Kriging and epistemic uncertainty: A critical discussion

Studies in Fuzziness and Soft Computing

Volumn 256, Issue , 2010, Pages 269-305

  Loquin, Kevin ^a   Dubois, Didier ^a  

^a IRIT   (France)

Author keywords

epistemic uncertainty; fuzzy subset; geostatistics; kriging; possibility theory; random function; variogram

Indexed keywords

EID: 77958559752     PISSN: 14349922     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-642-14755-5_11     Document Type: Article

References (59)

1. Armstrong, M.: Positive Definiteness is Not Enough. Math. Geol. 24, 135-143 (1990)
2. Armstrong, M., Jabin, R.: Variogram Models Must Be Positive Definite. Math. Geol. 13, 455-459 (1981)
3. Aumann, R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1-12 (1965)
4. Bardossy, A., Bogardi, I., Kelly, W.E.: Imprecise (fuzzy) information in geostatistics. Math. Geol. 20, 287-311 (1988)
5. Bardossy, A., Bogardi, I., Kelly, W.E.: Kriging with imprecise (fuzzy) variograms. I: Theory. Math. Geol. 22, 63-79 (1990)
6. Bardossy, A., Bogardi, I., Kelly, W.E.: Kriging with imprecise (fuzzy) variograms. II: Application. Math. Geol. 22, 81-94 (1990)
7. Berger, J.: An overview of robust Bayesian analysis [with Discussion]. Test 3, 5-124 (1994)
8. 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)
9. Chilés, J.P., Delfiner, P.: Geostatistics, Modeling Spatial Uncertainty. Wiley, New York (1999)
10. 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)
11. Cressie, N.A.C.: The origins of kriging. Math. Geol. 22, 239-252 (1990)
12. Cressie, N.A.C.: Statistics for Spatial Data. Revised edn. John Wiley & Sons, New York (1993)
13. de Finetti, B.: Theory of probability: a critical introductory treatment. John Wiley & Sons, New York (1974)
14. de Oliveira, V., Kedem, B., Short, D.A.: Bayesian prediction of transformed Gaussian random fields. Journal of the American Statistical Association 92, 1422-1433 (1997)
15. Diamond, P.: Fuzzy least squares. Information Sciences 46, 141-157 (1988)
16. Diamond, P.: Interval-valued random functions and the kriging of intervals. Math. Geol. 20, 145-165 (1988)
17. Diamond, P.: Fuzzy kriging. Fuzzy Sets and Systems 33, 315-332 (1989)
18. Dubois, D.: Possibility theory and statistical reasoning. Computational Statistics & Data Analysis 51, 47-69 (2006)
19. Dubois, D., Foulloy, L., Mauris, G., Prade, H.: Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities. Reliable Computing 10, 273-297 (2004)
20. Dubois, D., Prade, H.: Additions of interactive fuzzy numbers. IEEE Trans. on Automatic Control 26(4), 926-936 (1981)
21. Dubois, D., Prade, H.: Possibility Theory. Plenum Press, New York (1988)
22. Dubrule, O.: Comparing splines and kriging. Comp. Geosci. 10, 327-338 (1984)
23. Dubrule, O., Kostov, C.: An interpolation method taking into account inequality constraints: I. Methodology. Math. Geol. 18, 33-51 (1986)
24. Emery, X.: Disjunctive kriging with hard and imprecise data. Math. Geol. 35, 699-718 (2003)
25. Ferson, S., Ginzburg, L., Kreinovich, V., Longpŕe, L., Aviles, M.: Computing variance for interval data is NP-hard. SIGACT News 33, 108-118 (2002)
26. Gaudard, M., Karson, M., Linder, E., Sinha, D.: Bayesian spatial prediction. Environmental and Ecological Statistics 6, 147-171 (1999)
27. Goldstein, M., Wooff, D.: Bayes Linear Statistics. In: Theory and Methods, Wiley, Chichester (2007)
28. Goovaerts, P.: Geostatistics for Natural Resources Evaluation. Oxford Univ. Press, NewYork (1997)
29. Handcock, M.S., Stein, M.L.: A Bayesian analysis of kriging. Technometrics 35, 403-410 (1993)
30. Hanss, M.: The transformation method for the simulation and analysis of systems with uncertain parameters. Fuzzy Sets and Systems 130, 277-289 (2002)
