  This paper presents a Gaussian process (GP) model for estimating piecewise
continuous regression functions. In scientific and engineering applications of
regression analysis, the underlying regression functions are piecewise
continuous in that data follow different continuous regression models for
different regions of the data with possible discontinuities between the
regions. However, many conventional GP regression approaches are not designed
for piecewise regression analysis. We propose a new GP modeling approach for
estimating an unknown piecewise continuous regression function. The new GP
model seeks for a local GP estimate of an unknown regression function at each
test location, using local data neighboring to the test location. To
accommodate the possibilities of the local data from different regions, the
local data is partitioned into two sides by a local linear boundary, and only
the local data belonging to the same side as the test location is used for the
regression estimate. This local split works very well when the input regions
are bounded by smooth boundaries, so the local linear approximation of the
smooth boundaries works well. We estimate the local linear boundary jointly
with the other hyperparameters of the GP model, using the maximum likelihood
approach. Its computation time is as low as the local GP's time. The superior
numerical performance of the proposed approach over the conventional GP
modeling approaches is shown using various simulated piecewise regression
functions.
