Animal Movement Patterns In Spatial And Temporal Domains Biology Essay


Abstract: To efficiently identify behavioral processes and landscape patterns affecting movement of animals, it is imperative to undertake studies at spatial and temporal scales at which they affect the behavioral response. The aim and scope of the present study was thus to develop a novel method of integrating fractals and wavelets to identify animal movement patterns in across spatio-temporal scales. The GPS PTT tracking data of 11 Lesser black backed gulls (Larus fuscus) was used for the current study. Fractal analysis of individual birds during the wintering and breeding seasons consistently exhibited abrupt changes in Fractal Dimension (D) at scales of around 20km and a second change at around 80 km. These transitions define 2 distinct nested domains with indistinct sub domains. Wavelet analysis of the net displacement data in the first domain showed a dominant repetitive movement pattern of 1 or 2 cycles /day. Distinct daily temporal patterns indicate daily behavioral bouts such as foraging occurring within the 1st spatial domain. Wavelet analysis of higher domains (20-80km) showed dominant frequencies in the order of 10 days and greater indicating only transient bouts to such greater distances. Such responses in the movement patterns of animals indicate switches between three spatio-temporal scales 1) localized resources utilization (daily feeding bouts), 2) movement between habitat patches and 3) large scale movements driven by migration.

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A large number of studies since the 1980's have stressed the importance of spatial and temporal scales in relation to animal behavior and resource selection ((Johnson 1980; Wiens 1989; Levin 1992; Johnson, Parker et al. 2002).Within the paradigm of animal behavior studies a recent advancement in telemetry techniques has enabled scientist to gather empirical evidence about movement ecology in particular (Weimerskirch, Bonadonna et al. 2002; Fortin 2003; Brooks and Harris 2008). Detailed knowledge about the positional data of animals provides a huge repository to analyze movement (Weimerskirch, Bonadonna et al. 2002) across a range of spatial and temporal scales. Since adjustment in movements by animals is triggered by hierarchical spatial distribution of foraging resources and environmental factors (Bissonette 1997) in confluence with the physiological state of the animal it becomes important that such movement adjustments and their triggers are studied at the right scale as perceived by the animal (Johnson, Parker et al. 2002).

To efficiently identify behavioral processes and landscape patterns affecting movements, it is imperative to undertake studies at spatial and temporal scales at which they affect the behavioral response. Arbitrarily defined scales may result in failure to measure movement responses to variables relevant to a particular category of behavior. For example at very fine spatial resolution, say a variation in habitat types may result in disjointed patches which are meaningless for movement of animals. While at a very coarse resolution suitable and unsuitable patches may be clumped together resulting in errors in interpretation(Milne, Johnston et al. 1989). Thus the complexity of factors effecting movement patterns might be masked or misinterpreted if not analyzed at the right scale.

Techniques for analyzing movement patterns in space and time resolved domains are constantly being updated. Delving into behavioral ecology studies show that biologist for decades have been analyzing animal pathways at different scales. They have however relied on different data sources for different scales of analysis (Alerstam 1996; Fryxell, Hazell et al. 2008). When looking up methods for cross-scale analysis for movement data, however we find that studies focus primarily on biology rather than method (Laube 2010) which does not allow for statistical analysis of cross scale studies. Few methods such as Fractal and wavelet analysis are exceptional and have proved successful in analyzing movement across a range of spatial and temporal scales (respectively). We shall therefore be using these techniques to explore cross scale movement pathways in spatio-temporal domains.

Since its introduction to characterize tortuosity of animal trails (Dicke and Burrough 1988),the fractal dimension has been used to quantify patterns in movement pathways from invertebrates to vertebrates (Webb, Riffell et al. 2009). Fractal analysis has been widely used to describe movement patterns for a wide variety of organisms raging across a range of spatial scales. Mandelbrot (Mandelbrot 1982; Mandelbrot 1977)originally coined the term "fractal" to describe any geometric form that has some degree of self similarity. Exact fractals are perfectly self similar because at every spatial scale, the larger shape is composed of smaller and smaller versions of the exact same shape. Fractals in nature however tend to be imperfectly self similar and are thus called statistical fractals (Doerr and Doerr 2004). They show some degree of statistical self similarity over a limited range of scales. This existence of stepwise behavior (changes in fractal dimension when shifting between scales) implies that we observe partial self similarity over limited ranges of scales separated by transition zones.

