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Kriging with Ill-Known Variogram and Data

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Scalable Uncertainty Management (SUM 2010)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6379))

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Abstract

Kriging consists in estimating or predicting the spatial phenomenon at non sampled locations from an estimated random function. Although information necessary to properly select a unique random function model seems to be partially lacking, geostatistics in general, and the kriging methodology in particular, does not account for the incompleteness of the information that seems to pervade the procedure. On the one hand, the collected data may be tainted with errors that can be modelled by intervals or fuzzy intervals. On the other hand, the choice of parameter values for the theoretical variogram, an essential step, contains some degrees of freedom that is seldom acknowledged. In this paper we propose to account for epistemic uncertainty pervading the variogram parameters, and possibly the data set, by means of fuzzy interval uncertainty. We lay bare its impact on the kriging results, improving on previous attempts by Bardossy and colleagues in the late 1980’s.

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References

  1. Aumann, R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1–12 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bardossy, A., Bogardi, I., Kelly, W.E.: Imprecise (fuzzy) information in geostatistics. Math. Geol. 20, 287–311 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bardossy, A., Bogardi, I., Kelly, W.E.: Kriging with imprecise (fuzzy) variograms. I: Theory. Math. Geol. 22, 63–79 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bardossy, A., Bogardi, I., Kelly, W.E.: Kriging with imprecise (fuzzy) variograms. I: Application. Math. Geol. 22, 81–94 (1990)

    MATH  MathSciNet  Google Scholar 

  5. Baudrit, C., Dubois, D., Couso, I.: Joint propagation of probability and possibility in risk analysis: Towards a formal framework. International Journal of Approximate Reasoning 45(1), 82–105 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Baudrit, C., Dubois, D., Perrot, N.: Representing parametric probabilistic models tainted with imprecision. Fuzzy Sets and Systems 159, 1913–1928 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Berger, J.O., de Oliveira, V., Sanso, B.: Objective Bayesian analysis of spatially correlated data. Journal of the American Statistical Association 96, 1361–1374 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chilès, J.P., Delfiner, P.: Geostatistics, Modeling Spatial Uncertainty. Wiley, Chichester (1999)

    MATH  Google Scholar 

  9. Couso, I., Dubois, D.: On the variability of the concept of variance for fuzzy random variables. IEEE Transactions on Fuzzy Systems 17(5), 1070–1080 (2009)

    Article  Google Scholar 

  10. Cressie, N.A.C.: The origins of kriging. Math. Geol. 22, 239–252 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  11. Diamond, P.: Fuzzy least squares. Inf. Sci. 46, 141–157 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Diamond, P.: Interval-valued random functions and the kriging of intervals. Math. Geol. 20, 145–165 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  13. Diamond, P.: Fuzzy kriging. Fuzzy Sets and Systems 33, 315–332 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dubois, D.: Possibility theory and statistical reasoning. Computational Statistics & Data Analysis 51, 47–69 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Dubois, D., Foulloy, L., Mauris, G., Prade, H.: Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities. Reliable Computing 10, 273–297 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dubois, D., Kerre, E., Mesiar, R., Prade, H.: Fuzzy interval analysis. In: Dubois, D., Prade, H. (eds.) The Handbook of Fuzzy Sets. Fundamentals of Fuzzy Sets, vol. I, pp. 483–581. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  17. Dubois, D., Prade, H.: Additions of fuzzy interactive numbers. IEEE Transactions on Automatic Control 26, 926–936 (1981)

    Article  MathSciNet  Google Scholar 

  18. Dubois, D., Prade, H.: Possibility Theory. Plenum Press, New York (1988)

    MATH  Google Scholar 

  19. Dubois, D., Prade, H.: When upper probabilities are possibility measures. Fuzzy Sets and Systems 49, 65–74 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  20. Dubrule, O.: Comparing splines and kriging. Comp. Geosci. 10, 327–338 (1984)

    Article  Google Scholar 

  21. Ferson, S.: What Monte Carlo methods cannot do. Human and Ecological Risk Assessment: An International Journal 2(4), 990–1007 (1996)

    Google Scholar 

  22. Ferson, S., Troy Tucker, W.: Sensitivity analysis using probability bounding. Reliability Engineering and System Safety 91, 1435–1442 (2006)

    Article  Google Scholar 

  23. Fortin, J., Dubois, D., Fargier, H.: Gradual numbers and their application to fuzzy interval analysis. IEEE Transactions on Fuzzy Systems 16, 388–402 (2008)

