Abstract
This paper deals with a prediction problem of a new targeting variable corresponding to a new explanatory variable given a training dataset. To predict the targeting variable, we consider a model tree, which is used to represent a conditional probabilistic structure of a targeting variable given an explanatory variable, and discuss statistical optimality for prediction based on the Bayes decision theory. The optimal prediction based on the Bayes decision theory is given by weighting all the model trees in the model tree candidate set, where the model tree candidate set is a set of model trees in which the true model tree is assumed to be included. Because the number of all the model trees in the model tree candidate set increases exponentially according to the maximum depth of model trees, the computational complexity of weighting them increases exponentially according to the maximum depth of model trees. To solve this issue, we introduce a notion of meta-tree and propose an algorithm called MTRF (Meta-Tree Random Forest) by using multiple meta-trees. Theoretical and experimental analyses of the MTRF show the superiority of the MTRF to previous decision tree-based algorithms.
Highlights
Various studies in pattern recognition deal with a prediction problem of a targeting variable yn+1 corresponding to an explanatory variable xn+1 given pairs of explanatory and targeting variable {( xi, yi )}in=1
Under the assumption that the true model tree is in the restricted model tree candidate set represented by a meta-tree, Reference [13] proposed the optimal prediction based on the Bayes decision theory
As we have described above, if the true model tree is included in the model tree candidate set, the optimal prediction based on the Bayes decision theory is calculated
Summary
Various studies in pattern recognition deal with a prediction problem of a targeting variable yn+1 corresponding to an explanatory variable xn+1 given pairs of explanatory and targeting variable {( xi , yi )}in=1. If the true model tree is in a model tree candidate set represented by a meta-tree, the statistically optimal prediction—optimal prediction based on the Bayes decision theory—can be calculated. Under the assumption that the true model tree is in the restricted model tree candidate set represented by a meta-tree, Reference [13] proposed the optimal prediction based on the Bayes decision theory. As we have described above, if the true model tree is included in the model tree candidate set, the optimal prediction based on the Bayes decision theory is calculated. By using the model tree candidate set represented by multiple meta-trees, we predict yn+1 based on the Bayes decision theory. We call this proposed algorithm MTRF (MetaTree Random Forest).
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