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The traditional divider method to measure fractals can be adapted to measure the fractal dimension D over different ranges of scales. As mentioned earlier because different scales are often associated with different driving processes, the fractal dimension may have the desirable feature of only being constant over a finite range of measurement scales. Such changes in value of fractal dimension is then useful for (1) identifying characteristic scales of variability, and (2) comparing movements of organisms that may respond, for instance, to patchy structure of their environment at different absolute scales. Abrupt changes in the value of the fractal dimension with scale might indicate that a new set of environmental or behavioral processes is controlling movement behavior (for example, decreased influence of patch barriers or the effect or the effect of home range behavior) (Sugihara and M. May 1990; Doerr and Doerr 2004)

This property has widely been used in animal movement studies to identify spatial scales where the environmental constraints or properties acting upon organisms are changing rapidly. Using this concept Fritz et al (2003) was able to apply fractals to identify various scales and boundaries between scales where wandering albatrosses (Dimedea exulans) changed their foraging search movements across five orders of magnitude (10m to 1000km)

Models that incorporate temporal dependency range from complex state space models to those that use the strength of first order autocorrelation in the directionality of movement for foraging strategies (Polansky, Wittemyer et al. 2010) (Wittemyer, Polansky et al. 2008). A recent technique that successfully captures interesting patterns of temporal correlation of net displacement operating across a range of temporal scales is the time frequency method (Wavelet method). Testing the method on elephant movement patterns and later on a lion and African buffaloes, Wittemyer and Polansky have provided interesting evidence on the temporal dependencies in movement acting at multiple scales.

The concept behind using temporal correlation in movement pathways as statistical signatures across time domains is that such an oscillating movement behavior is a result of repetitive environmental(light, temperature, resource availability) and physiological (hunger, need for water, reproduction, memory) triggers affecting the animal (Wittemyer, Polansky et al. 2008). Time- frequency method of wavelet analysis is capable of providing compact summaries of temporal autocorrelation and therefore shows seasonal and diurnal based periodicities.

In ecological systems, non stationary movement data is difficult to analyze. The descriptive statistics (step length, turning angle, direction of movement, velocity etc) calculated from the tracking data of animals over a length of time has non stationary properties. The wavelet method is then extremely useful to analyze, visualize and manipulate such complex time based data. Wavelets are capable of 1) handling irregular data sets which are aperiodic, noisy and intermittent 2) represent complex structure without the knowledge of underlying mechanism and 3) finding scale dependent regularities

Although theoretical interpretation of movement patterns on a temporal scale has been done on a spatial context and vice versa (ref), most current techniques of animal movement analysis do not successfully incorporate an integrated analysis on both spatial and temporal scales. Fractals and wavelets are in themselves strong tools to analyze cross scale movement patterns in space and time respectively. Therefore in this paper we aim on developing an innovative technique of identifying dominant spatial domains using fractals and exploring dominant temporal signatures (using the wavelet method) in each of these spatial domains. The idea is to test the hypothesis that movements associated with small spatial scales are associated with daily behavioral patterns and can be expected to show such temporal signatures using wavelet analysis. While movement in higher spatial domains (to greater distances) are often associated with behavioral activities such as migration, stopovers, patch to patch movements, can be assumed to show temporal signatures in the range of a few days to months.

