Higher Order Vagueness in Geographical Information: Empirical Geographical Population of Type n Fuzzy Sets
- 189 Downloads
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.
Keywordsfuzzy sets mountains geomorphometry type 2 fuzzy sets vagueness uncertainty
We would like to thank Bob John of DeMontfort University for his comments on an earlier version of this paper. The Panorama DEM was supplied by Ordnance Survey through the EDINA service.
- 2.D.A. Bearhop. Munro’s Tables. Scottish Mountaineering Club: Edinburgh, Scotland, 1997.Google Scholar
- 4.D. Bennett (Ed.), The Munros: Hillwalker’s Guide. Volume 1. Scottish Mountaineering Club: Edinburgh, Scotland, 1999.Google Scholar
- 6.P.A. Burrough. Principles of Geographical Information Systems for Land Resources Assessment. Oxford University Press: Oxford, 1986.Google Scholar
- 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.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
- 10.P.F. Fisher. “Reconsideration of the viewshed function in terrain modelling,” Geographical Systems, Vol. 3:33–58, 1996.Google Scholar
- 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.
- 19.R. Keefe and P. Smith (Eds.), Vagueness: A Reader. MIT: Cambridge, MA, 1996.Google Scholar
- 20.G.J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall: Englewood Cliff, NJ, 1995.Google Scholar
- 21.R. Kruse, J. Gebhardt, and F. Klawonn. Foundations of Fuzzy Systems. Wiley: Chichester, 1994.Google Scholar
- 23.Y.C. Leung. “Locational choice: A fuzzy set approach,” Geographical Bulletin, Vol. 15:28–34, 1979.Google Scholar
- 25.Y.C. Leung. Spatial Analysis and Planning under Imprecision. North Holland: Amsterdam, The Netherlands, 1988.Google Scholar
- 26.J.M. Mendel. Uncertain Rule-based Fuzzy Logic Systems: Introduction and New Directions. Prentice-Hall: Upper Saddle River, NJ, 2001.Google Scholar
- 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
- 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.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
- 36.R.M. Sainsbury. Paradoxes. 2nd edition, Cambridge University Press: Cambridge, 1995.Google Scholar
- 37.J. Seymour. “GB Waypoints: Munros,” in http://www.itatwork.freeserve.co.uk/waypoints/munros/index.htm Accessed 20 April 2003, 2003.
- 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.T. Williamson. Vagueness. Routledge: London, 1994.Google Scholar
- 45.L. Wittgenstein. Remarks on Colour. Blackwell: Oxford, 1977.Google Scholar
- 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.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.J. Wood. LandSerf, Version 2.1, in http://www.landserf.org, 2004.