GeoInformatica

, Volume 11, Issue 3, pp 311–330

Higher Order Vagueness in Geographical Information: Empirical Geographical Population of Type n Fuzzy Sets

Article

Abstract

Fuzzy set theory has been suggested as a means for representing vague spatial phenomena, and is widely known for directly addressing some of the issues of vagueness such as the sorites paradox. Higher order vagueness is widely considered a necessary component of any theory of vagueness, but it is not so well known that it too is competently modelled by Type n Fuzzy sets. In this paper we explore the fuzzy representation of higher order vagueness with respect to spatial phenomena. Initially we relate the arguments on philosophical vagueness to Type n Fuzzy sets. As an example, we move on to an empirical generation of spatial Type 2 Fuzzy sets examining the spatial extent of mountain peaks in Scotland. We show that the Type 2 Fuzzy sets can be populated by using alternative parameterisations of a peak detection algorithm. Further ambiguities could also be explored using other parameters of this and other algorithms. We show some novel answers to interrogations of the mountain peaks of Scotland. The conclusion of this work is that higher order vagueness can be populated for Type 2 and higher fuzzy sets. It does not follow that it is always necessary to examine these higher order uncertainties, but a possible advantage in terms of the results of spatial inquiry is demonstrated.

Keywords

fuzzy sets mountains geomorphometry type 2 fuzzy sets vagueness uncertainty 

References

  1. 1.
    C. Arnot, P.F. Fisher, R. Wadsworth, and J. Wellens. “Landscape metrics with ecotones: Pattern under uncertainty,” Landscape Ecology, Vol. 19:181–195, 2004.CrossRefGoogle Scholar
  2. 2.
    D.A. Bearhop. Munro’s Tables. Scottish Mountaineering Club: Edinburgh, Scotland, 1997.Google Scholar
  3. 3.
    B. Bennett. “What is a forest? On the vagueness of certain geographic concepts,” Topoi, Vol. 20:189–201, 2001.CrossRefGoogle Scholar
  4. 4.
    D. Bennett (Ed.), The Munros: Hillwalker’s Guide. Volume 1. Scottish Mountaineering Club: Edinburgh, Scotland, 1999.Google Scholar
  5. 5.
    J.C. Bezdek, R. Ehrlich, and W. Full. “2FCM: The fuzzy c-means clustering algorithm,” Computers & Geosciences, Vol. 10:191–203, 1984.CrossRefGoogle Scholar
  6. 6.
    P.A. Burrough. Principles of Geographical Information Systems for Land Resources Assessment. Oxford University Press: Oxford, 1986.Google Scholar
  7. 7.
    M. Duckham and J. Sharp. “Uncertainty and geographic information: Computational and critical convergence,” in P.F. Fisher and D.J. Unwin (Eds.), Re-presenting GIS. Wiley: Chichester, 113–124, 2005.Google Scholar
  8. 8.
    I.S. Evans. An integrated system of terrain analysis and slope mapping. Final report on grant DA-ERO-591-73-G0040, University of Durham: Durham, England, 1979.Google Scholar
  9. 9.
    K. Fine. “Vagueness, truth and logic,” Synthese, Vol. 30:265–300, 1975.CrossRefGoogle Scholar
  10. 10.
    P.F. Fisher. “Reconsideration of the viewshed function in terrain modelling,” Geographical Systems, Vol. 3:33–58, 1996.Google Scholar
  11. 11.
    P.F. Fisher. “Propagating effects of database generalization on the viewshed,” Transactions in GIS, Vol. 1:69–81, 1996.CrossRefGoogle Scholar
  12. 12.
    P.F. Fisher. “Improved modelling of elevation error with geostatistics,” GeoInformatica, Vol. 2:215–233, 1998.CrossRefGoogle Scholar
