Long-range Dependence Property Research Articles

On the prediction of power outage length based on linear multifractional Lévy stable motion

In the power system, failure interruption hugely compromises the power system reliability. Therefore, an efficient method to correctly predict how long the power outage last would significantly improve the efficiency. To address this problem, we propose a method based on a suitable stochastic motion. First, we introduce linear multifractional Lévy stable motion (LMLSM). Then, by computing the global and local fractal characteristics we show that the LMLSM is multifractal and possesses the long-range dependence (LRD) property. Besides, the non-Gaussian characteristics of the LMLSM are analyzed, which means it can better describe the constant peak values in the stochastic time series. Furthermore, a discrete iterative prediction model is derived with fractional Black-Scholes differential equation. At last, a case study is provided based on the real failure interruption duration (FID) dataset in the power grid, by illustrating the validity of the proposed prediction method for power grid reliability.

Predicting the Remaining Useful Life of Turbofan Engines Using Fractional Lévy Stable Motion with Long-Range Dependence

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.

Generalized fractional negative binomial process

In this paper, we introduce a generalized fractional negative binomial process (GFNBP) by time changing the fractional Poisson process with an independent Mittag-Leffler (ML) Lévy subordinator. We study its distributional properties and its connection to PDEs. We examine the long-range dependence (LRD) property of the GFNBP and show that it is not infinitely divisible. The space-fractional and the non-homogeneous variants of the GFNBP are explored. Finally, simulated sample paths for the ML Lévy subordinator and the GFNBP are also presented.

On a time-changed variant of the generalized counting process

AbstractIn this paper, we time-change the generalized counting process (GCP) by an independent inverse mixed stable subordinator to obtain a fractional version of the GCP. We call it the mixed fractional counting process (MFCP). The system of fractional differential equations that governs its state probabilities is obtained using the Z transform method. Its one-dimensional distribution, mean, variance, covariance, probability generating function, and factorial moments are obtained. It is shown that the MFCP exhibits the long-range dependence property whereas its increment process has the short-range dependence property. As an application we consider a risk process in which the claims are modelled using the MFCP. For this risk process, we obtain an asymptotic behaviour of its finite-time ruin probability when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large. Later, we discuss some distributional properties of a compound version of the GCP.

Correlation Structure of Time-Changed Generalized Mixed Fractional Brownian Motion

The generalized mixed fractional Brownian motion (gmfBm) is a Gaussian process with stationary increments that exhibits long-range dependence controlled by its Hurst indices. It is defined by taking linear combinations of a finite number of independent fractional Brownian motions with different Hurst indices. In this paper, we investigate the long-time behavior of gmfBm when it is time-changed by a tempered stable subordinator or a gamma process. As a main result, we show that the time-changed process exhibits a long-range dependence property under some conditions on the Hurst indices. The time-changed gmfBm can be used to model natural phenomena that exhibit long-range dependence, even when the underlying process is not itself long-range dependent.

Performance degradation assessment for mechanical system based on semi-analytical solution of self-similar stable distribution process

To more accurately predict remaining useful life (RUL) and quantitatively evaluate the uncertainty of the predicted results, a performance degradation assessment framework based on semi-analytical solution of self-similar stable distribution process is proposed. The established performance degradation model based on adaptive fractional Lévy stable motion (AFLSM) is more flexible in revealing the long-range dependence, non-Gaussian, and heavy-tailed distribution properties of the incremental behavior. The corresponding stable distribution parameters are estimated through characteristic function method, and Hurst exponent is calculated based on the generalized Hurst exponent approach with narrower confidence interval. Aiming at the difficulties in solving the exact analytical solution and the excessive computation of the numerical solution in the whole process, based on Mellin-Stieltjes transform and direct integration, a semi-analytical solution of RUL distribution function is proposed, which can be readily implemented in practical equipment operations. The proposed performance degradation assessment framework is validated by the novel truck transmission dataset and the benchmark rolling bearing dataset. Experimental results indicate that the developed framework is more effective and superior than other state-of-the-art approaches in terms of RUL prediction.

Tempered space fractional negative binomial process

In this paper, we define a tempered space fractional negative binomial process (TSFNBP) by replacing the Poisson process by a tempered space fractional Poisson process (TSFPP) in the gamma subordinated form of the negative binomial process. We studied its distributional properties, long-range dependence (LRD) property and its connection with pde’s. The LRD property for the TSFPP process is also investigated. Finally, we present simulated sample paths for both the TSFPP and the TSFNBP.

