Search Results (11)

Our recent analysis of empirical limit order flow data in financial markets reveals a power-law distribution in limit order cancellation times. These times are modeled using a discrete probability mass function derived from the Tsallis q-exponential distribution, closely aligned with the second form of the Pareto distribution. We elucidate this distinctive power-law statistical property through the lens of agent heterogeneity in trading activity and asset possession. Our study introduces a novel modeling approach that combines fractional Lévy stable motion for limit order inflow with this power-law distribution for cancellation times, significantly enhancing the prediction of order imbalances. This model not only addresses gaps in current financial market modeling but also extends to broader contexts such as opinion dynamics in social systems, capturing the finite lifespan of opinions. Characterized by stationary increments and a departure from self-similarity, our model provides a unique framework for exploring long-range dependencies in time series. This work paves the way for more precise financial market analyses and offers new insights into the dynamic nature of opinion formation in social systems. Full article

The segmentation analysis of the Golding–Cox mRNA dataset clarifies the description of these trajectories as a Fractional Lévy Stable Motion (FLSM). The FLSM method has several important advantages. Using only a few parameters, it allows for the detection of jumps in segmented trajectories with non-Gaussian confined parts. The value of each parameter indicates the contribution of confined segments. Non-Gaussian features in mRNA trajectories are attributed to trajectory segmentation. Each segment can be in one of the following diffusion modes: free diffusion, confined motion, and immobility. When free diffusion segments alternate with confined or immobile segments, the mean square displacement of the segmented trajectory resembles subdiffusion. Confined segments have both Gaussian (normal) and non-Gaussian statistics. If random trajectories are estimated as FLSM, they can exhibit either subdiffusion or Lévy diffusion. This approach can be useful for analyzing empirical data with non-Gaussian behavior, and statistical classification of diffusion trajectories helps reveal anomalous dynamics. Full article

To enhance the economic viability of wind energy in cold regions and ensure the safe operational management of wind farms, this paper proposes a short-term wind turbine blade icing wind power prediction method that combines principal component analysis (PCA) and fractional Lévy stable motion (fLsm). By applying supervisory control and data acquisition (SCADA) data from wind turbines experiencing icing in a mountainous area of Yunnan Province, China, the model comprehensively considers long-range dependence (LRD) and self-similar features. Adopting a combined pattern of previous-day predictions and actual measurement data, the model predicts the power under near-icing conditions, thereby enhancing the credibility and accuracy of icing forecasts. After validation and comparison with other prediction models (fBm, CNN-Attention-GRU, XGBoost), the model demonstrates a remarkable advantage in accuracy, achieving an accuracy rate and F1 score of 96.86% and 97.13%, respectively. This study proves the feasibility and wide applicability of the proposed model, providing robust data support for reducing wind turbine efficiency losses and minimizing operational risks. Full article

Remaining useful life prediction guarantees a reliable and safe operation of turbofan engines. Long-range dependence (LRD) and heavy-tailed characteristics of degradation modeling make this method advantageous for the prediction of RUL. In this study, we propose fractional Lévy stable motion for degradation modeling. First, we define fractional Lévy stable motion simulation algorithms. Then, we demonstrate the LRD and heavy-tailed property of fLsm to provide support for the model. The proposed method is validated with the C-MAPSS dataset obtained from the turbofan engine. Principle components analysis (PCA) is conducted to extract sources of variance. Experimental data show that the predictive model based on fLsm with exponential drift exhibits superior accuracy relative to the existing methods. Full article

The capacity regeneration phenomenon is often overlooked in terms of prediction of the remaining useful life (RUL) of LIBs for acceptable fitting between real and predicted results. In this study, we suggest a novel method for quantitative estimation of the associated uncertainty with the RUL, which is based on adaptive fractional Lévy stable motion (AfLSM) and integrated with the Mellin–Stieltjes transform and Monte Carlo simulation. The proposed degradation model exhibits flexibility for capturing long-range dependence, has a non-Gaussian distribution, and accurately describes heavy-tailed properties. Additionally, the nonlinear drift coefficients of the model can be adaptively updated on the basis of the degradation trajectory. The performance of the proposed RUL prediction model was verified by using the University of Maryland CALEC dataset. Our forecasting results demonstrate the high accuracy of the method and its superiority over other state-of-the-art methods. Full article