  13. 13.
    P.F. Fisher. “Sorites paradox and vague geographies,” Fuzzy Sets and Systems, Vol. 113:7–18, 2000.CrossRefGoogle Scholar
  14. 14.
    P.F. Fisher, J. Wood, and T. Cheng. “Where is Helvellyn? Multiscale morphometry and the mountains of the English Lake District,” Transactions of the Institute of British Geographers, Vol. 29:106–128, 2004.CrossRefGoogle Scholar
  15. 15.
    P.F. Fisher, J. Wood, and T. Cheng. “Fuzziness and ambiguity in multi-scale analysis of landscape morphometry,” in F. Petry, V. Robinson, and M. Cobb (Eds.), Fuzzy Modeling with Spatial Information for Geographic Problems. Springer: Berlin Heidelberg New York, 209–232, 2005.CrossRefGoogle Scholar
  16. 16.
    S. Gale, “Inexactness fuzzy sets and the foundation of behavioral geography,” Geographical Analysis, Vol. 4:337–349, 1972.CrossRefGoogle Scholar
  17. 17.
    L. Herrington and G. Pellegrini. “An advanced shape of country classifier: Extraction of surface features from DEMs,” in Proceedings of the 4th International Conference on Integrating GIS and Environmental Modelling (GIS/EM4), http://www.colorado.edu/research/cires/banff/pubpapers/205/ consulted 25 March 2005, 2002.
  18. 18.
    R.I. John. “Type 2 fuzzy sets: An appraisal of theory and applications,” International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, Vol. 6:563–576, 1998.CrossRefGoogle Scholar
  19. 19.
    R. Keefe and P. Smith (Eds.), Vagueness: A Reader. MIT: Cambridge, MA, 1996.Google Scholar
  20. 20.
    G.J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall: Englewood Cliff, NJ, 1995.Google Scholar
  21. 21.
    R. Kruse, J. Gebhardt, and F. Klawonn. Foundations of Fuzzy Systems. Wiley: Chichester, 1994.Google Scholar
  22. 22.
    L. Kulik. “Spatial vagueness and second order vagueness,” Spatial Cognition and Computation, Vol. 3:157–183, 2003.CrossRefGoogle Scholar
  23. 23.
    Y.C. Leung. “Locational choice: A fuzzy set approach,” Geographical Bulletin, Vol. 15:28–34, 1979.Google Scholar
  24. 24.
    Y.C. Leung. “On the imprecision of boundaries,” Geographical Analysis, Vol. 19:125–151, 1987.CrossRefGoogle Scholar
  25. 25.
    Y.C. Leung. Spatial Analysis and Planning under Imprecision. North Holland: Amsterdam, The Netherlands, 1988.Google Scholar
  26. 26.
    J.M. Mendel. Uncertain Rule-based Fuzzy Logic Systems: Introduction and New Directions. Prentice-Hall: Upper Saddle River, NJ, 2001.Google Scholar
  27. 27.
    J.M. Mendel and R.I. John. “Type 2 fuzzy sets made simple,” IEEE Transactions on Fuzzy Systems, Vol. 10:117–127, 2002.CrossRefGoogle Scholar
  28. 28.
    M. Mizumuto and K. Tanka. “Some properties of fuzzy sets of Type 2,” Information and Control, Vol. 31:312–340, 1976.CrossRefGoogle Scholar
  29. 29.
    F. Petry, V. Robinson, and M. Cobb (Eds.), Fuzzy Modeling with Spatial Information for Geographic Problems. Springer: Berlin Heidelberg New York, 2005.Google Scholar
  30. 30.
    J.S. Pipkin. “Fuzzy sets and spatial choice,” Annals of the Association of American Geographers, Vol. 68:196–204, 1978.CrossRefGoogle Scholar
  31. 31.
    V.B. Robinson. “Some implications of fuzzy set theory applied to geographic databases,” Computers, Environment and Urban Systems, Vol. 12:89–98, 1988.CrossRefGoogle Scholar
  32. 32.
