over what the fractal dimension, or a
fractal dimension analysis, measures biologically
(Jelinek et al., 1998). In the
analysis of animal distribution there are several ways of applying fractal
geometry concepts (Laidre et al.,
2004; Nams, 1996;
Russell et al., 1992;
With, 1994). A common mistake
is that D measures sinuosity (Benhamou,
2004; Laidre et al.,
2004; Nams, 1996).
The track of an animal that turns 180° at every movement step will show
very high sinuosity but minimum D. Similarly, a track describing a circle and
a track describing a straight line both have a low D value (namely one) but
will have different sinuosity. A high D value will only result when a track's
convolutions lead to reasonably efficient coverage of an area in the plane. In
our study, D should be considered as an area-filling index, and is therefore
particularly suitable for the analysis of searching behavior. Fractal
dimension is classically used in studies interested in plane or volume filling
measurements (Chmel et al.,
2005; Kim and Kim,
2005; Phattaralerphong and
Sinoquet, 2005; Uttieri et
al., 2005). A recent study using fractal dimension in
three-dimensions highlighted the subtle differences between degree of
convolution and fractal dimension (Uttieri
et al., 2005).
Calculating a fractal dimension for non-fractal objects is also problematic. In our case, the range of length scales over which our calculation was done was always >10, which indicates that our segments were near-fractal. Although this was true in simulated intact tracks, this was artificial in corrected tracks and resulted from the interpolation process. The interpolation process was, however, crucial in order to obtain accurate estimates of D.
The main advantage of the fractal landscape method over others is that it provides a means to find and precisely describe each ARS zone separately. The method is an application of fractal geometry rather than a fractal analysis per se. It permits measurement of the variance in the characteristics of ARS zones, which is very informative and not possible with the previously available methods. Our method also produced results that were spatially explicit and quantitatively straightforward to interpret.
The principle of the method was focused on the animal's efficiency in covering some area of the plane. The places where this efficiency was relatively higher in the track were considered to be part of an ARS. It is important to note that this is different to focusing on the time the animal spends in some zones, as this time can be spent immobile or in rectilinear movement. The relationship between these two aspects can be easily confused as they are potentially linked. However, the link goes only one way: high plane coverage efficiency involves more time, but the reverse is not true. The reverse statement can, however, appear true when too much random inaccuracy affects the track. Together with our results, this indicates that the method's interest increases with the tracks' quality because the ability to discriminate between the effects of time and plane coverage increases. With the increasing usage of GPS tags, this method could potentially help in understanding subtle aspects of foraging behavior, such as the temporal optimization of searching efficiency (or how to search more space in a minimum amount of time). With lower quality tracks, the same can be achieved, but only at larger scales, for which track inaccuracy is negligible.
Comments on the concept of scale
Our method provides an efficient approach to finding and describing regions
of small-scale ARS but not in determining other, larger scales of movement. By
contrast, the first-passage time method and the traditional fractal analysis
method are more designed to identifying average scales at which animals modify
their behavior. In both methods, this is done in different ways, using and
outputting different measures. Nobody can tell which is more appropriate,
because there is no consensus on what really is a scale or an ARS. Therefore,
fundamental to this discussion is the question: what are the appropriate
scales of measurement of an animal track, and in which units should they be
measured?
Using a circle to approximate the size of an ARS region appears to be an
acceptable measure because, by definition, it is of small scale and it
generally looks circular. However, when an animal forages at larger scales, a
circle is generally not an appropriate descriptor of the region of movement
(Pinaud and Weimerskirch,
2005) and thus does not accurately describe the animal's movement.
The segment length used in classical fractal analysis appears more appropriate
in this respect. However, animal tracking data are by nature two-dimensional
and because filling the searched space as efficiently as possible is a key
component of searching behavior, we believe that the most relevant metric of
ARS is the amount of coverage of a plane (i.e. an area) (taking the earlier
example of the lost watch, this makes intuitive sense). For ARS, the area of
the circle could be used as an estimate of the smallest scale. For the larger
scale, techniques used in estimates of territory size and home range could be
used, such as the convex polygon method (example given in
Fig. 7) or a more sophisticated
kernel analysis that describes the distribution of ARS or track location. This
would allow accurate description of areas that match the shape of the animal
space-use pattern at all scales. Nonetheless, the fractal landscape method
derived here provides a measure of the ARS of small-scale events with good
precision, and with the added advantage of providing a measure of the time
spent searching within that localized region. Depending on the questions of
interest, different metrics could potentially be used to create a similar
landscape and to find associated zones (speed, sinuosity index, etc). For
example, as speed is often related to energy expense, a plot of speed against
time could be used to determine the portion of lower versus higher
speeds (i.e. potential energy expense). The principle is simple, intuitive and
different metrics can be adapted to the question initially asked.
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Acknowledgments |
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References |
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