Modeling financial markets based on empirical data poses challenges in selecting the most appropriate models. Despite the abundance of empirical data available, researchers often face difficulties in identifying the best fitting model. Long-range memory and self-similarity estimators, commonly used for this purpose, can yield inconsistent parameter values, as they are tailored to specific time series models. In our previous work, we explored order disbalance time series from the broader perspective of fractional L’evy stable motion, revealing a stable anti-correlation in the financial market order flow. However, a more detailed analysis of empirical data indicates the need for a more specific order flow model that incorporates the power-law distribution of limit order cancellation times. When considering a series in event time, the limit order cancellation times follow a discrete probability mass function derived from the Tsallis q-exponential distribution. The combination of power-law distributions for limit order volumes and cancellation times introduces a novel approach to modeling order disbalance in the financial markets. Moreover, this proposed model has the potential to serve as an example for modeling opinion dynamics in social systems. By tailoring the model to incorporate the unique statistical properties of financial market data, we can improve the accuracy of our predictions and gain deeper insights into the dynamics of these complex systems. Full article

This paper aims to propose a generalized fractional Fokker–Planck equation based on a stable Lévy stochastic process. To develop the general fractional equation, we will use the Lévy process rather than the Brownian motion. Due to the Lévy process, this fractional equation can provide a better description of heavy tails and skewness. The analytical solution is chosen to solve the fractional equation and is expressed using the H-function to demonstrate the indicator entropy production rate. We model market data using a stable distribution to demonstrate the relationships between the tails and the new fractional Fokker–Planck model, as well as develop an R code that can be used to draw figures from real data. Full article

We study the first-arrival (first-hitting) dynamics and efficiency of a one-dimensional random search model performing asymmetric Lévy flights by leveraging the Fokker–Planck equation with a $\delta$-sink and an asymmetric space-fractional derivative operator with stable index $\alpha$ and asymmetry (skewness) parameter $\beta$. We find exact analytical results for the probability density of first-arrival times and the search efficiency, and we analyse their behaviour within the limits of short and long times. We find that when the starting point of the searcher is to the right of the target, random search by Brownian motion is more efficient than Lévy flights with $\beta \le 0$ (with a rightward bias) for short initial distances, while for $\beta >0$ (with a leftward bias) Lévy flights with $\alpha \to 1$ are more efficient. When increasing the initial distance of the searcher to the target, Lévy flight search (except for $\alpha =1$ with $\beta =0$) is more efficient than the Brownian search. Moreover, the asymmetry in jumps leads to essentially higher efficiency of the Lévy search compared to symmetric Lévy flights at both short and long distances, and the effect is more pronounced for stable indices $\alpha$ close to unity. Full article

In the face of the upcoming 30th anniversary of econophysics, we review our contributions and other related works on the modeling of the long-range memory phenomenon in physical, economic, and other social complex systems. Our group has shown that the long-range memory phenomenon can be reproduced using various Markov processes, such as point processes, stochastic differential equations, and agent-based models—reproduced well enough to match other statistical properties of the financial markets, such as return and trading activity distributions and first-passage time distributions. Research has lead us to question whether the observed long-range memory is a result of the actual long-range memory process or just a consequence of the non-linearity of Markov processes. As our most recent result, we discuss the long-range memory of the order flow data in the financial markets and other social systems from the perspective of the fractional Lèvy stable motion. We test widely used long-range memory estimators on discrete fractional Lèvy stable motion represented by the auto-regressive fractionally integrated moving average (ARFIMA) sample series. Our newly obtained results seem to indicate that new estimators of self-similarity and long-range memory for analyzing systems with non-Gaussian distributions have to be developed. Full article

We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior. Full article

In this paper, a wind speed prediction method was proposed based on the maximum Lyapunov exponent (Le) and the fractional Levy stable motion (fLsm) iterative prediction model. First, the calculation of the maximum prediction steps was introduced based on the maximum Le. The maximum prediction steps could provide the prediction steps for subsequent prediction models. Secondly, the fLsm iterative prediction model was established by stochastic differential. Meanwhile, the parameters of the fLsm iterative prediction model were obtained by rescaled range analysis and novel characteristic function methods, thereby obtaining a wind speed prediction model. Finally, in order to reduce the error in the parameter estimation of the prediction model, we adopted the method of weighted wind speed data. The wind speed prediction model in this paper was compared with GA-BP neural network and the results of wind speed prediction proved the effectiveness of the method that is proposed in this paper. In particular, fLsm has long-range dependence (LRD) characteristics and identified LRD by estimating self-similarity index H and characteristic index α. Compared with fractional Brownian motion, fLsm can describe the LRD process more flexibly. However, the two parameters are not independent because the LRD condition relates them by αH > 1. Full article