Generalized Cauchy Process Research Articles

Interpretation of morphological descriptors on nanocellulose from oil palm frond fibers under weak acid, strong acid, and enzymatic treatments

Surface properties of the nanocellulose network from oil palm frond fibers have been modeled in this work by using a fractal model namely the generalized Cauchy process (GCP) to investigate the local fiber distribution and their geometrical effect on the loading capacity. Using the GCP model, the performance of the nanocellulose network under different treatment processes is quantified by interpreting the dual-fractal properties, namely power-law scaling in the local growth (variance) and correlation function, with independent scaling parameters. Our observations indicated a strong correlation between the fractal dimension of nanocellulose and the type and concentration of treatment chemicals, underlining its sensitivity to treatment variations. Additionally, we investigated the organization of cellulose nanocrystals within the morphological structure through simulated three-dimensional network topology, offering insights into pore distribution and aggregation formation. Furthermore, we delved into the influence of crosslinking on loading capacity. Our findings demonstrated a significant reduction in loading test result inconsistencies, emphasizing the enhanced stability and reliability of crosslinked nanocellulose. This research also highlighted the controlled release capabilities of crosslinked nanocellulose, positioning it as a promising candidate for tailored applications in sustained delivery systems.

Generalized Cauchy process based on heavy-tailed distribution and grey relational analysis for reliability predicting of distribution systems.

Failure interruption often causes large blackouts in power grids, severely impacting critical functions. Because of the randomness of power failure, it is difficult to predict the leading causes of failure. ASAI, an essential indicator of power-supply reliability, can be measured from the outage time series. The series is non-stationary stochastic, which causes some difficulty in analyzing power-supply reliability. Considering that the time series has long-range dependence (LRD) and self-similarity, this paper proposes the generalized Cauchy (GC) process for the prediction. The case study shows that the proposed model can predict reliability with a max absolute percentage error of 8.28%. Grey relational analysis (GRA) has proved to be an effective method for the degree of correlation between different indicators. Therefore, we propose the method, which combines both GC and GRA to obtain the correlation coefficients between different factors and ASAI and to get the main factors based on this coefficient. The case study illustrates the feasibility of this approach, which power enterprises can employ to predict power-supply reliability and its influencing factors and help them identify weaknesses in the grid to inform employees to take protective measures in advance.

An iterative model of the generalized Cauchy process for predicting the remaining useful life of lithium-ion batteries

The degradation process of lithium-ion batteries has memory, i.e. it has long-range dependence (LRD). In this paper, an iterative model of the generalized Cauchy (GC) process with LRD characteristics is proposed for the remaining useful life (RUL) prediction of lithium-ion batteries. The GC process uses two independent parameters, fractal dimension and Hurst exponent, to measure the LRD of the degradation process. The diffusion term of the GC iterative model is replaced by the increment of the GC time sequences, constructed via the autocorrelation function (ACF) to describe uncertainty and the LRD characteristics of the lithium-ion batteries capacity degradation. Linear and nonlinear drift terms are used to explain the degradation trend of the lithium-ion batteries capacity. A comparison is made with fractional Brownian motion (FBM) and long-short-term memory (LSTM) network models to show how the GC iterative model has the best performance in RUL prediction of lithium-ion batteries.

Application of Generalized Cauchy Process on Modeling the Long-Range Dependence and Self-Similarity of Sea Surface Chlorophyll Using 23 years of Remote Sensing Data

Understanding the temporal characteristics of sea surface chlorophyll (SSC) is helpful for marine environmental management. This study chose 10 time series of remote daily sea surface chlorophyll products from the European Space Agency during the period from July 29, 1998 to December 31, 2020. A generalized Cauchy model was employed to capture the local and global behaviors of sea surface chlorophyll from a fractal perspective; the fractal dimension D measures the local similarity while the Hurst parameter H measures the global long-range dependence. The generalized Cauchy model was fitted to the empirical autocorrelation function values of each SSC series. The results showed that the sea surface chlorophyll was multi-fractal in both space and time with the D values ranging from 1.0000 to 1.7964 and H values ranging from 0.6757 to 0.8431. Specifically, regarding the local behavior, 9 of the 10 series had low D values (<1.5), representing weak self-similarity; on the other hand, regarding the global behavior, high H values represent strong long-range dependence that may be a general phenomenon of daily sea surface chlorophyll.

