Calculation of information and complexity in time series

Analysis of the water budget of forestred catchments

Thesis of Frank Wolf

at the

Department of Ecological Modeling of the
Bayreuth Institute for Terrestrial Ecosystem Research (BITÖK)
University of Bayreuth, Germany

March 1996 - February 1999


[Deutsch]


Visit the extra page: calculation of information and complexity and required amount of data


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Table of contents


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Goals and Results

The goal of this work was the operationalization of measures of complexity for an application to ecosystem time series and the analysis of hydrological fluxes in forestred catchment areas (terrestrial ecosystems) according to the corresponding information fluxes. In particular:

  • required quality of data for application of the methods
  • effective measurement resolution from information theory's point of view
  • information fluxes inherent and through ecosystems
  • complexity of ecosystems
  • limits of predictability and modelling ecosystem time series
  • classification of ecosystems

The results are not exhaustive, but indicate the potential of the method. Total, 34 time series - each from a period of several (deci) years or - of precipiaion and runoff from 7 areas (21 watersheds) from Germany, Norway and the USA were investigated.

  • Development/Refinement of a new measure of complexity
  • Quantification of hydrological information fluxes. In principal, the precipitation information decreases while the water goes through a catchment.
  • Dependency of runoff information and vegetation
  • Estimation of an effective measurement resolution
  • Classification of water catchments (ecosystems).
  • Quantification of a decrease in predictability of runoff (e.g. floods) with increasing urbanization
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Information and Complexity in Time Series

The following measures of information and complexity were implemented into the new computer program SYMDYN [from: symbolic dynamics] for the analysis of experimental time series (including gaps):

  • Transinformation
  • Algorithmic Information
  • Rényi Entropies
  • Metric Entropy
  • Mean Information Gain
  • Mean Mutual Information
  • Effective Measure Complexity
  • Fluctuation Complexity
  • Rényi Complexity (New!)
  • Epsilon Komplexität
  • Meta-Statistical Complexities

The calculation of some selected measures is presented briefly on an extra page. There you can determine the required amount of data or maximum word length with the help of a JavaScript calculator.

3 types of time series of different information and complexity

The figure presents three types of time series to reveal the meaning of information and complexity. There is a kind of a parabolic dependency of Information (I) and complexity (C) as shown by the yellow icon, which is used for itemization on the left.

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Summary of the thesis

The main object of ecosystem research as performed at the BITÍK are so far the various processes related to forests. Here, time series of water related parameters (precipitation, matrix potential, and runoff) of forested catchments are investigated without a priori knowledge of specific processes. The aim was to formulate statements about system properties and predict-ability purely on the basis of the information inherent in the data. The restriction to water budgets in this initial study was due to their central role as a transport medium of dissolved matter and the quality and quantity of available data.

The proposed concept requires the quantification of information and complexity in experimen-tal time series. The applied methods are new in the field. They are presented in detail and investigated due to their applicability to experimental data. Therefore, the required amounts of data for a given mean accuracy were determined. This requirement and further considerations lead to an almost parameter-free procedure. The methods were programmed for a universal application to time series involving gaps. The program, SYMDYN, and operating instructions can be down-loaded from the internet free of charge by the public.

The percolation of the water through the investigated catchments was accopanied by a general decrease of the hydrologic information from precipitation to runoff. The runoff information decreases when the annual regularity of precipitation increases. It increases and thus contains more of the precipitation information for a declined (natural) forest stand. As was shown by a comparison with the raw data and the autocorrelation as a classical tool of time series analysis, the runoff information is a sensitive measure for judging the build up area or deforestation of a catchment.

The information of the soil matrix potential compared with the runoff information enables an estimation of the relevant percolation depth, of which the dynamics of precipitation is reduced to that of the runoff. This was possible without knowledge about soil structure and preferred flow paths. A maximum of complexity is interpreted as an index for an effective time scale. This allows the estimation of the redundancy and randomness and to judge the modellability and predictability of the measured data. Thus, future measurment campaigns should be planned for a resolution of the mean relevant dynamics of the observed parameter in the data records. Costs can be saved by avoiding non-reasonable high sampling rates. It was concluded that precipitation must be sampled hourly or more often to resolve its mean dynamic. Runoff can sufficiently be measured daily or even coarser. The estimated optimal sampling rates due to a maximum of complexity agree well with the heuristic experience of monitoring practice.

The data analysis of this study does not yield an exhaustive overview of the information inherent in the water budget of forested catchments, but illustrates what can be achieved by such an external viewpoint on the systems. This study provides the methodical requirements for a systematic analysis of the variety of hydrological monitoring data with respect to new indicators of changes. For this relevant aim in environmental research, SYMDYN was offered as a new tool for common usage.

Click here to download zip archive. Download program archive (219 kB):

Click on the leftside icon to download the time series analysis program (pc console application) symdyn.exe, german documentation (Word 97), example data and related files free of charge. The files are commented in a Readme.txt and provided in a zip archieve. The program and control file exists in english language.

Click here to download zipped postscript documentation. Download postscript documentation (375 kB):

Click on the leftside icon to download the control file documentation in postscript format. As mentioned above, this documentation is already included in the program zip archieve as a Microsoft Word 97 file.

SYMDYN icon Publications:

Currently, there exist only bfö publications.

Wolf, F; Lange, H; Hauhs, M: Ecosystem analysis by means of complexity theory. In: BITÖK (Ed.): BITÖK Forschungsbericht 1996. Bayreuther Forum Ökologie 1997, Band 41: 184-187.
Wolf, F; Lange, H; Hauhs, M: Ökosystem-Analysen mit komplexitätstheoretischen Ansätzen. In: Matzner, E (Ed.): BITÖK Forschungsbericht 1995-97. Bayreuther Forum Ökologie 1998, Band 56: 212-213.
Lange, H; Newig, J; Wolf, F: Comparison of complexity measures for time series from ecosystem research. In: Kastner-Maresch,A; Kurth, W; Sonntag, M; Breckling, B (Ed.): Individual-based structural and functional models in ecology. Contributions of a workshop helt at the University of Bayreuth. Bayreuth: Bayreuther Forum Ökologie 1998. Band 52: 99-116.
Wolf, F.: Berechnung von Information und Komplexität in Zeitreihen - Analyse des Wasserhaushaltes von bewaldeten Einzugsgebieten. Dissertation, Universität Bayreuth. Bayreuther Forum Ökologie (bfö) Band 65, 1999.
Get the printed thesis (in german) from the editor . See the pdf-file of the german thesis.

Last change: June, 8th 2008 Accesses since January, 31th 2000: