Long-range Dependence Property Research Articles

Convoluted Fractional Poisson Process

In this paper, we introduce and study a convoluted version of the time fractional Poisson process by taking the discrete convolution with respect to space variable in the system of fractional differential equations that governs its state probabilities. We call the introduced process as the convoluted fractional Poisson process (CFPP). The explicit expression for the Laplace transform of its state probabilities are obtained whose inversion yields its one-dimensional distribution. Some of its statistical properties such as probability generating function, moment generating function, moments etc. are obtained. A special case of CFPP, namely, the convoluted Poisson process (CPP) is studied and its time-changed subordination relationships with CFPP are discussed. It is shown that the CPP is a L\'evy process using which the long-range dependence property of CFPP is established. Moreover, we show that the increments of CFPP exhibits short-range dependence property.

Generalized Mittag-Leffler Lévy process and its connections to first passage times of Lévy subordinators

In this article, we introduce generalized Mittag-Leffler Lévy (GMLL) process. GMLL distribution is represented as a general Lévy subordinator delayed by a gamma process. We show various properties of this new process like it’s corresponding Lévy density function and the so-called long-range dependence property. We provide also an explicit representation for the cumulative density function (CDF) for such process which can be an inspiration for construction of various versions of the GMLL processes. Moreover we establish Fokker-Planck type equation for its one dimensional probability density functions (PDF). Our construction will allow us to link GMLL process to the first passage time of the Lévy subordinator.

Network traffic prediction based on INGARCH model

In this paper, we introduce the integer-valued generalized autoregressive conditional heteroscedasticity (INGARCH) as a network traffic prediction model. As the INGARCH is known as a non-linear analytical model that could capture the characteristics of network traffic such as Poisson packet arrival and long-range dependence property, INGARCH seems to be an adequate model for network traffic prediction. Based on the investigation for the traffic arrival process in various network topologies including IoT and VANET, we could confirm that assuming the Poisson process as packet arrival works for some networks and environments of networks. The prediction model is generated by estimating parameters of the INGARCH process and predicting the Poisson parameters of future-steps ahead process using the conditional maximum likelihood estimation method and prediction procedure, respectively. Its performance is compared with those of three different models; autoregressive integrated moving average, GARCH, and long short-term memory recurrent neural network. Anonymized passive traffic traces provided by the Center for Applied Internet Data Analysis are used in the experiment. Numerical results show that the proposed model predicts better than the three models in terms of measurements used in prediction models. Based on the study, we can conclude the followings: INGARCH can capture the characteristics of network traffic better than other statistic models, it is more tractable than neural networks (NNs) overcoming the black-box nature of NNs, and the performances of some statistical models are comparable or even superior to those of NNs, especially when the data is insufficient to apply deep NNs.

On the Long-Range Dependence of Mixed Fractional Poisson Process

In this paper, we show that the mixed fractional Poisson process (MFPP) exhibits the long-range dependence (LRD) property. It is proved by establishing an asymptotic result for the covariance of inverse mixed stable subordinator. Also, it is shown that the increments of the MFPP, namely, the mixed fractional Poissonian noise (MFPN) has the short-range dependence (SRD) property.

Stylized facts of the carbon emission market in China

This paper gives the first empirical analysis on the stylized facts of the returns of five carbon emission market in China. The results mainly show that: (1) there exist heavy tails in returns and weak autocorrelation in both raw and absolute returns; (2) returns show strong volatility clustering, leverage effect, but weak long-range dependent property; (3) returns and trading volume are power-law correlated. All these findings suggest that returns of carbon emission market display the similar stylized facts found in equity market.

