Measuring nonlinear dependence in hydrologic time series

  • H. S. KimEmail author
  • K. H. Lee
  • M. S. Kyoung
  • B. Sivakumar
  • E. T. Lee
Original Paper


It has been a common practice to employ the correlation dimension method to investigate the presence of nonlinearity and chaos in hydrologic processes. Although the method is generally reliable, potential limitations that exist in its applications to hydrologic data cannot be dismissed altogether. As for these limitations, two issues have dominated the discussions thus far: small data size and presence of noise. Another issue that is equally important, but less discussed in the literature, is the selection of delay time (τ d ) for reconstruction of the phase-space, which is an essential first step in the correlation dimension method, or any other chaos identification and prediction method for that matter. It has also been increasingly recognized that fixing the delay time window (τ w ) rather than just the delay time itself could be more appropriate, since the delay time window is the one that is of actual interest at the end to represent the dynamics. To this effect, Kim et al. (1998a) [Phys Rev E 58(5):5676–5682] developed a procedure for fixing the delay time window and demonstrated its effectiveness on three artificial chaotic series, and followed it up with the development of the C–C method to estimate both the delay time and the delay time window. The purpose of the present study is to test this procedure on real hydrologic time series and, hence, to assess their nonlinear deterministic characteristics. Three hydrologic time series are studied: (1) daily streamflow series from St. Johns near Cocoa, FL, USA; (2) biweekly volume time series from the Great Salt Lake, UT, USA; and (3) daily rainfall series from Seoul, South Korea. The results are also compared with those obtained using the conventional autocorrelation function (ACF) method.


Hydrologic time series Nonlinearity Chaos Correlation dimension Delay time Delay time window 



This study was supported by the 2007 SOC Project (07-GIBANGUCHUK-D03-01) through the Design Criteria Research Center of Abnormal Weather-Disaster Prevention in KICTTEP of MOCT. Bellie Sivakumar would like to thank the Korea Science and Technology Societies (for the Brainpool Fellowship) and Inha University that facilitated his stay at Inha University to collaborate on the above project.


  1. Abarbanel HDI, Lall U (1996) Nonlinear dynamics of the Great Salt Lake: system identification and prediction. Climate Dyn 12:287–297CrossRefGoogle Scholar
  2. Abarbanel HDI, Lall U, Moon YI, Mann M, Sangoyomi T (1996) Nonlinear dynamics and the Great Salk Lake: a predictable indicator of regional climate. Energy 21(7/8):655–666CrossRefGoogle Scholar
  3. Berndtsson R, Jinno K, Kawamura A, Olsson J, Xu S (1994) Dynamical systems theory applied to long-term temperature and precipitation time series. Trends Hydrol 1:291–297Google Scholar
  4. Brock WA, Hsieh DA, Lebaron B (1991) Nonlinear dynamics, chaos, and instability: statistical theory and economic evidence. MIT Press, CambridgeGoogle Scholar
  5. Brock WA, Dechert WA, Scheinkman JA, LeBaron B (1996) A test for independence based on the correlation dimension. Econ Rev 15(3):197–235CrossRefGoogle Scholar
  6. Cao L, Mees A, Judd K (1998) Dynamics from multivariate time series. Physica D 121:75–88CrossRefGoogle Scholar
  7. Casdagli M, Eubank S, Farmer JD, Gibson J (1991) State space reconstruction in the presence of noise. Physica D 51:52–98CrossRefGoogle Scholar
  8. Frazer AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 55:1134–1140CrossRefGoogle Scholar