Therefore the aim of the present study will be to develop a novel method of integrating fractals and wavelets to identify animal movement patterns across different spatio-temporal scales. We will be exploring behavioral changes and ecological implication of the changes in cross scale movement patterns in spatial and temporal domains

Material and method:

Lesser lack backed gull data type, collection and gaps:

The tracking data of Lesser black backed gulls (Larus fuscus) used for the current study was provided by Sovon Dutch Centre For Field Ornithology, Beek-Ubbergen, The Netherlands . The Lesser Black Backed Gulls were tracked using GPS PTT's (Platform Transmitting Terminal) as part of The European Space Agency (ESA) Flysafe Project (Ens et al. 2008).In May-June 2007, 14 Lesser black backed Gulls(male/female) were caught from their nest in Vlieland, The Netherlands, using the self operating fall trap. Each of these was equipped with solar powered Argos/GPS PTT manufactured by Microwave Telemetry Inc. The GPS PTT had a GPS accuracy of ±18m. Two kinds of GPS PTT's were used; 22 g recorded only the location, and the 30g PTT which also recorded the altitude above sea level (accuracy ±22m), heading (accuracy ±1°) and ground speed (accuracy ±1km/h).

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The Argos data for 11 (9 males and 2 females)Lesser Black backed gulls (LBB) was used for this study for a period of 3 years (2007-09). The data was found to have large number of missing fixes (23%). The reason for the missing fixes especially during the winters (for birds wintering in the north) was due to the fact that the GPS ptt's used were solar powered, and did not receive sufficient light to recharge during these months. Problems due to low battery power resulted in either the fix not being taken or the incapability to contact the Argos satellite for transmission.

The continuous position of the animal in time t obtained as spatial coordinates x(t) and y(t)was used to construct time series = Missing data values in due to missed GPS fixes needed an appropriate fix that did not artificially create time dependent signals. Such missing GPS fixes was estimated using expected values from a Kalman Smoother (Stoffer 2008) obtained from a state space model. Kalman smoothing is a standard technique used for signal processing, navigation systems and other engineering, econometrics and statistical applications (Anderson-Sprecher 1994; Sibert, Musyl et al. 2003; Royer and Lutcavage 2008; Patterson, McConnell et al. 2010). The state space model used was:

Lat obs (t) =lat true(t) + wt , lat true (t+1) = lat true (t) + v t

where wt and v t are each independent, identically distributed normal random variables both with mean 0 and variance σ2obs σ2proc respectively. Longitudinal positions are estimated similarly. The smoothing technique was done in R environment (R Development Core Team 2010)

Fractal Analysis:

We segregated the movement paths into breeding (months: May, June, July) and wintering (months: November, December, January) data. Fractal Dimension (D) for individual birds for each season was calculated using the program Fractal 3.16 (V. O. Nams, Nova Scotia Agricultural College, Truro, Nova Scotia, Canada). We used the Fractal Mean Estimator which uses the traditional divider method (Dicke and Burrough 1988) but also incorporating replication by re-measuring the path multiple times with each divider size, beginning from randomly selected points along the path. (Nams 2006).This corrects for truncation in measurement of gross distance using the divider method (Nams 2006).

The fractal Mean procedure was applied to a series of narrow windows of spatial scales to estimate D at various scales. To ensure that Fractals dimension was an effective measure of tortuosity (Doerr and Doerr 2004; Webb, Riffell et al. 2009), same range of spatial scales was used for all individuals ranging from 0-400 km. We estimated D at each position of the window by regressing log (path length) versus log (spatial scale). The +/- window used for d at each scale was 0.25.Window range is how wide a window is on the spatial scale axis to use for each fractal D estimate. This is a proportion. If it is 0.25, then the window of scales used goes from a minimum of (middle scale / 1.25) to a maximum of (middle scale x 1.25).

By plotting the fractal dimension for each sliding window, we could detect major and abrupt changes in D with spatial scale(Nams 1996; Fritz, Said et al. 2003).Scale domains was then defined as the first significant break in the slope of relationship between path length and divider size (Fritz, Said et al. 2003; Vilis and Maryse 2004). This helped in defining scale depended spatial domains where the animals change their movement patterns (for example adjust path tortuosity)

Wavelet analysis:

Descriptive movement parameters were first calculated (step-length, speed and net-displacement).We used net displacement (displacement from coordinates of a central location) as an indicator of behavioral response. Data was segregated into 2 major spatial domains (identified using fractal analysis) for each season (wintering and breeding). Wavelet analysis was conducted to discern dominant periods and temporal patterning within each spatial domain.

