Technical note: c-u-curve: A method to analyse, classify and compare dynamical systems by uncertainty and complexity
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)
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
Viewed (geographical distribution)
Review of “Technical note: c-u-curve: A method to analyse, classify and compare dynamical systems by uncertainty and complexity”
Summary: The authors resort to information theory and present the so-called c-u-curve to describe/quantify characteristic properties of hydrometeorological data. This c-u-curve displays graphically the relationship between what authors refer to as system uncertainty and complexity. This information is thought to be expressed and/or contained within spatial and/or temporal measurements of the data generating process of interest. System uncertainty is defined as the mean Shannon information entropy of many different time slices (time windows). The authors define system complexity as the ‘…uncertainty about uncertainty’ (P1, Line 11) and express this quantitatively as the entropy of the entropies of all time slices. As the two metrics depend strongly on the temporal extent (width) of the time window, the authors repeat their analysis for many different slice sizes. The c-u-curve is a graphical depiction of the relationship between the so-obtained system uncertainty (x-axis) and system complexity (y-axis), both of which have units of bits. The authors illustrate this idea by application to six different signals (time series), including simulated data of a (i) deterministic (horizontal line), (ii) random (normally distributed variates) and (iii) chaotic system (Lorenz attractor) and measured time series of (iv) precipitation and (v,vi) catchment discharge of the South Toe and Green Rivers in the United States. The authors conclude that the c-u-curve can be used to analyze, classify and compare dynamical systems.
Evaluation: The manuscript discusses an important topic in hydrology and complex systems analysis in general, namely the characterization of the dimensionality and complexity of dynamical systems. I enjoyed reading this manuscript. The document is well written and relatively easy to understand. Rationales and ideas are clearly presented. The six case studies demonstrate/showcase the potential use of the c-u-curve, inform readers about the methodology and how to interpret its results. I applaud the authors for their work, which I believe is very interesting. I do have serious concerns however about the mathematical and literature underpinning of the methodology, and the robustness and convergence properties of the c-u-curve. Based on these comments, I recommend a major revision.
The authors decided to present their work in the form of a technical note. This is an efficient way to rapidly disseminate new ideas. But technical notes have strict length requirements which can make it difficult to address all important aspects of the work presented. The ideas presented are very interesting, yet a full paper may do more justification to the ideas and work presented. I have several questions about the methodology, which I think should be addressed before readers can judge that what is presented is a substantial and/or important advance in our ability to analyze, classify and compare dynamical systems. Note that in my review below I use the word ‘signal’ for a measured or simulated time series of some quantity of interest. I also use the word ‘paper’ in reference to this technical note. This word is conveniently used and should certainly not imply that I was expecting a much longer manuscript.
I very much enjoyed reading this paper. From my comments above it is clear that I have concerns about the statistical/mathematical and literature underpinning of the methodology, the use of the words system uncertainty and system complexity, and the robustness and convergence properties of the methodology (= c-u curve). The additional studies I suggest will help answer important questions about the usefulness and diagnostic power of the c-u curve and its use in the analysis and classification of hydrometeorological time series. My comments are intended to help the authors further refine/improve their methodology for maximum exposure and use in the community.
PS. I did not proofread my review. Also, my comments are listed in a somewhat random order (with a c-u-curve that approaches a point) as a result of going back and forth in the paper.