DOCSLIB.ORG

Using the Rescaled Range Anal Ysis for the Stu Dy Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Using the Rescaled Range Anal Ysis for the Stu Dy Of Homage to Ramon Marga/ef; or Why there is such p/e asure in studyin g nature 21 (J. D. Ros & N. Prat, eds.). 1991 . Oec% gia aquatica. 10: 21-33 USING THE RESCALED RANGE ANALYSIS FOR THE STUDY OF HYDROLOGICAL RECORDS: THE RIVER TER AS AN EXAMPLE 1 1 2 JOAN ARMENGOL , SERGI SABATER , ANTONI VIDAL & FRANCESC 1 SABATER Departament d'Ecologia. Facultat de Biologia. Universitat de Barcelona. Av. Diagonal, 4 Barcelona. 1 6 5. 08028 Spain 2 Junta d' Aigües. Abastament d' Aigua del Ter. Cardedeu. 08440 Barcelona. Spain Received: November 1990 SUMMARY Water discharges into the Ter river have a strong influence on the water chemistry and on the dynamics of benthic cornmunities. For this reason, the temporal pattems followed by monthly runoff from 1954 to 1988 have been studied. From a statistical point of view, discharges show a very regular seasonal cycle, but in practice, climatic events introduce a source of variability on runoffs. To determine whether the series of 420 monthly discharges fluctuates with sorne regularity or at random, a rescaled range analilsis was made. The persistence (H) of the series of discharges was measured by the equation R / S = (T / 2) . In fact, H is a measure of the existence of clear trends or periodicities in the records of persistent stocastic processes. An additional measure of the persistence was obtained by means of the relationship D = 2 - H, where D is the local fractal dimensiono For comparative reasons, a random series of the same length, mean and standard deviation was generated and analysed in the same way as the Ter discharges were. The results were H = 0.68 and D = 1 .38 and H = 0.44 and D = 1.52, for the Ter data and for the random series respectively. These results show that the simulated series is a case of ordinary Brownian motion while the Ter discharges series has an intermediate value of persistence. Because of the value of H in the Ter series, there are periods whose values were estimated by the autocorrelation function and the periodogram of the series. The results show the existence of short fluctuations of 3, 6 and 12 months deterrnining the seasonality of the annual cycle, and large fluctuations with periods of 5.5, 8.6, 10.1 and 11.7 months which can be considered an expression of cycles of circa 7 and 1 1 years. KEYWORD S: River discharge, fractal dimension, Hurst exponent, fractional Brownian motion, persistence . In the case of the landscape, rain fa lling at random and moving from each point of the surface along the steepest profi le, through eros ion and convergence of runoffs, generates a secondary "hydrologic" relief Will the fractal quality be preserved (MARGALEF, ? 1988) INTRODUCTION seasonal c1imatic changes. Nevertheless, this pattern is usuall y modified by irregular In Mediterranen ecosystems , annual climatic fluctuations that cause sudden cyc1es are always described in terms of changes in their dynamics. Temporal ponds regular fluctuations associated with and rivers are among the most affected 22 ARMENGOL, SABATER, VI DAL & SABATER ecosystems because they are mainly taking place (SCHINDLER, 1988; controlled by the periodicity and intensity ARMENGOL, 1988). of the rains. The tumover time of the water In rivers, strong rains increase the acts as a key factor in most of the processes intensity of the advective transport of solid 250 A 200 (") 150 S :::c a 100 50 3 2 -2 -3 1955 1960 1965 1970 1975 1980 1985 3 I FIGURE 1. River Ter monthly discharges for the 1954 - 1988 periodo A) original series of data in Hm month- . B) normalized series by Y = . C) the standardized sequence. XO.22 RESCALED RANGE ANALYS IS OF THE TER RIVER 23 and dissolved substances. Because of this, (backward steps) (MARGALEF, 1977). the spates act as huge inputs of external In the Ter river, water flow is the most energy that simplify the structure of the important factor determining the annual river. Most of the organisms are carried variability in the environmental conditions down and the chemical composition of (SABATER & ARMENGOL, 1986) and in water beco mes more similar along the ri ver. the dynamics of phytobenthic (SABATER The annual cyc1e of the cornmunities can et al., 1988) and macroinvertebrate (PUIG be seen as a sequence of populations with et al., 1987) communities. These relation­ short life span, well adapted to seasonal ships are well established for short periods changes but able to reconstruct the of time, such as the annual cyc1e, but for cornmunity after