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Biases of correlation dimension estimates of streamflow data in the Canadian prairies
Author(s) -
Wang Qiang,
Gan Thian Yew
Publication year - 1998
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/98wr01379
Subject(s) - streamflow , mathematics , dimension (graph theory) , poisson distribution , statistics , cluster analysis , correlation , range (aeronautics) , sample (material) , physics , combinatorics , geography , geometry , cartography , drainage basin , materials science , composite material , thermodynamics
The correlation dimension ( D 2 ) of the unregulated streamflow data of six rivers in the Canadian prairies were estimated using the Hill estimate and Grassberger‐Procaccia algorithms. From both procedures and using an embedding phase space up to 10 the D 2 obtained for all cases are found to “saturate” at ∼3.0. However, the saturated D 2 generally increased to between 4 and 6 when the data were first randomly resampled prior to estimating the correlation dimension. Since the highest embedding dimension d used in this study is 10, theoretically, randomly resampled data should have a D 2 approximately equal to 10. Therefore there is a consistent underestimation of D 2 by an amount of 4–6. By accounting for the combined clustering effect of the streamflow data's underlying distribution (which is Gamma), the effect of sample size and possibly the choice of time delay ∇, the D 2 of the prairie streamflow data should be ∼7–9. The hypothesis that the clustering effect of unevenly distributed data contributed to the underestimation of D 2 was confirmed from the different D 2 obtained for random numbers of uniform, Gamma, and Poisson distributions. In addition, analysis of the uniformly distributed random numbers and streamflow data showed that increasing the sample size from, say 6500 to 17,000 marginally improved the estimated D 2 . However, for the range of time delay chosen for this study, 40–180 days, its effect on D 2 is less obvious. This may be partly because all the streamflow data tested exhibit long‐term persistence.