**Summary**

### Overview of FIS

Fuzzy inference systems are a method to deal with ambiguity in parameters seen in many real world problems. They are applied in a wide variety of fields with documented applications in environmental impact assessments, geology, engineering and many others (Shepard, R.B. 2005, Ross, T.J., Booker, J.M. et al 2002). The main building blocks of a FIS are fuzzy sets, fuzzy membership functions, fuzzy operation methods, rule implication methods, aggregation methods and defuzzification methods. This section will explore these concepts in more depth.#### Fuzzy Sets

The basic unit to a fuzzy inference system is the fuzzy set. Examples of fuzzy sets are shown in Figure 41 below. The domain or universe of discourse, which is represented by the x-axis, defines all possible values for a variable. The support set, also on the x-axis, is all possible values that can have a degree of membership to the fuzzy set. The y-axis in Figure 41 is the degree of membership to the fuzzy set. A values position in relation to the shape of the fuzzy set determines its degree of membership to that fuzzy set; similar to a function in classical mathematics.There are 4 main shapes that fuzzy sets take on: triangular, trapezoidal, Gaussian bell-shaped and singleton[1]. 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.

#### Fuzzy Operators

#### Aggregation

Due to the concept that a particular value can have degrees of membership in multiple fuzzy sets, many FIS rules can apply to the given inputs. To handle this, a FIS uses aggregation methods to combine all fuzzy sets created from the implication methods to create a final consequent fuzzy set. There are three main types: maximum, probabilistic and sum. The maximum method takes the maximum of each degree of membership and combines them. The probabilistic method acts as it did when used as an implication method, it adds all degrees of membership and from this subtracts there product. While the sum adds all degrees of membership to create the final output fuzzy set. This concept is explored in Figure 4. All of these aggregation methods create a range of values that need to be converted into a final single value, this is accomplished through defuzzification.#### Defuzzification

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.

[1] 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.

### Resources

- 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