each event (MARGALEF, longer periods it is assumed that c1imatic 1980). If longer periods of time are fluctuations have a similar effect. If we considered, changes in the communities can accept these relationships for the Ter river, be seen as a sequence of periods of the long-term study of a record of organisation (secondary successions) hydrological events as large as possible can followed by quick disruptive episodes provide us with sorne insight into how it A 3 2 3 B 2 o -1 -2 -3 1955 1960 1965 1970 1975 1980 1985 FIGURE 2. Simulated sequence of 420 values (35 years x 12 months) nonnally distributed with the same mean and standard deviation as the data of figure lB . B) the standardized series. 24 ARMENGOL, SABATER, VIDAL & SABATER functions from an ecological point of view. (MANDELB ROT & WALLIS, 1969); for For longer periods it is well established that this reason the existence of temporal the water discharges affect the chemical "cycles" has been studied here by mean s of pollution and the eutrophication levels in autocorrelation and spectral analysis. the reservoirs situated in the middle stretch of the river (VIDAL, 1977, 1987, 1989; ARMENGOL et al., 1986). MATERIAL AND METHODS Long-term records of hydrological events show erratic behaviour at different The record of hydrological data used in time scales (FEDER, 1988). These records this study is composed by the monthly can be studied by means of the rescaled discharges at Roda de Ter station from range analysis (HURST et al., 1965) and 1954 to 1988 (Fig. l). The data were characterized by an exponent H, called the obtained from the Dirección General de Hurst exponent, which is a measure of the Obras Hidráulicas gauging yearbooks persistence or "memory" of one series (D.G.O.H., 1972, 1973, 1974 and 1975; (MANDELBROT & VAN NESS, 1968). D.G.O.H., 1976) for the period 1965- 1977, MANDELBROT (1985) has shown that and were provided by the Servei Hidrologic this record is a self-affine fractal curve with de la Generalitat de Cata]unya for the a dimension D = 2 - H. For a practical periods 1954- 1 965 and 1978-1 988. perspective, the H exponent measures how Roda de Ter is situated in the middle far a temporal series of data departs from a stretch of the Ter river, 88.6 Km from the random series. source (river length 200,3 Km), where it 2 We have used a record of 35 years of has a drainage area of 1,476.8 Km (48. 11 monthly average discharges to analyse the % of the total). At this point the Ter is a persistence in the data sequence. The aim fourth order river. This station was selected of this study is to determine the temporal because it is situated aboye the reservoirs, trend followed by the series and how this thus excluding their regulatory effect, and key factor carried weight on the river at the same time the drainage area is large considered as an ecosystem. It is well enough to avoid the effect of sudden known that wet and dry years appear in discharge due to a local rainfall. clusters, the "Joseph effect" The hydrological characteristics of the 100 B A 75 75 ..c::..., s:: o 8 50 "'.......... 8 :r: 25 25 JF MAMJ JASo ND FIG URE Average annual cycle showing the mean monthly discharges with confidence interval: A) rol' 3. 95 % Ter discharges (Fig. lB) and B) for random series (Fig. 2 A). RESCALED RANGE ANALYSIS OF THE TER RIVER 25 3 Ter are typical of the Mediterranean region, for the period studied was 4l.3 3.74 Hm ±3 but the proximity of the Pyrenees also has with ranges between 23 l.4 Hm and 0.70 3 an influence. The annual cyc1e is Hm , while, if the data are grouped by 3 characterized by two periods of maximum years, a mean of 554.6 35.7 Hm is ± 3 water flow; the first, in May (28.6 6.0 obtained with a maximum of 1180 Hm in 3 1 ± 3 Hm month- , p <0.05 for all the data of 1972 and a rninimum of 258 Hm in the this paper), is the highest due to the effect year 1957. of the spring rains plus the thaw of the The Ter hydrological record does not snow accumulated in the Pyrenees, while follow a Gaussian distribution (the kurtosis the second, in November is less important is 4.9 12 and the skewness is 2.001). For ¡ 3 - (17.6 ± 5.7 Hm month ) and exc1usively this reason, it is not possible to use many of due to the autumn rains. the parametrical statistical methods. In Two minima are situated in between, the order to transform the original data into a 3 I lower in August (1 1.4 2.6 Hm month- new series with a near normality ± d and the other in February (14.2 3.9 Hm distribution, a power transformation has 1 ± month- ). The average monthly water flow been sought by means of the Box & Cox A ..c:: s:= 100 O E ...........