Wavelet analysis allows for local estimation of dominant frequencies correlated with by employing functions (wavelets) that are dilated or contracted versions of an analyzing wavelet (function) ψ, translated across the time series. The continuous wavelet transform of the discrete time series at scale a and time is defined by


where ψ* denotes the complex conjugate of the analyzing wavelet function ψ. We use the Morlet wave function


where controls the oscillation frequency and was chosen to be . The choice of scales (frequency) was chosen as a set of discrete values defined by where and Δk and K depend on the analyzing wavelet, length and resolution of the data (Torrence and Compo 1998). The estimated wavelet power spectrum of (also called the scalogram) defined as the data array of squared modulus values , j =0,1,…,N-1, k = 0,1,…,K, provides an estimate of the true wavelet spectrum. High values in the scalogram array distinguish the time where the frequency of the time series Xn matches with the Morlet wavelet of the specified frequency. Regions of significant scalogram values (temporal regions of significant cycling) was defined as those regions of modulus values greater than or equal to 0.95 sample quantile of 1000 bootstrapped scalogram of red noise null model fit to the data (Torrence and Compo 1998; Maraun and Kurths 2004; Maraun, Kurths et al. 2007; Polansky, Wittemyer et al. 2010). This procedure known as the "area-wise test" (Maraun, Kurths et al. 2007) removes spurious area of significant scalogram values deemed significant by a bootstrapping test (Maraun, Kurths et al. 2007; Polansky, Wittemyer et al. 2010). The wavelet analysis and the scalogram values helped in testing temporal auto-correlation within each identified spatial domains. Time specific frequency patterns were related to ecological and spatial patterning to understand changes in behavioral modes.


Fractal analysis:

Fractal analysis of the breeding and wintering trajectories of Lesser Black backed gulls consistently exhibit increase in the value of D with scale, with apparent discontinuity (abrupt changes in the value of D). Typically the value of fractal Dimension (D) ranges from 1.0 to 2.0 but in some cases can go beyond 2.0 when the movement trajectory crosses on itself a number of times creating an additional dimension from overlapping of lines ((Mandelbrot 1984; Webb, Riffell et al. 2009)(Figure:1). Fractal analysis of 11 individual birds undertaken for this present study ranged from 0 to 2.69 during the breeding season and 0 to 2.74 during the wintering season. Two major changes in the value of D (break in slope of the plot between log of gross distance and log of the spatial scales) was identified during the breeding and wintering season which defined transitional points separating the distinct nested domains (Figure: 2&3). Indistinct sub domains within these domains were also identified. Such sub-domains however did not show distinct temporal signatures and were therefore not considered for the wavelet analysis.

During the breeding season the 1st domain was identified up to a spatial distance of 18.68 km (std =7.51km) and a second domain from 18.68 km to 85.0 km (std=24.27km). We found the break points in the value of fractal dimension (D) during the wintering season to be centered at mean values of 19.91 km (std =7.31 km) and a second at 87.09km (std= 30.96km).

Wavelet analysis:

Wavelet analysis of net displacement (without segregating into domains) did not reveal any distinct temporal patterning or behavioral changes. Signal processing of net displacement data segregated into two domains(Table 1), in each season, however revealed complex and non-stationary auto-correlative patterning in the data (Figure 4&5). Such distinct auto-correlative structure in the movement pathway is also a function of the distance that an animal moves from a central location (net displacement) and together they can be used to distinguish movement modes.

Wavelet analysis of the net displacement within the first domain (I)during the breeding as well as the wintering season, indicated dominant time frequency patches around period=1 and 2 cycles /day. Diurnal patterns in movement of birds indicate phases of periods of roosting followed by phases of directional movement bouts. Significant frequencies in the wavelet scalogram appear to be related to the time periods of distinct movement bouts during the morning and evening. Evidence of this daily cyclic behavior is reiterated by the box plot of step-length and velocity to the time of the day (time budget). The time budget of step length of all the birds during the breeding as well as the wintering season shows cyclic periods of movements to greater distances with higher velocity followed by periods of rest(roosting) (Figure:6). These daily phases of movement occur between 4-10 am and 4-8 pm as seen by the time budget. This correspond to movement differences between time periods of significant frequencies in the scalogram values and those lacking significant frequencies values (i.e. with no temporal auto-correlation).