  9. Frison T (1994) Nonlinear data analysis techniques. In: Deboeck GJ (ed) Trading on the edge: neural, genetic, and fuzzy systems for chaotic financial markets. Wiley, New York, pp 280–296Google Scholar
  10. Ghilardi P, Rosso R (1990) Comment on “Chaos in Rainfall”. Water Resour Res 26(8):1837–1839Google Scholar
  11. Graf KE, Elbert T (1990) Dimensional analysis of the waking EEG. In: Basar D (ed) Chaos in brain function. Springer, New YorkGoogle Scholar
  12. Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 7:153–180CrossRefGoogle Scholar
  13. Havstad JW, Ehlers CL (1989) Attractor dimension of nonstationary dynamical systems from small data sets. Phys Rev A 39(2):845–853CrossRefGoogle Scholar
  14. Henon M (1976) A two-dimensional mapping with a strange attractor. Commun Math Phys 50:69–77CrossRefGoogle Scholar
  15. Holzfuss J, Mayer-Kress G (1986) An approach to error-estimation in the application of dimension algorithms. In: Mayer-Kress G (ed) Dimensions and entropies in chaotic systems. Springer, New YorkGoogle Scholar
  16. Hossain F, Sivakumar B (2006) Spatial pattern of arsenic contamination in shallow wells of Bangladesh: regional geology and nonlinear dynamics. Stoch Environ Res Risk Assess 20(1–2):66–76CrossRefGoogle Scholar
  17. Hurst HE (1951) Long term storage capacities of reservoirs. Trans ASCE 116:776–808Google Scholar
  18. Islam MN, Sivakumar B (2002) Characterization and prediction of runoff dynamics: a nonlinear dynamical view. Adv Water Resour 25(2):179–190CrossRefGoogle Scholar
  19. Jayawardena AW, Gurung AB (2000) Noise reduction and prediction of hydrometeorological time series: dynamical systems approach vs. stochastic approach. J Hydrol 228:242–264CrossRefGoogle Scholar
  20. Jayawardena AW, Lai F (1994) Analysis and prediction of chaos in rainfall and stream flow time series. J Hydrol 153:23–52CrossRefGoogle Scholar
  21. Jayawardena AW, Li WK, Xu P (2002) Neighborhood selection for local modeling and prediction of hydrological time series. J Hydrol 258:40–57CrossRefGoogle Scholar
  22. Jeong GD, Rao AR (1996) Chaos characteristics of tree ring series. J Hydrol 182:239–257CrossRefGoogle Scholar
  23. Jinno K, Xu S, Berndtsson R, Kawamura A, Matsumoto M (1995) Prediction of sunspots using reconstructed chaotic system equations. J Geophys Res 100(A8):14773–14781CrossRefGoogle Scholar
  24. Kantz H, Schreiber T (1997) Nonlinear time series analysis. Cambridge University Press, CambridgeGoogle Scholar
  25. Khan S, Ganguly AR, Saigal S (2005) Detection and predictive modeling of chaos in finite hydrological time series. Nonlinear Processes Geophys 12:41–53Google Scholar
  26. Kim HS, Eykholt R, Salas JD (1998a) Delay time window and plateau onset of the correlation dimension for small data sets. Phys Rev E 58(5):5676–5682CrossRefGoogle Scholar
  27. Kim HS, Park JU, Kim JH (1998b) Hurst phenomenon in hydrologic time series. J Korean Soc Civil Eng 18(II-6):571–582Google Scholar
  28. Kim HS, Eykholt R, Salas JD (1999) Nonlinear dynamics, delay times, and embedding windows. Physica D 127(1–2):48–60CrossRefGoogle Scholar
  29. Lall U, Sangoyomi T, Abarbanel HDI (1996) Nonlinear dynamics of the Great Salt Lake: nonparametric short-term forecasting. Water Resour Res 32(4):975–985CrossRefGoogle Scholar
  30. Liebert W, Schuster HG (1989) Proper choice of the time delay for the analysis of chaotic time series. Phys Lett A 141:386–390CrossRefGoogle Scholar
  31. Lisi F, Villi V (2001) Chaotic forecasting of discharge time series: a case study. J Am Water Resour Assoc 37:271–279CrossRefGoogle Scholar