Load more

Recommended publications
  • Rescaled Range Analysis and Detrended Fluctuation Analysis: Finite Sample Properties and Confidence Intervals
    AUCO Czech Economic Review 4 (2010) 236–250 Acta Universitatis Carolinae Oeconomica Rescaled Range Analysis and Detrended Fluctuation Analysis: Finite Sample Properties and Confidence Intervals Ladislav Kristoufekˇ ∗ Received 14 September 2009; Accepted 1 March 2010 Abstract We focus on finite sample properties of two mostly used methods of Hurst exponent H estimation—rescaled range analysis (R/S) and detrended fluctuation analysis (DFA). Even though both methods have been widely applied on different types of financial assets, only seve- ral papers have dealt with the finite sample properties which are crucial as the properties differ significantly from the asymptotic ones. Recently, R/S analysis has been shown to overestimate H when compared to DFA. However, we show that even though the estimates of R/S are truly significantly higher than an asymptotic limit of 0.5, for random time series with lengths from 29 to 217, they remain very close to the estimates proposed by Anis & Lloyd and the estimated standard deviations are lower than the ones of DFA. On the other hand, DFA estimates are very close to 0.5. The results propose that R/S still remains useful and robust method even when compared to newer method of DFA which is usually preferred in recent literature. Keywords Rescaled range analysis, detrended fluctuation analysis, Hurst exponent, long-range dependence, confidence intervals JEL classification G1, G10, G14, G15 ∗ 1. Introduction Long-range dependence and its presence in the financial time series has been discussed in several recent papers (Czarnecki et al. 2008; Grech and Mazur 2004; Carbone et al. 2004; Matos et al.
    [Show full text]
  • Improved Estimates for the Rescaled Range and Hurst
    App ears in Neural Networks in the Capital Markets Pro ceedings of the Third International Conference London Octob er A Refenes Y AbuMostafa J Mo o dy and A Weigend eds World Scientic London IMPROVED ESTIMATES FOR THE RESCALED RANGE AND HURST EXPONENTS John Mo o dy and Lizhong Wu Computer Science Dept Oregon Graduate Institute Portland OR Email mo o dycseogiedu lwucseogiedu ABSTRACT Rescaled Range RS analysis and Hurst Exp onents are widely used as measures of longterm memory structures in sto chastic pro cesses Our empirical studies show however that these statistics can incorrectly indicate departures from random walk b ehavior on short and intermediate time scales when very short term correlations are present A mo dication of rescaled range estimation RS analysis intended to correct bias due to shortterm dep endencies was prop osed by Lo Weshow however that Los RS statistic is itself biased and intro duces other problems including distortion of the Hurst exp onents We prop ose a new statistic RS that corrects for mean bias in the range Rbut do es not suer from the short term biases of RS or Los RS We supp ort our conclusions with exp eriments on simulated random walk and AR pro cesses and exp eriments using high frequency interbank DEM USD exchange rate quotes We conclude that the DEM USD series is mildly trending on time scales of to ticks and that the mean reversion suggested on these time RS analysis is spurious scales by RS or Intro duction and Overview There are three widely used metho ds for longterm dep endence
    [Show full text]
  • Time Series Analysis: Mandelbrot Theory at Work in Economics M. F. Guiducci and M. I. Loflredo Dipartimento Di Matematica, Unive
    Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533 Time Series Analysis: Mandelbrot Theory at work in Economics M. F. Guiducci and M. I. Loflredo Dipartimento di Matematica, Universita di Siena, Siena, Italy Abstract The consequences of the Gaussian hypothesis, which leads to the Efficient Market Hypothesis, are investigated in the framework of time teries tnalysis in economics. The validity of an alternative model, based on Mandelbrot theory, is discussed using the Rescaled Range technique. Hurst exponents related to the underlying fractional Brownian motion are evaluated. 