There are 4 main shapes that fuzzy sets take on: triangular, trapezoidal, Gaussian bell-shaped and singleton. The slope of the lines that make up the sides of the fuzzy set are the fuzzy membership function and provide a means to determine the degree of membership a value has in that set. Specifically steep slope would represent an exclusive fuzzy set with few values having high membership while a gradual slope is more inclusive and may have many values that have high degrees of membership. For example a triangular shape represents a phenomenon that has one ideal value and depending on the degree of slope of its sides will decrease in degree of membership rapidly with steep slopes or more gradually with more gradual sloping sides. While the presence of a horizontal line at the top of a trapezoid means this membership function has a larger range of values that can have full membership when compared to a triangular membership function. The Gaussian bell-shape is similar to the trapezoidal membership function but has a sharper decline from the high grade membership values. A singleton is a special membership function represented by a vertical line; it has one value that defines its support set and is said to function as a crisp value within a fuzzy logic system (Shepard 2005,Matlab FIS Documentation). These concepts are visually explored in Figure 1.
The final step is the conversion of the aggregated fuzzy set into a single value is known as defuzzification. There are 5 main options for defuzzification: centroid, bisector, middle of maximum, largest of maximum, and smallest of maximum. Middle of maximum, largest of maximum and smallest of maximum are all quite similar; they identify the maximum or highest degree of membership in the output fuzzy set and define the final value as the vertical middle value, largest value and smallest value respectively. If there is only maximum within the final output fuzzy set then the middle of maximum, largest of maximum, and smallest of maximum will all defuzzify to the same value. The bisector method outputs the value that if a vertical line were drawn from it would divide the region into two equal area sub-regions . The centroid method is the most widely used defuzzification method. It takes the aggregated fuzzy set and finds a center of mass to calculate the final single value (Shepard 2005, Matlab FIS Documentation). The bisector and centroid methods are quite similar and generally their output is quite similar and can often be the same value. These defuzzification methods and a comparison of their final output are presented in Figure 5.
Defuzzification is the final step of using a FIS as the final value is in a single value that can be more easily used rather than a range of values or an ambiguous understanding of a value. The main pieces of a FIS; a fuzzy set, membership functions, fuzzy operators, implication methods, aggregation methods, and defuzzification methods have been presented in this section. To connect all of these concepts in a fluid example all of these pieces are presented in the context of a functioning FIS used in a geomorphic change detection study in Figure 46. The purpose of this FIS is the same as in this paper to provide a structure to quantify the spatial parameters of slope and surface roughness and their influence on the final uncertainty of a surface produced from surveying under these conditions.
 There are many other membership functions, however these 4 are the most commonly used and these membership functions support many value ranges observed in earth sciences.
- Fuzzy Inference Systems for Modelling DEM Error - GCD Concept Reference
- See pages 97-108 of:
- Chapter 4 of Wheaton JM. 2008. Uncertainty in Morphological Sediment Budgeting of Rivers. Unpublished PhD Thesis, University of Southampton, Southampton, 412 pp.
- See page 142-146 of:
- Wheaton JM, Brasington J, Darby SE and Sear D. 2010. Accounting for Uncertainty in DEMs from Repeat Topographic Surveys: Improved Sediment Budgets. Earth Surface Processes and Landforms. 35 (2): 136-156. DOI: 10.1002/esp.1886.
- Matlab Fuzzy Logic Toolbox Documentation