Search results for: uncertain time series

Authors: Nabilah Filzah Mohd Radzuan, Zalinda Othman, Azuraliza Abu Bakar, Abdul Razak Hamdan

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Uncertain data is believed to be an important issue in building up a prediction model. The main objective in the time series uncertainty analysis is to formulate uncertain data in order to gain knowledge and fit low dimensional model prior to a prediction task. This paper discusses the performance of a number of techniques in dealing with uncertain data specifically those which solve uncertain data condition by minimizing the loss of compression properties.

Keywords: compression properties, uncertainty, uncertain time series, mining technique, weather prediction

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Authors: Zhongmin Wang, Lin Wei, Hengshan Zhang, Tianhua Chen, Yimin Zhou

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The assembly line balancing problems are signficant in mass production systems. In order to deal with the uncertainties that practically exist but barely mentioned in the literature, this paper develops a mathematic model with an optimisation algorithm to solve the assembly line balancing problem with uncertainty operation time. The developed model is able to work with a variable number of workstations under the uncertain environment, aiming to obtain the minimal number of workstation and minimal idle time for each workstation. In particular, the proposed approach first introduces the concept of protection time that closely works with the uncertain operation time. Four dominance rules and the mechanism of determining up and low bounds are subsequently put forward, which serve as the basis for the proposed branch and bound algorithm. Experimental results show that the proposed work verified on a benchmark data set is able to solve the uncertainties efficiently.

Keywords: assembly lines, SALBP-UOT, uncertain operation time, branch and bound algorithm.

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Authors: Tak-Chung Fu, Ying-Kit Hung, Fu-Lai Chung

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In the field of time series data mining, the concept of the Perceptually Important Point (PIP) identification process is first introduced in 2001. This process originally works for financial time series pattern matching and it is then found suitable for time series dimensionality reduction and representation. Its strength is on preserving the overall shape of the time series by identifying the salient points in it. With the rise of Big Data, time series data contributes a major proportion, especially on the data which generates by sensors in the Internet of Things (IoT) environment. According to the nature of PIP identification and the successful cases, it is worth to further explore the opportunity to apply PIP in time series ‘Big Data’. However, the performance of PIP identification is always considered as the limitation when dealing with ‘Big’ time series data. In this paper, two distributed versions of PIP identification based on the Specialized Binary (SB) Tree are proposed. The proposed approaches solve the bottleneck when running the PIP identification process in a standalone computer. Improvement in term of speed is obtained by the distributed versions.

Keywords: distributed computing, performance analysis, Perceptually Important Point identification, time series data mining

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Authors: Sandeep Kaur, Jenny Vuong, Marcel Julliard, Sean O'Donoghue

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Time-series data are useful for modelling as they can enable model-evaluation. However, when reconstructing models from phosphoproteomic data, often non-exact methods are utilised, as the knowledge regarding the network structure, such as, which kinases and phosphatases lead to the observed phosphorylation state, is incomplete. Thus, such reactions are often hypothesised, which gives rise to uncertainty. Here, we propose a framework, implemented via a web-based tool (as an extension to Minardo), which given time-series phosphoproteomic datasets, can generate κ models. The incompleteness and uncertainty in the generated model and reactions are clearly presented to the user via the visual method. Furthermore, we demonstrate, via a toy EGF signalling model, the use of algorithmic verification to verify κ models. Manually formulated requirements were evaluated with regards to the model, leading to the highlighting of the nodes causing unsatisfiability (i.e. error causing nodes). We aim to integrate such methods into our web-based tool and demonstrate how the identified erroneous nodes can be presented to the user via the visual method. Thus, in this research we present a framework, to enable a user to explore phosphorylation proteomic time-series data in the context of models. The observer can visualise which reactions in the model are highly uncertain, and which nodes cause incorrect simulation outputs. A tool such as this enables an end-user to determine the empirical analysis to perform, to reduce uncertainty in the presented model - thus enabling a better understanding of the underlying system.