  32. Liu Q, Islam S, Rodriguez-Iturbe I, Le Y (1998) Phase-space analysis of daily streamflow: characterization and prediction. Adv Water Resour 21:463–475CrossRefGoogle Scholar
  33. Martinerie JM, Albano AM, Mees AI, Rapp PE (1992) Mutual information, strange attractors, and the optimal estimation of dimension. Phys Rev A 45:7058–7064CrossRefGoogle Scholar
  34. Nerenberg MAH, Essex C (1990) Correlation dimension and systematic geometric effects. Phys Rev A 42(12):7065–7074CrossRefGoogle Scholar
  35. Olsson J, Niemczynowicz J, Berndtsson R (1993) Fractal analysis of high-resolution rainfall time series. J Geophys Res 98(D12):23265–23274CrossRefGoogle Scholar
  36. Osborne AR, Provenzale A (1989) Finite correlation dimension for stochastic systems with power-law spectra. Physica D 35:357–381CrossRefGoogle Scholar
  37. Packard NH, Crutchfield JD, Farmer JD, Shaw RS (1980) Geometry from a time series. Phys Rev Lett 45(9):712–716CrossRefGoogle Scholar
  38. Phoon KK, Islam MN, Liaw CY, Liong SY (2002) A practical inverse approach for forecasting of nonlinear time series analysis. ASCE J Hydrol Eng 7(2):116–128CrossRefGoogle Scholar
  39. Porporato A, Ridolfi L (1996) Clues to the existence of deterministic chaos in river flow. Int J Mod Phys B 10:1821–1862CrossRefGoogle Scholar
  40. Porporato A, Ridolfi L (1997) Nonlinear analysis of river flow time sequences. Water Resour Res 33(6):1353–1367CrossRefGoogle Scholar
  41. Porporato A, Ridolfi L (2001) Multivariate nonlinear prediction of river flows. J Hydrol 248(1–4):109–122CrossRefGoogle Scholar
  42. Puente CE, Obregon N (1996) A deterministic geometric representation of temporal rainfall: results for a storm in Boston. Water Resour Res 32(9):2825–2839CrossRefGoogle Scholar
  43. Regonda S, Sivakumar B, Jain A (2004) Temporal scaling in river flow: can it be chaotic? Hydrol Sci J 49(3):373–385CrossRefGoogle Scholar
  44. Regonda S, Rajagopalan B, Lall U, Clark M, Moon YI (2005) Local polynomial method for ensemble forecast of time series. Nonlinear Processes Geophys 12:397–406Google Scholar
  45. Rodriguez-Iturbe I, De Power BF, Sharifi MB, Georgakakos KP (1989) Chaos in rainfall. Water Resour Res 25(7):1667–1675CrossRefGoogle Scholar
  46. Rössler OE (1976) An equation for continuous chaos. Phys Lett A 57:397–398CrossRefGoogle Scholar
  47. Sangoyomi TB, Lall U, Abarbanel HDI (1996) Nonlinear dynamics of the Great Salt Lake: dimension estimation. Water Resour Res 32(1):149–159CrossRefGoogle Scholar
  48. Schertzer D, Tchiguirinskaia I, Lovejoy S, Hubert P, Bendjoudi H (2002) Which chaos in the rainfall-runoff process? A discussion on ‘Evidence of chaos in the rainfall-runoff process’ by Sivakumar et al. Hydrol Sci J 47(1):139–147Google Scholar
  49. Schreiber T, Kantz H (1996) Observing and predicting chaotic signals: is 2% noise too much? In: Kravtsov YuA, Kadtke JB (eds) Predictability of complex dynamical systems. Springer Series in Synergetics. Springer, Berlin, pp 43–65Google Scholar
  50. Schuster HG (1988) Deterministic chaos. VCH, WeinheimGoogle Scholar
  51. Sivakumar B (2000) Chaos theory in hydrology: important issues and interpretations. J Hydrol 227(1–4):1–20CrossRefGoogle Scholar
  52. Sivakumar B (2001) Rainfall dynamics at different temporal scales: a chaotic perspective. Hydrol Earth Syst Sci 5(4):645–651Google Scholar
  53. Sivakumar B (2002) A phase-space reconstruction approach to prediction of suspended sediment concentration in rivers. J Hydrol 258:149–162CrossRefGoogle Scholar
  54. Sivakumar B (2004a) Chaos theory in geophysics: past, present and future. Chaos, Solitons Fractals 19(2):441–462CrossRefGoogle Scholar
  55. Sivakumar B (2004b) Dominant processes concept in hydrology: moving forward. Hydrol Processes 18(12):2349–2353CrossRefGoogle Scholar
