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Analyzing Uncertainty in Civil Engineering pp 51–72Cite as

The mathematics of uncertainty: models, methods and interpretations

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This article discusses various mathematical theories that have been put forth as tools for modelling uncertainty, among them probability, interval arithmetic, random sets, and fuzzy sets. After recalling the definitions, we stress their interpretations (semantics), axioms, interrelations as well as numerical procedures and demonstrate how the concepts are applied in practice.

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References

  1. H. Adeli, B. Möller (Eds.). Special Issue: Fuzzy Modeling. Vol. 14(2) of Computer Aided Civil and Infrastructure Engineering, 1999.

    Google Scholar 

  2. G. Auvinet, J. L. González. Three-dimensional reliability analysis of earth slopes. Computers and Geotechnics 26 (2000), 247–261.

    Article  Google Scholar 

  3. B. M. Ayyub (Ed.). Uncertainty Modeling and Analysis in Civil Engineering. CRC Press, Boca Raton 1998.

    Google Scholar 

  4. H. Bandemer. Ratschläge zum mathematischen Umgang mit Ungewißheit — Reasonable Computing. Teubner, Leipzig 1997.

    Google Scholar 

  5. H. Bandemer, S. Gottwald. Einführung in Fuzzy-Methoden. 4. Aufl., Akademie Verlag, Berlin 1993.

    Google Scholar 

  6. H. Bandemer, W. Näther. Fuzzy Data Analysis. Kluwer Acad. Publ., Dordrecht 1992.

    Google Scholar 

  7. A. Bárdossy, L. Duckstein. Fuzzy Rule-Based Modeling with Applications to Geophysical, Biological and Engineering Systems. CRC Press, Boca Raton 1995.

    Google Scholar 

  8. T. Bedford, R. Cooke. Probabilistic Risk Analysis. Cambridge University Press, Cambridge 2001.

    Google Scholar 

  9. A. Bernardini. Qualitative and quantitative measures in seismic damage assessment and forecasting of masonry buildings. In: A, Bernardini (Ed.). Seismic Damage to Masonry Buildings. A. A. Balkema, Rotterdam 1999, 169–177.

    Google Scholar 

  10. C. Bertoluzza, M.Á. Gil, D. A. Ralescu (Eds.). Statistical Modeling, Analysis and Management of Fuzzy Data. Physica-Verlag, Heidelberg 2002.

    Google Scholar 

  11. D. Blockley, I. Ferraris. Managing the quality of a project. In: Seguridad en Ingenieráa, Academia Nacional de Ingeniería, Editorial Dunken, Buenos Aires 2000, 69–89.

    Google Scholar 

  12. V. V. Bolotin. Statistical Methods in Structural Mechanics. Holden-Day, Inc., San Francisco, 1969.

    Google Scholar 

  13. J. J. Buckley, E. Eslami, T. Feuring. Fuzzy Mathematics in Economics and Engineering. Physica-Verlag, Heidelberg 2002.

    Google Scholar 

  14. G. Choquet. Theory of capacities. Annales de l’Institut Fourier 5 (1953/54), 131–295.

    MathSciNet  Google Scholar 

  15. G. De Cooman, T. Fine, T. Seidenfeld (Eds.). ISIPTA’01, Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications. Shaker Publishing BV, Maastricht, 2001.

    Google Scholar 

  16. A. P. Dempster. Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematical Statistics 38:325–339, 1967.

    MathSciNet  MATH  Google Scholar 

  17. D. Denneberg. Non-additive Measure and Integral. Kluwer Acad. Publ., Dordrecht 1994.

    Google Scholar 

  18. R. Diedrich. Entscheidungen bei Ungewißheit. Physica-Verlag, Heidelberg 1999.

    Google Scholar 

  19. O. Ditlevsen, N. J. Tarp-Johansen, H. Denver. Bayesian soil assessments combining prior with posterior censored samples. Computers and Geotechnics, 26 (2000), 187–198.

    Article  Google Scholar 

  20. D. Dubois, H. Prade. Possibility Theory. Plenum Press, New York, 1988.

    Google Scholar 

  21. D. Dubois, H. Prade, R. Sabbadin. Decision-theoretic foundations of qualitative possibility theory. European J. Operations Res. 128 (2001), 459–478.