Keywords: κ-models, model verification, time-series phosphoproteomic datasets, uncertainty and error visualisation

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Authors: Ognjen Vukovic

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Chern-Simons equation represents the cornerstone of quantum physics. The question that is often asked is if the aforementioned equation can be successfully applied to the interaction in international financial markets. By analysing the time series in financial theory, it is proved that Chern-Simons equation can be successfully applied to financial time-series. The aforementioned statement is based on one important premise and that is that the financial time series follow the fractional Brownian motion. All variants of Chern-Simons equation and theory are applied and analysed. Financial theory time series movement is, firstly, topologically analysed. The main idea is that exchange rate represents two-dimensional projections of three-dimensional Brownian motion movement. Main principles of knot theory and topology are applied to financial time series and setting is created so the Chern-Simons equation can be applied. As Chern-Simons equation is based on small particles, it is multiplied by the magnifying factor to mimic the real world movement. Afterwards, the following equation is optimised using Solver. The equation is applied to n financial time series in order to see if it can capture the interaction between financial time series and consequently explain it. The aforementioned equation represents a novel approach to financial time series analysis and hopefully it will direct further research.

Keywords: Brownian motion, Chern-Simons theory, financial time series, econophysics

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Authors: Farhad Asadi, Mohammad Javad Mollakazemi

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In this paper, Bayesian online inference in models of data series are constructed by change-points algorithm, which separated the observed time series into independent series and study the change and variation of the regime of the data with related statistical characteristics. variation of statistical characteristics of time series data often represent separated phenomena in the some dynamical system, like a change in state of brain dynamical reflected in EEG signal data measurement or a change in important regime of data in many dynamical system. In this paper, prediction algorithm for studying change point location in some time series data is simulated. It is verified that pattern of proposed distribution of data has important factor on simpler and smother fluctuation of hazard rate parameter and also for better identification of change point locations. Finally, the conditions of how the time series distribution effect on factors in this approach are explained and validated with different time series databases for some dynamical system.

Keywords: time series, fluctuation in statistical characteristics, optimal learning, change-point algorithm

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Authors: Behnam Tavakkol

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Uncertain data mining algorithms use different ways to consider uncertainty in data such as by representing a data object as a sample of points or a probability distribution. Fuzzy methods have long been used for clustering traditional (certain) data objects. They are used to produce non-crisp cluster labels. For uncertain data, however, besides some uncertain fuzzy k-medoids algorithms, not many other fuzzy clustering methods have been developed. In this work, we develop a fuzzy kernel k-medoids algorithm for clustering uncertain data objects. The developed fuzzy kernel k-medoids algorithm is superior to existing fuzzy k-medoids algorithms in clustering data sets with non-linearly separable clusters.

Keywords: clustering algorithm, fuzzy methods, kernel k-medoids, uncertain data

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Authors: M. J. Shofa, A. Hidayatno, O. M. Armand

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MRP as a production control system is appropriate for the deterministic environment. Unfortunately, most production systems such as customer demands are stochastic. Demand-Driven MRP (DDMRP) is a new approach for inventory control system, and it deals with demand uncertainty. The objective of this paper is to compare the MRP and DDMRP work for a long lead time and uncertain demand in terms of on-hand inventory levels. The evaluation is conducted through a discrete event simulation using purchased part data from an automotive company. The result is MRP gives 50,759 pcs / day while DDMRP gives 34,835 pcs / day (reduce 32%), it means DDMRP is more effective inventory control than MRP in terms of on-hand inventory levels.

Keywords: Demand-Driven MRP, long lead time, MRP, uncertain demand

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Authors: C. Slim

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Traditional techniques for analyzing time series are based on the notion of stationarity of phenomena under study, but in reality most economic and financial series do not verify this hypothesis, which implies the implementation of specific tools for the detection of such behavior. In this paper, we study nonstationary non-seasonal time series tests in a non-exhaustive manner. We formalize the problem of nonstationary processes with numerical simulations and take stock of their statistical characteristics. The theoretical aspects of some of the most common unit root tests will be discussed. We detail the specification of the tests, showing the advantages and disadvantages of each. The empirical study focuses on the application of these tests to the exchange rate (USD/TND) and the Consumer Price Index (CPI) in Tunisia, in order to compare the Power of these tests with the characteristics of the series.

