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Part V Long Memory Modelling
Long Memory Modelling
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PART V LONG MEMORY MODELLING The Hurst phenomenon cnated one of the m s t interesting, controversial and long-lasting scientific debates ever to arise in the field of hydrology. The genesis of the Hurst phenomenon took place over forty years ago in Egypt Just after World War II. a British scientist by the name of H a d d Edwin Hurst became deeply involved in studying how the Nile River could be optimally controlled and utilized for the benefit of both Egypt and Sudan. As D k t o r G e n e r a l of the Physical Department in the Ministry of Public Works in Cairo, Egypt,Hurst was particularly interested in the long-term storage requirements of the Nile River. In addition to annual riverflow series, Hurst analyzed a wide variety of other yearly geophysical time series in order to examine the statistical properties of some specific statistics that are closely related to long term storage. These statistical studies led Hurst to develop an empirical law upon which the definition of the Hurst phenomenon is b u d . The fact that the Hurst phenomenon arose from scientific work carried out in Egypt provided the controversy with an aura of mystery and intrigue. Was the Hurst phenomenon more difficult to solve than the riddle of the Sphinx? Indeed, a range of explanations has been put forward for solving the Hurst phenomenon. Furthermore, in the process of studying the Hurst phenomenon, many original contributions have been made to the fields of hydrology and statistics. In Chapter 10, the Hurst phenomenon is defined and both theoretical and empirical work related to this phenomenon are described. One spinoff From research connected to Hurst’s work is the development of a stochastic model called fractional Gaussian noise (FGN). This model possesses long memory (see Section 2.5.3) and was designed for furnishing an explanation to the Hurst phenomenon. As demonstrated in Chapter 10, this long memory model fails to solve the Hunt riddle. Nevertheless, the introduction of FGN into the field of hydrology initiated major theoretical and practical developments in long memory modelling by not only hydrologists but also by statisticians and economists. Probably the most flexible and comprehensive type of long memory model is the fractional autoregressive-moving average or FARMA model presented in Chapter 11. In fact, the FARMA family of models is a dinct extension of the ARMA class of models defined in Chapter 3. If FGN modelling cannot provide a reasonable solution to the Hurst phenomenon, then wherein Lies the answer? The solution to Hurst’s riddle is put foward in Section 10.6 of the next chapter. Simulation experiments demonstrate that when the most appropriate ARMA models are fitted to a wide variety of annual natural time series. a statistic called tbe H u n t coefiicient is “statistically preserved” by the calibrated ARMA models. Therefore, although the Hurst coefficient and other related statistics are not directly incorporated as model parameters in the design of an ARMA model, these statistics can still be indirectly accounted for or modelled by ARMA models.
325
PART V LONG MEMORY MODELLING The Hurst phenomenon cnated one of the m s t interesting, controversial and long-lasting scientific debates ever to arise in the field of hydrology. The genesis of the Hurst phenomenon took place over forty years ago in Egypt Just after World War II. a British scientist by the name of H a d d Edwin Hurst became deeply involved in studying how the Nile River could be optimally controlled and utilized for the benefit of both Egypt and Sudan. As D k t o r G e n e r a l of the Physical Department in the Ministry of Public Works in Cairo, Egypt,Hurst was particularly interested in the long-term storage requirements of the Nile River. In addition to annual riverflow series, Hurst analyzed a wide variety of other yearly geophysical time series in order to examine the statistical properties of some specific statistics that are closely related to long term storage. These statistical studies led Hurst to develop an empirical law upon which the definition of the Hurst phenomenon is b u d . The fact that the Hurst phenomenon arose from scientific work carried out in Egypt provided the controversy with an aura of mystery and intrigue. Was the Hurst phenomenon more difficult to solve than the riddle of the Sphinx? Indeed, a range of explanations has been put forward for solving the Hurst phenomenon. Furthermore, in the process of studying the Hurst phenomenon, many original contributions have been made to the fields of hydrology and statistics. In Chapter 10, the Hurst phenomenon is defined and both theoretical and empirical work related to this phenomenon are described. One spinoff From research connected to Hurst’s work is the development of a stochastic model called fractional Gaussian noise (FGN). This model possesses long memory (see Section 2.5.3) and was designed for furnishing an explanation to the Hurst phenomenon. As demonstrated in Chapter 10, this long memory model fails to solve the Hunt riddle. Nevertheless, the introduction of FGN into the field of hydrology initiated major theoretical and practical developments in long memory modelling by not only hydrologists but also by statisticians and economists. Probably the most flexible and comprehensive type of long memory model is the fractional autoregressive-moving average or FARMA model presented in Chapter 11. In fact, the FARMA family of models is a dinct extension of the ARMA class of models defined in Chapter 3. If FGN modelling cannot provide a reasonable solution to the Hurst phenomenon, then wherein Lies the answer? The solution to Hurst’s riddle is put foward in Section 10.6 of the next chapter. Simulation experiments demonstrate that when the most appropriate ARMA models are fitted to a wide variety of annual natural time series. a statistic called tbe H u n t coefiicient is “statistically preserved” by the calibrated ARMA models. Therefore, although the Hurst coefficient and other related statistics are not directly incorporated as model parameters in the design of an ARMA model, these statistics can still be indirectly accounted for or modelled by ARMA models.