    MathSciNet  Google Scholar 

  22. D. Dubois, H. Prade, S. Sandri. On possibility/probability transformations. In: R. Lowen, M. Roubens (Eds.). Fuzzy Logic: State of the Art. Kluwer Academic Publishers, Dordrecht 1993, 103–112.

    Google Scholar 

  23. H. H. Einstein. Quantifying uncertain engineering geologic information. Felsbau, 19(5) (2001), 72–84.

    Google Scholar 

  24. F. Eisenführ, M. Weber. Rationales Entscheiden. 3rd Ed., Springer-Verlag, Berlin 1999.

    Google Scholar 

  25. I. Elishakoff (Ed.). Whys and Hows in Uncertainty Modelling: Probability, Fuzziness and Anti-Optimization. Springer-Verlag, Wien 1999.

    Google Scholar 

  26. I. Elishakoff. What may go wrong with probabilistic methods? In I. Elishakoff (Ed.). Whys and Hows in Uncertainty Modelling: Probability, Fuzziness and Anti-Optimization Springer-Verlag, Wien 1999, 265–283.

    Google Scholar 

  27. W. Fellin. Assessment of characteristic soil parameters and its implication in geotechnical design. In this volume.

    Google Scholar 

  28. S. Ferson, V. Kreinovich, L. Ginzburg, D. S. Myers, K. Sentz. Constructing probability boxes and Dempster-Shafer structures. SAND REPORT SAND 2002-4015, Sandia National Laboratories, Albuquerque, 2003.

    Google Scholar 

  29. Th. Fetz. Sets of joint probability measures generated by weighted marginal focal sets. In: G. De Cooman, T. Fine, T. Seidenfeld (Eds.). ISIPTA’01, Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications. Shaker Publishing BV, Maastricht, 2001, 171–178.

    Google Scholar 

  30. Th. Fetz. Multi-parameter models: rules and computational methods for combining uncertainties. In this volume.

    Google Scholar 

  31. Th. Fetz, J. Jäger, D. Köll, G. Krenn, H. Lessmann, M. Oberguggenberger, R. Stark. Fuzzy models in geotechnical engineering and construction management. In this volume.

    Google Scholar 

  32. Th. Fetz, M. Oberguggenberger. Propagation of uncertainty through multivariate functions in the framework of sets of probability measures. Reliability Engineering and Systems Safety, 85 (2004), 75–87.

    Google Scholar 

  33. T. L. Fine. Theories of Probability. An Examination of Foundations. Academic Press, New York 1973.

    Google Scholar 

  34. A. B. Gelman, J. S. Carlin, H. S. Stern, D. B. Rubin. Bayesian Data Analysis. Chapman & Hall/CRC, Boca Raton 1997.

    Google Scholar 

  35. A. Gheorghe, J.-P. Krause, R. Mock. Integration von Fuzzy-Logic für eine regionale Umweltrisiko-Analyse. Hochschulverlag an der ETH, Zürich 1996.

    Google Scholar 

  36. I. R. Goodman, H. T. Nguyen. Fuzziness and randomness. In: C. Bertoluzza, M.Á. Gil, D. A. Ralescu (Eds.), Statistical Modeling Analysis and Management of Fuzzy Data. Physica-Verlag, Heidelberg 2002, 3–21.

    Google Scholar 

  37. J. Goutsias, R. P. S. Mahler, H. T. Nguyen (Eds.). Random Sets. Theory and Applications. Springer-Verlag, New York 1997.

    Google Scholar 

  38. J. Hall, J. Lawry. Imprecise probabilities of engineering system failure from random and fuzzy set reliability analysis. In: G. De Cooman, T. Fine, T. Seidenfeld (Eds.). ISIPTA’01, Proceedings of the Second International Symposium on Imprecise Probabilities and Their Applications. Shaker Publishing BV, Maastricht, 2001, 195–204.

    Google Scholar 

  39. J. Hall, E. Rubio, M. Anderson. Random sets of probability measures in slope hydrology and stability analysis. Zeitschrift f. Angewandte Mathematik u. Mechanik, to appear.

