Technical note: Complexity–uncertainty curve (c-u-curve) – a method to analyse, classify and compare dynamical systems

We propose and provide a proof of concept of a method to analyse, classify and compare dynamical systems of arbitrary dimensions 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) characteristic timescale expressed by the width of the time slice for which maximum complexity occurs. We demonstrate the method with 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-curve clearly reflect the particular characteristics of each time series. ... mehrFor 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 pre-condition for regionalization, and it can be used as a diagnostic measure and 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-variate to multivariate systems, and (iv) that it is applicable to both deterministic and probabilistic value representations, permitting e.g. application to ensemble model predictions.