Preservation of the rescaled adjusted range: 1. A reassessment of the Hurst Phenomenon

Previous research related to the controversial Hurst phenomenon is reviewed and evaluated. Because of the inherent statistical properties of the rescaled adjusted range (RAR) statistic it is suggested that research primarily be devoted to this statistic rather than to the various definitions of the Hurst coefficient. Simulation studies reveal that for independently distributed random variables the RAR does not significantly depend on the underlying distribution of the random variables but is a function of the sample size. For modeling correlated data the statistical attributes of a discrete fractional Gaussian noise (FGN) process are studied and also improved. An efficient maximum likelihood estimation technique is developed for the FGN model, and it is shown how the residuals of the fitted model can be calculated and then subjected to diagnostic checks. An exact simulation procedure is developed for simulating FGN in such a way that synthetic traces from the model lie outside the Brownian domain. The Akaike information criterion (AIC) is suggested as a method for choosing between a FGN and a Box-Jenkins model. For the six annual river flow series that are considered the AIC selects the best Box-Jenkins model in preference to the FGN process for each data set. Because Box-Jenkins models can be shown to preserve the historical RAR, in many practical applications it may be advantageous to use a Box-Jenkins model instead of a FGN process.

[1]  W. Feller.  The Asymptotic Distribution of the Range of Sums of Independent Random Variables , 1951 .

[2]  H. E. Hurst,et al.  Long-Term Storage Capacity of Reservoirs , 1951 .

[3]  ON THE RANGE OF PARTIAL SUMS OF A FINITE NUMBER OF INDEPENDENT NORMAL VARIATES , 1953 .

[4]  THE VARIANCE OF THE MAXIMUM OF PARTIAL SUMS OF A FINITE NUMBER OF INDEPENDENT NORMAL VARIATES , 1955 .

[5]  H. Hurst.  METHODS OF USING LONG-TERM STORAGE IN RESERVOIRS. , 1956 .

[6]  ON THE MOMENTS OF THE MAXIMUM OF PARTIAL SUMS OF A FINITE NUMBER OF INDEPENDENT NORMAL VARIATES , 1956 .

[7]  THE MEAN AND VARIANCE OF THE MAXIMUM OF THE ADJUSTED PARTIAL SUMS OF A FINITE NUMBER OF INDEPENDENT NORMAL VARIATES , 1957 .

[8]  H. Hurst,et al.  A Suggested Statistical Model of some Time Series which occur in Nature , 1957, Nature.

[9]  P. A. P. Moran,et al.  The Theory of Storage , 1961 .

[10]  S. Berman.  Limiting Distribution of the Maximum of a Diffusion Process , 1964 .

[11]  P. A. P. Moran,et al.  On the range of cumulative sums , 1964 .

[12]  Calyampudi Radhakrishna Rao,et al.  Linear Statistical Inference and its Applications , 1967 .

[13]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[14]  M. J. R. Healy,et al.  Algorithm as 6: Triangular Decomposition of a Symmetric Matrix , 1968 .

[15]  J. R. Wallis,et al.  Noah, Joseph, and Operational Hydrology , 1968 .

[16]  J. R. Wallis,et al.  Computer Experiments with Fractional Gaussian Noises: Part 3, Mathematical Appendix , 1969 .

[17]  J. R. Wallis,et al.  Computer Experiments With Fractional Gaussian Noises: Part 1, Averages and Variances , 1969 .

[18]  J. R. Wallis,et al.  Computer Experiments with Fractional Gaussian Noises: Part 2, Rescaled Ranges and Spectra , 1969 .

[19]  J. R. Wallis,et al.  Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence , 1969 .

[20]  J. R. Wallis,et al.  Some long‐run properties of geophysical records , 1969 .

[21]  Adrian E. Scheidegger,et al.  Stochastic Models in Hydrology , 1970 .

[22]  J. R. Wallis,et al.  Small Sample Properties of H and K—Estimators of the Hurst Coefficient h , 1970 .