Keywords: stationarity, unit root tests, economic time series, ADF tests

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Authors: Vineetha Bettaiah, Heggere S. Ranganath

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This paper presents a Hierarchical Piecewise Linear Approximation (HPLA) for the representation of time series data in which the time series is treated as a curve in the time-amplitude image space. The curve is partitioned into segments by choosing perceptually important points as break points. Each segment between adjacent break points is recursively partitioned into two segments at the best point or midpoint until the error between the approximating line and the original curve becomes less than a pre-specified threshold. The HPLA representation achieves dimensionality reduction while preserving prominent local features and general shape of time series. The representation permits course-fine processing at different levels of details, allows flexible definition of similarity based on mathematical measures or general time series shape, and supports time series data mining operations including query by content, clustering and classification based on whole or subsequence similarity.

Keywords: data mining, dimensionality reduction, piecewise linear representation, time series representation

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Authors: Ioannis Andrianakis, Vasileios Gkatas, Nikos Eleftheriadis, Alexios Ellinidis, Ermioni Avramidou

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pscmsForecasting is an open-source web service that implements a variety of time series forecasting algorithms and exposes them to the user via the ubiquitous HTTP protocol. It allows developers to enhance their applications by adding time series forecasting functionalities through an intuitive and easy-to-use interface. This paper provides some background on time series forecasting and gives details about the implemented algorithms, aiming to enhance the end user’s understanding of the underlying methods before incorporating them into their applications. A detailed description of the web service’s interface and its various parameterizations is also provided. Being an open-source project, pcsmsForecasting can also be easily modified and tailored to the specific needs of each application.

Keywords: time series, forecasting, web service, open source

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Authors: Dong Sang Yoo

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We consider nonlinear uncertain systems such that a priori information of the uncertainties is not available. For such systems, we assume that the upper bound of the uncertainties is represented as a Fredholm integral equation of the first kind and we propose an adaptation law that is capable of estimating the upper bound and design a continuous robust control which renders nonlinear uncertain systems ultimately bounded.

Keywords: adaptive control, estimation, Fredholm integral, uncertain system

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Authors: Muhammad Tariq, Hammad Tahir, Fawwad Mahmood Butt

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Purpose: This study attempts to examine the best forecasting methodologies in the time series. The time series forecasting models such as VAR, ARCH and the ARMA are considered for the analysis. Methodology: The Bench Marks or the parameters such as Adjusted R square, F-stats, Durban Watson, and Direction of the roots have been critically and empirically analyzed. The empirical analysis consists of time series data of Consumer Price Index and Closing Stock Price. Findings: The results show that the VAR model performed better in comparison to other models. Both the reliability and significance of VAR model is highly appreciable. In contrary to it, the ARCH model showed very poor results for forecasting. However, the results of ARMA model appeared double standards i.e. the AR roots showed that model is stationary and that of MA roots showed that the model is invertible. Therefore, the forecasting would remain doubtful if it made on the bases of ARMA model. It has been concluded that VAR model provides best forecasting results. Practical Implications: This paper provides empirical evidences for the application of time series forecasting model. This paper therefore provides the base for the application of best time series forecasting model.

Keywords: forecasting, time series, auto regression, ARCH, ARMA

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Authors: O. Delage, H. Bencherif, T. Portafaix, A. Bourdier

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The main purpose of time series analysis is to learn about the dynamics behind some time ordered measurement data. Two approaches are used in the literature to get a better knowledge of the dynamics contained in observation data sequences. The first of these approaches concerns time series decomposition, which is an important analysis step allowing patterns and behaviors to be extracted as components providing insight into the mechanisms producing the time series. As in many cases, time series are short, noisy, and non-stationary. To provide components which are physically meaningful, methods such as Empirical Mode Decomposition (EMD), Empirical Wavelet Transform (EWT) or, more recently, Empirical Adaptive Wavelet Decomposition (EAWD) have been proposed. The second approach is to reconstruct the dynamics underlying the time series as a trajectory in state space by mapping a time series into a set of Rᵐ lag vectors by using the method of delays (MOD). Takens has proved that the trajectory obtained with the MOD technic is equivalent to the trajectory representing the dynamics behind the original time series. This work introduces the singular spectrum decomposition (SSD), which is a new adaptive method for decomposing non-linear and non-stationary time series in narrow-banded components. This method takes its origin from singular spectrum analysis (SSA), a nonparametric spectral estimation method used for the analysis and prediction of time series. As the first step of SSD is to constitute a trajectory matrix by embedding a one-dimensional time series into a set of lagged vectors, SSD can also be seen as a reconstruction method like MOD. We will first give a brief overview of the existing decomposition methods (EMD-EWT-EAWD). The SSD method will then be described in detail and applied to experimental time series of observations resulting from total columns of ozone measurements. The results obtained will be compared with those provided by the previously mentioned decomposition methods. We will also compare the reconstruction qualities of the observed dynamics obtained from the SSD and MOD methods.