Multifractional Brownian Motion and Quantum-Behaved Partial Swarm Optimization for Bearing Degradation Forecasting

Gradual degradation of the bearing vibration signal is usually studied as a nonstationary stochastic time series. Roller bearings are working at high speed in a heavy load environment so that the combination of bearing faults gradually degraded during the rotation might lead to unpredicted catastrophic accidents. The degradation process has the property of long-range dependence (LRD), so that the fractional Brownian motion (fBm) is taken into account for a prediction model. Because of the dramatic changes in the bearing degradation process, the Hurst exponent that describes the fBm will change during the degradation process. A priori Hurst value of the conventional fBm in the prediction is fixed, thus inducing a minor accuracy of the prediction. To avoid this problem, we propose an improved prediction method. Based on the following steps, at the initial data processing, a skip-over factor is selected as the characteristics parameter of the bearing degradation process. A multifractional Brownian motion (mfBm) replaces the fBm for the degradation modeling. We will show that also our mfBm has the same property of long-range dependence as the fBm. Moreover, a time-varying Hurst exponent H(t) is taken to replace the constant H in fBm. Finally, we apply the quantum-behaved partial swarm optimization (QPSO) to optimize H(t) for a finite interval. Some tests and corresponding experimental results will show that our model QPSO + mfBm have a much better performance on the prediction effect than fBm.

Long-range dependence, multi-fractality and volume-return causality of Ether market.

In spite of the increasing popularity of Ethereum, market analysis of the corresponding cryptocurrencies Ether is relatively unexplored until now. This paper is devoted to filling in the research gap of Ether market analysis, the purpose being to provide useful insights on Ether investment. In particular, we first employ the detrended fluctuation analysis and the asymmetric multifractal detrended fluctuation analysis to investigate the properties of long-range dependence, multifractality, and its asymmetry. After that, we study the causality between returns and volume of Ether to find how the activity of investors influences returns based on a nonparametric causality-in-quantiles test. Besides, by making a comparison with the Bitcoin market, we further uncover some unique properties of the Ether market.

Correlation properties of continuous-time autoregressive processes delayed by the inverse of the stable subordinator

We define the delayed Lévy-driven continuous-time autoregressive process via the inverse of the stable subordinator. We derive correlation structure for the observed non-stationary delayed Lévy-driven continuous-time autoregressive processes of order p, emphasizing low orders, and we show they exhibit long-range dependence property. Distributional properties are discussed as well.

Stochastic Duffing equation in modelling of financial time series

In this study, we aim to introduce a new stochastic modelling and forecasting method for the chaotic noisy Duffing map series and then for the financial time series with similar properties. We examine the fractal properties of the noisy Duffing map by correlation dimension, wavelet analysis and MFDFA analysis. Then, it is shown that the non-linear deterministic multifractal Duffing map with additive white noise process exhibits the telegraph process modulated with the fractional Gaussian noise. The combination of these stochastic processes and the fractional Gaussian noise fit the Duffing map series with white noise which has regime switching and long-range dependent properties and then the real-world financial time series with similar properties. Numerical experiments show that the proposed model generates remarkable forecast result in different horizons when it is compared with the actual outcomes. The results put forward that the chaotic Duffing equation can be a fitting model to certain financial time series or the time series with the same characteristics.

Non-homogeneous space-time fractional Poisson processes

The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto (2012), is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson process (SFPP). We study the fractional generalization of the non-homogeneous Poisson process and call it the non-homogeneous space-time fractional Poisson process (NHSTFPP). We compute their pmf and generating function and investigate the associated differential equation. The limit theorems for the NHSTFPP process are studied. We study the distributional properties, the asymptotic expansion of the correlation function of the non-homogeneous time fractional Poisson process (NHTFPP) and subsequently investigate the long-range dependence (LRD) property of a special NHTFPP. We investigate the limit theorem for the fractional non-homogeneous Poisson process (FNHPP) studied by Leonenko et al. (2014). Finally, we present some simulated sample paths of the NHSTFPP process.

