Prediction of structural responses with the aid of fuzzy stochastic time series
Final Report Abstract
Mathematical methods for predicting uncertain structural responses with the aid of fuzzy time series are developed in the context of this research project. Uncertain measurements of structural loads and responses over time are considered as time series comprised of fuzzy data. Time series comprised of fuzzy data are regarded as realizations of a fuzzy random process. Methods for identifying and quantifying the underlying fuzzy random process are developed. For the purpose of forecasting time series comprised of fuzzy data, optimal forecasts, fuzzy forecast intervals, and fuzzy random forecasts are defined. With the aid of forecasted measurements it is possible to estimate future structural responses in a model-free manner or with the aid of a mechanical model. In contrast to the common definition and application of fuzzy time series, the fuzzy numbers in this case are approximated by a new incremental representation. This incremental representation is the sine qua non for a direct description, modeling and forecasting, whereby "direct" implies that the sequence of fuzzy numbers is retained during the description, modeling and forecasting stages. Up to the present forecasting of sequences of fuzzy data without defuzzification and refuzzification, e.g. under retention of fuzziness was unsolved. With the aid of the developed methods the preconditions are created, to enhance further random process models in order to describe time series comprised of fuzzy data. Exemplary the following process models may be mentioned; nonlinear autoregressive processes (NLAR processes), autoregressive processes with random coefficients (ARCH and GARCH processes) or condition-based models. Methods of artificial intelligence may be used in order to avoid special process models. The developed multilayer perceptron for fuzzy variables enables the application of further artifical neural networks for the mapping of fuzzy variables. E.g., artificial neural networks with partial recurrent architectures (Jordan and Elman neural networks) or artificial neural networks with time history. The application of the enhanced artificial neural networks is not limited to the analysis and forecasting of fuzzy time series. Further fields of application are nonlinear structural analysis with uncertain input data or problems of the system identification (e.g. the determination of uncertain damping properties). Yet another potential subsequent field of application is the investigation of multivariate time series comprised of fuzzy data. Especially the development of multivariate fuzzy random process models and the investigation of artificial neural networks for the analysis and forecasting of multivariate fuzzy time series are further tasks.
Publications
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Application of fuzzy randomness to time-dependent reliability. In: Augusti, G., Schueller, G. I., Ciampoli, M. (eds.) Safety and Reliability of Engineering Systems and Structures - Proceedings of the 9th Int. Conference on Structural Safety and Reliability. Millpress, Rotterdam, pp. 1709-1716, 2005
Sickert, J.-U.; Graf, W.; Reuter, U.
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Prediction of structural responses using time series with fuzzy data. In: Hackl, K., Meschke, G., Reese, S. (eds.) 1st GACM - Colloquium for Young Scientist on Computational Mechanics. Ruhr-Univarsität Bochum, p. 51, 2005
Reuter, U.
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Theoratical Basics of Fuzzy Randomness - Application to Time Series with Fuzzy Data. In: Augusti, G., Schueller, G. I., Ciampoli, M. (ads.) Safety and Reliability of Engineering Systems and Structures - Proceedings of the 9th Int. Conference on Structural Safety and Reliability. Millpress, Rotterdam, pp. 1701-1707, 2005
Möller, B.; Beer, M.; Reuter, U.
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Analyse und Prognose von Zeitreihen mit Fuzzy-Daten zur Prädiktion von Strukturantworten. Dissertation, TU Dresden, Veröffentlichungen Institut für Statik und Dynamik der Tragwerke, Heft 10, 2006
Reuter, U.
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Analyse und Prognose von Zeitreihen mit Fuzzy-Daten zur Prädiktion von Strukturantworten. In: Hofstetter, G. (Hrsg.) Baustatik - Baupraxis, Forschungskolloquium 2006, Universität Innsbruck, Österreich, S. 34, 2006
Reuter, U.
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Forecasting of structural responses using fuzzy time series. In: Liu, W.K., Chen, J.S. (eds.) 7th World Congress on Computational Mechanics. Kaynote lecture, Los Angeles, paper 674, 2006
Möller, B.; Rauter, U.
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Numerisches Tragwerksmonitoring und Prognose. In: Ruge, P., Graf, W. (Hrsg.) 10. Dresdner Baustatik-Seminar Neue Bauweisen -Trends in Statik und Dynamik, Lehrstuhl für Statik, Technische Universität Dresden, S. 147-156, 2006
Graf, W.; Bartzsch, M.; Beer, M.; Liebscher, M.; Reuter, U.
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Prediction of uncertain structural responses with fuzzy time series. In: Muhanna, R.L. und R.L. Mullan (eds.) 2nd Workshop on Reliable Engineering Computing. Kluwar Academic Publishers, Savannah, GA, pp. 419-440, 2006
Möller, B.; Reuter, U.
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Random processes with uncertain data. In: Spanos, P.D. (ed.) 5th Workshop Computational Stochastic Mechanics. Rhodes, 9 pp., 2006
Möller, B.; Reuter, U.
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Uncertain data caused by extreme hazards - modeling by fuzzy randomness In: Ibrahimbegovic, A., Kozar, I. (eds.) Proceedings of NATO Advanced Research Workshop - Extreme Man-Made and Natural Hazards in Dynamics of Structures. Faculty of Civil Engineering Rijeka, pp. 427-431, 2006
Möller, B.; Reuter, U., Liebscher, M.
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Numerical simulation based on fuzzy stochastic analysis. Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis, Vol. 13, Issue 4, pp. 349-364, 2007
Möller, B.; Graf, W.; Sickert, J.-U.; Reuter, U.
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Uncertainty Forecasting in Engineering. Springer Verlag, 2007
Möller, B.; Reuter, U.
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Prediction of uncertain structural responses using fuzzy time series. Computers and Structures, Vol. 86(10), pp. 1123-1139, 2008
Möller, B.; Reuter, U.
