More sophisticated membership functions

For example, if we have class allocations concerning individual points/pixels of a set, and we are interested in the whole set, we may apportion the membership of the whole set, according to the fraction of its elements that belong to each class. This is the situation of identifying the slope of a region using grid data stored in a GIS: if 40% of the points have slope in class "gentle" and 60% in class "average", we may say that the membership function of the region to class "gentle" slope is 40% and to class "average" slope is 60%. This is a straight forward application of the philosophy of Fuzzy membership to a set.

An alternative way of thinking, is to go down to the measurement that was made to decide the class of an individual point. Any measurement is subject to errors. Let us assume that the errors are Gaussianly distributed. Then if we measure a slope of 19%, there is a probability that in reality is 20% or 18%, a smaller probability that it is 2% or 17%, etc., according to how wide the distribution of errors is. This approach was carefully investigated by Sasikala, Petrou and Kittler in a paper they published in EARSeL Advances in Remote Sensing,Vol 4(4), pp 97-105.(Click here to obtain an acrobat version of their paper). If the measurement errors are assumed to be uniformly distributed over an interval, the membership functions take the forms of triangular and trapezium functions, like those defined in the previous pages.

 Both the above approaches to defining the membership functions were used by Sasikala and Petrou in a paper published in the journal Fuzzy Sets and Systems. This paper was concerned with predicting the risk of desertification of burned forests in the Mediterranean region. (Click here for an acrobat version of the paper). However, the paper will be easier to understand  if you go on first to learn how you combine values of the membership functions to draw conclusions.