Wavelet analysis in the second domain (II) showed dominant time frequency patches in the order of 10 day cycles and greater. However significant patches in this domain were scarce indicating only transient bouts to such greater distances.


Our results indicate that given the large number of drivers that act as plausible triggers to movement in animals, the integrated approach using fractals and wavelets is immensely useful as an initial statistical probe into the scales at which behavioral processes and bio-physical factors could possibly be affecting movements (Dalziel et al. 2008). Studies such as these facilitate in identifying spatial and temporal scales, as perceived by the animal rather than scales defined and perceived by humans (Ferguson and Elkie 2004).

The marked changes in the fractal D, 'transitions', indicate that the patterns of movement of the animal change across scales. The spatial range of an animal may be divided into regions, 'domains', where different aspects of the animal's biology are important. These changes in fractal D with scales also suggests that the organisms change the way in which they view and interact with its environment at that scale. Searching for resources, traversing the home range or dispersing to new habitats are all likely to be very different types of movement conducted in different "domains" of scale (Wiens 1989). The fractal analysis shows that it is possible to test this idea and identify transitions between domains by calculating D for a series of narrow ranges of divider size, then plotting the midpoint of each range vs. D, and looking for abrupt changes (Nams 2005). This method defines scales at which constraints or behavioral decision operates.

Fractal analysis gives only an indication that movement modes are changing with spatial scales but not about the behavioral ecology in operation in each domain. Testing the auto-correlative properties in movement pathway in the spatial domain is a straightforward and rapid method of identifying behavioral changes occurring in these domains. The time frequency method of wavelet analysis is effective in identifying significant periods of auto correlative movement patterns versus random movements in each spatial scale. The theory behind such movement signatures is that highly repetitive movements may offer best utilization strategies of resources and space use when ecological conditions are seasonally static(Conradt, Bodsworth et al. 2000; Weimerskirch, Guionnet et al. 2000). In these situations high degree of auto correlated movements results in risk reduction and energy conservation (Wittemyer, Polansky et al. 2008).

The results from the wavelet analysis clearly provided statistical description to the auto-correlative patterns of movement in each spatial domain. In the first spatial domain daily cycles of movement and rest periods at evident. Such behavior is consistent with observation of lesser black backed gulls made in the field. There is an apparent oriented movement towards the feeding grounds during the early morning and evening hours followed by prolonged periods of roosting. Although the temporal signatures in the second domain were not consistent with all individuals, however a strong inclination towards temporal signatures in the range of 10 days and more indicate that the gulls do make transient bouts to distances beyond 80 km. Lesser black backed gulls nesting on the island Vlieland, are known to make occasional long distance movements to the mainland which could correspond to the movement signature obtained in the second domain. However it is difficult to confirm the exact behavioral implication of these long distance bouts and requires active exploration into the habitat and environmental variables connected to it. An interesting pattern however worth mentioning is that the lesser black backed gulls have been observed to make frequent long range flights just before they embark on their migratory flight(Ens B.J. 2008). The reason for this kind of movement is unknown but likely to be governed by fuel deposition rates, flight mechanics and migration strategies of the birds (Alerstam 2001).

Due to changes in habitat on a seasonal scale and internal state of animals, responses in the movement patterns of the gulls based on the results presented allows us to infer that the gulls show switches between three spatio temporal scales: 1) localized resources utilization (daily feeding bouts) 2) movement between habitat patches and 3)large scale movements driven by migration. We believe that such information obtained from these initial statistical analyses of movement pathways is critical in designing monitoring techniques for prolonged periods of time to garner information on foraging specialization, area restricted search patterns, and critical habitats for animals. For instance the knowledge that lesser black backed gulls is making daily cyclic movements in the range of 0-20 km around a place of central location corroborated with knowledge about the general behavioral biology of the birds can help us deduce that this buffer area is exclusively being used for the purpose of foraging. Extended studies can then be conducted exclusively in this area on the accessibility to foraging patches, foraging resources and potential threats to such areas. Therefore context specific decisions can then be taken regarding at which scale to undertake studies related to foraging strategy, socio-spatial processes and landscape properties; each at a scale as perceived and prioritized by the animal under study.

