Incorporating Expert Knowledge Continued from page 11 Membership Function Linear Description A linear increasing or decreasing membership between two inputs. A linearized sigmoid shape. where min and max are user inputs Sigmoid shape where large inputs have large membership Definition Large Small Sigmoid shape where small inputs have large membership MS Large Sigmoid shape defined by the mean and standard deviation where large inputs have large memberships. Sigmoid shape defined by the mean and standard deviation, where small inputs have large memberships. A curved peak of membership over an intermediate value. where m = mean, s = standard deviation and b and a are multipliers provided by the user. MS Small where m = mean, s = standard deviation, and b and a are multipliers provided by the user. Near Gaussian A Gaussian peak of membership over an intermediate value. The experts can visualize the membership values displayed on the map. Each named class is assigned a membership value by the expert. Applied to slightly adjust a membership function. Applied to slightly adjust a membership function. Membership is defined based on the classes in the symbolization in the Map document table of contents. Membership is defined by entering the values times a multiplier into a reclassification table. Square root of membership. Membership squared. Table of Contents (TOC) Categorical Somewhat Very Table 2: Summary of fuzzy membership functions implemented in the Fuzzy Membership tool in ArcGIS 9.4. In addition, there are two hedges (Somewhat and Very) that qualify the membership. These functions have been found most useful in spatial modeling problems. The first five membership functions produce a sigmoid shape of the membership, which is used commonly in many fuzzy logic applications. Experience with these functions can be gained rapidly by implementing them in a spreadsheet and adjusting the parameters. 12 ArcUser Spring 2010 www.esri.com