Inferring Lévy walks from curved trajectories: A rescaling method

An important problem in the study of anomalous diffusion and transport concerns the proper analysis of trajectory data. The analysis and inference of Lévy walk patterns from empirical or simulated trajectories of particles in two and three-dimensional spaces (2D and 3D) is much more difficult than in 1D because path curvature is nonexistent in 1D but quite common in higher dimensions. Recently, a new method for detecting Lévy walks, which considers 1D projections of 2D or 3D trajectory data, has been proposed by Humphries et al. The key new idea is to exploit the fact that the 1D projection of a high-dimensional Lévy walk is itself a Lévy walk. Here, we ask whether or not this projection method is powerful enough to cleanly distinguish 2D Lévy walk with added curvature from a simple Markovian correlated random walk. We study the especially challenging case in which both 2D walks have exactly identical probability density functions (pdf) of step sizes as well as of turning angles between successive steps. Our approach extends the original projection method by introducing a rescaling of the projected data. Upon projection and coarse-graining, the renormalized pdf for the travel distances between successive turnings is seen to possess a fat tail when there is an underlying Lévy process. We exploit this effect to infer a Lévy walk process in the original high-dimensional curved trajectory. In contrast, no fat tail appears when a (Markovian) correlated random walk is analyzed in this way. We show that this procedure works extremely well in clearly identifying a Lévy walk even when there is noise from curvature. The present protocol may be useful in realistic contexts involving ongoing debates on the presence (or not) of Lévy walks related to animal movement on land (2D) and in air and oceans (3D).

Figure 1

2D curved trajectories of random walks generated using models I and II, with total lengths (a) 500 and (b) $\approx {10}^5$ steps (see text for details). Model I represents a non-Markvoian CRW built on the backbone of a Lévy walk, whereas model II is a short-range correlated Markovian walk. Both models share the same pdfs of step sizes and turning angles. But model I has long-range order and is superdiffusive at long times, whereas model II only has short-range order and is diffusive at long times.

Figure 2

Joint pdf in $x$ and $y$ directions of step length vectors (or velocity vectors) for (a) $\lambda =1$ (original data sets), and rescaled data using (b) $\lambda =20$ and (c) $\lambda =100$. In the original data set (a) all step lengths are fixed at ${\ell }_n={\ell }_0=1$ for both models I (left column) and II (right column). As rescalings under larger values of $\lambda$ are applied, the difference between the distributions of models I and II becomes remarkable. The presence of long steps in model I is indicated by the dense filling close to the circle edge. In contrast, the rescaled data of model II is similar to a Gaussian, with a high density of short steps and a rapidly decreasing probability for large displacements. The important point to note is that the rescaling (or renormalization) brings out the difference between the two models. Without the rescaling, the models look indistinguishable, but after rescaling the models can be easily distinguished.

Figure 3

Double-log plot of the rms length, ${\ell }_{\text{rms}}={\langle {\ell }^2-{\langle \ell \rangle }^2\rangle }^{1/2}$, of models I (black line) and II (blue line) as a function of the rescaling parameter $\lambda$. Here $\ell$ are the values of the step lengths shown in Fig. 2. In model II the crossover to the Brownian behavior, ${\ell }_{\text{rms}}\sim {\lambda }^H$, with $H=0.5$ (red line as a guide to the eyes), starts around $\lambda \approx 50$. Below this value a transient superdiffusive behavior is set. In contrast, the superdiffusivity of model I, with $H>0.5$, does not show any signal of convergence to the Brownian regime in the interval studied. The rescaling method thus yields the values of the exponent for the two models. The models are easily distinguished by their exponents. Only model I is superdiffusive.

Figure 4

Evolution with the number of steps $t_n$ of the coordinates $(x,y)$ of the 2D curved random walks shown in Fig. 1 generated using models I and II: (a) original (i.e., nonrescaled, $\lambda =1$) data sets; (b) rescaled data using $\lambda =100$. The difference between (a) and (b) lies mainly in the timescale. The insets show details of the correspondence under rescaling between parts of the plots (a) and (b).

Figure 5

Histograms of distances between successive turnings projected onto the axes $x$ [(a) and (c)] and $y$ [(b) and (d)] of the time series shown in Fig. 4 (nonrescaled data, $\lambda =1$). (a), (b) Model I; (c), (d) model II. Practically no difference is observed. This lack of difference demonstrates the difficulty of distinguishing the two models.

Figure 6

Histograms of the distances between successive turnings projected onto the axes $x$ [(a) and (c)] and $y$ [(b) and (d)] of the time series shown in Fig. 4, now rescaled data using $\lambda =100$. (a), (b) Model I; (c), (d) model II. The Gaussian fits, shown in dashed lines, describe model II reasonably well, but completely fail at the tails for model I. The rescaling has led to a noticeable difference between the models, when compared with the nonrescaled analysis shown in the previous figure.

Figure 7

Semilog plots of the histograms displayed in Fig. 6 for the rescaled data using $\lambda =100$, including the best fits to a Gaussian pdf shown in dashed lines. (a), (b) Model I; (c), (d) model II. Notice how model I has heavy tails, whereas model II is basically Gaussian.

Figure 8

Double-log plots of the histograms displayed in Figs. 5 and 6 for (a), (b) the original (i.e., nonrescaled, $\lambda =1$) data, and (c), (d) the rescaled data using $\lambda =100$. Black circles (blue squares) depict the results for the model I (model II). Notice how models I and II are very different at the tails, which are approximately described by a (truncated) power law. This difference only arises due to the rescaling. Without rescaling, the models are indistinguishable, as shown in Fig. 5. The key point to note is that the projection method together with the rescaling method makes it possible to reveal that model I is a (curved) Lévy walk.