1 Introduction Representing time series for financial markets as non-linear dynamical systems with many degrees of freedom, within the framework of the theory of chaos and fractionals, can be traced back to papers published by Mandelbrot [1,2]. Starting from the discrepancy between the natural consequences of the Efficient Market Hypothesis and the real behavior of financial time series, he took into account a wider choice of underlying probability spaces whose distribution functions could be possible candidates to describe real data. In particular, the description based on fractional Brownian motion is taken as an acceptable alternative. The use of fractional noise can be considered as a generalization of the usual description based on pure Brownian motion and is based mostly on a property of scale invariance of the distributions of returns. This property is typical of fractional objects in nature and is widely used in the study of percolative and aggregation phenomena in complex systems, such as Transactions on Engineering Sciences vol 15, © 1997 WIT Press, www.witpress.com, ISSN 1743-3533 194 Innovation In Mathematics phase transitions, when intrinsic scales of length cease to exist.
    [Show full text]
  • Understanding the Limitations of Estimation Methods for Long-Range Dependence
    Understanding the Limitations of Estimation Methods for Long-Range Dependence Thomas Karagiannis, Mart Molle, Michalis Faloutsos Department of Computer Science & Engineering University of California, Riverside {tkarag,mart,michalis}@cs.ucr.edu Abstract Over the last ten years, long-range dependence (LRD) has become a key concept in modeling networking phenom- ena. The research community has undergone a mental shift from Poisson and memoryless processes to LRD and bursty processes. Despite its popularity, LRD analysis is hindered by two main problems: a) it cannot be used by nonexperts easily, and b) the identification of LRD is often questioned and disputed. The main cause for both these problems is the absence of a systematic and unambiguous way to identify the existence of LRD. This paper has two main thrusts. First, we explore the (lack of) accuracy and robustness in LRD estimation. We find that the current estimation meth- ods can often be inaccurate and unreliable, reporting LRD erroneously. We search for the source of such problems and identify a number of caveats and common mistakes. For example, some of the methods misinterpret short-range correlations for LRD. Second, we develop methods to improve the robustness of the estimation. Through case studies, we demonstrate the effectiveness of our methods in overcoming most known caveats. Finally, we integrate all required functionality and methods in an easy to use software tool. Our work is a first step towards a systematic approach and a comprehensive tool for the reliable estimation of LRD. I. INTRODUCTION The concepts of long-range dependence and self-similarity have revolutionized network modeling.
    [Show full text]
  • Fractional Brownian Motions and Their Multifractal Analysis Applied to Parana River Flow
    Fractional Brownian Motions and their multifractal analysis applied to Parana river flow M. N. Piacquadio Losada, R. Seoane, A. de la Barra, L. F. Caram Facultad de Ingeniería, Universidad de Buenos Aires, Argentina) Abstract: A number of different analysis techniques have been used to analyze long-term time series data from different rivers, starting with the determination of the Hurst coefficient. We summarize the concept of fractals, multifractals and Fractional Brownian Motion (FBM), and apply some such techniques to daily stream flow data from the Paraná River recorded at Corrientes, Argentina, for 106 years. After determining the Hurst coefficient for the entire data set (H = 0.76), we analyze the data for each of four seasons and draw the corresponding FBM graphs and their multifractal spectra (MFS). Three of the seasons are similar, but autumn is very different for both FBM and MFS. Based on the MFS results, we propose a number of indices for measuring variations in stream flow, and determine the values of the indices for the three “similar” seasons. The indices are based on important parameters of the multifractal spectra : , ( , ͚( )) ($) , , and . The geometry of the spectra as well as the indices all (3 ͚($) , ͚(3 (͚(3 ) indicate that Winter is the most stable season. This is in contrast to the Boxplot of seasonal stream flow data where Winter shows the largest variation. Thus, these indices provide insight into river flow stability, not detected in—and indeed contradictory to— that from basic statistical analysis. Introduction There exists an interest in studying the properties of times series that describe hydrological processes such as precipitation and stream flows.
