The Bump Hunting Method Using the Genetic Algorithm with the Extreme-Value Statistics

Abstract

In difficult classification problems of the z-dimensional points into two groups giving 0-1 responses due to the messy data structure, we try to find the denser regions for the favorable customers of response 1, instead of finding the boundaries to separate the two groups. Such regions are called the bumps, and finding the boundaries of the bumps is called the bump hunting. The main objective of this paper is to find the largest region of the bumps under a specified ratio of the number of the points of response 1 to the total. Then, we may obtain a trade-off curve between the number of points of response 1 and the specified ratio. The decision tree method with the Gini's index will provide the simple-shaped boundaries for the bumps if the marginal density for response 1 shows a rather simple or monotonic shape. Since the computing time searching for the optimal trees will cost much because of the NP-hardness of the problem, some random search methods, e.g., the genetic algorithm adapted to the tree, are useful. Due to the existence of many local maxima unlike the ordinary genetic algorithm search results, the extreme-value statistics will be useful to estimate the global optimum number of captured points; this also guarantees the accuracy of the semi-optimal solution with the simple descriptive rules. This combined method of genetic algorithm search and extreme-value statistics use is new. We apply this method to some artificial messy data case which mimics the real customer database, showing a successful result. The reliability of the solution is discussed.