16 Mar 2022
16 Mar 2022
Status: this preprint is currently under review for the journal HESS.

Technical note: c-u-curve: A method to analyse, classify and compare dynamical systems by uncertainty and complexity

Uwe Ehret1 and Pankaj Dey2 Uwe Ehret and Pankaj Dey
  • 1Institute of Water Resources and River Basin Management, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
  • 2Interdisciplinary Centre for Water Research, Indian Institute of Science, Bangalore, India

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.

Uwe Ehret and Pankaj Dey

Status: final response (author comments only)

Comment types: AC – author | RC – referee | CC – community | EC – editor | CEC – chief editor | : Report abuse
  • RC1: 'Comment on hess-2022-16', Jasper Vrugt, 23 Mar 2022
    • AC1: 'Reply on RC1', Uwe Ehret, 29 Apr 2022
  • RC2: 'Comment on hess-2022-16', Anonymous Referee #2, 07 Sep 2022
    • AC2: 'Reply on RC2', Uwe Ehret, 29 Sep 2022

Uwe Ehret and Pankaj Dey

Uwe Ehret and Pankaj Dey


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Short summary
We propose the “c-u-curve” method to characterize dynamical (time-variable) systems of all kinds. “U” is for uncertainty and expresses how well a system can be predicted in a given period of time. “C” is for complexity and expresses how predictability differs between different periods, i.e. how well predictability itself can be predicted. The method helps to better classify and compare dynamical systems across a wide range of disciplines, thus facilitating scientific collaboration.