The Local Piecewisely Linear Kernel Smoothing Procedure for Fitting Jump Regression Surfaces

It is known that a surface fitted by conventional local smoothing procedures is not statistically consistent at the jump locations of the true regression surface. In this paper, a procedure is suggested for modifying conventional local smoothing procedures such that the modified procedures can fit the surface with jumps preserved automatically. Taking the local linear kernel smoothing procedure as an example, in a neighborhood of a given point, we fit a bivariate piecewisely linear function with possible jumps along the boundaries of four quadrants. The fitted function provides four estimators of the surface at the given point, which are constructed from observations in the four quadrants, respectively. When the difference among the four estimators is smaller than a threshold value, the given point is most likely a continuous point and the surface at that point is then estimated by the average of the four estimators. When the difference is larger than the threshold value, the given point is likely a jump point and at least one of the four estimators estimates the surface well under some regularity conditions. By comparing the weighted residual sums of squares of the four estimators, the best one is selected to define the surface estimator at the given point. Like most conventional estimators, the current surface estimator has an explicit mathematical formula. Therefore it is easy to compute and convenient to use. It can be applied directly to image reconstruction problems and other jump surface estimation problems including mine surface estimation in geology and equi-temperature surface estimation in meteorology and oceanography.