Fuzzy Output Error

Many training algorithms use mean square error (MSE) for training when we have numeric data, with MSE also used to indicate the quality of the results. In many real world applications we are also interested in the number of samples which are “close enough” to the correct value and would also use the number of samples which could be mapped to correct classifications in reporting our results. We extend our previous work to use fully fuzzy output values, and demonstrate the benefits of our approach which allows choice in the shape of fuzzy output error membership functions which can be used to specialize the approach to particular domains or adapt to particular kinds of data sets. We show how we can extend our approach using fuzzy true positive, fuzzy true negative, fuzzy false positive, fuzzy false negative identifications as well as regression line weighted variants of these. Previous work Fuzzy Classification Error In our previous work (Mendis and Gedeon, 2008) we formulated the Sum of Fuzzy Classification Error (SYCLE) in the following way. We call it Fuzzy as it considers transition between good and bad classifications using several categories of error. First, we specify that both desired output and predicted output of an experiment are in the range [0, 1]. Next, we define a set of rules for the classification and these rules are visualised in the following figure. According to Figure 1, there are 3 categories of classifications that can occur, they are Good, Bad, and Very Bad (V. Bad). Now we assume the pair of predicted and desired values, of the ith input, respectively taken as X and Y coordinates of the point Pi on the 2 dimensional fuzzy classification error rule space, Figure 1. The fuzzy classification error of an arbitrary point Pi can be written as, FYCLE Pi ( ) = 0 if Pi ÎGood 0.5 if Pi ÎBad 1 if Pi ÎVeryBad ì í ï î ï Let us consider the 4 straight lines, B1, B2, G1, and G2, in Figure 1. In this experiment, they are equivalent to, B1 ≡ y − x − 0.5 G1 ≡ y − x − 0.2 G2 ≡ y − x + 0.2 B2 ≡ y−x+0.5 Now, The fuzzy classification error of an arbitrary point Pi can be calculated as, FYCLE Pi ( ) = 0 if G1 Pi ( ) £ 0 AND G2 Pi ( ) 3 0 0.5 if B1 Pi ( ) £ 0 AND G1 Pi ( ) > 0 ( ) OR G2 Pi ( ) < 0 AND B1 Pi ( ) 3 0 ( ) æ è ç ç ö ø ÷ ÷ 1 if B1 Pi ( ) > 0 AND B2 Pi ( ) < 0 ì