, Volume 31, Issue 1, pp 87–101 | Cite as

Modelling hydrological data with and without long memory

  • Paolo Burlando
  • Alberto Montanari
  • Renzo Rosso


The paper focuses on the problem of long-range dependence when analysing time series of hydrological data. Three time series are analysed: the monthly rainfall in the town of Florence, Italy; the daily minimum temperatures in the same town; and, finally, the daily water inflow to Lake Maggiore, Italy. Heuristic methods and maximum likelihood estimation of a parametric model are used to investigate the Hurst phenomena and to detect whether long-range dependence is present in any of the time series. We found that long-range dependence is not present in the first series but it is present in the last two. The daily water inflow to Lake Maggiore was modelled by a fractionally differenced arima model (farima) which contains a long-range dependence component. It is shown that the fit is much better than the one provided by more traditional arima models that do not have such a component.

Key words

Time series Hurst phenomenon Long-range dependence farima Hydrometeorology 


Lo studio considera il problema dell'identificazione dei fenomeni di dipendenza a lungo termine (long range dependence) nelle serie temporali di dati idrologici. Allo scopo sono state analizzate tre serie temporali, rispettivamente quella dei totali mensili di precipitazione rilevati alla stazione dell'Osservatorio Ximeniano de Firenze, quella delle temperature minime giornaliere per la stessa stazione e quella degli afflussi giornalieri al Lago Maggiore. Per identificare la presenza di fenomeni di dipendenza long range, attraverso la valutazione della consistenza del fenomeno di Hurst, sono stati utilizzati sia metodi euristici, sia la stima a massima verosimiglianza di un modello parametrico. Due delle tre serie analizzate sono risultate caratterizzate da tale dipendenza. Per la serie degli afflussi al Lago Maggiore, si è inoltre proceduto alla simulazione attraverso un modello arima a differenziazione frazionaria (farima), la cui struttura contiene una componente long-range. I risultati ottenuti, mostrano che tale modello fornisce risultati significativamente migliori dei tradizionali modelli arima, privi di tale componente.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beran J., ‘A test of location for data with a slowly decaying serial correlations’, Biometrika, 76 (1989) 261–269.Google Scholar
  2. 2.
    Beran J., Statistic for Long-Memory Processes, Monographs on Statistics and Applied Probability 61, Chapman & Hall, New York, 1994.Google Scholar
  3. 3.
    Bhattacharya R.N., Gupta V.K., and Waymire E., ‘The Hurst effect under trends’, J. of Appl. Prob., 20 (1983) 649–662.Google Scholar
  4. 4.
    Boes D.C., and Salas J.D., ‘On the expected range and expected adjusted range of partial sums of exchangeable random variables’, J. of Appl. Prob., 10 (1978) 671–677.Google Scholar
  5. 5.
    Box G.E.P., and Jenkins G.M., Time Series Analysis: Forecasting and Control, revised edn., Holden Day, San Francisco, California, 1976.Google Scholar
  6. 6.
    Brockwell P.J., and Davis R.A., Time Series: Theory and Method, Springer-Verlag, New York, 2nd edn., 1991.Google Scholar
  7. 7.
    Fox R., and Taqqu M.S., ‘Large sample properties of parameter estimates for strongly dependent stationary Gaussian time series’, Ann. Statist., 14 (1986) 517–532.Google Scholar
  8. 8.
    Giraitis L., and Surgailis D., ‘A central limit theorem for quadratic forms in strongly dependent linear variables and application to asymptotical normality of Whittle's estimate’, Prob. Th. Rel. Fields, 86 (1990) 87–104.Google Scholar
  9. 9.
    Granger C.W.J., and Joyeux R., ‘An introduction to long-range time series models and fractional differencing,’ J. of Time Series Analysis, 1 (1980) 15–30.Google Scholar
  10. 10.
    Hosking J.R.M., ‘Fractional differencing’, Biometrika, 68 (1981) 165–176.Google Scholar
  11. 11.
    Hurst H.E., ‘Long term storage capacity of reservoirs’, Trans. Am. Soc. Civil Engineers, 116 (1951) 770–799.Google Scholar
  12. 12.
    Hurst H.E., ‘A suggested statistical model of some time series which occur in nature’, Nature, 180 (1957) 494.Google Scholar
  13. 13.
    Klemes V., ‘The Hurst phenomenon-a puzzle?’, Water Resour. Res., 10 (1974) 675–688.Google Scholar
  14. 14.
    Mandelbrot B.B., ‘Limit theorems of the self-normalized range for weakly and strongly dependent processes’, Z. Wahr. verw. Geb., 31 (1975) 271–285.Google Scholar
  15. 16.
    Mandelbrot B.B., and Wallis J.R., ‘Computer experiments with fractional gaussian noises, Parts 1, 2 and 3’, Water Resour. Res. 5 (1969a) 228–267.Google Scholar
  16. 17.
    Mandelbrot B.B., and Wallis J.R., ‘Some long-run properties of geophysical records’, Water Resour. Res., 5 (1969b) 321–340.Google Scholar
  17. 18.
    Potter K.W., ‘Evidence for non-stationarity as a physical explanation of the Hurst phenomenon’, Water Resour. Res., 12 (1976) 1047–1052.Google Scholar
  18. 19.
    Samorodnitsky G., and Taqqu M.S., Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance, Chapman & Hall, New York, 1994.Google Scholar
  19. 20.
    Taqqu, M.S., Teverovsky, V., and Willinger W., ‘Estimators for long-range dependence: an empirical study’, Preprint (1995).Google Scholar
  20. 21.
    Teverovsky, V., and Taqqu, M. S., ‘Testing for long-range dependence in the presence of shifting means or a slowly declining trend, using a variance-type estimator’, Preprint (1995).Google Scholar
  21. 22.
    Wallis J.R., and Matalas N.C., ‘Small sample properties of H and K-estimators of the Hurst coefficient h’, Water Resour. Res., 6 (1970) 1583–1594.Google Scholar
  22. 23.
    Whittle P., ‘Estimation and information in stationary time series’, Ark. Mat. 2 (1953) 423–434.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Paolo Burlando
    • 1
  • Alberto Montanari
    • 1
  • Renzo Rosso
    • 1
  1. 1.Politecnico di MilanoD.I.I.A.R.Milano(Italy)

Personalised recommendations