Step Length Distribution Research Articles

Animal space use: distinguishing a two‐level superposition of scale‐specific walks from scale‐free Lévy walk

How to differentiate between scale‐free space use like Lévy walk and a two‐level scale‐specific process like composite random walk (mixture of intra‐ and inter‐patch habitat movement) is surrounded by controversy. Composite random walk may under some parameter conditions appear Lévy walk‐like from the perspective of the path’s distribution of step lengths due to superabundance of very long steps relative to the expectation from a classic (single‐level) random walk. However, a more explicit focus on the qualitative differences between studying movement at a high resolution mechanistic (behavioral) level and the more coarse‐grained statistical mechanical level may contribute to resolving both this and other issues related to scaling complexity. Specifically, a re‐sampling of a composite random walk at larger time lags than the micro‐level unit time step for the simulation makes a Lévy‐look‐alike step length distribution re‐shaping towards a Brownian motion‐like pattern. Conversely, a true Levy walk maintains its scaling characteristics upon re‐sampling. This result illustrates how a confusing pattern at the mechanistic level may be resolved by changing observational scale from the micro level to the coarser statistical mechanical meso‐ or macro‐scale. The instability of the composite random walk pattern under rescaling is a consequence of influence of the central limit theorem. I propose that a coarse‐graining test – studying simulated animal paths at a coarsened temporal scale by re‐sampling a series – should be routinely performed prior to comparing theoretical results with those patterns generated from GPS data describing animal movement paths. Fixes from terrestrial mammals are often collected at hourly intervals or larger, and such a priori coarse‐grained series may thus comply better with the statistical mechanical meso‐ or macro‐level of analysis than the behavioral mechanics observed at finer resolutions typically in the range of seconds and minutes. If fixes of real animals are collected at this high frequency, coarse graining both the simulated and real series is advised in order to bring the analysis into a temporal scale domain where analytical methods from statistical mechanics can be applied.

Measuring gravel transport and dispersion in a mountain river using passive radio tracers

ABSTRACTRandom walk models of fluvial sediment transport recognize that grains move intermittently, with short duration steps separated by rests that are comparatively long. These models are built upon the probability distributions of the step length and the resting time. Motivated by these models, tracer experiments have attempted to measure directly the steps and rests of sediment grains in natural streams. This paper describes results from a large tracer experiment designed to test stochastic transport models. We used passive integrated transponder (PIT) tags to label 893 coarse gravel clasts and placed them in Halfmoon Creek, a small alpine stream near Leadville, Colorado, USA. The PIT tags allow us to locate and identify tracers without picking them up or digging them out of the streambed. They also enable us to find a very high percentage of our rocks, 98% after three years and 96% after the fourth year. We use the annual tracer displacement to test two stochastic transport models, the Einstein–Hubbell–Sayre (EHS) model and the Yang–Sayre gamma‐exponential model (GEM). We find that the GEM is a better fit to the observations, particularly for slower moving tracers and suggest that the strength of the GEM is that the gamma distribution of step lengths approximates a compound Poisson distribution. Published in 2012. This article is a US Government work and is in the public domain in the USA.

Rationality, irrationality and escalating behavior in lowest unique bid auctions.

Information technology has revolutionized the traditional structure of markets. The removal of geographical and time constraints has fostered the growth of online auction markets, which now include millions of economic agents worldwide and annual transaction volumes in the billions of dollars. Here, we analyze bid histories of a little studied type of online auctions – lowest unique bid auctions. Similarly to what has been reported for foraging animals searching for scarce food, we find that agents adopt Lévy flight search strategies in their exploration of “bid space”. The Lévy regime, which is characterized by a power-law decaying probability distribution of step lengths, holds over nearly three orders of magnitude. We develop a quantitative model for lowest unique bid online auctions that reveals that agents use nearly optimal bidding strategies. However, agents participating in these auctions do not optimize their financial gain. Indeed, as long as there are many auction participants, a rational profit optimizing agent would choose not to participate in these auction markets.

