Kinetic equation of linear fractional stable motion and applications to modeling the scaling of intermittent bursts

Goodness-of-fit test for $$\alpha$$-stable distribution based on the quantile conditional variance statistics

The class of $$\alpha$$ α -stable distributions is ubiquitous in many areas including signal processing, finance, biology, physics, and condition monitoring. In particular, it allows efficient noise modeling and incorporates distributional properties such as asymmetry and heavy-tails. Despite the popularity of this modeling choice, most statistical goodness-of-fit tests designed for $$\alpha$$ α -stable distributions are based on a generic distance measurement methods. To be efficient, those methods require large sample sizes and often do not efficiently discriminate distributions when the corresponding $$\alpha$$ α -stable parameters are close to each other. In this paper, we propose a novel goodness-of-fit method based on quantile (trimmed) conditional variances that is designed to overcome these deficiencies and outperforms many benchmark testing procedures. The effectiveness of the proposed approach is illustrated using extensive simulation study with focus set on the symmetric case. For completeness, an empirical example linked to plasma physics is provided.

Pattern formation in reaction-diffusion systems in the presence of non-Markovian diffusion

We study reaction-diffusion systems beyond the Markovian approximation to take into account the effect of memory on the formation of spatiotemporal patterns. Using a non-Markovian Brusselator model as a paradigmatic example, we show how to use reductive perturbation to investigate the formation and stability of patterns. Focusing in detail on the Hopf instability and short-term memory, we derive the corresponding complex Ginzburg-Landau equation that governs the amplitude of the critical mode and we establish the explicit dependence of its parameters on the memory properties. Numerical solution of this memory-dependent complex Ginzburg-Landau equation as well as direct numerical simulation of the non-Markovian Brusselator model illustrates that memory changes the properties of the spatiotemporal patterns. Our results indicate that going beyond the Markovian approximation might be necessary to study the formation of spatiotemporal patterns even in systems with short-term memory. At the same time, our work opens up a new window into the control of these patterns using memory.

Consequences of anomalous diffusion in disordered systems under cyclic forcing

We study the particle scale response of a 2D frictionless disk system to bulk forcing via cyclic shear with reversal amplitude γ_{r}. We find a subdiffusive γ_{r}-dependent regime, which is consistent with models of anomalous diffusion with scale-invariant cage dynamics, and a crossover to diffusive grain motion at high γ_{r}. Analysis of local displacements of a particle relative to its cage of neighbors reveals a key distinction from thermal systems. Particles are moved by fluctuations of their cage of neighbors rather than rattling in their cage, indicating a distinct cage-breaking mechanism.

Incorrect likelihood methods were used to infer scaling laws of marine predator search behaviour

Background

Ecologists are collecting extensive data concerning movements of animals in marine ecosystems. Such data need to be analysed with valid statistical methods to yield meaningful conclusions.

Principal Findings

We demonstrate methodological issues in two recent studies that reached similar conclusions concerning movements of marine animals (Nature 451∶1098; Science 332∶1551). The first study analysed vertical movement data to conclude that diverse marine predators (Atlantic cod, basking sharks, bigeye tuna, leatherback turtles and Magellanic penguins) exhibited “Lévy-walk-like behaviour”, close to a hypothesised optimal foraging strategy. By reproducing the original results for the bigeye tuna data, we show that the likelihood of tested models was calculated from residuals of regression fits (an incorrect method), rather than from the likelihood equations of the actual probability distributions being tested. This resulted in erroneous Akaike Information Criteria, and the testing of models that do not correspond to valid probability distributions. We demonstrate how this led to overwhelming support for a model that has no biological justification and that is statistically spurious because its probability density function goes negative. Re-analysis of the bigeye tuna data, using standard likelihood methods, overturns the original result and conclusion for that data set. The second study observed Lévy walk movement patterns by mussels. We demonstrate several issues concerning the likelihood calculations (including the aforementioned residuals issue). Re-analysis of the data rejects the original Lévy walk conclusion.

Conclusions

We consequently question the claimed existence of scaling laws of the search behaviour of marine predators and mussels, since such conclusions were reached using incorrect methods. We discourage the suggested potential use of “Lévy-like walks” when modelling consequences of fishing and climate change, and caution that any resulting advice to managers of marine ecosystems would be problematic. For reproducibility and future work we provide R source code for all calculations.

Recognition of stable distribution with Lévy index α close to 2

We address the problem of recognizing α-stable Lévy distribution with Lévy index close to 2 from experimental data. We are interested in the case when the sample size of available data is not large, thus the power law asymptotics of the distribution is not clearly detectable, and the shape of the empirical probability density function is close to a Gaussian. We propose a testing procedure combining a simple visual test based on empirical fourth moment with the Anderson-Darling and Jarque-Bera statistical tests and we check the efficiency of the method on simulated data. Furthermore, we apply our method to the analysis of turbulent plasma density and potential fluctuations measured in the stellarator-type fusion device and demonstrate that the phenomenon of the L-H transition from low confinement, L mode, to a high confinement, H mode, which occurs in this device is accompanied by the transition from Lévy to Gaussian fluctuation statistics.

Fractional Lévy stable motion can model subdiffusive dynamics

We show in this paper that the sample (time average) mean-squared displacement (MSD) of the fractional Lévy α -stable motion behaves very differently from the corresponding ensemble average (second moment). While the ensemble average MSD diverges for α<2 , the sample MSD may exhibit either subdiffusion, normal diffusion, or superdiffusion. Thus, H -self-similar Lévy stable processes can model either a subdiffusive, diffusive or superdiffusive dynamics in the sense of sample MSD. We show that the character of the process is controlled by a sign of the memory parameter d=H-1/α . We also introduce a sample p -variation dynamics test which allows to distinguish between two models of subdiffusive dynamics. Finally, we illustrate a subdiffusive behavior of the fractional Lévy stable motion on biological data describing the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells, but it may concern many other fields of contemporary experimental physics.