Spatiotemporal BME characterization and mapping of sea surface chlorophyll in Chesapeake Bay (USA) using auxiliary sea surface temperature data

Improving the spatiotemporal coverage of remote sensing (RS) products, such as sea surface chlorophyll concentration (SSCC), can offer a better understanding of the spatiotemporal SSCC distribution for ocean management purposes. In the first part of this work, 834 in-situ SSCC measurements of the SeaBASS-NASA (National Aeronautics and Space Administration) during 2002–2016 served as the empirical dataset. A moving window with ±3 days and ±0.5° centered at each of the in-situ SSCC measurements established a search neighborhood for Moderate Resolution Imaging Spectroradiometer Level 2 (MODIS L2) SSCC and MODIS L2 sea surface temperature (SST) data, and the matched SSCC and SST data were used for building a linear SSCC-SST relationship. The unmatched SST was introduced to the linear model for generating soft SSCC data with uniform distributions. The inherent spatiotemporal dependency of the SSCC distribution was then represented by the Bayesian maximum entropy (BME) method, which incorporated the soft SSCC data as auxiliary variable for SSCC estimation and mapping purposes. The results showed that a 75.3% accuracy improvement of remote SSCC retrieval in terms of R2 can be achieved by BME-based method compared to the original MODIS L2 product. Subsequently, the BME-based method was applied to obtain daily SSCC dataset in Chesapeake Bay (USA) during the period 2010–2019. It was found that the SSCC distribution exhibited a decreasing spatial trend from the upper bay to the outer bay, whereas decreasing and increasing temporal trends were detected during the periods 2011–2014 and 2016–2019, respectively. The generalized Cauchy process was used to quantitatively describe the autocorrelation SSCC function in the Chesapeake Bay. The results showed that the outer bay exhibited the strongest long-range dependence among the four sub-regions, whereas the middle bay exhibited the weakest long-range dependence. Finally, one-point and two-point stochastic site indicators (SSIs) were employed to explore the spatiotemporal SSCC characteristics in Chesapeake Bay. The one-point SSI results showed that nearly 100% of the upper, middle and the lower bay areas experienced a high SSCC level (>5 mg/m3) during the entire study period. The area with SSCC >5 mg/m3 in the outer bay increased a lot during the winter season, but the area with SSCC >10 or 20 mg/m3 decreased significantly in the upper, middle and lower bay. Simultaneously, the SSCC dispersion in these areas was rather small during the winter season. On the other hand, the two-point SSI results showed that although the SSCC levels differ among the four sub-regions, but the SSCC connectivity structures between pairs of points also displayed some similarities in terms of their spatiotemporal dependency. In conclusion, the proposed BME-based method was shown to be a promising remote SSCC mapping technique that exhibited a powerful ability to improve both accuracy and coverage of RS products. The SSIs can be also used to explore the spatiotemporal characteristics of a variety of natural attributes in waters.

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.

Long-range dependence and self-similarity of teletraffic with different protocols at the large time scale of day in the duration of 12 years: Autocorrelation modeling

Teletraffic (traffic in short) modeling is crucial in the analysis and design of the infrastructure of cyber-physical network systems from a view of traffic engineering. However, reports regarding traffic modeling at the large time scale of day in the duration of years are rarely seen. This paper addresses our finding in autocorrelation function modeling in the closed form of traffic at the time scale of day (daily traffic for short) in the duration of 12 years with different protocols. We shall show that the autocorrelation function of daily traffic takes the autocorrelation function form of the generalized Cauchy process based on studying the autocorrelation function modeling of daily traffic with real traffic data. Thus, the long-range dependence and local self-similarity of daily traffic are in general uncorrelated according to the theory of the generalized Cauchy process. In addition, we will exhibit that the concrete values of long-range dependence measure and local self-similarity one of daily traffic may relate to protocol types.

Multi-fractional generalized Cauchy process and its application to teletraffic

The contributions given in this paper are in two aspects. The first is to introduce a novel random function, which we call the multi-fractional generalized Cauchy (mGC) process. The second is to dissertate its application to network traffic for studying the multi-fractal behavior of traffic on a point-by-point basis. The introduced mGC process is with the time varying fractal dimension D(t) and the time varying Hurst parameter H(t). The representations of the autocorrelation function (ACF) and the power spectrum density (PSD) of the mGC process are proposed. Besides, the asymptotic expressions of the ACF and PSD of the mGC process are presented. The computation formula of D(t) is given. The mGC model may be a new tool to describe the multi-fractal behavior of traffic. Precisely, it may be used to reveal the local irregularity or local self-similarity (LSS), which is a small-time scale behavior of traffic, and global long-term persistence or long-range dependence (LRD), which is a large-time scale behavior of traffic, on a point-by-point basis. The cast study with real traffic traces exhibits that the variance of D(t) is much greater than that of H(t). Thus, the present mGC model may provide a novel way to explain the fact that traffic has highly local irregularity while its LRD is robust.

Fractional generalized Cauchy process.