Individual Analyses of Lévy Walk in Semi-Free Ranging Tonkean Macaques (Macaca tonkeana)

Animals adapt their movement patterns to their environment in order to maximize their efficiency when searching for food. The Lévy walk and the Brownian walk are two types of random movement found in different species. Studies have shown that these random movements can switch from a Brownian to a Lévy walk according to the size distribution of food patches. However no study to date has analysed how characteristics such as sex, age, dominance or body mass affect the movement patterns of an individual. In this study we used the maximum likelihood method to examine the nature of the distribution of step lengths and waiting times and assessed how these distributions are influenced by the age and the sex of group members in a semi free-ranging group of ten Tonkean macaques. Individuals highly differed in their activity budget and in their movement patterns. We found an effect of age and sex of individuals on the power distribution of their step lengths and of their waiting times. The males and old individuals displayed a higher proportion of longer trajectories than females and young ones. As regards waiting times, females and old individuals displayed higher rates of long stationary periods than males and young individuals. These movement patterns resembling random walks can probably be explained by the animals moving from one location to other known locations. The power distribution of step lengths might be due to a power distribution of food patches in the enclosure while the power distribution of waiting times might be due to the power distribution of the patch sizes.

Transport and scaling in quenched two- and three-dimensional Lévy quasicrystals

We consider correlated Lévy walks on a class of two- and three-dimensional deterministic self-similar structures, with correlation between steps induced by the geometrical distribution of regions, featuring different diffusion properties. We introduce a geometric parameter α, playing a role analogous to the exponent characterizing the step-length distribution in random systems. By a single-long-jump approximation, we analytically determine the long-time asymptotic behavior of the moments of the probability distribution as a function of α and of the dynamic exponent z associated with the scaling length of the process. We show that our scaling analysis also applies to experimentally relevant quantities such as escape-time and transmission probabilities. Extensive numerical simulations corroborate our results which, in general, are different from those pertaining to uncorrelated Lévy-walk models.

Linking animal movement to site fidelity

Site fidelity, the recurrent visit of an animal to a previously occupied area is a wide-spread behavior in the animal kingdom. The relevance of site fidelity to territoriality, successful breeding, social associations, optimal foraging and other ecological processes, demands accurate quantification. Here we generalize previous theory that connects site fidelity patterns to random walk parameters within the framework of the space-time fractional diffusion equation. In particular, we describe the site fidelity function in terms of animal movement characteristics via the Lévy exponent, which controls the step-length distribution of the random steps at each turning point, and the waiting time exponent that controls for how long an animal awaits before actually moving. The analytical results obtained will provide a rigorous benchmark for empirically driven studies of animal site fidelity.

Balance and Gait Analysis Following Uncoupled Cranks Cycling: A Pilot Study

PURPOSE: Older adults suffer from muscle imbalances and coordination deficits that impact static balance and gait. Uncoupled cranks, such as PowerCranks™, have been shown to improve cycling efficiency and coordination through increased balance of agonist/antagonist muscle groups of the hips and lower extremities for active populations. The objective of this pilot study was to assess the effects of short duration, uncoupled cranks training on static balance, step length and gait distribution time. METHODS: 19 college-aged individuals (8 male, 11 female, 23.5 ± 3.7 years) were divided into two groups, an uncoupled cranks intervention group (UCG, n=10) and a traditional cranks control group (CG, n=9). Both groups underwent baseline and post-intervention balance assessments using the Biodex Balance System Clinical Test for Sensory Integration of Balance (CTSIB), and gait analysis using the Biodex Gait Trainer 2. The intervention required both groups to cycle for 6 sessions at a cadence of 60 rpm for 5 min / session. The UCG used a Rev Master Spin Cycle equipped with PowerCranks™ and the CG rode a Monark™ stationary bike equipped with traditional cranks. RESULTS: No significant difference was observed for static balance on a firm surface, eyes closed or eyes open, nor on a foam surface, eyes closed or eyes open, between baseline and post-intervention measurements (p=0.05). No significant changes in step length for either leg occurred (p>0.05), however, a significant difference between pre- and post-intervention distribution of time spent on each foot was detected (p<0.05). No significant differences occurred between UCG or CG for balance assessments or gait analysis (p>0.05). CONCLUSION: Results indicate the biomechanics of cycling, through either use of uncoupled or traditional cranks, impact gait distribution time. Neuromuscular adaptations in the hips and lower extremities may explain this, however, further research involving a larger sample population and subjects suffering from abnormal gaits due to muscular imbalances are warranted. Although the results for balance revealed no significant results, older adults suffering from balance impairments should also be assessed to determine if cycling is an appropriate method for improving static balance.