    Article  Google Scholar 

  24. Goovaerts, P.: Geostatistics for Natural Resources Evaluation. Oxford Univ. Press, New-York (1997)

    Google Scholar 

  25. Handcock, M.S., Stein, M.L.: A Bayesian analysis of kriging. Technometrics 35, 403–410 (1993)

    Article  Google Scholar 

  26. Hanss, M.: The transformation method for the simulation and analysis of systems with uncertain parameters. Fuzzy Sets and Systems 130, 277–289 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Helton, J.C., Oberkampf, W.L.: Alternative representations of epistemic uncertainty. Reliability Engineering and System Safety 85, 1–10 (2004)

    Article  Google Scholar 

  28. Journel, A.G., Huijbregts, C.J.: Mining Geostatistics. Academic Press, New York (1978)

    Google Scholar 

  29. Journel, A.G.: The deterministic side of geostatistics. Math. Geol. 17, 1–15 (1985)

    Article  MathSciNet  Google Scholar 

  30. Journel, A.G.: Geostatistics: Models and Tools for the Earth Sciences. Math. Geol. 18, 119–140 (1986)

    Article  MathSciNet  Google Scholar 

  31. Krige, D.G.: A statistical approach to some basic mine valuation problems on the Witwatersrand. Journal of the Chemical, Metallurgical and Mining Society of South Africa 52, 119–139 (1951)

    Google Scholar 

  32. Loquin, K., Dubois, D.: Kriging and epistemic uncertainty: a critical discussion. In: Jeansoulin, R., Papini, O., Prade, H., Schockaert, S. (eds.) Methods for Handling Imperfect Spatial Information. Springer, Heidelberg (2009)

    Google Scholar 

  33. Loquin, K., Strauss, O., Crouzet, J.F.: Possibilistic signal processing: How to handle noise? International Journal of Approximate Reasoning, special issue selected papers ISIPTA 2009 (2009) (in Press)

    Google Scholar 

  34. Matheron, G., Blondel, F.: Traité de géostatistique appliquée. Editions Technip (1962)

    Google Scholar 

  35. Matheron, G.: Le krigeage Transitif. Unpublished note, Centre de Morphologie Mathmatique de Fontainebleau (1967)

    Google Scholar 

  36. Matheron, G.: Estimer et choisir: essai sur la pratique des probabilités. ENSMP (1978)

    Google Scholar 

  37. Puri, M.L., Ralescu, D.A.: Fuzzy random variables. J. Math. Anal. Appl. 114, 409–422 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  38. Shafer, G.: A mathematical theory of evidence. Princeton University Press, Princeton (1976)

    MATH  Google Scholar 

  39. Shafer, G., Vovk, V.: Probability and Finance: It’s Only a Game! Wiley, New York (2001)

    Book  Google Scholar 

  40. Srivastava, R.M.: Philip and Watson–Quo vadunt? Math. Geol. 18, 141–146 (1986)

    Article  MathSciNet  Google Scholar 

  41. Taboada, J., Rivas, T., Saavedra, A., Ordóñez, C., Bastante, F., Giráldez, E.: Evaluation of the reserve of a granite deposit by fuzzy kriging. Engineering Geol. 99, 23–30 (2008)

    Article  Google Scholar 

  42. Walley, P.: Statistical reasoning with imprecise probabilities. Chapman and Hall, Boca Raton (1991)

    MATH  Google Scholar 

  43. Watson, G.S.: Smoothing and interpolation by kriging and with splines. Math. Geol. 16, 601–615 (1984)

    Article  Google Scholar 

  44. Yaglom, A.M.: An introduction to the theory of stationary random functions. Courier Dover Publications (2004)

    Google Scholar 

  45. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. TSI department, Institut TELECOM ParisTech, 46, rue Barrault, 75013, Paris, France

    Kevin Loquin

  2. RPDMP department, Institut de Recherche en Informatique de Toulouse, 118 Route de Narbonne, 31062, Toulouse Cedex 9, France

    Didier Dubois

Authors
  1. Kevin Loquin
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  2. Didier Dubois
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Maryland, MD 20910, College Park, USA

    Amol Deshpande

  2. Department of Computer Science, University College London, Gower Street, WC1E 6BT, London, UK

    Anthony Hunter

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Loquin, K., Dubois, D. (2010). Kriging with Ill-Known Variogram and Data. In: Deshpande, A., Hunter, A. (eds) Scalable Uncertainty Management. SUM 2010. Lecture Notes in Computer Science(), vol 6379. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15951-0_23

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  • DOI: https://doi.org/10.1007/978-3-642-15951-0_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15950-3

  • Online ISBN: 978-3-642-15951-0

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