Classification and Categorical Inputs with Treed Gaussian Process Models
Recognizing the successes of treed Gaussian process (TGP) models as an interpretable and thrifty model for nonparametric regression, we seek to extend the model to classification. Both treed models and Gaussian processes (GPs) have, separately, enjoyed great success in application to classification problems. An example of the former is Bayesian CART. In the latter, real-valued GP output may be utilized for classification via latent variables, which provide classification rules by means of a softmax function. We formulate a Bayesian model averaging scheme to combine these two models and describe a Monte Carlo method for sampling from the full posterior distribution with joint proposals for the tree topology and the GP parameters corresponding to latent variables at the leaves. We concentrate on efficient sampling of the latent variables, which is important to obtain good mixing in the expanded parameter space. The tree structure is particularly helpful for this task and also for developing an efficient scheme for handling categorical predictors, which commonly arise in classification problems. Our proposed classification TGP (CTGP) methodology is illustrated on a collection of synthetic and real data sets. We assess performance relative to existing methods and thereby show how CTGP is highly flexible, offers tractable inference, produces rules that are easy to interpret, and performs well out of sample.
KeywordsTreed models Gaussian process, Bayesian model averaging Latent variable
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- ABRAHAMSEN, P. (1997), “A Review of Gaussian Random Fields and Correlation Functions”, Technical Report 917, Norwegian Computing Center, Oslo, Norway.Google Scholar
- ASUNCION,A., and NEWMAN,D.J. (2007), ”UCIMachine Learning Repository”, School of Information and Computer Sciences, University of California, Irvine, http://www.ics.uci.edu/~mlearn/MLRepository.html.
- DIMITRIADOU, E., HORNIK, K., LEISCH, F., MEYER, D., and WEINGESSEL, A. (2010), “e1071: Misc Functions of the Department of Statistics (e1071), TU Wien”, Version 1.5-24, CRAN repository, maintained by Friedrich Leisch, obtained from http://cran.r-project.org/web/packages/e1071/e1071.pdf.
- GRAMACY, R.B. (2005), Bayesian Treed Gaussian Process Models, Santa Cruz: University of California.Google Scholar
- GRAMACY, R.B. (2007), ”tgp: An R Package for Bayesian Nonstationary, Semiparametric Nonlinear Regression and Design by Treed Gaussian Process Models”, Journal of Statistical Software, 19(9), 1548–7660.Google Scholar
- GRAMACY, R.B., and TADDY, M.A. (2008), “tgp: Bayesian Treed Gaussian Process Models”, R Package Version 2.1-2, http://www.ams.ucsc.edu/rbgramacy/tgp.html.
- GRAMACY, R.B., and TADDY, M.A. (2009), “Categorical Inputs, Sensitivity Analysis, Optimization and Importance Tempering with tgp”, Version 2, an R Package for Treed Gaussian Process Models, University of Cambridge, http://www.cran.r-project.org/web/packages/tgp/vignettes/tgp2.pdf, submitted to the Journal of Statistical Software.
- MAT´ERN, B. (1986), Spatial Variation (2nd ed.), New York: Springer-Verlag.Google Scholar
- NEAL, R.M. (1997), “Monte Carlo Implementation of Gaussian Process Models for Bayesian Regression and Classification”, Technical Report 9702, Department of Statistics, University of Toronto.Google Scholar
- NEAL, R.M. (1998), “Regression and Classification Using Gaussian Process Priors (with Discussion)”, in Bayesian Statistics 6, eds. J.M. Bernardo, et al., Oxford: Oxford University Press, pp. 476–501.Google Scholar
- RIPLEY, B. (2009), “Feed-forward Neural Networks and Multinomial Log-Linear Models”, Version 7.3-1, CRAN repository, maintained by Brian Ripley, obtained from http://cran.r-project.org/web/packages/nnet/nnet.pdf.
- R DEVELOPMENT CORE TEAM (2008), “R: A Language and Environment for Statistical Computing”, R Foundation for Statistical Computing, Vienna, ISBN 3-900051-00-3, http://www.R-project.org.
- THERNEAU, T.M., and ATKINSON, B. (2010), “rpart: Recursive Partitioning”, Version 3.1-46, CRAN repository, maintained by Brian Ripley, obtained from http://cran.rproject.org/web/packages/rpart/index.html.