Keywords: time series analysis, adaptive time series decomposition, wavelet, phase space reconstruction, singular spectrum analysis

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Authors: Mohammad H. Fattahi

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Chaotic analysis has been performed on the river flow time series before and after applying the wavelet based de-noising techniques in order to investigate the noise content effects on chaotic nature of flow series. In this study, 38 years of monthly runoff data of three gauging stations were used. Gauging stations were located in Ghar-e-Aghaj river basin, Fars province, Iran. The noise level of time series was estimated with the aid of Gaussian kernel algorithm. This step was found to be crucial in preventing removal of the vital data such as memory, correlation and trend from the time series in addition to the noise during de-noising process.

Keywords: chaotic behavior, wavelet, noise reduction, river flow

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Authors: Changhong Guo, Shaomei Fang, Yong He

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In this paper, fractional Black-Scholes models for the European option pricing were established based on the fractional G-Brownian motion (fGBm), which generalizes the concepts of the classical Brownian motion, fractional Brownian motion and the G-Brownian motion, and that can be used to be a tool for considering the long range dependence and uncertain volatility for the financial markets simultaneously. A generalized fractional Black-Scholes equation (FBSE) was derived by using the Taylor’s series of fractional order and the theory of absence of arbitrage. Finally, some explicit option pricing formulas for the European call option and put option under the FBSE were also solved, which extended the classical option pricing formulas given by F. Black and M. Scholes.

Keywords: European option pricing, fractional Black-Scholes equations, fractional g-Brownian motion, Taylor's series of fractional order, uncertain volatility

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Authors: Lars Laußat, Manfred Helmus, Kamil Szczesny, Markus König

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As one result of the project “Reactive Construction Project Scheduling using Real Time Construction Logistic Data and Simulation”, a procedure for using data about uncertain resource availability assumptions in reactive scheduling processes has been developed. Prediction data about resource availability is generated in a formalized way using real-time monitoring data e.g. from auto-ID systems on the construction site and in the supply chains. The paper focuses on the formalization of the procedure for monitoring construction logistic processes, for the detection of disturbance and for generating of new and uncertain scheduling assumptions for the reactive resource constrained simulation procedure that is and will be further described in other papers.

Keywords: auto-ID, construction logistic, fuzzy, monitoring, RFID, scheduling

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Authors: Manlika Ratchagit

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This paper is concerned with robust mean square stability of uncertain stochastic switched discrete time-delay systems. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the robust mean square stability for the uncertain stochastic discrete time-delay system is designed via linear matrix inequalities. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes.

Keywords: robust mean square stability, discrete-time stochastic systems, hybrid systems, interval time-varying delays, Lyapunov functional, linear matrix inequalities

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Authors: Manlika Rajchakit

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This paper is concerned with robust mean square stability of uncertain stochastic switched discrete time-delay systems. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the robust mean square stability for the uncertain stochastic discrete time-delay system is designed via linear matrix inequalities. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes.

Keywords: robust mean square stability, discrete-time stochastic systems, hybrid systems, interval time-varying delays, lyapunov functional, linear matrix inequalities

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Authors: Valeria Bondarenko

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We proposed algorithms for: estimation of parameters fBm (volatility and Hurst exponent) and for the approximation of random time series by functional of fBm. We proved the consistency of the estimators, which constitute the above algorithms, and proved the optimal forecast of approximated time series. The adequacy of estimation algorithms, approximation, and forecasting is proved by numerical experiment. During the process of creating software, the system has been created, which is displayed by the hierarchical structure. The comparative analysis of proposed algorithms with the other methods gives evidence of the advantage of approximation method. The results can be used to develop methods for the analysis and modeling of time series describing the economic, physical, biological and other processes.

