Correlation dimension estimation of hydrological series and data size requirement: myth and reality/Estimation de la dimension de corrélation de séries hydrologiques et taille nécessaire du jeu de données: mythe et réalité

Abstract The issue of data size (length) requirement for correlation dimension estimation continues to be the nucleus of criticisms on the (low) correlation dimensions reported for hydrological series. The present study addresses this issue from the viewpoints of both the existing theoretical guidelines and the practical reality. For this purpose, correlation dimension analysis is carried out for various data sizes from each of three types of series: (a) stochastic series (artificially generated using a random number generation technique); (b) chaotic series (artificially generated using the Henon map equation); and (c) hydrological series (real flow data observed on the Göta River in Sweden). The outcomes of the analysis of the (artificial) stochastic and chaotic series are used as a basis for interpreting the outcomes of the hydrological series. It is found that reliable dimension results for the stochastic and chaotic series are obtained even when the data size is only a few hundred points (i.e. no underestimation of dimension for small data sizes is visible), with no significant change in the scaling regimes (of the dimension plots) with respect to data size. This implies that the dimension results obtained for the hydrological series even with a few hundred points are also close to the actual ones. The insignificant difference in the scaling regimes for the various data sizes further supports this point. These results lead to the conclusions that: (1) the issue of data size requirement for correlation dimension estimation is more of a myth than reality; (2) the dimension estimates reported thus far for hydrological series could indeed be close to the actual ones (unless influenced by factors other than data size, e.g. delay time, noise, zeros, intermittency).