Efficient algorithms for computing the best subset regression models for large-scale problems

Abstract

Several strategies for computing the best subset regression models are proposed. Some of the algorithms are modified versions of existing regression-tree methods, while others are new. The first algorithm selects the best subset models within a given size range. It uses a reduced search space and is found to outperform computationally the existing branch-and-bound algorithm. The properties and computational aspects of the proposed algorithm are discussed in detail. The second new algorithm preorders the variables inside the regression tree. A radius is defined in order to measure the distance of a node from the root of the tree. The algorithm applies the preordering to all nodes which have a smaller distance than a certain radius that is given a priori. An efficient method of preordering the variables is employed. The experimental results indicate that the algorithm performs best when preordering is employed on a radius of between one quarter and one third of the number of variables. The algorithm has been applied with such a radius to tackle large-scale subset-selection problems that are considered to be computationally infeasible by conventional exhaustive-selection methods. A class of new heuristic strategies is also proposed. The most important of these is one that assigns a different tolerance value to each subset model size. This strategy with different kind of tolerances is equivalent to all exhaustive and heuristic subset-selection strategies. In addition the strategy can be used to investigate submodels having noncontiguous size ranges. Its implementation provides a flexible tool for tackling large scale models.