Local and average behaviour in inhomogeneous superdiffusive media

We consider a random walk on one-dimensional inhomogeneous graphs built from Cantor fractals. Our study is motivated by recent experiments that demonstrated superdiffusion of light in complex disordered materials, thereby termed Lévy glasses. We introduce a geometric parameter α which plays a role analogous to the exponent characterising the step length distribution in random systems. We study the large-time behaviour of both local and average observables; for the latter case, we distinguish two different types of averages, respectively over the set of all initial sites and over the scattering sites only. The 'single long-jump approximation” is applied to analytically determine the different asymptotic behaviour as a function of α and to understand their origin. We also discuss the possibility that the root of the mean square displacement and the characteristic length of the walker distribution may grow according to different power laws; this anomalous behaviour is typical of processes characterised by Lévy statistics and here, in particular, it is shown to influence average quantities.

One size does not fit all: flexible models are required to understand animal movement across scales

1. Large data sets containing precise movement data from free-roaming animals are now becoming commonplace. One means of analysing individual movement data is through discrete, random walk-based models. 2. Random walk models are easily modified to incorporate common features of animal movement, and the ways that these modifications affect the scaling of net displacement are well studied. Recently, ecologists have begun to explore more complex statistical models with multiple latent states, each of which are characterized by a distribution of step lengths and have their own unimodal distribution of turning angles centred on one type of turn (e.g. reversals). 3. Here, we introduce the compound wrapped Cauchy distribution, which allows for multimodal distributions of turning angles within a single state. When used as a single state model, the parameters provide a straightforward summary of the relative contributions of different turn types. The compound wrapped Cauchy distribution can also be used to build multiple state models. 4. We hypothesize that a multiple state model with unimodal distributions of turning angles will best describe movement at finer resolutions, while a multiple state model using our multimodal distribution will better describe movement at intermediate temporal resolutions. At coarser temporal resolutions, a single state model using our multimodal distribution should be sufficient. We parameterize and compare the performance of these models at four different temporal resolutions (1, 4, 12 and 24 h) using data from eight individuals of Loxodonta cyclotis and find support for our hypotheses. 5. We assess the efficacy of the different models in extrapolating to coarser temporal resolution by comparing properties of data simulated from the different models to the properties of the observed data. At coarser resolutions, simulated data sets recreate many aspects of the observed data; however, only one of the models accurately predicts step length, and all models underestimate the frequency of reversals. 6. The single state model we introduce may be adequate to describe movement data at many resolutions and can be interpreted easily. Multiscalar analyses of movement such as the ones presented here are a useful means of identifying inconsistencies in our understanding of movement.

Density profiles in open superdiffusive systems

We numerically solve a discretized model of Lévy random walks on a finite one-dimensional domain with a reflection coefficient r and in the presence of sources. At the domain boundaries, the steady-state density profile is nonanalytic. The meniscus exponent μ, introduced to characterize this singular behavior, uniquely identifies the whole profile. Numerical data suggest that μ = α/2 + r(α/2 - 1), where α is the Lévy exponent of the step-length distribution. As an application, we show that this model reproduces the temperature profiles obtained for a chain of oscillators displaying anomalous heat conduction. Remarkably, the case of free-boundary conditions in the chain corresponds to a Lévy walk with negative reflection coefficient.

Multiple Scattering of Light in Superdiffusive Media

Light transport in superdiffusive media of finite size is studied theoretically. The intensity Green's function for a slab geometry is found by discretizing the fractional diffusion equation and employing the eigenfunction expansion method. Truncated step length distributions and complex boundary conditions are considered. The profile of a coherent backscattering cone is calculated in the superdiffusion approximation.

Accurate particle size distribution determination by nanoparticle tracking analysis based on 2-D Brownian dynamics simulation

A physical model is presented to simulate the average step length distribution during nanoparticle tracking analysis experiments as a function of the particle size distribution and the distribution of the number of steps within the tracks. Considering only tracks of at least five steps, numerical simulation could be replaced by a normal distribution approximation. Based on this model, simulation of a step length distribution allows obtaining a much more reliable estimation of the particle size distribution, thereby reducing the artificial broadening of the distribution, as is typically observed by direct conversion of step length to particle size data. As this fitting procedure also allowed including data from particles that were followed for a relatively low number of steps, the measurement time could be reduced for particles that are known to be monodisperse. Whereas the inversion is less sensitive towards the particle size distribution width, still similar values were obtained for both the average diameter and standard deviation of a polystyrene latex sample irrespective of the track length, provided that the latter included at least five steps.