The limitation of the present study was that data was not available at a higher resolution, which could have resulted in more accurate results of changes in spatial scales and temporal signatures in these spatial domains. However even high resolution data is not sufficient to efficiently predict precise boundaries in the spatial and temporal domains where behavioural modes change. For such accurate breakdown of spatio temporal domains it is essential that environmental covariates and expert knowledge about the species is supplemented and these variables be as comprehensive as the trajectory data used.

(R Development Core Team 2010). R: A Language and Environment for Statistical Computing. Vienna, Austria, R Foundation for Statistical Computing.

Alerstam, T. (1996). "The geographical scale factor in orientation of migrating birds." J Exp Biol 199(1): 9-19.

Alerstam, T. (2001). "Detours in Bird Migration." Journal of Theoretical Biology 209(3): 319-331.

Anderson-Sprecher, R. (1994). "Robust Estimates of Wildlife Location Using Telemetry Data." Biometrics 50(2): 406-416.

Bissonette, J. A. (1997). Wildlife and Landscape Ecology: Effects of Pattern and Scale. New York, Springer-Verlag.

Brooks, C. J. and S. Harris (2008). "Directed movement and orientation across a large natural landscape by zebras, Equus burchelli antiquorum." Animal Behaviour 76(2): 277-285.

Conradt, L., E. J. Bodsworth, et al. (2000). "Non-Random Dispersal in the Butterfly Maniola jurtina: Implications for Metapopulation Models." Proceedings: Biological Sciences 267(1452): 1505-1510.

Dalziel, Benjamin D., Juan M. Morales, et al. (2008). "Fitting Probability Distributions to Animal Movement Trajectories: Using Artificial Neural Networks to Link Distance, Resources, and Memory." The American Naturalist 172(2): 248-258.

Dicke, M. and P. A. Burrough (1988). "USING FRACTAL DIMENSIONS FOR CHARACTERIZING TORTUOSITY OF ANIMAL TRAILS." Physiological Entomology 13(4): 393-398.

Doerr, V. A. J. and E. D. Doerr (2004). "Fractal Analysis Can Explain Individual Variation in Dispersal Search Paths." Ecology 85(5): 1428-1438.

Ens B.J., B. F., Camphuysen C.J., Boer P. de, Exo K.-M., Gallego N., Hoye B., Klaassen R., Oosterbeek K., Shamoun-Baranes J., Jeugd H. van der & Gasteren H. van (2008). Tracking of individual birds. Report on WP 3230 (bird tracking sensor characterization) and WP 4130 (sensor adaptation and calibration for bird tracking system) of the FlySafe basic activities project. Nederland, Beek-Ubbergen, SOVON Dutch Centre for Field Ornithology.

Ferguson, S. H. and P. C. Elkie (2004). "Seasonal movement patterns of woodland caribou (Rangifer tarandus caribou)." Journal of Zoology 262(2): 125-134.

Fortin, D. (2003). "Searching behavior and use of sampling information by free-ranging bison (Bos bison)." Behavioral Ecology and Sociobiology 54(2): 194-203.

Fritz, H., S. Said, et al. (2003). "Scale-Dependent Hierarchical Adjustments of Movement Patterns in a Long-Range Foraging Seabird." Proceedings: Biological Sciences 270(1520): 1143-1148.

Fryxell, J. M., M. Hazell, et al. (2008). "Multiple movement modes by large herbivores at multiple spatiotemporal scales." Proceedings of the National Academy of Sciences 105(49): 19114-19119.

Johnson, C. J., K. L. Parker, et al. (2002). "Movement Parameters of Ungulates and Scale-Specific Responses to the Environment." Journal of Animal Ecology 71(2): 225-235.

Johnson, D. H. (1980). "The Comparison of Usage and Availability Measurements for Evaluating Resource Preference." Ecology 61(1): 65-71.

Laube, P. P., R S (2010). Cross-scale movement trajectory analysis. Proceedings of the GIS Research UK 18th Annual Conference GISRUK 2010, London, UK, Haklay, M [et al.]

