Fuzzy partitioning of continuous attributes through discretization methods to construct fuzzy decision tree classifiers

Abstract

Generating Membership Functions (MFs) from data is one of the fundamental challenges associated with the applications of fuzzy set theory. This paper proposes a new two-step algorithm, which uses discretization methods for initial partitioning, to generate MFs from data. In the first step, discretization algorithm divides domain of attributes to several partitions, and then, in the second step, an MF is defined on each partition. Four different methods are proposed to define MFs in the second step: the first method is based on partition width, the second is based on standard deviation of examples, the third is based on Coverage Rate of Neighbor Partitions (NPCR) and the last one is based on Coverage Rate of Partition (PCR). Coverage rate of partition and coverage rate of neighbor partition are two new introduced parameters, which can be used for MF generation. In addition, this paper proposes a new MF generation algorithm, called Fuzzy Entropy Based Fuzzy Partitioning (FEBFP), which is a specific version of the proposed two-step algorithm with some modifications. FEBFP uses fuzzy entropy of partition to generate MFs and involves the parameters of MFs in the process of MF generation to combine two steps of the algorithm. Non-parametric statistical tests are used to compare Fuzzy Decision Trees (FDTs) induced using the MFs generated by the proposed methods (employing different discretization algorithms as well as four MF generation methods). Experimental results show that eight methods outperform the others in terms of both accuracy and number of nodes. Among them, trapezoidal MFs that are defined by PCR on partitions generated by Zeta discretization algorithm, outperform the others when the accuracy and complexity of FDT have the same degree of importance. Moreover, the results show that the PCR and NPCR MF definition methods perform better than the other ones.