Option Pricing in Stochastic Volatility Models Driven by Fractional Jump-Diffusion Processes

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

Degradation Trend Prediction for Rotating Machinery Using Long-Range Dependence and Particle Filter Approach

Timely maintenance and accurate fault prediction of rotating machinery are essential for ensuring system availability, minimizing downtime, and contributing to sustainable production. This paper proposes a novel approach based on long-range dependence (LRD) and particle filter (PF) for degradation trend prediction of rotating machinery, taking the rolling bearing as an example. In this work, the degradation prediction is evaluated based on two health indicators time series; i.e., equivalent vibration severity (EVI) time series and kurtosis time series. Specifically, the degradation trend prediction issues here addressed have the following two distinctive features: (i) EVI time series with weak LRD property and (ii) kurtosis time series with sharp transition points (STPs) in the forecasted region. The core idea is that the parameters distribution of the LRD model can be updated recursively by the particle filter algorithm; i.e., the parameters degradation of the LRD model are restrained, and thus the prognostic results could be generated real-time, wherein the initial LRD model is designed randomly. The prediction results demonstrate that the significant improvements in prediction accuracy are obtained with the proposed method compared to some state-of-the-art approaches such as the autoregressive–moving-average (ARMA) model and the fractional order characteristic (FOC) model, etc.

Remaining Useful Life Prediction for Degradation Processes With Long-Range Dependence

A prerequisite for the existing remaining useful life prediction methods based on stochastic processes is the assumption of independent increments. However, this is in sharp contrast to some practical systems including batteries and blast furnace walls, in which the degradation processes have the property of long-range dependence. Based on the fractional Brownian motion, we adopt a degradation process with long-range dependence to predict the remaining useful life of the above systems. Because the degradation process with long-range dependence is neither a Markovian process nor a semimartingale, the exact analytical first passage time is difficult to derive directly. To address this problem, a weak convergence theorem is first adopted to approximately transform a fractional Brownian motion-based degradation process into a Brownian motion-based one with a time-varying coefficient. Then, with a space-time transformation, the first passage time of the degradation process with long-range dependence can be obtained in a closed form. Unknown parameters in the degradation model can be identified using discrete dyadic wavelet transform and maximum likelihood estimation. Numerical simulations and a practical example of a blast furnace wall are given to verify the effectiveness of the proposed method.

Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse

In this paper, we study the fractional Poisson process (FPP) time-changed by an independent Levy subordinator and the inverse of the Levy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property, and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. The bivariate distributions of the TCFPP-II are derived. Some specific examples for both the processes are discussed. Finally, we present simulations of the sample paths of these processes.

On the long-range dependence properties of annual precipitation using a global network of instrumental measurements

Abstract The long-range dependence (LRD) is considered an inherent property of geophysical processes, whose presence increases uncertainty. Here we examine the spatial behaviour of LRD in precipitation by regressing the Hurst parameter estimate of mean annual precipitation instrumental data which span from 1916–2015 and cover a big area of the earth's surface on location characteristics of the instrumental data stations. Furthermore, we apply the Mann-Kendall test under the LRD assumption (MKt-LRD) to reassess the significance of observed trends. To summarize the results, the LRD is spatially clustered, it seems to depend mostly on the location of the stations, while the predictive value of the regression model is good. Thus when investigating for LRD properties we recommend that the local characteristics should be considered. The application of the MKt-LRD suggests that no significant monotonic trend appears in global precipitation, excluding the climate type D (snow) regions in which positive significant trends appear.

An Evaluation of ARFIMA (Autoregressive Fractional Integral Moving Average) Programs

Strong coupling between values at different times that exhibit properties of long range dependence, non-stationary, spiky signals cannot be processed by the conventional time series analysis. The autoregressive fractional integral moving average (ARFIMA) model, a fractional order signal processing technique, is the generalization of the conventional integer order models—autoregressive integral moving average (ARIMA) and autoregressive moving average (ARMA) model. Therefore, it has much wider applications since it could capture both short-range dependence and long range dependence. For now, several software programs have been developed to deal with ARFIMA processes. However, it is unfortunate to see that using different numerical tools for time series analysis usually gives quite different and sometimes radically different results. Users are often puzzled about which tool is suitable for a specific application. We performed a comprehensive survey and evaluation of available ARFIMA tools in the literature in the hope of benefiting researchers with different academic backgrounds. In this paper, four aspects of ARFIMA programs concerning simulation, fractional order difference filter, estimation and forecast are compared and evaluated, respectively, in various software platforms. Our informative comments can serve as useful selection guidelines.