Convoluted fractional Poisson process of order k

In this article, we define a convoluted fractional Poisson process of order k (CFPPoK), which is governed by the discrete convolution operator in the system of fractional differential equations. Next, we obtain its one-dimensional distribution by using the Laplace transform of its state probabilities. Various distributional properties, such as probability generating function, moment generating function and moments, are derived. A special case of CFPPoK, (say) convoluted Poisson process of order k (CPPoK) is studied and also established Martingale characterization for CPPoK. We further derive the covariance structure of CFPPoK and investigate the long-range dependence property.

Some Compound Fractional Poisson Processes

In this paper, we introduce and study fractional versions of the Bell–Touchard process, the Poisson-logarithmic process and the generalized Pólya–Aeppli process. The state probabilities of these compound fractional Poisson processes solve a system of fractional differential equations that involves the Caputo fractional derivative of order 0<β<1. It is shown that these processes are limiting cases of a recently introduced process, namely, the generalized counting process. We obtain the mean, variance, covariance, long-range dependence property, etc., for these processes. Further, we obtain several equivalent forms of the one-dimensional distribution of fractional versions of these processes.

Asymptotic normality of wavelet covariances and multivariate wavelet Whittle estimators

Multivariate processes with long-range dependence properties can be encountered in many fields of application. Two fundamental characteristics in such frameworks are long-range dependence parameters and correlations between component time series. We consider multivariate long-range dependent linear processes, not necessarily Gaussian. We show that the covariances between the wavelet coefficients in this setting are asymptotically Gaussian. We also study the asymptotic distributions of the estimators of the long-range dependence parameter and the long-run covariance by a wavelet-based Whittle procedure. We prove the asymptotic normality of the estimators, and we provide an explicit expression for the asymptotic covariances. An empirical illustration of this result is proposed on a real dataset of rat brain connectivity.

Skellam and time-changed variants of the generalized fractional counting process

In this paper, we study a Skellam type variant of the generalized counting process (GCP), namely, the generalized Skellam process. Some of its distributional properties such as the probability mass function, probability generating function, mean, variance and covariance are obtained. Its fractional version, namely, the generalized fractional Skellam process (GFSP) is considered by time-changing it with an independent inverse stable subordinator. It is observed that the GFSP is a Skellam type version of the generalized fractional counting process (GFCP) which is a fractional variant of the GCP. It is shown that the one-dimensional distributions of the GFSP are not infinitely divisible. An integral representation for its state probabilities is obtained. We establish its long-range dependence property by using its variance and covariance structure. Also, we consider two time-changed versions of the GFCP. These are obtained by time-changing the GFCP by an independent Lévy subordinator and its inverse. Some particular cases of these time-changed processes are discussed by considering specific Lévy subordinators.

Testing Day-Of-The-Week Persistence and Seasonality in Spanish Electricity Energy Prices

Day-of-the-week persistence and seasonality of electricity prices in Spain, spanning 01/01/2006 to 04/11/2021, are investigated by employing updated fractional persistence frameworks in non-linear settings. The results show marginal higher persistence in electricity prices during the working days (Tuesday, Wednesday, Thursday, and Friday) compared to weekend days. In all cases, electricity prices are mean-reverting with long-range dependence properties. Results also show that the monthly electricity price series contain no seasonal effect.

ARMA–GARCH model with fractional generalized hyperbolic innovations

In this study, a multivariate ARMA–GARCH model with fractional generalized hyperbolic innovations exhibiting fat-tail, volatility clustering, and long-range dependence properties is introduced. To define the fractional generalized hyperbolic process, the non-fractional variant is derived by subordinating time-changed Brownian motion to the generalized inverse Gaussian process, and thereafter, the fractional generalized hyperbolic process is obtained using the Volterra kernel. Based on the ARMA–GARCH model with standard normal innovations, the parameters are estimated for the high-frequency returns of six U.S. stocks. Subsequently, the residuals extracted from the estimated ARMA–GARCH parameters are fitted to the fractional and non-fractional generalized hyperbolic processes. The results show that the fractional generalized hyperbolic process performs better in describing the behavior of the residual process of high-frequency returns than the non-fractional processes considered in this study.

A Generalization of Multifractional Brownian Motion

In this article, some properties of multifractional Brownian motion (MFBM) are discussed. It is shown that it has persistence of signs long range dependence (LRD) and persistence of magnitudes LRD properties. A generalization called here nth order multifractional Brownian motion (n-MFBM) that allows to take the functional parameter H(t) values in the range (n−1,n) is discussed. Two representations of the n-MFBM are given and their relationship with each other is obtained.

