On Speed of Stochastic CART Model Search

Authors

  • Márton Ispány University of Debrecen, Hungary
  • Ilona Krasznahorkay University of Debrecen, Hungary

DOI:

https://doi.org/10.17713/ajs.v36i1.318

Abstract

Decision trees have proved to be commonly used nonlinear tools for supervised learning. This technique is a way to divide the space of the predictor variables into bricks in order to achieve as homogeneous partitions as possible. We improved the CART method proposed by Breiman et al. (1984) using a stochastic search, first suggested by Chipman et al. (1998) in the Bayesian framework. In this paper estimates are given for the rate of
convergence and the mixing time of the MCMC method defined on decision trees.

References

Breiman, L., Friedman, J. H., Olsen, A. O., and Stone, C. J. (1984). Classification and Regression Trees. Wadsworth International Group.

Chipman, H. A., George, E. I., and McCulloch, R. E. (1998). Bayesian CART model search. Journal of the American Statistical Association, 93, 935-960.

Cormen, T. H., Leierson, C. E., and Rives, L. R. (1990). Introduction to Algorithms. The Massachusetts Institute of Technology.

Hastie, T., Tibshirani, R., and Friedman, J. (2001). The Elements of Statistical Learning. Data Mining, Inference and Prediction. New York: Springer.

Hoeffgen, K. U., Simon, H. U., and Van Horn, K. S. (1995). Robust trainability of single neurons. Journal of Computer System Sciences, 50, 114-125.

Hyafil, L., and Rivest, R. L. (1976). Constructing optimal binary trees is NP-complete. Information Processing Letters, 5, 15-17.

Jerrum, M. (1998). Mathematical foundations of the markov chain monte carlo method. In Probabilistic methods for algorithmic discrete mathematics.

Jerrum, M. (2003). Counting, Sampling and Integrating: Algorithms and Complexity. Basel: Birkhäuser.

Jerrum, M., and Sinclair, A. (1996). The Markov chain Monte Carlo method: An approach to approximate counting and integration. In D. S. Hochbaum (Ed.), Approximation Algorithm for NP-hard Problems (p. 482-520). Boston.

Kurzynski, M. W. (1983). The optimal strategy of a tree classifier. Pattern Recognition, 16, 81-87.

Loh, W. Y., and Vanichsetakal, N. (1988). Tree-structured classification via generalized discriminant analysis. Journal of the American Statistical Association, 83, 715-725.

Murphy, P. M., and McCraw, R. L. (1991). Designing storage efficient decision trees. IEEE Transactions on Computers, 40, 315-320.

Murthy, S. K. (1998). Automatic construction of decision trees from data: A multidisciplinary survey. In Data Mining and Knowledge Discovery, 2. Boston: Kluwer Academic Publishers.

Roberts, G. O. (1996). Markov chain concepts related to sampling algorithms. In W. R. Gilks, S. Richardson, and D. J. Spiegelhalter (Eds.), Markov Chain Monte Carlo in Practice (p. 45-57). London: Chapman & Hall/CRC.

Safavian, S. R., and Landgrebe, D. (1991). A survey of decision tree classifier methodology. IEEE Transaction on Systems, Man and Cybernetics, 21, 660-674.

Saloff-Coste, L. (1997). Lectures on finite Markov chains. In P. Bernard (Ed.), Lectures on Probability Theory and Statistics (p. 301-413). Berlin: Springer.

Sinclair, A. (1992). Improved bounds for mixing rates of Markov chains and multicommodity flow. Combinatorics, Probability and Computing, 1, 351-370.

Downloads

Published

2016-04-03

Issue

Section

Articles

How to Cite

On Speed of Stochastic CART Model Search. (2016). Austrian Journal of Statistics, 36(1), 27–40. https://doi.org/10.17713/ajs.v36i1.318