Stable Levy Motion Research Articles

Linear multifractional stable motion for modeling of fluid-filled regions in retinal optical coherence tomography images

Fluid appears in several retinal disorders can be visualized by optical coherence tomography (OCT) images. Since manual survey of OCT images is challenging, automatic fluid detection is desirable. This paper develops a generalized multifractal framework to model the local scale-invariant property of retinal OCT signals and accordingly proposes a simple methodology for automatic localization of fluid-filled regions. The proposed framework comprises both local self-similarity and heavy tailed distribution of OCT signals by generalization of fractional Levy stable motion (fLsm) to the Riemann-Liouville multifractional Levy stable motion (RL-mLsm). In order to measure the multifractality appears in RL-mLsm, the scaling properties of Levy flights and stochastic differential equations (SDEs) are used to develop a fractional momentbased algorithm namely, multi-fractional Levy fluctuation analysis (MLFA) algorithm. Here, we apply MLFA to model OCT images. Modeling results indicate the significant difference between the multifractal bandwidth in non-fluid and fluid regions. In addition, by utilizing various surrogate series of the signals, we discover the source of multifractality of OCT signals. As an application of the proposed multifractal modeling, we propose a new method, based on simple clustering of estimated local scaling exponent of modeled OCT signals, to localize fluid regions in OCT images. The localization method is validated, and the experimental results reveal the superiority of proposed method. Overall, the proposed multifractal framework has full advantages on excellent improvement of fluid localization, which promises the proposed model is suitable for effective description of self-similar nature of normal and abnormal OCT images independent of capturing device.

Short-term photovoltaic power prediction based on fractional Levy stable motion

Accurate prediction of photovoltaic (PV) power generation is the key to daily dispatch management and safe and stable grid operation. In order to improve the accuracy of the prediction, a finite iterative PV power prediction model with long range dependence (LRD) characteristics was developed using fractional Lévy stable motion (fLsm) and applied to a real dataset collected in the DKASC photovoltaic system in Alice Springs, Australia. The LRD prediction model considers the influence of current and past trends in the stochastic series on the future trends. Firstly, the calculation of the maximum steps prediction was introduced based on the maximum Lyapunov. The maximum prediction steps could provide the prediction steps for subsequent prediction models. Secondly, the order stochastic differential equation (FSDE) which describes the fLsm can be obtained. The parameters of the FSDE were estimated by using a novel characteristic function method. The PV power forecasting model with the LRD characteristics was obtained by discretization of FSDE. By comparing statistical performance indicators such as root max error, mean square error with Conv-LSTM, BiLSTM, and GA-LSTM models, the performance of the proposed fLsm model has been demonstrated. The proposed methods can provide better theoretical support for the stable and safe operation of PV grid connection. They have high reference value for grid dispatching department.

Multifractional and long-range dependent characteristics for remaining useful life prediction of cracking gas compressor

Cracking gas compressor (CGC) is a complex equipment used in ethylene production facilities. For the reliable and safe operation of CGC, the prediction of its remaining useful life (RUL) of relevance. The degradation process of a CGC from a normal state to a failure state has long-range dependence (LRD) with nonlinear and multifractal features. Concurrently, the increment of the degradation process obeys a non-Gaussian distribution. In this study, a degradation model for RUL prediction of CGC is developed. The model is based on a nonlinear drift function and Linear Multifractional Levy Stable Motion (LMSM). The drift function describes the nonlinear characteristics of the degradation process, whereas the LMSM allows accounting for its LRD, multifractal and non-Gaussian characteristics. The LRD features reflect the slowness of the degradation process, the multifractional features allow capturing local irregularities due to degenerate data fluctuations, and can specifically describe degenerate sequences. Finally, a RUL prediction framework for CGC is proposed and, then, verified with real observation data collected from an operating CGC.

Fractional Levy Stable and Maximum Lyapunov Exponent for Wind Speed Prediction

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.

Nonlocal Dynamics for Non-Gaussian Systems Arising in Biophysical Modeling

The aim of this article is to review our recent work on nonlocal dynamics of non-Gaussian systems arising in a gene regulatory network. We have used the mean exit time, escape probability and maximal likely trajectory to quantify dynamical behaviors of a stochastic differential system with non-Gaussian $$\alpha$$-stable Levy motions, to examine how the non-Gaussianity index and noise intensity affect the gene transcription processes.

