TITLE:
An Improved Treed Gaussian Process
AUTHORS:
John Guenther, Herbert K. H Lee
KEYWORDS:
Bayesian Statistics, Treed Gaussian Process, Gaussian Process, Emulator, Binary Tree
JOURNAL NAME:
Applied Mathematics,
Vol.11 No.7,
July
22,
2020
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.