Biases of correlation dimension estimates of streamflow data in the Canadian prairies

Abstract

The correlation dimension (D2) 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 D2 obtained for all cases are found to “saturate” at ∼3.0. However, the saturated D2 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 D2 approximately equal to 10. Therefore there is a consistent underestimation of D2 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 D2 of the prairie streamflow data should be ∼7–9. The hypothesis that the clustering effect of unevenly distributed data contributed to the underestimation of D2 was confirmed from the different D2 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 D2. However, for the range of time delay chosen for this study, 40–180 days, its effect on D2 is less obvious. This may be partly because all the streamflow data tested exhibit long‐term persistence.