Fuzzy Handling of Continuous-Valued Attributes in Decision Trees

Classical crisp decision trees (DT) are widely applied to classiication tasks. Nevertheless, there are still a lot of problems especially when dealing with numerical (continuous-valued) attributes. Some of those problems can be solved using fuzzy decision trees (FDT). This paper proposes a method for handling continuous-valued attributes with automatically generated (as opposed to user deened) membership functions. An example will be given at the end. The aim of this paper is to combine decision trees (DT) with fuzziness, i.e. to use fuzziness to solve problems in the eld of DT. Most real-world applications of classiication learning involve continuous-valued attributes. However, Top Down Induction of Decision Tree (TDIDT) Qui86] algorithms only use nominal attributes. Therefore continuous-valued attributes have to be discretized before they are selected, typically by partitioning the range of the attribute into subranges. In principle, a discretization is simply a logical condition (using one or more attributes) that serves to partition the data into at least two subsets. The main question is where to set the so-called cut points to partition the numerical attributes. Fayyad and Irani FI92a] FI92b] suggested a method for selecting the cut point as the midpoint between each successive pair of attribute values which has later been improved by the same authors and by Seidelmann Sei93]. The base for the algorithm are logical conditions like "A(x) < T", which means: a threshold T is determined and the test A(x) < T is assigned to the left branch while A(x) T is assigned to the right branch. Further investigations tried to reduce the number of cut points (boundary points) that always separate two classes FI93]. By using entropy (as known from the construction of decision trees Qui86]) these boundaries decide whether such a point separates the attribute values having maximum information gain (cut point). While constructing the decision tree, algorithms do not only have to nd the testing attribute for each node, but also the "best" separation of this attribute. Moreover, Fayyad and Irani FI93] generalize the method and show that the separation is not necessarilly binary. In our opinion, there is one weak point in the method sketched above: the ""nal" position of the cut points is exactly halfway between two numerical attribute values. Unseen examples that have to be classiied later might be near to this threshold value and therefore classiied incorrectly. A possible way to overcome this diiculty is handling the numerical attributes …