Generalized moment estimators for $$\alpha $$-stable Ornstein–Uhlenbeck motions from discrete observations

We study the parameter estimation problem for discretely observed Ornstein–Uhlenbeck processes driven by $$\alpha $$-stable Levy motions. A method of moments via ergodic theory and via sample characteristic functions is proposed to estimate all the parameters involved in the Ornstein–Uhlenbeck processes. We obtain the strong consistency and asymptotic normality of the proposed joint estimators when the sample size $$n \rightarrow \infty $$ while the sampling time step h remains arbitrarily fixed. We also design a procedure to select the grid points in the characteristic functions in certain optimal way for the proposed estimators.

Stable Lévy Motion with Values in the Skorokhod Space: Construction and Approximation

In this article, we introduce an infinite-dimensional analogue of the $$\alpha $$-stable Levy motion, defined as a Levy process $$Z=\{Z(t)\}_{t \ge 0}$$ with values in the space $${\mathbb {D}}$$ of cadlag functions on [0, 1], equipped with Skorokhod’s $$J_1$$ topology. For each $$t \ge 0$$, Z(t) is an $$\alpha $$-stable process with sample paths in $${\mathbb {D}}$$, denoted by $$\{Z(t,s)\}_{s\in [0,1]}$$. Intuitively, Z(t, s) gives the value of the process Z at time t and location s in space. This process is closely related to the concept of regular variation for random elements in $${\mathbb {D}}$$ introduced in de Haan and Lin (Ann Probab 29:467–483, 2001) and Hult and Lindskog (Stoch Proc Appl 115:249–274, 2005). We give a construction of Z based on a Poisson random measure, and we show that Z has a modification whose sample paths are cadlag functions on $$[0,\infty )$$ with values in $${\mathbb {D}}$$. Finally, we prove a functional limit theorem which identifies the distribution of this modification as the limit of the partial sum sequence $$\{S_n(t)=\sum _{i=1}^{[nt]}X_i\}_{t\ge 0}$$, suitably normalized and centered, associated with a sequence $$(X_i)_{i\ge 1}$$ of i.i.d. regularly varying elements in $${\mathbb {D}}$$.

Convergence of Euler-Maruyama Method for Stochastic Differential Equations Driven by alpha-stable Levy Motion

In the literature, the Euler-Maruyama (EM) method for approximation purposes of stochastic differential Equations (SDE) driven by alph -stable Levy motions is reported. Convergence in probability of this method was proven but it is surrounded by some ambiguities. To accomplish the method without ambiguities, this article has derived convergence in probability of numerical EM method based on diffusion given by semimartingales for SDEs driven by alpha -stable processes. Some examples are provided, their numerical solution are obtained and theoretical results are reconfirmed. The adopted method could be applied to other subclasses of semimartingales.

Discriminating between scaled and fractional Brownian motion via p-variation statistics

In recent years the anomalous diffusion processes have found various applications including polymers. The most common anomalous diffusion processes are continuous time random walk and fractional Brownian motion. However in the literature one can find different systems like fractional Levy stable motion or scaled Brownian motion. A most common tool for anomalous diffusion behavior recognition is the time-averaged mean square displacement. However, this statistics has the same behavior for fractional and scaled Brownian motions. Therefore there is need to apply different approach in order to discriminate between those two anomalous diffusion processes. One of the possibility is the p-variation statistics which demonstrates different behavior for fractional and scaled Brownian motions. In this paper we prove the formula for p-variation of scaled Brownian motion and compare it to the known formula of p-variation for fractional Brownian motion. Moreover, we show how to apply them to real trajectories analysis in order to recognize the proper anomalous diffusion model. The theoretical results are supported by simulated trajectories.

Asymptotics of random processes with immigration I: Scaling limits

Let $(X_{1},\xi_{1}),(X_{2},\xi_{2}),\ldots$ be i.i.d. copies of a pair $(X,\xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\infty)$ and $\xi$ is a positive random variable. Define $S_{k}:=\xi_{1}+\cdots+\xi_{k}$, $k\in\mathbb{N}_{0}$ and $Y(t):=\sum_{k\geq0}X_{k+1}(t-S_{k})\mathbf{1}_{\{S_{k}\leq t\}}$, $t\geq0$. We call the process $(Y(t))_{t\geq0}$ random process with immigration at the epochs of a renewal process. We investigate weak convergence of the finite-dimensional distributions of $(Y(ut))_{u>0}$ as $t\to\infty$. Under the assumptions that the covariance function of $X$ is regularly varying in $(0,\infty)\times(0,\infty)$ in a uniform way, the class of limiting processes is rather rich and includes Gaussian processes with explicitly given covariance functions, fractionally integrated stable Levy motions and their sums when the law of $\xi$ belongs to the domain of attraction of a stable law with finite mean, and conditionally Gaussian processes with explicitly given (conditional) covariance functions, fractionally integrated inverse stable subordinators and their sums when the law of $\xi$ belongs to the domain of attraction of a stable law with infinite mean.