Generalized Fractional Counting Process

In this paper, we obtain additional results for a fractional counting process introduced and studied by Di Crescenzo et al. [8]. For convenience, we call it the generalized fractional counting process (GFCP). It is shown that the one-dimensional distributions of the GFCP are not infinitely divisible. Its covariance structure is studied, using which its long-range dependence property is established. It is shown that the increments of GFCP exhibit the short-range dependence property. Also, we prove that the GFCP is a scaling limit of some continuous time random walk. A particular case of the GFCP, namely, the generalized counting process (GCP), is discussed for which we obtain a limiting result and a martingale result and establish a recurrence relation for its probability mass function. We have shown that many known counting processes such as the Poisson process of order k, the Pólya-Aeppli process of order k, the negative binomial process and their fractional versions etc., are other special cases of the GFCP. An application of the GCP to risk theory is discussed.

Fractionally Integrated Moving Average Stable Processes With Long-Range Dependence

Long memory processes driven by L\'evy noise with finite second-order moments have been well studied in the literature. They form a very rich class of processes presenting an autocovariance function which decays like a power function. Here, we study a class of L\'evy process whose second-order moments are infinite, the so-called $\alpha$-stable processes. Based on Samorodnitsky and Taqqu (2000), we construct an isometry that allows us to define stochastic integrals concerning the linear fractional stable motion using Riemann-Liouville fractional integrals. With this construction, follows naturally an integration by parts formula. We then present a family of stationary $S\alpha S$ processes with the property of long-range dependence, using a generalized measure to investigate its dependence structure. In the end, the law of large number's result for a time's sample of the process is shown as an application of the isometry and integration by parts formula.

H.264 and H.265 video traffic modeling using neural networks

As video has become the dominant type of traffic over wired and wireless networks, the efficient transmission of video streams is of paramount importance. Hence, especially for wireless networks, the optimum utilization of the available bandwidth while preserving the users’ Quality of Service and Quality of Experience requirements is crucial. Towards this goal, the accurate prediction of upcoming video frame sizes can play a significant role. This work focuses on achieving such an accurate prediction for videos encoded with H.264 and H.265, which are the major state-of-the-art standards based on their current market share. Unlike previous studies, we use single-step and multi-step approaches to capture the long-range dependence and short-range dependence properties of variable bit rate video traces through neural networks-based modeling. We evaluate the accuracy of Long Short Term Memory, Convolutional Neural Networks and Sequence-to-Sequence models and compare them with existing approaches. Our models show significantly higher accuracy for a variety of videos. We also provide a case study on how our model can be used for traffic policing purposes.

Fractional non-homogeneous Poisson and Pólya-Aeppli processes of order k and beyond

We introduce two classes of point processes: a fractional non-homogeneous Poisson process of order k and a fractional non-homogeneous Pólya-Aeppli process of order k. We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property.

Mixed fractional risk process

In this paper, we introduce a compound version of the mixed fractional Poisson process (MFPP). We obtain its mean, variance and the system of fractional differential equations that governs its state probabilities. By using the compound MFPP as claim process, we introduce a risk process, namely, the mixed fractional risk process (MFRP). It is shown that the MFRP is a martingale with respect to a suitable filtration. Its variance and covariance are obtained using which its long-range dependence property is established. Also, we show that its increments exhibit the short-range dependence property. Later, we consider a different fractional risk process based on the compound MFPP. We call it the mixed fractional risk process-II (MFRP-II) and it differs from the MFRP in terms of premium. For exponentially distributed claim sizes the probability density function of the ruin time of MFRP-II is obtained. An asymptotic behaviour of its finite time ruin probability is derived when the claim sizes are subexponentially distributed and the initial capital is arbitrarily large.

Generalized Cauchy Process: Difference Iterative Forecasting Model

The contribution of this article is mainly to develop a new stochastic sequence forecasting model, which is also called the difference iterative forecasting model based on the Generalized Cauchy (GC) process. The GC process is a Long-Range Dependent (LRD) process described by two independent parameters: Hurst parameter H and fractal dimension D. Compared with the fractional Brownian motion (fBm) with a linear relationship between H and D, the GC process can more flexibly describe various LRD processes. Before building the forecasting model, this article demonstrates the GC process using H and D to describe the LRD and fractal properties of stochastic sequences, respectively. The GC process is taken as the diffusion term to establish a differential iterative forecasting model, where the incremental distribution of the GC process is obtained by statistics. The parameters of the forecasting model are estimated by the box dimension, the rescaled range, and the maximum likelihood methods. Finally, a real wind speed data set is used to verify the performance of the GC difference iterative forecasting model.