Influence of fuzzy norms and other heuristics on "Mixed fuzzy rule formation"

In Mixed Fuzzy Rule Formation [Int. J. Approx. Reason. 32 (2003) 67] a method to extract mixed fuzzy rules from data was introduced. The underlying algorithm’s performance is influenced by the choice of fuzzy t-norm and t-conorm, and a heuristic to avoid conflicts between patterns and rules of different classes throughout training. In the following addendum to [Int. J. Approx. Reason. 32 (2003) 67], we discuss in more depth how these parameters affect the generalization performance of the resulting fuzzy rule models. 2003 Elsevier Inc. All rights reserved. 1. Mixed fuzzy rule formation The training method described in [1] is based on an iterative algorithm. During each learning epoch, i.e. presentation of all training patterns, new fuzzy rules are introduced when necessary and existing ones are adjusted whenever a conflict occurs. For each pattern three main steps are executed. Firstly, if a new training pattern lies inside the support-region of an existing fuzzy rule of the * Corresponding author. Address: Department of Computer and Information Science, University of Konstanz, P.O. Box M712, 78457 Konstanz, Germany. E-mail addresses: [email protected] (T.R. Gabriel), [email protected] (M.R. Berthold). 0888-613X/$ see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.ijar.2003.10.004 www.elsevier.com/locate/ijar International Journal of Approximate Reasoning 35 (2004) 195–202 correct class, its core-region is extended in order to cover the new pattern (cover). Secondly, if the new pattern is not yet covered, a new fuzzy rule of the correct class is introduced (commit). The new example is assigned to its core, whereas the support-region is initialized ‘‘infinite’’, that is, the new fuzzy rule covers the entire domain. Lastly, if a new pattern is incorrectly covered by an existing fuzzy rule, the fuzzy point’s support-region is reduced so that the conflict is avoided (shrink). This heuristic for conflict avoidance aims to minimize the loss in volume [1]. In Section 2 three different heuristics to determine the loss in volume are compared in more detail. As discussed in [1], the algorithm terminates after only few iterations over the set of example patterns. The resulting set of fuzzy rules can then be used to classify new patterns by computing the overall fuzzy membership degree. The accumulated membership degrees over all input dimensions and across multiple rules are calculated using fuzzy t-norm resp. t-conorm. Again, [1] does not discuss different fuzzy norms, thus we present some choices in Section 3 in more detail and show how they can affect the classification accuracy. 2. Shrink heuristics As mentioned above, the training procedure relies on a heuristic which affects the strategy to avoid conflicts. We have several different choices for this conflict avoidance heuristic. One common approach is to shrink the fuzzy rule in dimension i (16 i6 n) that minimizes the loss in volume: imin 1⁄4 argmini1⁄41;...;nfVig: The loss in volume Vi of a fuzzy rule R (using trapezoid membership functions with parameters ha; b; c; di where ða; bÞ and ðc; dÞ bound the supportregion, and 1⁄2b; c the fuzzy rule’s core-region) is then: Vi 1⁄4 d i ð~x;RÞ Yn j1⁄41;j 61⁄4i d j ð~x;RÞ; where d i ð Þ (16 i6 n) is the distance between example pattern~x and the border (coreor support-region) of a fuzzy rule R in dimension i, and d j ð Þ (16 j6 n) indicates the distance to fuzzy rule R in dimension j. Later in this section, these two functions are defined more precisely. Furthermore, the loss in volume is normalized with respect to the overall volume: V norm i 1⁄4 d i ð~x;RÞ Qn j1⁄41;j 61⁄4i d j ð~x;RÞ Qn j1⁄41 d j ð~x;RÞ 1⁄4 d i ð~x;RÞ d i ð~x;RÞ: 196 T.R. Gabriel, M.R. Berthold / Internat. J. Approx. Reason. 35 (2004) 195–202