Truncated Levy Flight Research Articles

Performance analysis of diffusive molecular timing channels

In this study, the authors consider a diffusive molecular communication channel where the information to be transmitted is the time of release of the information molecules. Using the truncated Levy distribution to model the first passage time of a molecule, the authors develop some general statistics for a random variable described by a truncated Levy distribution. The authors then consider a multi-particle molecular communication system, where the average arrival time of the information particles can be represented by a truncated Levy flight, and the number of molecules required for the cross-over from the Levy regime to the Gaussian regime to occur is calculated. Furthermore, they use these results to analyse single as well as multi-particle molecular communication channels in terms of capacity bounds and symbol error probability.

Chile2015: Lévy Flight and Long-Range Correlation Analysis of Earthquake Magnitudes in Chile

The stochastic Truncated Levy Flight model and detrended fluctuation analysis (DFA) are used to investigate the temporal distribution of earthquake magnitudes in Chile. We show that Levy Flight is appropriated for modeling the time series of the magnitudes of the earthquakes. Furthermore, DFA shows that these events present memory effects, suggesting that the magnitude of impeding earthquakes depends on the magnitude of previous earthquakes. Based on this dependency, we use a non-linear regression to estimate the magnitude of the 2015, M8.3 Illapel earthquake based on the magnitudes of the previous events.

Levy Flight Algorithm for Optimization Problems - A Literature Review

This paper presents a literature review on applications of Levy flight. Nowadays, Levy flight laws has been used for a broad class of processes such as in physical, chemical, biological, statistical and also in financial. From the review, Levy flight technique has been applied mostly in physics area where the researchers use Levy flight technique to solve and optimize the problem regarding diffusive, scaling and transmission. This paper also reviews the latest researches using modified Levy flight technique such as truncated, smoothly truncated and gradually truncated Levy Flight for optimization. Finally, future trends of Levy flight are discussed.

Arbitrary truncated Levy flight: Asymmetrical truncation and high-order correlations

The generalized correlation approach, which has been successfully used in statistical radio physics to describe non-Gaussian random processes, is proposed to describe stochastic financial processes. The generalized correlation approach has been used to describe a non-Gaussian random walk with independent, identically distributed increments in the general case, and high-order correlations have been investigated. The cumulants of an asymmetrically truncated Levy distribution have been found. The behaviors of asymmetrically truncated Levy flight, as a particular case of a random walk, are considered. It is shown that, in the Levy regime, high-order correlations between values of asymmetrically truncated Levy flight exist. The source of high-order correlations is the non-Gaussianity of the increments: the increment skewness generates threefold correlation, and the increment kurtosis generates fourfold correlation.

ON THE ENSEMBLE AVERAGE OF GENERALIZED TRUNCATED RANDOM WALK

A new analytical solution is developed for the one- and two- dimensional generalized discrete time random walk with discrete and finite length of jumping. It is shown that the ensemble average of the mean square displacement has a linear relation with time. This result is general and independent of mathematical form of distribution function but magnitude of diffusion coefficient depends on the distribution function. This result confirms our previous numerical work. The truncated Levy flight is studied as a special case and the diffusion coefficient for different maximum length of jump is calculated numerically. The obtained results are in good agreement with the analytical solution.

Normalized truncated Levy walk applied to flexible pavement performance

Government agencies and consulting companies face the challenge of pavement management aiming to maintain pavements in serviceable condition in the long term. This paper analyzed the evolution of the performance of flexible pavements to forecast the change in the serviceability level offered by the structure. Data from the Minnesota Road Research Project (MnROAD) road test corresponding to the low-volume traffic loop were used in the study. The data analysis centered on the normalized truncated Levy walk model. It was concluded that the truncated Levy distribution optimally describes the decrease of pavement serviceability caused by traffic and climatic conditions specific to the infrastructure location and traffic. The Truncated Levy Flight (TLF) provided a better approximation over the full range of the normalized pavement serviceability index than the Gaussian distribution for dataset considered.

Cumulant approach of arbitrary truncated Levy flight

The problem of an arbitrary truncated Levy flight description using the method of cumulant approach has been solved. The set of cumulants of the truncated Levy distribution given the assumption of arbitrary truncation has been found. The influence of truncation shape on the truncated Levy flight properties in the Gaussian and the Levy regimes has been investigated.

Study of memory effects in international market indices

Abstract Long term memory effects in stock market indices that represent internationally diversified stocks are analyzed in this paper and the results are compared with the S&P 500 index. The Hurst exponent and the Detrended fluctuation analysis (DFA) technique are the tools used for this analysis. The financial time-series data of these indices are tested with the Normalized Truncated Levy Flight to check whether the evolution of these indices is explained by the TLF . Some features that seem to be specific for international indices are discovered and briefly discussed. In particular, a potential investor seems to be faced with new investment opportunities in emerging markets during and especially after a crisis.