Keywords: mathematical model, random process, Wiener process, fractional Brownian motion

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Authors: Plessen Mogens Graf

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Within the context of multivariate time series segmentation, this paper proposes a method inspired by a posteriori optimal trading. After a normalization step, time series are treated channelwise as surrogate stock prices that can be traded optimally a posteriori in a virtual portfolio holding either stock or cash. Linear transaction costs are interpreted as hyperparameters for noise filtering. Trading signals, as well as trading signals obtained on the reversed time series, are used for unsupervised channelwise labeling before a consensus over all channels is reached that determines the final segmentation time instants. The method is model-free such that no model prescriptions for segments are made. Benefits of proposed approach include simplicity, computational efficiency, and adaptability to a wide range of different shapes of time series. Performance is demonstrated on synthetic and real-world data, including a large-scale dataset comprising a multivariate time series of dimension 1000 and length 2709. Proposed method is compared to a popular model-based bottom-up approach fitting piecewise affine models and to a recent model-based top-down approach fitting Gaussian models and found to be consistently faster while producing more intuitive results in the sense of segmenting time series at peaks and valleys.

Keywords: time series segmentation, model-free, trading-inspired, multivariate data

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Authors: Wajahat Ali, Shakeel Javaid

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In this study, we have developed a mathematical programming model for a solid transportation problem with three objective functions arranged in hierarchical order. The mathematical programming models with more than one objective function to be solved in hierarchical order is termed as a multi-level programming model. Our study explores a Multi-Level Solid Transportation Problem with Uncertain Parameters (MLSTPWU). The proposed MLSTPWU model consists of three objective functions, viz. minimization of transportation cost, minimization of total transportation time, and minimization of deterioration during transportation. These three objective functions are supposed to be solved by decision-makers at three consecutive levels. Three constraint functions are added to the model, restricting the total availability, total demand, and capacity of modes of transportation. All the parameters involved in the model are assumed to be uncertain in nature. A solution method based on fuzzy logic is also discussed to obtain the compromise solution for the proposed model. Further, a simulated numerical example is discussed to establish the efficiency and applicability of the proposed model.

Keywords: solid transportation problem, multi-level programming, uncertain variable, uncertain environment

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Authors: Sukhveer Singh, Sandeep Singh

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A bi-objective fuzzy transportation problem with the objectives to minimize the total fuzzy cost and fuzzy time of transportation without according priorities to them is considered. To the best of our knowledge, there is no method in the literature to find efficient solutions of the bi-objective transportation problem under uncertainty. In this paper, a bi-objective transportation problem in an uncertain environment has been formulated. An algorithm has been proposed to find efficient solutions of the bi-objective transportation problem under uncertainty. The proposed algorithm avoids the degeneracy and gives the optimal solution faster than other existing algorithms for the given uncertain transportation problem.

Keywords: uncertain transportation problem, efficient solution, ranking function, fuzzy transportation problem

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Authors: Anita Setianingrum, Oki S. Jaya, Zuherman Rustam

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Forecasting stock price is a challenging task due to the complex time series of the data. The complexity arises from many variables that affect the stock market. Many time series models have been proposed before, but those previous models still have some problems: 1) put the subjectivity of choosing the technical indicators, and 2) rely upon some assumptions about the variables, so it is limited to be applied to all datasets. Therefore, this paper studied a novel Adaptive Neuro-Fuzzy Inference System (ANFIS) time series model based on Support Vector Regression (SVR) for forecasting the stock market. In order to evaluate the performance of proposed models, stock market transaction data of TAIEX and HIS from January to December 2015 is collected as experimental datasets. As a result, the method has outperformed its counterparts in terms of accuracy.

Keywords: ANFIS, fuzzy time series, stock forecasting, SVR

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Authors: Morteza Aien, Masoud Rashidinejad, Mahmud Fotuhi-Firuzabad

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As energetic and environmental issues are getting more and more attention all around the world, the penetration of distributed energy resources (DERs) mainly those harvesting renewable energies (REs) ascends with an unprecedented rate. This matter causes more uncertainties to appear in the power system context; ergo, the uncertainty analysis of the system performance is an obligation. The uncertainties of any system can be represented probabilistically or possibilistically. Since sufficient historical data about all the system variables is not available, therefore, they do not have a probability density function (PDF) and must be represented possibilistiacally. When some of system uncertain variables are probabilistic and some are possibilistic, neither the conventional pure probabilistic nor pure possibilistic methods can be implemented. Hence, a combined solution is appealed. The first of this two-paper series formulates a new possibilistic-probabilistic tool for the load flow uncertainty assessment. The proposed methodology is based on the evidence theory and joint propagation of possibilistic and probabilistic uncertainties. This possibilistic- probabilistic formulation is solved in the second companion paper in an uncertain load flow (ULF) study problem.