  56. Sivakumar B (2005) Correlation dimension estimation of hydrologic series and data size requirement: myth and reality. Hydrol Sci J 50(4):591–604CrossRefGoogle Scholar
  57. Sivakumar B, Liong SY, Liaw CY, Phoon KK (1999a) Singapore rainfall behavior: chaotic? ASCE J Hydrol Eng 4(1):38–48CrossRefGoogle Scholar
  58. Sivakumar B, Phoon KK, Liong SY, Liaw CY (1999b) A systematic approach to noise reduction in chaotic hydrological time series. J Hydrol 219(3–4):103–135CrossRefGoogle Scholar
  59. Sivakumar B, Berndttson R, Olsson J, Jinno K (2001a) Evidence of chaos in the rainfall-runoff process. Hydrol Sci J 46(1):131–145Google Scholar
  60. Sivakumar B, Berndtsson R, Persson M (2001b) Monthly runoff prediction using phase-space reconstruction. Hydrol Sci J 46(3):377–387CrossRefGoogle Scholar
  61. Sivakumar B, Sorooshian S, Gupta HV, Gao X (2001c) A chaotic approach to rainfall disaggregation. Water Resour Res 37(1):61–72CrossRefGoogle Scholar
  62. Sivakumar B, Berndtsson R, Olsson J, Jinno K (2002a) Reply to ‘which chaos in the rainfall-runoff process? Hydrol Sci J 47(1):149–158Google Scholar
  63. Sivakumar B, Jayawardena AW, Fernando TMGH (2002b) River flow forecasting: use of phase-space reconstruction and artificial neural networks approaches. J Hydrol 265:225–245CrossRefGoogle Scholar
  64. Sivakumar B, Persson M, Berndtsson R, Uvo CB (2002c) Is correlation dimension a reliable indicator of low-dimensional chaos in short hydrological time series? Water Resour Res 38(2). doi: 10.1029/2001WR000333
  65. Sivakumar B, Wallender WW, Puente CE, Islam MN (2004) Streamflow disaggregation: a nonlinear deterministic approach. Nonlinear Processes Geophys 11:383–392Google Scholar
  66. Sivakumar B, Jayawardena AW, Li WK (2007) Hydrologic complexity and classification: a simple data reconstruction approach. Hydrol Processes 21(20):2713–2728CrossRefGoogle Scholar
  67. Smith LA (1988) Intrinsic limits on dimension calculations. Phys Lett A 133(6):283–288CrossRefGoogle Scholar
  68. Takens F (1981) Detecting strange attractors in turbulence. In: Rand DA, Young LS (eds) Dynamical systems and turbulence, Lecture Notes in Mathematics 898. Springer, Berlin, pp 366–381Google Scholar
  69. Tsonis AA (1992) Chaos: from theory to applications. Plenum Press, New YorkGoogle Scholar
  70. Tsonis AA, Elsner JB (1988) The weather attractor over short timescales. Nature 333:545–547CrossRefGoogle Scholar
  71. Tsonis AA, Elsner JB, Georgakakos KP (1993) Estimating the dimension of weather and climate attractors: important issues about the procedure and interpretation. J Atmos Sci 50:2549–2555CrossRefGoogle Scholar
  72. Tsonis AA, Triantafyllou GN, Elsner JB, Holdzkom JJ II, Kirwan AD Jr (1994) An investigation of the ability of nonlinear methods to infer dynamics from observables. Bull Am Meteor Soc 75:1623–1633CrossRefGoogle Scholar
  73. Wang Q, Gan TY (1998) Biases of correlation dimension estimates of streamflow data in the Canadian prairies. Water Resour Res 34(9):2329–2339CrossRefGoogle Scholar
  74. Wilcox BP, Seyfried MS, Matison TH (1991) Searching for chaotic dynamics in snowmelt runoff. Water Resour Res 27(6):1005–1010CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • H. S. Kim
    • 1
    Email author
  • K. H. Lee
    • 1
  • M. S. Kyoung
    • 1
  • B. Sivakumar
    • 2
  • E. T. Lee
    • 3
  1. 1.Department of Civil EngineeringInha UniversityIncheonSouth Korea
  2. 2.Department of Land, Air and Water ResourcesUniversity of CaliforniaDavisUSA
  3. 3.Department of Civil EngineeringKyungHee UniversitySuwonSouth Korea

Personalised recommendations