LÉvy Flights and Fractal Modeling of Internet Traffic

The relation between burstiness and self-similarity of network traffic was identified in numerous papers in the past decade. These papers suggested that the widely used Poisson based models were not suitable for modeling bursty, local-area and wide-area network traffic. Poisson models were abandoned as unrealistic and simplistic characterizations of network traffic. Recent papers have challenged the accuracy of these results in today's networks. Authors of these papers believe that it is time to reexamine the Poisson traffic assumption. The explanation is that as the amount of Internet traffic grows dramatically, any irregularity of the network traffic, such as burstiness, might cancel out because of the huge number of different multiplexed flows. Some of these results are based on analyses of particular OC48 Internet backbone connections and other historical traffic traces. We analyzed the same traffic traces and applied new methods to characterize them in terms of packet interarrival times and packet lengths. The major contribution of the paper is the application of two new analytical methods. We apply the theory of smoothly truncated Levy flights and the linear fractal model in examining the variability of Internet traffic from self-similar to Poisson. The paper demonstrates that the series of interarrival times is still close to a self-similar process, but the burstiness of the packet lengths decreases significantly compared to earlier traces.

Long correlations and Normalized Truncated Levy Models applied to the study of Indian Market Indices in comparison with other emerging markets

This work is devoted to the study of long correlations and other statistical properties of the Indian Market Indices in comparison to other emerging market indices. We verified that the behavior of the return is compatible with a Normalized Truncated Levy Flight. We also detected long-range correlations in the absolute value of the return. Finally, we concluded that the statistical behavior of emerging markets is similar to the behavior of developed economies.

Extreme events as foundation of Lévy walks with varying velocity

Abstract In this work we study the role of extreme events [E.W. Montroll, B.J. West, in: J.L. Lebowitz, E.W. Montrell (Eds.), Fluctuation Phenomena, SSM, vol. VII, North-Holland, Amsterdam, 1979, p. 63; J.-P. Bouchaud, M. Potters, Theory of Financial Risks from Statistical Physics to Risk Management, Cambridge University Press, Cambridge, 2001; D. Sornette, Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and Disorder: Concepts and Tools, Springer, Berlin, 2000] in determining the scaling properties of Levy walks with varying velocity. This model is an extension of the well-known Levy walks one [J. Klafter, G. Zumofen, M.F. Shlesinger, in M.F. Shlesinger, G.M. Zaslavsky, U. Frisch (Eds.), Levy Flights and Related Topics ion Physics, Lecture Notes in Physics, vol. 450, Springer, Berlin, 1995, p. 196; G. Zumofen, J. Klafter, M.F. Shlesinger, in: R. Kutner, A. Pȩkalski, K. Sznajd-Weron (Eds.), Anomalous Diffusion. From Basics to Applications, Lecture Note in Physics, vol. 519, Springer, Berlin, 1999, p. 15] introduced in the context of chaotic dynamics where a fixed value of the walker velocity is assumed for simplicity. Such an extension seems to be necessary when the open and/or complex system is studied. The model of Levy walks with varying velocity is spanned on two coupled velocity–temporal hierarchies: the first one consisting of velocities and the second of corresponding time intervals which the walker spends between the successive turning points. Both these hierarchical structures are characterized by their own self-similar dimensions. The extreme event, which can appear within a given time interval, is defined as a single random step of the walker having largest length. By finding power-laws which describe the time-dependence of this displacement and its statistics we obtained two independent diffusion exponents, which are related to the above-mentioned dimensions and which characterize the extreme event kinetics. In this work we show the principal influence of extreme events on the basic quantities (one-step distributions and moments as well as two-step correlation functions) of the continuous-time random walk formalism. Besides, we construct both the waiting-time distribution and sojourn probability density directly in a real space and time in the scaling form by proper component analysis which takes into account all possible fluctuations of the walker steps in contrast to the extreme event analysis. In this work we pay our attention to the basic quantities, since the summarized multi-step ones were already discussed earlier [Physica A 264 (1999) 107; Comp. Phys. Commun. 147 (2002) 565]. Moreover, we study not only the scaling phenomena but also, assuming a finite number of hierarchy levels, the breaking of scaling and its dependence on control parameters. This seems to be important for studying empirical systems the more so as there are still no closed formulae describing this phenomenon except the one for truncated Levy flights [Phys. Rev. Lett. 73 (1994) 2946]. Our formulation of the model made possible to develop an efficient Monte Carlo algorithm [Physica A 264 (1999) 107; Comp. Phys. Commun. 147 (2002) 565] where no MC step is lost.

Generalized Lotka-Volterra (GLV) Models of Stock Markets

The Generalized Lotka-Volterra (GLV) model: [Formula: see text] provides a general method to simulate, analyze and understand a wide class of phenomena that are characterized by power-law probability distributions: [Formula: see text] and truncated Levy flights fluctuations [Formula: see text]. We show how the model applies to economic systems.

Econophysics: Scaling and its breakdown in finance

We discuss recent empirical results obtained by analyzing high-frequency data of a stock market index, the Standard and Poor’s 500. We focus on the scaling properties and on its breakdown of the index dynamics. A simple stochastic model, the truncated Levy flight, is illustrated. Successes and limitations of this model are presented. A discussion about similarities and differences between the scaling properties observed in financial markets and in fully developed turbulence is also provided.