Global Induction of Decision Trees

Decision trees are, besides decision rules, one of the most popular forms of knowledge representation in Knowledge Discovery in Databases process (Fayyad, Piatetsky-Shapiro, Smyth & Uthurusamy, 1996) and implementations of the classical decision tree induction algorithms are included in the majority of data mining systems. A hierarchical structure of a tree-based classifier, where appropriate tests from consecutive nodes are subsequently applied, closely resembles a human way of decision making. This makes decision trees natural and easy to understand even for an inexperienced analyst. The popularity of the decision tree approach can also be explained by their ease of application, fast classification and what may be the most important, their effectiveness. Two main types of decision trees can be distinguished by the type of tests in non-terminal nodes: univariate and multivariate decision trees. In the first group, a single attribute is used in each test. For a continuousvalued feature usually an inequality test with binary outcomes is applied and for a nominal attribute mutually exclusive groups of attribute values are associated with outcomes. As a good representative of univariate inducers, the well-known C4.5 system developed by Quinlan (1993) should be mentioned. In univariate trees a split is equivalent to partitioning the feature space with an axis-parallel hyper-plane. If decision boundaries of a particular dataset are not axis-parallel, using such tests may lead to an overcomplicated classifier. This situation is known as the “staircase effect”. The problem can be mitigated by applying more sophisticated multivariate tests, where more than one feature can be taken into account. The most common form of such tests is an oblique split, which is based on a linear combination of features (hyper-plane). The decision tree which applies only oblique tests is often called oblique or linear, whereas heterogeneous trees with univariate, linear and other multivariate (e.g., instance-based) tests can be called mixed decision trees (Llora & Wilson, 2004). It should be emphasized that computational complexity of the multivariate induction is generally significantly higher than the univariate induction. CART (Breiman, Friedman, Olshen & Stone, 1984) and OC1 (Murthy, Kasif & Salzberg, 1994) are well known examples of multivariate systems.