Streamflow simulation: 2. The broken line process as a potential model for hydrologic simulation

Mej­ía, José; Rodrı́guez-Iturbe, Ignacio; Dawdy, David R.

doi:10.1029/wr008i004p00931

Cited by 67 publications

(31 citation statements)

References 4 publications

(1 reference statement)

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“…Several types of models, such as fractional Gaussian noise (FGN) models (Mandelbrot, 1965;Mandelbrot & Wallis, 1969a,b,c), fast fractional Gaussian noise models (Mandelbrot, 1971), broken line models (Ditlevsen, 1971;Mejia et al, 1972), fractional autoregressive integrated moving-average models (Hosking, 1981(Hosking, , 1984, and symmetric moving average models based on a generalized autocovariance structure (Koutsoyiannis, 2000), have been proposed to reproduce the Hurst phenomenon when generating synthetic time series (see also Bras & Rodriguez-Iturbe, 1985).…”

Section: Open For Discussion Until 1 February 2003mentioning

confidence: 99%

The Hurst phenomenon and fractional Gaussian noise made easy

Koutsoyiannis¹

2002

Hydrological Sciences Journal

248

225

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The Hurst phenomenon, which characterizes hydrological and other geophysical time series, is formulated and studied in an easy manner in terms of the variance and autocorrelation of a stochastic process on multiple temporal scales. In addition, a simple explanation of the Hurst phenomenon based on the fluctuation of a hydrological process upon different temporal scales is presented. The stochastic process that was devised to represent the Hurst phenomenon, i.e. the fractional Gaussian noise, is also studied on the same grounds. Based on its studied properties, three simple and fast methods to generate fractional Gaussian noise, or good approximations of it, are proposed.

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Section: Open For Discussion Until 1 February 2003mentioning

confidence: 99%

The Hurst phenomenon and fractional Gaussian noise made easy

Koutsoyiannis¹

2002

Hydrological Sciences Journal

248

225

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“…However, by choosing parameters carefully, it can be shown that it is possible to replicate the observed Hurst phenomenon over a large range of n. O'Connell [33] was an early exponent of this idea; specifically, he used an ARMA(1, 1) model which could (roughly) preserve a given first-lag auto-correlation as well as h. For completeness, we mention that other modelling approaches were investigated to try and replicate the Hurst phenomenon. One such model was the so-called "broken-line" process detailed by Rodriguez-Iturbe et al [62], Garcia et al [63], and Mejia et al [64,65] which sought to preserve a twice differentiable spectrum. This was criticised by Mandelbrot [66] and did not prosper.…”

Section: Reactions To Mandelbrot's Modelmentioning

confidence: 99%

A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951–1980

Graves

Gramacy

Watkins

et al. 2017

Entropy

107

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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. We will also elucidate the sometimes contrasting approaches to long memory in different scientific communities.Keywords: long-range dependence; Hurst effect; fractionally differenced models; Mandelbrot IntroductionIn many fields, there is strong evidence that a phenomenon called "long memory" plays a significant role, with implications for forecast skill, low frequency variations, and trends. In a stationary time series, the term "long memory"-sometimes "long range dependence" (LRD) or "long term persistence"-implies that there is non-negligible dependence between the present and all points in the past. To dispense quickly with some technicalities, we clarify here that our presentation follows the usual convention in statistics [1,2] and define a stationary finite variance process to have long memory when its two-sided autocorrelation function (ACF) diverges:lim N→∞ ∑ N k=−N ρ(k) → ∞. This is equivalent to its power spectrum having a pole at zero frequency [1,2]. In practice, this means the ACF and the power spectrum both follow a power-law, because the underlying process does not have any characteristic decay timescale. This is in striking contrast to many standard (stationary) stochastic processes where the effect of each data point decays so fast that it rapidly becomes indistinguishable from noise. The study of long memory processes is important because they exhibit nonintuitive properties where many familiar mathematical results fail to hold, and because of the numerous datasets [1,2] where evidence for long memory has been found. In this paper, we will give a historical account of three key aspects of long memory: (1) The environmetric observations in the 1950s which first

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“…Ces modèles à mémoire longue incluent les modèles « bruit Gaussien » (MANOELBROT et WALLIS, 1969a , b, c), de « ligne brisée»» (MEJIA et al, 1972), «niveau variable» (BOES et SALAS, 1978;BALLERINI et BOES, 1985), et « autorégressif et moyennes mobiles sur différen-ces fractionnaires »» (GRANGER, 1980 ;GRANGER et JOYEUX, 1980 ;HOSKING, 1981HOSKING, , 1984HOSKING, , 1985Ll et McLEOD, 1986). Plusieurs de ces modèles à mémoire longue sont aussi présentés dans des ouvrages d'hydrologie stochastique (KOTTEGODA, 1980 ;SALAS et al, 1980 ;BRASS et RODRIGUEZ-ITURBE, 1984 ; HIPEL et MCLEOD, 1990).…”

Section: Introductionunclassified

Développements récents dans la modélisation de la persistance à long terme

Jimenez¹,

Hipel²,

McLeod³

2005

rseau

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Afin de modéliser efficacement la persistance dans les séries chronologiques rencontrées en hydrologie, des développements récents autour du modèle fractionnaire auto-régressif à moyenne mobile (FARMA) (fractional autoregressive-moving average model) sont présentés. On s'intéresse particulièrement ici à de nouvelles procédures permettant d'estimer les paramètres du modèle FARMA d'une manière efficace au point de vue calcul. Pour obtenir les distributions d'échantillons des estimateurs des paramètres à partir de petits échantillons, une technique faisant appel au bootstrap peut être utilisée. Des applications pratiques à des séries de débits en rivière, de précipitations et de températures, montrent l'utilité des modèles FARMA.In order to model effectively persistence in hydrologic tune series, recent developments in fractional autoregressive-moving average (FARMA) models are presented. A time series possesses persistence or long memory if it has an autocorrelation structure that attenuates slowly to zero with increasing lags. Based on the controversy surrounding the Hurst phenomenon, some hydrologists claim that it is important to employ stochastic models which have the ability to model long memory when it is present in a given time series. Fractional Gaussian noise models and approximations thereof were developed within the field of hydrology in order to be able to model long memory. However, a particularly flexible set of models having the capability to describe long memory is the FARMA family of models, which constitutes a direct generalization of autoregressive integrated moving average (ARIMA) models.In particular, like an ARIMA model, a FARMA model contains autoregressive and moving average parameters. Whereas the differencing operator d is restricted to be zero or take on positive integer values in an ARIMA model, the parameter d in a FARMA model can have real values and is estimated along with the other model parameters. For a specified range of values for the d parameter, a FARMA model has long memory. Besides reviewing the background and main theoretical properties of FARMA models, simulation and forecasting techniques are presented. Additionally, procedures for estimating the parameters of a FARMA model are given and a bootstrapping technique is described to obtain the small sample distributions of the estimated parameters.To explain how to apply FARMA models in practice and demonstrate their usefulness, they are fitted to riverflow, precipitation and temperature time series

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Streamflow simulation: 2. The broken line process as a potential model for hydrologic simulation

Abstract: This paper deals with the theory of the broken line process and its applications in the simulation of stochastic sequences.

Cited by 67 publications

References 4 publications

The Hurst phenomenon and fractional Gaussian noise made easy

The Hurst phenomenon and fractional Gaussian noise made easy

A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951–1980

Développements récents dans la modélisation de la persistance à long terme

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