Time-changed generalized mixed fractional Brownian motion and application to arithmetic average Asian option pricing

In this paper we present a novel model to analyze the behavior of random asset price process under the assumption that the stock price pro-cess is governed by time-changed generalized mixed fractional Brownian motion with an inverse gamma subordinator. This model is con-structed by introducing random time changes into generalized mixed fractional Brownian motion process. In practice it has been observed that many different time series have long-range dependence property and constant time periods. Fractional Brownian motion provides a very general model for long-term dependent and anomalous diffusion regimes. Motivated by this facts in this paper we investigated the long-range dependence structure and trapping events (periods of prices stay motionless) of CSCO stock price return series. The constant time periods phenomena are modeled using an inverse gamma process as a subordinator. Proposed model include the jump behavior of price process because the gamma process is a pure jump Levy process and hence the subordinated process also has jumps so our model can be capture the random variations in volatility. To show the effectiveness of proposed model, we applied the model to calculate the price of an average arithmetic Asian call option that is written on Cisco stock. In this empirical study first the statistical properties of real financial time series is investigated and then the estimated model parameters from an observed data. The results of empirical study which is performed based on the real data indicated that the results of our model are more accuracy than the results based on traditional models.

Resonant behavior of a harmonic oscillator with fluctuating mass driven by a Mittag-Leffler noise

The resonant behavior of a generalized Langevin equation (GLE) in the presence of a Mittag-Leffler noise is studied analytically in this paper. Considering that a GLE with a Mittag-Leffler friction kernel is very useful for modeling anomalous diffusion processes with long-memory and long-range dependence, and the surrounding molecules do not only collide with the Brownian particle but also adhere to the Brownian particle for random time. Thus, we consider the Brownian particle with fluctuating mass, and the fluctuations of the mass are modelled as a dichotomous noise. Applying the stochastic averaging method, we obtain the exact expression of the output amplitude gain of the system. By studying the impact of the driving frequency and the noise parameters, we find the non-monotonic behaviors of the output amplitude gain. The results indicate that the bona fide SR, the wide sense SR and the conventional SR phenomena occur in the proposed harmonic oscillator with fluctuating mass driven by Mittag-Leffler noise. It is found that when we consider the output amplitude gain versus the driving frequency, the phenomena of stochastic multi-resonance (SMR) with two, three and four peaks are observed, and the quadruple-peaks SR phenomenon had never been observed in previous literature. Besides, when we investigate the dependence of output amplitude gain on the memory exponent, the inverse stochastic resonance (ISR) phenomenon takes place, in contrast to the well-known phenomenon of stochastic resonance. Furthermore, we compare the corresponding ordinary harmonic oscillator without memory to our generalized model, and found that the properties of long-memory and long-range dependence endows our generalized model with more abundant dynamic behaviors than the ordinary harmonic oscillator without memory.

Generalized fractional Laplace motion

In this paper, a new stochastic process called generalized fractional Laplace motion (GFLM) is introduced. This process is obtained by superposition of nth-order fractional Brownian motion (n-FBM) as outer process and gamma process as inner process. It is shown that n-FBM process has long-range-dependence (LRD), persistence of signs LRD and persistence of magnitudes LRD properties. Distributional properties of GFLM such as probability density function, moments, covariance structure are established. The fractional partial differential equation governed by the GFLM density is obtained. Finally, it is shown that GFLM has LRD property for all H∈(n−1,n), similar to the n-FBM case.

On the long-range dependence of fractional Poisson and negative binomial processes

Abstract We discuss the short-range dependence (SRD) property of the increments of the fractional Poisson process, called the fractional Poissonian noise. We also establish that the fractional negative binomial process (FNBP) has the long-range dependence (LRD) property, while the increments of the FNBP have the SRD property. Our definitions of the SRD/LRD properties are similar to those for a stationary process and different from those recently used in Biard and Saussereau (2014).