    [Show full text]
  • Hurst, Mandelbrot and the Road to ARFIMA, 1951–1980
    entropy Article A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951–1980 Timothy Graves 1,†, Robert Gramacy 1,‡, Nicholas Watkins 2,3,4,* ID and Christian Franzke 5 1 Statistics Laboratory, University of Cambridge, Cambridge CB3 0WB, UK; [email protected] (T.G.); [email protected] (R.G.) 2 Centre for Fusion, Space and Astrophysics, University of Warwick, Coventry CV4 7AL, UK 3 Centre for the Analysis of Time Series, London School of Economics and Political Sciences, London WC2A 2AE, UK 4 Faculty of Science, Technology, Engineering and Mathematics, Open University, Milton Keynes MK7 6AA, UK 5 Meteorological Institute, Center for Earth System Research and Sustainability, University of Hamburg, 20146 Hamburg, Germany; [email protected] * Correspondence: [email protected]; Tel.: +44-(0)2079-556-015 † Current address: Arup, London W1T 4BQ, UK. ‡ Current address: Department of Statistics, Virginia Polytechnic and State University, Blacksburg, VA 24061, USA. Received: 24 May 2017; Accepted: 18 August 2017; Published: 23 August 2017 Abstract: Long memory plays an important role in many fields by determining the behaviour and predictability of systems; for instance, climate, hydrology, finance, networks and DNA sequencing. In particular, it is important to test if a process is exhibiting long memory since that impacts the accuracy and confidence with which one may predict future events on the basis of a small amount of historical data. A major force in the development and study of long memory was the late Benoit B. Mandelbrot. Here, we discuss the original motivation of the development of long memory and Mandelbrot’s influence on this fascinating field.
    [Show full text]
  • Fractal Analysis of Time Series and Distribution Properties of Hurst Exponent
    Fractal Analysis of Time Series and Distribution Properties of Hurst Exponent Malhar Kale² Ferry Butar Butar³ Abstract Fractal analysis is done by conducting rescaled range (R/S) analysis of time series. The Hurst exponent and the fractal (fractional) dimension of a time series can be estimated with the help of R/S analysis. The Hurst exponent can classify a given time series in terms of whether it is a random, a persistent, or an anti-persistent process. Simulation study is run to study the distribution properties of the Hurst exponent using first-order autoregressive process. If time series data are randomly generated from a normal distribution then the estimated Hurst Exponents are also normally distributed. Fractals: Concepts and Analysis Introduction The term fractal was coined by Benot Mandelbrot in 1975, from the Latin fractus or ªfraction/ brokenº. The concept of fractals is that they have a large degree of self similarity within themselves. They are copies of themselves buried deep within the original. They also reveal infinite detail. Definition 1. A Fractal is ªa geometric shape that is self-similar and has fractional dimensionsº. Peters (1994) stated, ªFractal geometry is the geometry of the Demiurge. Unlike Euclidean geometry, it thrives on roughness and symmetryº. Objects are infinitely complex. The more closely one examines them, the more detail they reveal. A fir tree, for example, is a fractal form. It can be visualized as a cone sitting atop a rectangle. Yet, fir trees are not cones and rectangles. They are a complex network of branches, no branch being the same as any other.