Stochastic modelling of animal movement

Modern animal movement modelling derives from two traditions. Lagrangian models, based on random walk behaviour, are useful for multi-step trajectories of single animals. Continuous Eulerian models describe expected behaviour, averaged over stochastic realizations, and are usefully applied to ensembles of individuals. We illustrate three modern research arenas. (i) Models of home-range formation describe the process of an animal 'settling down', accomplished by including one or more focal points that attract the animal's movements. (ii) Memory-based models are used to predict how accumulated experience translates into biased movement choices, employing reinforced random walk behaviour, with previous visitation increasing or decreasing the probability of repetition. (iii) Lévy movement involves a step-length distribution that is over-dispersed, relative to standard probability distributions, and adaptive in exploring new environments or searching for rare targets. Each of these modelling arenas implies more detail in the movement pattern than general models of movement can accommodate, but realistic empiric evaluation of their predictions requires dense locational data, both in time and space, only available with modern GPS telemetry.

Turn designation, sampling rate and the misidentification of power laws in movement path data using maximum likelihood estimates

Many authors have claimed to observe animal movement paths that appear to be Levy walks, i.e. a random walk where the distribution of move lengths follows an inverse power law. A Levy walk is known to be the optimal search strategy of a particular class of random walks in certain environments; hence, it is important to distinguish correctly between Levy walks and other types of random walks in observed animal movement paths. Evidence of a power law distribution in the step length distribution of observed animal movement paths is often used to classify a particular movement path as a Levy walk. However, there is some doubt about the accuracy of early studies that apparently found Levy walk behaviour. A recently accepted method to determine whether a movement path truly exhibits Levy walk behaviour is based on an analysis of move lengths with a maximum likelihood estimate using Akaike weights. Here, we show that simulated (non-Levy) random walks representing different types of animal movement behaviour (a composite correlated random walk; pooled data from a set of random walks with different levels of correlation and three-dimensional correlated random walks projected into one dimension) can all show apparent power law behaviour typical of Levy walks when using the maximum likelihood estimation method. The probability of the movement path being identified as having a power law step distribution is related to both the sampling rate used by the observer and the way that ‘turns’ or ‘reorientations’ in the movement path are designated. However, identification is also dependent on the nature and properties of the simulated path, and there is currently no standard method of observation and analysis that is robust for all cases. Our results indicate that even apparently robust maximum likelihood methods can lead to a mismatch between pattern and process, as paths arising from non-Levy walks exhibit Levy-like patterns.

Normal and anomalous diffusion of gravel tracer particles in rivers

One way to study the mechanism of gravel bed load transport is to seed the bed with marked gravel tracer particles within a chosen patch and to follow the pattern of migration and dispersal of particles from this patch. In this study, we invoke the probabilistic Exner equation for sediment conservation of bed gravel, formulated in terms of the difference between the rate of entrainment of gravel into motion and the rate of deposition from motion. Assuming an active layer formulation, stochasticity in particle motion is introduced by considering the step length (distance traveled by a particle once entrained until it is deposited) as a random variable. For step lengths with a relatively thin (e.g., exponential) tail, the above formulation leads to the standard advection‐diffusion equation for tracer dispersal. However, the complexity of rivers, characterized by a broad distribution of particle sizes and extreme flood events, can give rise to a heavy‐tailed distribution of step lengths. This consideration leads to an anomalous advection‐diffusion equation involving fractional derivatives. By identifying the probabilistic Exner equation as a forward Kolmogorov equation for the location of a randomly selected tracer particle, a stochastic model describing the temporal evolution of the relative concentrations is developed. The normal and anomalous advection‐diffusion equations are revealed as its long‐time asymptotic solution. Sample numerical results illustrate the large differences that can arise in predicted tracer concentrations under the normal and anomalous diffusion models. They highlight the need for intensive data collection efforts to aid the selection of the appropriate model in real rivers.

Fractional dispersion in a sand bed river

Random walk models of fluvial bed load transport use probability distributions to describe the distance a grain travels during an episode of transport and the time it rests after deposition. These models typically employ probability distributions with finite first and second moments, reflecting an underlying assumption that all the factors that influence sediment transport tend to combine in such a way that the length of a step or the duration of a rest can be characterized by a mean value surrounded by a specific amount of variability. The observation that many transport systems exhibit apparent scale‐dependent behavior and non‐Fickian dispersion suggests that this assumption is not always valid. We revisit a nearly 50 year old tracer experiment in which the tracer plume exhibits the hallmarks of dispersive transport described by a step length distribution with a divergent second moment and no characteristic dispersive size. The governing equation of this type of random walk contains fractional‐order derivatives. We use the data from the experiment to test two versions of a fractional‐order model of dispersive fluvial bed load transport. The first version uses a heavy‐tailed particle step length distribution with a divergent second moment to reproduce the anomalously high fraction of tracer mass observed in the downstream tail of the spatial distribution. The second version adds a feature that partitions mass into a detectable mobile phase and an undetectable, immobile phase. This two‐phase transport model predicts other features observed in the data: a decrease in the amount of detected tracer mass over the course of the experiment and enhanced particle retention near the source. The fractional‐order models match the observed plume shape and growth rates better than prior attempts with classical models.