    Google Scholar 

  40. P. R. Halmos. Measure Theory. Van Nostrand, New York 1950.

    Google Scholar 

  41. J. Hanisch, W. Struck. Charakteristischer Wert einer Boden-oder Materialeigenschaft aus Stichprobenergebnissen und zusätzlicher Information. Bautechnik 10 (1985), 338–348.

    Google Scholar 

  42. J. A. Hartigan. Bayes Theory. Springer-Verlag, New York 1983.

    Google Scholar 

  43. J. C. Helton, F. J. Davis. Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. SAND REPORT SAND 2001-0417, Sandia National Laboratories, Albuquerque, 2002.

    Google Scholar 

  44. I. Herle. Difficulties related to numerical predictions of deformations. In this volume.

    Google Scholar 

  45. H. Hirsch. Interview by H. Hochner. In: ZIB3, Austrian Television, May 5, 2001.

    Google Scholar 

  46. P. J. Huber. Robust Statistics. John Wiley & Sons, New York 1981.

    Google Scholar 

  47. A. Kandel. Fuzzy Mathematical Techniques with Applications. Addison-Wesley, Reading, 1986.

    Google Scholar 

  48. O. Klingmüller, U. Bourgund. Sicherheit und Risiko im Konstruktiven Ingenieurbau. Verlag Vieweg, Braunschweig, 1992.

    Google Scholar 

  49. G. J. Klir, M. J. Wiermann. Uncertainty-Based Information. Elements of Generalized Information Theory. Physica-Verlag, Heidelberg 1998.

    Google Scholar 

  50. A. Kolmogoroff. Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer-Verlag, Berlin 1933.

    Google Scholar 

  51. V. Kreinovich. Random sets unify, explain, and aid known uncertainty methods in expert systems. In: J. Goutsias, R. P. S. Mahler, H. T. Nguyen (Eds.). Random Sets. Theory and Applications. Springer-Verlag, New York 1997.

    Google Scholar 

  52. E. Kriegler, H. Held. Global mean temperature projections for the 21st century using random sets. In: J.-M. Bernard, T. Seidenfeld, M. Zaffalon (Eds.). ISIPTA’03. Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications. Proceedings in Informatics 18, Carlton Scientific 2003.

    Google Scholar 

  53. H. Lehar, G. Niederwanger, G. Hofstetter. FE ultimate load analyses of pilesupported pipelines — tackling uncertainty in a real design problem. In this volume.

    Google Scholar 

  54. K. Lesny, A. Kisse, W. Richwien: Proof of foundation stability using a consistent failure model. In: R. Pöttler, H. Klapperich, H. F. Schweiger (Eds.). Probabilistics in GeoTechnics: Technical and Economic Risk Estimation. United Engineering Foundation, Verlag Glückauf, Essen, 2002, 95–103.

    Google Scholar 

  55. H. Lessmann, J. Mühlögger, M. Oberguggenberger. Netzplantechnik mit unscharfen Methoden. Bauingenieur 69 (1994) 469–478.

    Google Scholar 

  56. R.-J. Li, E. S. Lee. Analysis of fuzzy queues. Computers Math. Appl., 17 (1989) 1143–1147.

    MathSciNet  Google Scholar 

  57. B. S. L. P. de Lima, N. F. F. Ebecken. A comparison of models for uncertainty analysis by the finite element method. Finite Elements in Analysis and Design, 34 (2000), 211–232.

    Google Scholar 

  58. M. Loève. Probability Theory II. 4th Ed., Springer-Verlag, New York 1978.

    Google Scholar 

  59. C. R. Manski. Partial Identification of Probability Distributions. Springer-Verlag, New York 2003.

    Google Scholar 

  60. M. A. Meyer, J. M. Booker. Eliciting and Analyzing Expert Judgement. SIAM, Philadelphia PA, 2001.

    Google Scholar 

  61. B. Möller. Fuzzy-Modellierung in der Baustatik. Bauingenieur, 72 (1997) 75–84.

    Google Scholar 

  62. B. Möller. Fuzzy randomness — a contribution to imprecise probability. Zeitschrift f. Angewandte Mathematik u. Mechanik, to appear.