    [Show full text]
  • Gold Price Data Analysis Using Rescaled Range (R/S) Analysis
    International Journal of Economics, Commerce and Research (IJECR) ISSN(P): 2250-0006; ISSN(E): 2319-4472 Vol. 4, Issue 1, Feb 2014, 7-14 © TJPRC Pvt. Ltd. GOLD PRICE DATA ANALYSIS USING RESCALED RANGE (R/S) ANALYSIS SHAIKH YUSUF H1, KAPSAE S. K.2, KHAN A R3 & G. RABBANI4 1Shivaji Arts, Commerce and Science College, Kannad, India 2S.B. Science College, Aurangabad, Maharashtra, India 3,4Maulana Azad College Dr. Rafiq Zakaria Campus, Dr. Rafiq Zakaria Marg, Aurangabad, Maharashtra, India ABSTRACT Gold has been at centre of attraction from various angles, particularly for value and financial importance and served as a reference. There has been a continuous rise in the international gold prices since for ages. Gold is a major part of savings of investors, this has caused concern as to whether any correction in gold prices will have destabilizing implications on the financial markets. With this backdrop, we make an attempt to analyze the gold prices over a period of more than last 32 years to explore the variability in the gold price in India. The gold market has recently attracted a lot of attention and the global price of gold is relatively higher than its historical changing pattern and trend. There has been a drastic increase in international gold prices in the last few years, though there was one large correction in 2008. The gold price data for about 32 years is divided into four sets for the purpose of implementation of Rescaled Range (R/S) Analysis. R/S analysis is an effective technique for analyzing random data in the form of time series in order to bring out trends and any memory effects present in the behavior of the changing data.
    [Show full text]
  • Fractional Brownian Motion and Predictability Index in Financial Market
    Global Journal of Mathematical Sciences: Theory and Practical. ISSN 0974-3200 Volume 5, Number 3 (2013), pp. 197-203 © International Research Publication House http://www.irphouse.com Fractional Brownian Motion and Predictability Index in Financial Market S. J. Bhatt1, H. V. Dedania2 and Vipul R Shah3 1Department of Mathematics, Sardar Patel University, VallabhVidyanagar Email: [email protected] 2Department of Mathematics, Sardar Patel University, VallabhVidyanagar Email: [email protected] 3G H Patel College of Engineering & Technology, VallabhVidyanagar Email: [email protected] Abstract A predictability index for volatility in financial market is proposed. Market volatility of a particular day is suggested to be represented by a 4–vector consisting of opening value, highest value, lowest value and closing value of the volatility index (VIX) of that day. Regarding the stochastic market dynamics following fractional Brownian motion, the predictability is quantified using fractal dimension analysis of the time series for each of these volatility values. The predictability index is calculated for each of India VIX with underlying NIFTY options prices and CBOE with underlying S&P 500 stock index options prices. These are shown to follow fractional Brownian motion. Keywords: Brownian motion, Fractal Brownian motion, volatility, Hurst exponent, Fractal dimension, predictability index. 1. Fractional Brownian motion A discrete Brownian motion (BM) is a real–valued stochastic process → () over discrete time values, = 0,1,2, … such that: (i) () = 0 (ii) for 0 ≤ < ≤ < the increments − () and () − () are independent random variables and (iii) for 0 < < the increments − () are normal random variables with mean zero and variance ~ − . Taking BM as the source of randomness, the celebrated Black–Scholes- Merton theory (1973) develops option pricing formula involving stochastic calculus which is based on integration with respect to the (Brownian motion), the underlying assumption being that the market follows a Brownian path.
