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Heteroscedastic BART via Multiplicative Regression Trees

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posted on 22.11.2019 by M. T. Pratola, H. A. Chipman, E. I. George, R. E. McCulloch

Bayesian additive regression trees (BART) has become increasingly popular as a flexible and scalable nonparametric regression approach for modern applied statistics problems. For the practitioner dealing with large and complex nonlinear response surfaces, its advantages include a matrix-free formulation and the lack of a requirement to prespecify a confining regression basis. Although flexible in fitting the mean, BART has been limited by its reliance on a constant variance error model. Alleviating this limitation, we propose HBART, a nonparametric heteroscedastic elaboration of BART. In BART, the mean function is modeled with a sum of trees, each of which determines an additive contribution to the mean. In HBART, the variance function is further modeled with a product of trees, each of which determines a multiplicative contribution to the variance. Like the mean model, this flexible, multidimensional variance model is entirely nonparametric with no need for the prespecification of a confining basis. Moreover, with this enhancement, HBART can provide insights into the potential relationships of the predictors with both the mean and the variance. Practical implementations of HBART with revealing new diagnostic plots are demonstrated with simulated and real data on used car prices and song year of release. Supplementary materials for this article are available online.

Funding

This research was partially supported by the U.S. National Science Foundation grants DMS-1106862, 1106974, and 1107046, the STATMOS research network on Statistical Methods in Oceanic and Atmospheric Sciences. H. A. Chipman acknowledges support from the Natural Sciences and Engineering Research Council of Canada. E. I. George acknowledges support from NSF grant DMS-1406563 and from a Simons Fellowship from the Isaac Newton Institute at the University of Cambridge.

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