Movement and ranging patterns of a tropical rat (Leopoldamys sabanus) in logged and unlogged rain forests

Abstract Knowledge of how animals move through the environment is important for predicting effects of habitat change on faunal distributions. Logging of tropical rain forests produces habitat changes on multiple scales that may affect movement and habitat use by small mammals. To explore the effects of such habitat changes, we compared movement and ranging patterns of the long-tailed giant rat (Leopoldamys sabanus) in logged and unlogged rain forests in Borneo. On a small scale, movement was quantified using spool-and-line tracks; on a larger scale, movement was quantified via radiotracking. At the small scale, paths (49 tracks of 55.2 m ± 20.7 SD each) were relatively straight, with similar step (straight-line section) length distributions in both forest types. At the larger scale, the rats (16 individuals tracked for 4 nights each, X = 1,443 ± 991 m of movement per night) moved with similar speed through both forest types (mean distance covered per 10-min interval = 32 ± 45 m). Based on telemetry data,...

Optimal foraging: Lévy pattern or process?

Many different species have been suggested to forage according to a Lévy walk in which the distribution of step lengths is heavy-tailed. Theoretical research has shown that a Lévy exponent of approximately 2 can provide a higher foraging efficiency than other exponents. In this paper, a composite search model is presented for non-destructive foraging behaviour based on Brownian (i.e. non-heavy-tailed) motion. The model consists of an intensive search phase, followed by an extensive phase, if no food is found in the intensive phase. Quantities commonly observed in the field, such as the distance travelled before finding food and the net displacement in a fixed time interval, are examined and compared with the results of a Lévy walk model. It is shown that it may be very difficult, in practice, to distinguish between the Brownian and the Lévy models on the basis of observed data. A mathematical expression for the optimal time to switch from intensive to extensive search mode is derived, and it is shown that the composite search model provides higher foraging efficiency than the Lévy model.

An effective trajectory generation method for bipedal walking

This paper presents the virtual height inverted pendulum mode (VHIPM), which is a simple and effective trajectory generation method for the stable walking of biped robots. VHIPM, which is based on the inverted pendulum mode (IPM), can significantly reduce the zero moment point (ZMP) error by adjusting the height in the inverted pendulum. We show the relationship between VHIPM and other popular trajectory generation methods, and compare the ZMP errors in walking when trajectories are generated by various methods including VHIPM. We also investigate the sensitivity of the ZMP error in VHIPM to the step length, walking period and mass distribution of a robot. The simulation results show that VHIPM significantly reduces the ZMP errors compared to other methods under various circumstances.

Probability-based prediction of multi-mode vibration response to walking excitation

In vibration serviceability checks of footbridges, a force induced by a single person walking is usually modelled as a harmonic force having a frequency that matches one of the footbridge natural frequencies. This approach assumes that, among the infinite number of harmonics a walking force is composed of, only a single harmonic is important for a vibration serviceability check. Another usual assumption is that the footbridge can be modelled as an SDOF system, implying that only vibration in a single mode is of interest. In addition, due to the deterministic nature of this approach, it cannot take into account inter- and intra-subject variabilities in the walking force that are now well documented in the literature. To account for these variabilities, a novel probabilistic approach to carry out a vibration serviceability check is developed in this paper. Factors such as the probability distribution of walking frequencies, step lengths and amplitude of walking force for its five lowest harmonics and subharmonics are taken into account. Using walking force time histories measured on a treadmill, the frequency content of the force was investigated, resulting in the formulation of a multi-harmonic force model. This model can be used to estimate the multi-mode response in footbridges. This was verified successfully on an as-built catenary footbridge structure. Although only the vibration response of footbridges was analysed in this paper, the force model proposed has the potential to be implemented in the estimation of floor vibration as well, where multi-mode response occurs more frequently. The model is easily programmable and as such could present a powerful tool for estimating efficiently the probability of various levels of vibration response due to single person walking. Therefore, the proposed probability-based methodology has the potential to revolutionise the philosophy of the current codes of practice dealing with vibration serviceability of structures under human-induced vibration.