This paper presents a fractional generalized Cauchy process (FGCP) with an additive and a multiplicative Gaussian white noise for describing subordinated anomalous fluctuations. The FGCP displays intermittent dynamics on random time durations, whose analytical representation is given by the Ito[over ̂] stochastic integral. The associated probability density function is given by a generalized Cauchy distribution at the stationary state. A fractional Feynman-Kac formula is utilized to show that weak ergodicity breaking of the FGCP depends on the existence of the subordinator and/or the divergence of variance.

Performance modelling for data monitoring services in smart grid: A network calculus-based approach

This paper focuses on solving the modeling issues of monitoring system service performance based on the network calculus theory. First, we formulate the service model of the smart grid monitoring system. Then, we derive the flow arrival curve based on the incremental process related functions. Next, we develop flow arrival curves for the case of the incremental process being a fractional Gaussian process, and then we obtain the generalized Cauchy process. Three technical theorems related to network calculus are presented as our main results. Mathematically, the variance of arrival flow for the continuous time case is derived. Assuming that the incremental process of network flow is a Gaussian stationary process, and given the auto-correlation function of the incremental process with violation probability, the formula of the arrival curve is derived. In addition, the overall flow variance under the discrete time case is explicitly derived. The theoretical results are evaluated in smart grid applications. Simulations indicate that the generalized Cauchy process outperforms the fractional Gaussian process for our considered problem.

Random fractal surface analysis of disordered organic thin films

Study of surface morphologies reveals useful information about interfacial contacts and materials properties such as charge carrier mobility, optical absorption and adhesion that directly affect system performance. The total effective interface area and root mean square (rms) of surface height fluctuations are among the common parameters often used to corroborate device performance upon surface modification. In this paper, random fractal surface analysis is used to characterize geometrical features of microstructure morphologies in disordered organic nickel tetrasulfonated phthalocyanine (NiTsPc) films under different solvent treatment. The different but complementary approaches are used, namely the power spectral density analysis, variogram method based on generalized Cauchy process (GCP) and grayscale fractal box counting approach. It is shown that each of these approaches offers certain perspective about the complex morphologies often found in disordered thin films and thus joint interpretation of these approaches offer better characterization of the surface properties, filling the gap left by others. The morphologies are also interpreted in the context of percolation network supported by the photocurrent density measurements.

Superstatistics and System Identification for a Class of Generalized Cauchy Processes

Superstatistics and System Identification for a Class of Generalized Cauchy Processes

Morphological image interpretation of organic nickel(II) phthalocyanine-tetrasulfonic acid tetrasodium film using fractal analysis

Surface morphologies are often used to characterize various interface processes including charge transfer and surface contact between layers in thin films. In fractal surface analysis, Hurst exponent is used as descriptor of surface roughness and the spectral exponent as long-range correlation between two spatially separated points. In this paper, an alternative surface model known as generalized Cauchy process (GCP) is introduced to provide dual description of fractal and long-range correlation properties simultaneously. The GCP model is used to characterize surface morphologies of organic nickel(II) phthalocyanine-tetrasulfonic acid tetrasodium thin film. The fractal and correlation exponents estimated from the AFM images of the thin film are then used to explain the variation in the electrical conductivity of thin films treated with solvent for different time durations that modified the surface properties. Higher electric conductivity is observed for thin films with higher fractal dimension but also enhanced when there exists spatially correlated morphologies in the form of network enhances charge transport at interfaces.

Long-Memory and Features of Fluctuation in A Fractional Generalized Cauchy Process

Long-Memory and Features of Fluctuation in A Fractional Generalized Cauchy Process

Equivalent continuous and discrete realizations of Lévy flights: A model of one-dimensional motion of an inertial particle

The paper is devoted to the relationship between the continuous Markovian description of Lévy flights developed previously (see, e.g., I.A. Lubashevsky, Truncated Lévy flights and generalized Cauchy processes, Eur. Phys. J. B 82 (2011) 189–195 and references therein) and their equivalent representation in terms of discrete steps of a wandering particle, a certain generalization of continuous time random walks. To simplify understanding the key points of the technique to be created, our consideration is confined to the one-dimensional model for continuous random motion of a particle with inertia. Its dynamics governed by stochastic self-acceleration is described as motion on the phase plane {x,v} comprising the position x and velocity v=dx/dt of the given particle. A notion of random walks inside a certain neighborhood L of the line v=0 (the x-axis) and outside it is developed. It enables us to represent a continuous trajectory of particle motion on the plane {x,v} as a collection of the corresponding discrete steps. Each of these steps matches one complete fragment of the velocity fluctuations originating and terminating at the “boundary” of L. As demonstrated, the characteristic length of particle spatial displacement is mainly determined by velocity fluctuations with large amplitude, which endows the derived random walks along the x-axis with the characteristic properties of Lévy flights. Using the developed classification of random trajectories a certain parameter-free core stochastic process is constructed. Its peculiarity is that all the characteristics of Lévy flights similar to the exponent of the Lévy scaling law are no more than the parameters of the corresponding transformation from the particle velocity v to the related variable of the core process. In this way the previously found validity of the continuous Markovian model for all the regimes of Lévy flights is explained. Based on the obtained results an efficient “single-peak” approximation is constructed. In particular, it enables us to calculate the basic characteristics of Lévy flights using the probabilistic properties of extreme velocity fluctuations and the shape of the most probable trajectory of particle motion within such extreme fluctuations.