    V.B. Robinson and A.H. Strahler. “Issues in designing geographic information systems under conditions of inexactness,” in Proceedings of the 10th International Symposium on Machine Processing of Remotely Sensed Data, pp. 198–204, Purdue University: Lafayette, 1984.Google Scholar
  33. 33.
    V.B. Robinson and D. Thongs. “Fuzzy set theory applied to the mixed pixel problem of multispectral landcover databases,” in B. Opitz (Ed.), Geographic Information Systems in Government A. Deerpak: Hampton, 871–885, 1986.Google Scholar
  34. 34.
    R.M. Sainsbury. “What is a vague object?,” Analysis, Vol. 49:99–103, 1989CrossRefGoogle Scholar
  35. 35.
    R.M. Sainsbury. “Is there higher order vagueness?,” Philosophical Quarterly, Vol. 41:167–182, 1991.CrossRefGoogle Scholar
  36. 36.
    R.M. Sainsbury. Paradoxes. 2nd edition, Cambridge University Press: Cambridge, 1995.Google Scholar
  37. 37.
    J. Seymour. “GB Waypoints: Munros,” in http://www.itatwork.freeserve.co.uk/waypoints/munros/index.htm Accessed 20 April 2003, 2003.
  38. 38.
    B. Smith and D.M. Mark. “Do mountains exist? Towards an ontology of landforms,” Environment and Planning B: Environment and Design, Vol. 30:411–427, 2003.CrossRefGoogle Scholar
  39. 39.
    R.A. Sorensen. “An argument for the vagueness of the ‘vague’,” Analysis, Vol. 45:134–137, 1985.CrossRefGoogle Scholar
  40. 40.
    A.C. Varzi. “Vagueness in geography,” Philosophy and Geography, Vol. 4:49–65, 2001.CrossRefGoogle Scholar
  41. 41.
    A.C. Varzi. “Higher-order vagueness and the vagueness of ‘vague’,” Mind, Vol. 112:295–298, 2003.CrossRefGoogle Scholar
  42. 42.
    J. Verstraete, G. de Tré, R. de Caluwe, and A. Hallez.“ Field based method for the modelling of fuzzy spatial data,” in F. Petry, V. Robinson, and M. Cobb (Eds.), Fuzzy Modeling with Spatial Information for Geographic Problems. Springer: Berlin Heidelberg New York, 41–69, 2005.CrossRefGoogle Scholar
  43. 43.
    T. Williamson. Vagueness. Routledge: London, 1994.Google Scholar
  44. 44.
    T. Williamson. “On the structure of higher order vagueness,” Mind, Vol. 108:127–143, 1999.CrossRefGoogle Scholar
  45. 45.
    L. Wittgenstein. Remarks on Colour. Blackwell: Oxford, 1977.Google Scholar
  46. 46.
    J. Wood. “Scale-based characterisation of digital elevation models,” in D. Parker (Ed.), Innovations in GIS 3. Taylor & Francis: London, 163–175, 1996.Google Scholar
  47. 47.
    J. Wood. “Modelling the continuity of surface form using digital elevation models,” in Proceedings of the 8th International Symposium on Spatial Data Handling, pp. 725–736, Simon Fraser University: Burnaby, British Columbia, 1998.Google Scholar
  48. 48.
    J. Wood. LandSerf, Version 2.1, in http://www.landserf.org, 2004.
  49. 49.
    C. Wright. “Is higher order vagueness coherent?,” Analysis, Vol. 52:129–139, 1992.CrossRefGoogle Scholar
  50. 50.
    R.R. Yager. “Fuzzy subsets of type II in decisions,” Journal of Cybernetics, Vol. 10:137–159, 1980.CrossRefGoogle Scholar
  51. 51.
    L. Zadeh. “Fuzzy sets,” Information and Control, Vol. 8:338–353, 1965.CrossRefGoogle Scholar
  52. 52.
    L. Zadeh. “The concept of a linguistic variable and its application to approximate reasoning—1,” Information Sciences, Vol. 8:199–249, 1975.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Information ScienceCity UniversityLondonUK
  2. 2.Department of Land Surveying and Geo-InformaticsThe Hong Kong Polytechnic UniversityKowloonHong Kong
  3. 3.Department of Geomatic EngineeringUniversity College LondonLondonUK

Personalised recommendations