Functional Convergence of Linear Processes with Heavy-Tailed Innovations

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients, necessary and sufficient conditions for the finite dimensional convergence to an \(\alpha \)-stable Levy Motion are given. The conditions lead to new, tractable sufficient conditions in the case \(\alpha \le 1\). In the functional setting, we complement the existing results on \(M_1\)-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (Ann Probab 20:483–503, 1992) and improved by Louhichi and Rio (Electr J Probab 16(89), 2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the \(S\) topology, introduced by Jakubowski (Electr J Probab 2(4), 1997).

An accurate algorithm to calculate the Hurst exponent of self-similar processes

Abstract In this paper, we introduce a new approach which generalizes the GM2 algorithm (introduced in Sanchez-Granero et al. (2008) [52] ) as well as fractal dimension algorithms (FD1, FD2 and FD3) (first appeared in Sanchez-Granero et al. (2012) [51] ), providing an accurate algorithm to calculate the Hurst exponent of self-similar processes. We prove that this algorithm performs properly in the case of short time series when fractional Brownian motions and Levy stable motions are considered. We conclude the paper with a dynamic study of the Hurst exponent evolution in the S&P500 index stocks.

Mean Exit Time and Escape Probability for Dynamical Systems Driven by Lévy Noises

The mean first exit time and escape probability are utilized to quantify dynamical behaviors of stochastic differential equations with non-Gaussian $\alpha$-stable type Levy motions. An efficient and accurate numerical scheme is developed and validated for computing the mean exit time and escape probability from the governing differential-integral equation. An asymptotic solution for the mean exit time is given when the pure jump measure in the Levy motion is small. From both the analytical and numerical results, it is observed that the mean exit time depends strongly on the domain size and the value of $\alpha$ in the $\alpha$-stable Levy jump measure. The mean exit time and escape probability could become discontinuous at the boundary of the domain, when the value of $\alpha$ is in (0,1).

Modeling network traffic by a cluster Poisson input process with heavy and light-tailed file sizes

We consider a cluster Poisson model with heavy-tailed interarrival times and cluster sizes as a generalization of an infinite source Poisson model where the file sizes have a regularly varying tail distribution function or a finite second moment. One result is that this model reflects long-range dependence of teletraffic data. We show that depending on the heaviness of the file sizes, the interarrival times and the cluster sizes we have to distinguish different growths rates for the time scale of the cumulative traffic. The mean corrected cumulative input process converges to a fractional Brownian motion in the fast growth case. However, in the intermediate and the slow growth case we can have convergence to a stable Levy motion or a fractional Brownian motion as well depending on the heaviness of the underlying distributions. These results are contrary to the idea that cumulative broadband network traffic converges in the slow growth case to a stable process. Furthermore, we derive the asymptotic behavior of the cluster Poisson point process which models the arrival times of data packets and the individual input process itself.

Finite-size Lyapunov exponent for Levy processes

The finite-size Lyapunov exponent (FSLE) is the exponential rate at which two particles separate from a distance of r to a x r (a>1) and provides a measure of dispersive mixing in chaotic systems. It is shown analytically that for particle trajectories governed by symmetric alpha -stable Levy motion, the FSLE is proportional to the diffusion coefficient and inversely proportional to r(alpha). This power law provides an easy method to determine the parameters for Levy processes and hence has applications to superdiffusion in the atmospheric, oceanic, and terrestrial sciences.

Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights

Let \(U_{j} ,\;j \in \mathbb{N}\) be independent and identically distributed random variables with heavy-tailed distributions. Consider a sequence of random weights \({\left\{ {W_{j} } \right\}}_{{j \in \mathbb{N}}}\), independent of \({\left\{ {U_{j} } \right\}}_{{j \in \mathbb{N}}}\) and focus on the weighted sums \({\sum\nolimits_{j = 1}^{{\left[ {nt} \right]}} {W_{j} {\left( {U_{j} - \mu } \right)}} }\), where μ involves a suitable centering. We establish sufficient conditions for these weighted sums to converge to non-trivial limit processes, as n→∞, when appropriately normalized. The convergence holds, for example, if \({\left\{ {W_{j} } \right\}}_{{j \in \mathbb{N}}}\) is strictly stationary, dependent, and W 1 has lighter tails than U 1. In particular, the weights W j s can be strongly dependent. The limit processes are scale mixtures of stable Levy motions. We establish weak convergence in the Skorohod J 1-topology. We also consider multivariate weights and show that they converge weakly in the strong Skorohod M 1-topology. The M 1-topology, while weaker than the J 1-topology, is strong enough for the supremum and infimum functionals to be continuous.

