Multi-Dimensional Interpolations with Fuzzy Sets

Geographic phenomena are continuous and dynamic but are often represented with data that are static, discrete and crisp. Interpolation is a technique that uses discrete sample data to generate a continuous spatial representation of geographic phenomena. Further, fuzzy set theory represents one of the avenues to overcome the problems of static and crisp data representations. This chapter explores the benefits of integrating fuzzy sets theory and spatio-temporal interpolation techniques within geographical information systems (GIS) to address the multidimensionality of geographic phenomena. The fundamental theory of spatial interpolation using geographic sample data, together with an assessment of the inexactness of such data is presented. Moreover, fuzzy interpolation methods that use the concepts of fuzzy data, fuzzy numbers and fuzzy arithmetic to generate fuzzy surfaces are elaborated. Four case studies that use GIS-based fuzzy set reasoning to build multidimensional spatial or spatio-temporal interpolation methods are discussed.