An Improved Treed Gaussian Process

Abstract

Many black box functions and datasets have regions of different variability. Models such as the Gaussian process may fall short in giving the best representation of these complex functions. One successful approach for modeling this type of nonstationarity is the Treed Gaussian process [1] , which extended the Gaussian process by dividing the input space into different regions using a binary tree algorithm. Each region became its own Gaussian process. This iterative inference process formed many different trees and thus, many different Gaussian processes. In the end these were combined to get a posterior predictive distribution at each point. The idea was that when the iterations were combined, smoothing would take place for the surface of the predicted points near tree boundaries. We introduce the Improved Treed Gaussian process, which divides the input space into a single main binary tree where the different tree regions have different variability. The parameters for the Gaussian process for each tree region are then determined. These parameters are then smoothed at the region boundaries. This smoothing leads to a set of parameters for each point in the input space that specify the covariance matrix used to predict the point. The advantage is that the prediction and actual errors are estimated better since the standard deviation and range parameters of each point are related to the variation of the region it is in. Further, smoothing between regions is better since each point prediction uses its parameters over the whole input space. Examples are given in this paper which show these advantages for lower-dimensional problems.