    [Show full text]
  • 5.2.13 the Hurst Exponent and Rescaled Range Analysis
    A measure of persistence in daily pound exchange rates Article Accepted Version Brooks, C. (1995) A measure of persistence in daily pound exchange rates. Applied Economics Letters, 2 (11). pp. 428- 431. ISSN 1466-4291 doi: https://doi.org/10.1080/135048595356998 Available at http://centaur.reading.ac.uk/35993/ It is advisable to refer to the publisher’s version if you intend to cite from the work. See Guidance on citing . Published version at: http://dx.doi.org/10.1080/135048595356998 To link to this article DOI: http://dx.doi.org/10.1080/135048595356998 Publisher: Taylor & Francis All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in the End User Agreement . www.reading.ac.uk/centaur CentAUR Central Archive at the University of Reading Reading’s research outputs online This is an Author's Accepted Manuscript of an article published in Applied Economics Letters (1995) [copyright Taylor & Francis], available online at: http://www.tandfonline.com/10.1080/135048595356998 A Measure of Persistence in Daily Pound Exchange Rates by Chris Brooks Department of Economics, Faculty of Letters and Social Sciences, The University of Reading, P.O. Box 218, Whiteknights, Reading, Berks. RG6 6AA Fax Number: (+44) 1734 316 533 Abstract An alternative procedure to that of Lo (1991) is proposed for assessing whether there is significant evidence of persistence in time series. The technique estimates the Hurst exponent itself, and significance testing is based on an application of bootstrapping using surrogate data.
    [Show full text]
  • R/S Analysis in Mobile Social Networks
    Advances in Computer Science Research, volume 44 3rd International Conference on Wireless Communication and Sensor Network (WCSN 2016) R/S Analysis in Mobile Social Networks Wei Zheng School of Software Kang Zhao Nanchang Hangkong University School of Software Nanchang 330063, China Nanchang Hangkong University School of Mathematics and Statistics Nanchang 330063, China Xidian University, Xi’an 710071 E-mail: [email protected] Abstract-In this paper, a R/S analysis method based on method to analyze some hydrometric station near 6 branches clustering coefficient time series is presented to testify the of the Yangtze River [6]. Zhang et al. do a research on the mobile social networks (MSNs) is fractal and organized. It is trend of underground water level of Xing Ping city [7]. found that MSNs are fractal in the scale of clustering Wang et al. find that Chinese Security Markets is nonlinear, coefficient. Then use the Hurst exponent to make some simple and it also has the character of persistence and forecasts if the clustering coefficient of the network is breaking out in the next period. The result of the experiment verifies anti-persistence [8]. Fen et al. try to forecast the possibility that the clustering coefficient time series of complex networks of the typhoon landing in Zhejiang province [9]. Zhao et al. (MSNs) have the characteristics of fractal, and R/S analysis use R/S analysis method to analyze some basic characters of method proves that the cluster coefficient time series is bias climate in Lanzhou city, and they find persistence in random walk, and the results on forecasting the outburst of the urbanization heating [10].
    [Show full text]
  • Simulation of Fractional Brownian Motion
    Simulation of fractional Brownian motion Ton Dieker [email protected] CWI P.O. Box 94079 1090 GB Amsterdam The Netherlands and University of Twente Department of Mathematical Sciences P.O. Box 217 7500 AE Enschede The Netherlands Preface In recent years, there has been great interest in the simulation of long-range dependent processes, in par- ticular fractional Brownian motion. Motivated by applications in communications engineering, I wrote my master's thesis on the subject in 2002. Since many people turned out to be interested in various aspects of fractional Brownian motion, I decided to update my thesis and make it publicly available. Some references are added and the section on spectral simulation is rewritten according to the paper [22]. Fractional Brownian motion is not only of interest for communications engineers. Its properties have been investigated by researchers in theoretical physics, probability, statistics, hydrology, biology, and many others. As a result, the techniques that have been used to study this Gaussian process are quite diverse, and it may take some effort to study them. Undoubtedly, this also makes the field more interesting. This report gives an introduction to generation and estimation of fractional Brownian motion. However, as the literature on the subject is quite extensive (see, for instance, [24]), it has not been my goal to write a complete introduction. Running the risk of satisfying nobody, it is my hope that this report provides some help to find a way through the literature. Since it is written on the level of a master's student, limited background is required. I would like to take this opportunity to thank my thesis advisor, Michel Mandjes, for many discussions and for his help to prepare this manuscript.
    [Show full text]