    Google Scholar 

  63. P.-A. Monney. A Mathematical Theory for Statistical Evidence. Physica-Verlag, Heidelberg 2003.

    Google Scholar 

  64. A. Neumaier. Interval Methods for Systems of Equations. Cambridge University Press, Cambridge 1990.

    Google Scholar 

  65. A. Neumaier. Fuzzy modeling in terms of surprise. Fuzzy Sets and Systems 135, 21–38 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  66. A. Neumaier. Clouds, fuzzy sets and probability intervals. Reliable Computing, 10 (2004), 249–272.

    Article  MathSciNet  MATH  Google Scholar 

  67. H. T. Nguyen, T. Wang. Belief functions and random sets. In: J. Goutsias, R. P. S. Mahler, H. T. Nguyen (Eds.). Random Sets. Theory and Applications. Springer-Verlag, New York 1997.

    Google Scholar 

  68. M. Oberguggenberger. Queueing models with fuzzy data in construction management. In this volume.

    Google Scholar 

  69. M. Oberguggenberger, W. Fellin. The fuzziness and sensitivity of failure probabilities. In this volume.

    Google Scholar 

  70. M. Oberguggenberger, F. Russo. Fuzzy, probabilistic and stochastic modelling of an elastically bedded beam. In this volume.

    Google Scholar 

  71. M. Oberguggenberger, G. I. Schuëller, K. Marti (Eds.). Special Issue: Fuzzy Sets, Imprecise Probability and Stochastics in Engineering. Zeitschrift f. Angewandte Mathematik u. Mechanik, to appear.

    Google Scholar 

  72. W. L. Oberkampf, J. C. Helton. Investigation of Evidence Theory for Engineering Applications. American Institute of Aeronautics and Astronautics, Reston VA, AIAA 2002-1569, Non-Deterministic Approaches Forum, Seattle, WA (2002).

    Google Scholar 

  73. W. L. Oberkampf, J. C. Helton, K. Sentz. Mathematical Representation of Uncertainty. American Institute of Aeronautics and Astronautics, Reston VA, AIAA 2001-1645, Non-Deterministic Approaches Forum,Seattle, WA (2001).

    Google Scholar 

  74. A. O’Hagan. Eliciting expert beliefs in substantial practical applications. The Statistician 47 (1998), 21–35.

    Google Scholar 

  75. A. Ostermann. Sensitivity analysis. In this volume.

    Google Scholar 

  76. K. Popper. The propensity interpretation of the calculus of probability, and the quantum theory. In: S. Körner ((Ed.), Observation and Interpretation, London 1957.

    Google Scholar 

  77. K. Popper. Logik der Forschung. 10. Aufl., J. C. B. Mohr (Paul Siebeck), Tübingen 1994.

    Google Scholar 

  78. R. Rackwitz. Reviewing probabilistic soils modelling. Computers and Geotechnics 26 (2000), 199–223

    Article  Google Scholar 

  79. T. J. Ross, J. M. Booker, W. J. Parkinson (Eds.). Fuzzy Logic and Probability Applications — Bridging the Gap. SIAM, Philadelphia PA, ASA, Alexandria VA, 2002.

    Google Scholar 

  80. L. J. Savage. The Foundations of Statistics. Wiley, New York 1954.

    Google Scholar 

  81. H. R. Schneider. Definition and determination of characteristic soil properties. In: Proceedings of the Fourteenth International Conference on Soil Mechanics and Foundation Engineering, Hamburg. A. A. Balkema, Rotterdam 1997, 2271–2274.

    Google Scholar 

  82. G. I. Schuëller, H. J. Pradlwarter, P. S. Koutsourelakis. A critical appraisal of reliability estimation procedures for high dimensions. Probabilistic Engineering Mechanics, to appear.

    Google Scholar 

  83. R. Seising (Ed.). Fuzzy Theorie und Stochastik. Vieweg, Braunschweig 1999.

    Google Scholar 

  84. G. L. S. Shackle. Decision — order and time in human affairs. Cambridge University Press, Cambridge 1969.

    Google Scholar 

  85. G. Shafer. A Mathematical Theory of Evidence. Princeton University Press, Princeton, NJ 1976.

    Google Scholar 

  86. F. Tonon. On the use of random set theory to bracket the results of Monte Carlo simulations. Reliable Computing, 10 (2004), 107–137.