Keywords: probabilistic uncertainty modeling, possibilistic uncertainty modeling, uncertain load flow, wind turbine generator

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Authors: Rama Debbarma

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The present study deals with the performance of Linear base isolation system to mitigate seismic response of structures characterized by random system parameters. This involves optimization of the tuning ratio and damping properties of the base isolation system considering uncertain system parameters. However, the efficiency of base isolator may reduce if it is not tuned to the vibrating mode it is designed to suppress due to unavoidable presence of system parameters uncertainty. With the aid of matrix perturbation theory and first order Taylor series expansion, the total probability concept is used to evaluate the unconditional response of the primary structures considering random system parameters. For this, the conditional second order information of the response quantities are obtained in random vibration framework using state space formulation. Subsequently, the maximum unconditional root mean square displacement of the primary structures is used as the objective function to obtain optimum damping parameters Numerical study is performed to elucidate the effect of parameters uncertainties on the optimization of parameters of linear base isolator and system performance.

Keywords: linear base isolator, earthquake, optimization, uncertain parameters

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Authors: Sumit Sen, Sohan Khedekar, Umang Shinde, Shivam Bhargava

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This study has been undertaken to investigate whether the deep learning models are able to predict the future stock prices by training the model with the historical stock price data. Since this work required time series analysis, various models are present today to perform time series analysis such as Recurrent Neural Network LSTM, ARIMA and Facebook Prophet. Applying these models the movement of stock price of stocks are predicted and also tried to provide the future prediction of the stock price of a stock. Final product will be a stock price prediction web application that is developed for providing the user the ease of analysis of the stocks and will also provide the predicted stock price for the next seven days.

Keywords: Autoregressive Integrated Moving Average, Deep Learning, Long Short Term Memory, Time-series

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Authors: Tushnik Sarkar, Mofazzal H. Khondekar, Subrata Banerjee

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This paper investigates the nature of the fluctuation of the daily average Solar wind speed time series collected over a period of 2492 days, from 1^st January, 1997 to 28^th October, 2003. The degree of self-similarity and scalability of the Solar Wind Speed signal has been explored to characterise the signal fluctuation. Multi-fractal Detrended Fluctuation Analysis (MFDFA) method has been implemented on the signal which is under investigation to perform this task. Furthermore, the singularity spectra of the signals have been also obtained to gauge the extent of the multifractality of the time series signal.

Keywords: detrended fluctuation analysis, generalized hurst exponent, holder exponents, multifractal exponent, multifractal spectrum, singularity spectrum, time series analysis

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Authors: Maria C. Mariani, Md Al Masum Bhuiyan, Osei K. Tweneboah, Hector G. Huizar

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This work is devoted to the study of modeling geophysical time series. A stochastic technique with time-varying parameters is used to forecast the volatility of data arising in geophysics. In this study, the volatility is defined as a logarithmic first-order autoregressive process. We observe that the inclusion of log-volatility into the time-varying parameter estimation significantly improves forecasting which is facilitated via maximum likelihood estimation. This allows us to conclude that the estimation algorithm for the corresponding one-step-ahead suggested volatility (with ±2 standard prediction errors) is very feasible since it possesses good convergence properties.

Keywords: Augmented Dickey Fuller Test, geophysical time series, maximum likelihood estimation, stochastic volatility model

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Authors: Valeria Bondarenko

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In this paper, we propose two problems related to fractal Brownian motion. First problem is simultaneous estimation of two parameters, Hurst exponent and the volatility, that describe this random process. Numerical tests for the simulated fBm provided an efficient method. Second problem is approximation of the increments of the observed time series by a power function by increments from the fractional Brownian motion. Approximation and estimation are shown on the example of real data, daily deposit interest rates.

Keywords: fractional Brownian motion, Gausssian processes, approximation, time series, estimation of properties of the model

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