Truncated Lévy flights and generalized Cauchy processes

A continuous Markovian model for truncated Levy random walks is proposed. It generalizes the approach developed previously by Lubashevsky et al. Phys. Rev. E 79, 011110 (2009); 80, 031148 (2009), Eur. Phys. J. B 78, 207 (2010) allowing for nonlinear friction in wondering particle motion and saturation of the noise intensity depending on the particle velocity. Both the effects have own reason to be considered and individually give rise to truncated Levy random walks as shown in the paper. The nonlinear Langevin equation governing the particle motion was solved numerically using an order 1.5 strong stochastic Runge-Kutta method and the obtained numerical data were employed to calculate the geometric mean of the particle displacement during a certain time interval and to construct its distribution function. It is demonstrated that the time dependence of the geometric mean comprises three fragments following one another as the time scale increases that can be categorized as the ballistic regime, the Levy type regime (superballistic, quasiballistic, or superdiffusive one), and the standard motion of Brownian particles. For the intermediate Levy type part the distribution of the particle displacement is found to be of the generalized Cauchy form with cutoff. Besides, the properties of the random walks at hand are shown to be determined mainly by a certain ratio of the friction coefficient and the noise intensity rather then their characteristics individually.

A generalized Cauchy process having cubic nonlinearity

A generalized Cauchy process with a cubic nonlinear term (a nonlinear friction) is studied under the influence of independent multiplicative and additive Gaussian-white noises. Three methods of parameter estimation (i.e. the maximum likelihood, the moment and the log-amplitude moment) are presented in detail. The effect of nonlinearity-noise mterplay associated with the nonlinear friction under the influences of both multiplicative and additive noises are discussed in conjunction with fluctuation-dissipation theorem. The physical significance of nonlinear friction is demonstrated with the use of time series data in economics and fluid turbulence.

Transition on the relationship between fractal dimension and Hurst exponent in the long-range connective sandpile models

The relationships between the Hurst exponent H and the power-law scaling exponent B in a new modification of sandpile models, i.e. the long-range connective sandpile (LRCS) models, exhibit a strong dependence upon the system size L. As L decreases, the LRCS model can demonstrate a transition from the negative to positive correlations between H- and B-values. While the negative and null correlations are associated with the fractional Gaussian noise and generalized Cauchy processes, respectively, the regime with the positive correlation between the Hurst and power-law scaling exponents may suggest an unknown, interesting class of the stochastic processes.

Power spectrum of generalized Cauchy process

The generalized Cauchy process was recently applied to modeling long-range dependent traffic. However, the closed form of the power spectrum density of the generalized Cauchy process is a problem that remains unsolved. This paper gives the solution to that problem. The property of the power-law-type power spectrum density for the generalized Cauchy process is pointed out and discussed.

Fractal morphological analysis of Bacteriorhodopsin (bR) layers deposited onto Indium Tin Oxide (ITO) electrodes

Uniform Bacteriorhodopsin layers for the purpose of fabricating Bacteriorhodopsin-based biosensors were prepared by allowing drying of the layers under a constant electric field. To properly observe and understand the “electric field effect” on the protein Bacteriorhodopsin, the electric and non-electric field influenced Bacteriorhodopsin layers prepared using a manual syringe-deposition method applied onto Indium Tin Oxide electrodes were structurally investigated using Scanning Electron Microscopy and Atomic Force Microscopy. The results yield obvious morphological differences between the electric and non-electric field assisted Bacteriorhodopsin layers and brings to attention the occurrence of the so-called “coffee-ring” effect in the latter case. We applied stochastic fractal method based on the generalized Cauchy process to describe the morphological features surrounding the void. Fractal dimension is used to characterize the local regularity of the Bacteriorhodopsin clusters and the correlation exponent is used to describe the long-range correlation between the clusters. It is found that the Bacteriorhodopsin protein tends to exhibit with strong spatial correlation in the presence of external electric field compared to in absence of the electric field. Long-range correlation in the morphological feature may be associated to the enhancement of aggregation process of Bacteriorhodopsin protein in the presence of electric field, thereby inhibiting the formation of the so-called “coffee-ring” effect. As such, the observations discussed in this work suggest some amount of control of surface uniformity when forming layers.