written by Grant Hamilton

## Introduction

Fuzzy logic or fuzzy math is a branch of mathematics developed by L.A. Zadeh. Fuzzy math differs from conventional mathematics primarily in the area of set theory. For example in a conventional AND statement both statements must be true for the statement to be true. However, in fuzzy logic statements are not always true or false, they merely have varying levels of confidence.

**With fuzzy logic, membership in a category such as age is measured in degrees. Zadeh (2008) explains that “in fuzzy logic everything is or is allowed to be granulated, with a granule being a clump of attribute-values drawn together by indistinguishability, similarity, proximity or functionality.” **

Another classic application of fuzzy math is a fan thermostat that regulates the fan blades’ revolutions per minute based on the temperature. Using conventional mathematics, the fan could only be programmed to turn on when the temperature reached a certain level (“hot”) and then turn off if the temperature dropped below that level (“cold”). Using fuzzy logic the fan can be programmed to operate at a higher rpm as the temperature increases and operate at a lower rpm as the temperature decreases.

## GIS Applications

Conventional mathematics places limits the ability of a GIS to model the real world. Modelling degrees of a phenomenon (e.g. urbanization) presents a challenge with the traditional raster data model. Binary classes or “crisp” sets (e.g. “developed” or 1 and “undeveloped” or 0) are easier to model than degrees of development (e.g. undeveloped, rural, suburban, high-density urban, and everything in between) (DeMers, 2002, pp. 170 – 171).

ArcGIS offers several tools to incorporate fuzzy logic into models. The Fuzzy Overlay Tool allows users to classify data into classes that are not restricted to “either or” categories. For example multiple variables can be used to determine the degree of suitability for development of an area using landcover, soil type, distance from roads, and slope as inputs. The variables can also be weighted by importance to more realistically model suitability. The Fuzzy Overlay Tool results in a raster dataset that ranks suitability on a scale of 0 or completely unsuitable to 1 or most suitable (ESRI, 2012).

**Weighted Overlay is one of four tools that utilizes fuzzy logic within the Overlay toolset. It is located within the Spatial Analyst Tools toolbox.**

## Remote Sensing Applications

Sharp delineations between ecological zones, landcover types, or elevation classes rarely occur in nature. Fuzzy transitional zones can more realistically model what really occurs in natural landscapes (Tso and Mather, 2009, pp. 155 – 156). With fuzzy logic, “membership grade values are assigned that describe how close a pixel measurement is to the means of all classes” (Lillesand and Kiefer, p. 565). Fuzzy supervised classification uses training sites that are heterogeneous containing mixed pixels. The software can then assign membership grades in landcover categories such as “0.68 for class ‘forest,’ 0.29 for ‘street,’ and 0.03 for ‘grass'” – the sum of the grades must total 1 (Lillesand, p. 565). Object-based classification is one method of image classification that utilizes fuzzy logic. Sub-pixel classification is another remote sensing application of fuzzy logic. Spectral mixture analysis and multiple endmember spectral mixture analysis are two sub-pixel classification methods that leverage fuzzy logic.

## References

DeMers, M.N. (2002). *GIS modeling in raster*. New York: John Wiley and Sons.

ESRI. (2012). Applying fuzzy logic to overlay rasters. ArcGIS Help 10.1. http://resources.arcgis.com/en/help/main/10.1/index.html#//009z000000rv000000

Lillesand, T.M. and Kiefer, R.W. (2007). *Remote sensing and image interpretation*. New York: John Wily and Sons.

Tso, B. and Mather, P.M. (2009). *Classification methods for remotely sensed data*. Boca Raton, FL: CRC Press.

Wang, F. (1990). Fuzzy supervised classification of remote sensing images. *IEEE Transactions on Geoscience and Remote Sensing*. 28 (2), pp.194,201, March. doi: 10.1109/36.46698

- Description of fuzzy supervised classification using Landsat MSS imagery.

Zadeh, L.A. (1965). Fuzzy sets. *Information and Control* 8(3), pp. 338–353. doi:10.1016/S0019-9958(65)90241-X

Zadeh, L.A. (2008). Fuzzy logic. Scholarpedia, 3(3):1766. http://www.scholarpedia.org/article/Fuzzy_logic