Levin, S. A. (1992). "The Problem of Pattern and Scale in Ecology: The Robert H. MacArthur Award Lecture." Ecology 73(6): 1943-1967.

Mandelbrot, B. B., Ed. (1982). The Fractal geometry of nature W. H. Freeman, New York, USA.

Mandelbrot, B. B. (1984). "Fractals in physics: Squig clusters, diffusions, fractal measures, and the unicity of fractal dimensionality." Journal of Statistical Physics 34(5): 895-930.

Mandelbrot, B. B., Freeman and Co., San Francisco, Calif, Ed. ( 1977). Fractals -- Form, chance and dimension: . Earth-Science Reviews.

Maraun, D. and J. Kurths (2004). "Cross wavelet analysis: significance testing and pitfalls." Nonlinear Processes in Geophysics 11(4): 505-514.

Maraun, D., J. Kurths, et al. (2007). "Nonstationary Gaussian processes in wavelet domain: Synthesis, estimation, and significance testing." Physical Review E 75(1).

Milne, B. T., K. M. Johnston, et al. (1989). "Scale-dependent proximity of wildlife habitat in a spatially-neutral Bayesian model." Landscape Ecology 2(2): 101-110.

Nams, V. (2006). "Improving Accuracy and Precision in Estimating Fractal Dimension of Animal movement paths." Acta Biotheoretica 54: 1-11.

Nams, V. O. (1996). "The VFractal: a new estimator for fractal dimension of animal movement paths." Landscape Ecology 11(5): 289-297.

Nams, V. O. (2005). "Using animal movement paths to measure response to spatial scale." Oecologia 143(2): 179-188.

Patterson, T. A., B. J. McConnell, et al. (2010). "Using GPS data to evaluate the accuracy of state-space methods for correction of Argos satellite telemetry error." Ecology 91(1): 273-285.

Polansky, L., G. Wittemyer, et al. (2010). "From moonlight to movement and synchronized randomness: Fourier and wavelet analyses of animal location time series data." Ecology 91(5): 1506-1518.

Royer, F. and M. Lutcavage (2008). "Filtering and interpreting location errors in satellite telemetry of marine animals." Journal of Experimental Marine Biology and Ecology 359(1): 1-10.

Sibert, J. R., M. K. Musyl, et al. (2003). "Horizontal movements of bigeye tuna (Thunnus obesus) near Hawaii determined by Kalman filter analysis of archival tagging data." Fisheries Oceanography 12(3): 141-151.

Stoffer, R. H. S. a. D. S. (2008). Time series analysis and its applications with R examples, 2nd edn, Springer Berlin / Heidelberg.

Sugihara, G. and R. M. May (1990). "Applications of fractals in ecology." Trends in Ecology & Evolution 5(3): 79-86.

Torrence, C. and G. P. Compo (1998). "A Practical Guide to Wavelet Analysis." Bulletin of the American Meteorological Society 79(1): 61-78.

Vilis, O. N. and B. Maryse (2004). "Fractal analysis measures habitat use at different spatial scales: an example with American marten." Canadian Journal of Zoology 82: 1738-1747.

Webb, S. L., S. K. Riffell, et al. (2009). "Using Fractal Analyses to Characterize Movement Paths of White-Tailed Deer and Response to Spatial Scale." Journal of Mammalogy 90(5): 1210-1217.

Weimerskirch, H., F. Bonadonna, et al. (2002). "GPS Tracking of Foraging Albatrosses." Science 295(5558): 1259-.

Weimerskirch, H., T. Guionnet, et al. (2000). "Fast and Fuel Efficient? Optimal Use of Wind by Flying Albatrosses." Proceedings: Biological Sciences 267(1455): 1869-1874.

Wiens, J. A. (1989). "Spatial scaling in Ecology." Functional Ecology 3(4): 385-397.

Wittemyer, G., L. Polansky, et al. (2008). "Disentangling the effects of forage, social rank, and risk on movement autocorrelation of elephants using Fourier and wavelet analyses." Proceedings of the National Academy of Sciences 105(49): 19108-19113.