31. Hartigan, J.A.: Linear Bayesian methods. J. Royal Stat. Soc. Ser. B 31, 446-454 (1969)
32. Helton, J.C., Oberkampf, W.L.: Alternative representations of epistemic uncertainty. Reliability Engineering and System Safety 85, 1-10 (2004)
33. Hukuhara, M.: Integration des applications measurables dont la valeur est un compact convexe. Funkcialaj Ekvacioj 10, 205-223 (1967)
34. Journel, A.G., Huijbregts, C.J.: Mining Geostatistics. Academic Press, New York (1978)
35. Journel, A.G.: The deterministic side of geostatistics. Math. Geol. 17, 1-15 (1985)
36. Journel, A.G.: Geostatistics: Models and Tools for the Earth Sciences. Math. Geol. 18, 119-140 (1986)
37. Journel, A.G.: Constrained interpolation and qualitative information - The soft kriging approach. Math. Geol. 18, 269-286 (1986)
38. Kostov, C., Dubrule, O.: An interpolation method taking into account inequality constraints: II. Practical approach. Math. Geol. 18, 53-73 (1986)
39. Krige, D.G.: A statistical approach to some basic mine valuation problems on theWitwatersrand. Journal of the Chemical, Metallurgical and Mining Society of South Africa 52, 119-139 (1951)
40. Loquin, K., Dubois, D.: Kriging with ill-known variogram and data. In: Deshpande, A., Hunter, A. (eds.) SUM 2010. LNCS (LNAI), vol. 6379, pp. 219-232. Springer, Heidelberg (2010)
41. Mallet, J.L.: Ŕegression sous contraintes lińeaires: application au codage des variables aĺeatoires. Revue de Statistiques appliqúees 28, 57-68 (1980)
42. Matheron, G., Blondel, F.: Trait́e de ǵeostatistique appliqúee. Editions Technip, Paris (1962)
43. Matheron, G.: Le krigeage universel. Cahiers du Centre de Morphologie Math́ematique de Fontainebleau, Fasc. 1, École des Mines de Paris (1969)
44. Matheron, G.: Random Sets and Integral Geometry. JohnWiley & Sons, NewYork (1975)
45. Matheron, G.: Estimer et choisir: essai sur la pratique des probabilit́es. École des Mines de Paris (1978)
46. Matheron, G.: Suffit-il pour une covariance d'âetre de type positif? Études Ǵeostatistiques V, SÉminaire CFSG sur la Ǵeostatistique, Fontainebleau, Sciences de la Terre Informatiques (1987)
47. Matheron, G.: The Internal Consistency of Models in Geostatistics. In: Armstrong, M. (ed.) Geostatistics, Proceedings of the Third International Geostatistics, Avignon, pp. 21-38. Kluwer Academic Publishers, Dordrecht (1989)
48. Omre, H.: Bayesian Kriging - merging observations and qualified guesses in kriging. Math. Geol. 19, 25-39 (1987)
49. Puri, M.L., Ralescu, D.A.: Fuzzy random variables. J. Math. Anal. Appl. 114, 409-422 (1986)
50. Rios Insua, D., Ruggieri, F.: Robust Bayesian Analysis. Springer, Berlin (2000)
51. Shafer, G.: A mathematical theory of evidence. Princeton University Press, Princeton (1976)
52. Shafer, G., Vovk, V.: Probability and Finance: It's Only a Game!Wiley, New York (2001)
53. Srivastava, R.M.: Philip and Watson-Quo vadunt? Math. Geol. 18, 141-146 (1986)
54. Taboada, J., Rivas, T., Saavedra, A., Ord́oñez, C., Bastante, F., Giŕaldez, E.: Evaluation of the reserve of a granite deposit by fuzzy kriging. Engineering Geol. 99, 23-30 (2008)
55. Walley, P.: Statistical reasoning with imprecise probabilities. Chapman and Hall, London (1991)
56. Watson, G.S.: Smoothing and interpolation by kriging and with splines. Math. Geol. 16, 601-615 (1984)
57. Yaglom, A.M.: An introduction to the theory of stationary random functions. Courier Dover Publications (2004)
58. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338-353 (1965)
59. Zadeh, L.A.: Fuzzy sets as a basis for a theory of ossibility. Fuzzy Sets and Systems 1, 3-28 (1978)