On sampling of stationary increment processes

Let $X(t)$, $0\leq t\leq 1$, be a stochastic process with stationary increments. Let $q_1(t)$, $q_2(t)$ and $w(t)$, $t>0$, be positive functions of $t$ which tend to $0$ for $t\to 0$, and define the random variables $$Z_1(t)=\{X(t)-X([t/q_1(\epsilon)]q_1(\epsilon))-\epsilon\}/w(\epsilon),$$ and $$Z_2(t)={|X(t)-X([t/q_2(\epsilon)]q_2(\epsilon))|-\epsilon}/w(\epsilon),\text{ for }\epsilon>0.$$ Under a technical condition of eight parts it is shown that $$H(x)=\lim_{\epsilon\downarrow 0}P\{\sup_{0\leq t\leq 1}Z_i(t)\leq x}, i=1,2,$$ exists for $x\in J$, for some open interval $J$, and $H$ is identified. The functions $q_i$, $w$ and $H$ are themselves defined in the formulation of the conditions in the assumptions of the theorem stating the result. Some of these conditions are shown to be similar to those previously used in extreme value theory. Examples with strictly $\alpha$-stable Levy motions are given. The main result is also shown to hold for a totally skewed linear fractional $\alpha$-stable motion.

Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed

Consider M independent and identically distributed renewal-reward processes with heavy-tailed renewals and rewards that have either finite variance or heavy tails. Let W * (Ty,M),y∈[0,1] , denote the total reward process computed as the sum of all rewards in M renewal-reward processes over the time interval [0,T]. If T→∞ and then M→∞, Taqqu and Levy have shown that the properly normalized total reward process W * (T⋅,M) converges to the stable Levy motion, but, if M→∞ followed by T→∞, the limit depends on whether the tails of the rewards are lighter or heavier than those of renewals. If they are lighter, then the limit is a self-similar process with stationary and dependent increments. If the rewards have finite variance, this self-similar process is fractional Brownian motion, and if they are heavy-tailed rewards, it is a stable non-Gaussian process with infinite variance. We consider asymmetric rewards and investigate what happens when M and T go to infinity jointly, that is, when M is a function of T and M=M(T)→∞ as T→∞. We provide conditions on the growth of M for the total reward process W * (T⋅,M(T)) to converge to any of the limits stated above, as T→∞. We also show that when the tails of the rewards are heavier than the tails of the renewals, the limit is stable Levy motion as M=M(T)→∞, irrespective of the function M(T).

Small and Large Time Scale Analysis of a Network Traffic Model

Empirical studies of the internet and WAN traffic data have observed multifractal behavior at time scales below a few hundred milliseconds. There have been some attempts to model this phenomenon, but there is no model to connect the small time scale behavior with behavior observed at large time scales of bigger than a few hundred milliseconds. There have been separate analyses of models for high speed data transmissions, which show that appropriate approximations to large time scale behavior of cumulative traffic are either fractional Brownian motion or stable Levy motion, depending on the input rates assumed. This paper tries to bridge this gap and develops and analyzes a model offering an explanation of both the small and large time scale behavior of a network traffic model based on the infinite source Poisson model. Previous studies of this model have usually assumed that transmission rates are constant and deterministic. We consider a nonconstant, multifractal, random transmission rate at the user level which results in cumulative traffic exhibiting multifractal behavior on small time scales and self-similar behavior on large time scales.

Time Change Representation of Stochastic Integrals

By the Dambis--Dubins--Schwarz theorem, any stochastic integral $M:=\int_0^\cdot H_sdW_s$ of Brownian motion can be written as a time-changed Brownian motion, i.e., $M=({\widehat{W}}_{\widehat{T_t}})_{t\in\bf{R}_+}$ for some Brownian motion $({\widehat{W}}_\theta)_{\theta\in\bf{R}_+}$ and some time change $({\widehat{T_t}})_{t\in\bf{R}_+}$. In [J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin--Heidelberg, 1987] and [O. Kallenberg, Stochastic Process. Appl., 40 (1992), pp.~199--223] it is shown that in this statement Brownian motion can be replaced with (symmetric) $\alpha$-stable Levy motion}. Using the cumulant process of a semimartingale, we give new short proofs. Moreover, we show that the statement cannot be extended to any other Levy processes.