[23]  M. Taqqu.  Note on evaluations of R/S for fractional noises and geophysical records , 1970 .

[24]  B. Mandelbrot.  A Fast Fractional Gaussian Noise Generator , 1971 .

[25]  J. R. Wallis,et al.  Statistical Properties of Multivariate Fractional Noise Processes , 1971 .

[26]  B. Mandelbrot.  Broken line process derived as an approximation to fractional noise , 1972 .

[27]  J. M. Mejía,et al.  Long memory monthly streamflow simulation by a broken line model , 1972 .

[28]  Clive W. J. Granger,et al.  “Infinite Variance” and Research Strategy in Time Series Analysis , 1972 .

[29]  Jose M. Mejia,et al.  Streamflow Simulation: 1. A new look at Markovian Models, fractional Gaussian noise, and Crossing Theory , 1972 .

[30]  Ignacio Rodriguez-Iturbe,et al.  Streamflow simulation: 2. The broken line process as a potential model for hydrologic simulation , 1972 .

[31]  ON THE EXPECTED RANGE AND EXPECTED ADJUSTED RANGE OF PARTIAL SUMS OF EXCHANGEABLE RANDOM VARIABLES , 1973 .

[32]  G. C. Tiao,et al.  Bayesian inference in statistical analysis , 1973 .

[33]  J. R. Wallis,et al.  FIRM RESERVOIR YIELD—HOW RELIABLE ARE HISTORIC HYDROLOGICAL RECORDS? , 1973 .

[34]  V. Klemeš.  The Hurst Phenomenon: A puzzle? , 1974 .

[35]  H. Akaike.  A new look at the statistical model identification , 1974 .

[36]  Stochastic modelling of long-term persistence in streamflow sequences , 1974 .

[37]  Expected range and adjusted range of hydrologic sequences , 1974 .

[38]  A. I. McLeod,et al.  Intervention Analysis in Water Resources , 1975 .

[39]  Skew inputs and the Hurst effect , 1975 .

[40]  Ian McLeod.  Derivation of the Theoretical Autocovariance Function of Autoregressive–Moving Average Time Series , 1975 .

[41]  Range and deficit analysis using Markov chains , 1975 .

[42]  G. K. Young,et al.  Modeling monthly hydrologic persistence , 1976 .

[43]  Skew inputs and the Hurst effect — A comment , 1976 .

[44]  E. J. Hannan,et al.  Vector linear time series models , 1976, Advances in Applied Probability.

[45]  M. M. Siddiqui.  The Asymptotic Distribution of the Range and Other Functions of Partial Sums of Stationary Processes , 1976 .

[46]  Kenneth W. Potter,et al.  Evidence for nonstationarity as a physical explanation of the Hurst Phenomenon , 1976 .

[47]  E. H. Lloyd,et al.  The expected value of the adjusted rescaled Hurst range of independent normal summands , 1976 .

[48]  Keith W. Hipel,et al.  Advances in Box‐Jenkins modeling: 2. Applications , 1977 .

[49]  N. T. Kottegoda,et al.  Stochastic Modelling of Riverflow Time Series , 1977 .

[50]  A. I. McLeod,et al.  Advances in Box-Jenkins modeling: 1. Model construction , 1977 .

[51]  Dennis P. Lettenmaier,et al.  Operational assessment of hydrologic models of long-term persistence , 1977 .

[52]  A. I. McLeod,et al.  Stochastic modelling of the effects of reservoir operation , 1977 .

[53]  A. I. Moleod.  Improved Box-Jenkins estimators , 1977 .

[54]  Keith W. Hipel,et al.  Simulation procedures for Box‐Jenkins Models , 1978 .

[55]  A. I. McLeod,et al.  On the Distribution of Residual Autocorrelations in Box–Jenkins Models , 1978 .

[57]  A. I. McLeod,et al.  Preservation of the rescaled adjusted range: 3. Fractional Gaussian noise algorithms , 1978 .