Decision Tree Induction

Decision Tree Induction (DTI) is a tool to induce a classification or regression model from (usually large) datasets characterized by n objects (records), each one containing a set x of numerical or nominal attributes, and a special feature y designed as its outcome. Statisticians use the terms “predictors” to identify attributes and “response variable” for the outcome. DTI builds a model that summarizes the underlying relationships between x and y. Actually, two kinds of model can be estimated using decision trees: classification trees if y is nominal, and regression trees if y is numerical. Hereinafter we refer to classification trees to show the main features of DTI. For a detailed insight into the characteristics of regression trees see Hastie et al. (2001). As an example of classification tree, let us consider a sample of patients with prostate cancer on which data Figure 1. The prostate cancer dataset such as those summarized in Figure 1 have been collected. Suppose a new patient is observed and we want to determine if the tumor has penetrated the prostatic capsule on the basis of the other available information. Posing a series of questions about the characteristic of the patient can help to predict the tumor’s penetration. DTI proceeds in such a way, inducing a series of follow- up (usually binary) questions about the attributes of an unknown instance until a conclusion about what is its most likely class label is reached. Questions and their alternative answers can be represented hierarchically in the form of a decision tree, such as the one depicted in Figure 2.