    Article  MathSciNet  MATH  Google Scholar 

  87. F. Tonon, A. Bernardini, I. Elishakoff. Hybrid Analysis of uncertainty: probability, fuzziness and anti-optimization Chaos, Solitons and Fractals 12 (2001), 1403–1414.

    Article  Google Scholar 

  88. F. Tonon, A. Bernardini, A. Mammino. Determination of parameters range in rock engineering by means of random set theory. Reliabilty Engineering & System Safety 70, 241–261 (2000).

    Google Scholar 

  89. F. Tonon, A. Bernardini, A. Mammino. Reliability analysis of rock mass response by means of random set theory. Reliabilty Engineering & System Safety 70, 263–282 (2000).

    Google Scholar 

  90. R. Viertl. Einführung in die Stochastik. Mit Elementen der Bayes-Statistik und der Analyse unscharfer Information. 3. Aufl., Springer-Verlag, Wien 2003.

    Google Scholar 

  91. P. Walley. Statistical Reasoning with Imprecise Probabilities. Chapman & Hall, London 1991.

    Google Scholar 

  92. P. Walley. Towards a unified theory of imprecise probability. International Journal of Approximate Reasoning 24 (2000), 125–148.

    Article  MathSciNet  MATH  Google Scholar 

  93. Z. Wang, G. J. Klir. Fuzzy Measure Theory. Plenum Press, New York, 1992.

    Google Scholar 

  94. K. Weichselberger. Elementare Grundbegriffe einer allgemeinen Wahrscheinlichkeitsrechnung I. Physica-Verlag, Heidelberg 2001.

    Google Scholar 

  95. O. Wolkenhauer. Data Engineering. Fuzzy Mathematics in Systems Theory and Data Analysis. John Wiley & Sons, New York 2001.

    Google Scholar 

  96. L. A. Zadeh. Fuzzy sets. Information and Control, 8 (1965) 338–353.

    Article  MathSciNet  MATH  Google Scholar 

  97. L. A. Zadeh. The concept of a linguistic variable and its applications to approximate reasoning, Part 1-3. Information Sci. 8 (1975), 199–250; 301-357; 9 (1975), 43-80.

    MathSciNet  MATH  Google Scholar 

  98. L. A. Zadeh. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1 (1978), 3–28.

    Article  MathSciNet  MATH  Google Scholar 

  99. H.-J. Zimmermann. Fuzzy Sets, Decision Making, and Expert Systems. Kluwer Academic Publishers, Boston 1987.

    Google Scholar 

  100. H.-J. Zimmermann. Fuzzy Set Theory — and Its Applications. 2nd Ed., Kluwer Academic Publishers, Boston 1991.

    Google Scholar 

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Authors and Affiliations

  1. Institut für Technische Mathematik, Geometrie und Bauinformatik, Universität Innsbruck, Austria

    Michael Oberguggenberger

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  1. Michael Oberguggenberger
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Editors and Affiliations

  1. Institut für Geotechnik und Tunnelbau, Universität Innsbruck, Technikerstr. 13, 6020, Innsbruck, Austria

    Wolfgang Fellin

  2. Starkenbühel 304, 6073, Sistrans, Austria

    Heimo Lessmann

  3. Institut für Technische Mathematik, Geometrie und Bauinformatik, Universität Innsbruck, Technikerstr. 13, 6020, Innsbruck, Austria

    Michael Oberguggenberger

  4. Vieider Ingenieur GmbH, Rebschulweg 1/E, 39052, Kaltern an der Weinstraße, Italy

    Robert Vieider

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Oberguggenberger, M. (2005). The mathematics of uncertainty: models, methods and interpretations. In: Fellin, W., Lessmann, H., Oberguggenberger, M., Vieider, R. (eds) Analyzing Uncertainty in Civil Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26847-2_4

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  • DOI: https://doi.org/10.1007/3-540-26847-2_4

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  • Print ISBN: 978-3-540-22246-0

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