The Bump Hunting by the Decision Tree with the Genetic Algorithm

In difficult classification problems of the z-dimensional points into two groups giving 0–1 responses due to the messy data structure, it is more favorable to search for the denser regions for the response 1 points than to find the boundaries to separate the two groups. For such problems which can often be seen in customer databases, we have developed a bump hunting method using probabilistic and statistical methods as shown in the previous study. By specifying a pureness rate in advance, a maximum capture rate will be obtained. In finding the maximum capture rate, we have used the decision tree method combined with the genetic algorithm. Then, a trade-off curve between the pureness rate and the capture rate can be constructed. However, such a trade-off curve could be optimistic if the training data set alone is used. Therefore, we should be careful in assessing the accuracy of the tradeoff curve. Using the accuracy evaluation procedures such as the cross validation or the bootstrapped hold-out method combined with the training and test data sets, we have shown that the actually applicable trade-off curve can be obtained. We have also shown that an attainable upper bound trade-off curve can be estimated by using the extreme-value statistics because the genetic algorithm provides many local maxima of the capture rates with different initial values. We have constructed the three kinds of trade-off curves; the first is the curve obtained by using the training data; the second is the return capture rate curve obtained by using the extreme-value statistics; the last is the curve obtained by using the test data. These three are indispensable like the Trinity to comprehend the whole figure of the trade-off curve between the pureness rate and the capture rate. This paper deals with the behavior of the trade-off curve from a statistical viewpoint.