Comment on hess-2022-16
Uwe Ehret, Pankaj Dey · 2022
Abstract. We propose and provide a proof-of-concept of a method to analyse, classify and compare dynamical systems of arbitrary dimension by the two key features uncertainty and complexity. It starts by subdividing the system’s time-trajectory into a number of time slices. For all values in a time slice, the Shannon information entropy is calculated, measuring within-slice variability. System uncertainty is then expressed by the mean entropy of all time slices. We define system complexity as “uncertainty about uncertainty”, and express it by the entropy of the entropies of all time slices. Calculating and plotting uncertainty u and complexity c for many different numbers of time slices yields the c-u-curve. Systems can be analysed, compared and classified by the c-u-curve in terms of i) its overall shape, ii) mean and maximum uncertainty, iii) mean and maximum complexity, and iv) its characteristic time scale expressed by the width of the time slice for which maximum complexity occurs. We demonstrate the method at the example of both synthetic and real-world time series (constant, random noise, Lorenz attractor, precipitation and streamflow) and show that the shape and properties of the respective c-u-curves clearly reflect the particular characteristics of each time series. For the hydrological time series we also show that the c-u-curve characteristics are in accordance with hydrological system understanding. We conclude that the c-u-curve method can be used to analyse, classify and compare dynamical systems. In particular, it can be used to classify hydrological systems into similar groups, a precondition for regionalization, and it can be used as a diagnostic measure which can be used as an objective function in hydrological model calibration. Distinctive features of the method are i) that it is based on unit-free probabilities, thus permitting application to any kind of data, ii) that it is bounded, iii) that it naturally expands from single- to multivariate systems, and iv) that it is applicable to both deterministic and probabilistic value representations, permitting e.g. application to ensemble model predictions.