基于双边核估计的保持跳跃曲线回归过程

A Jump-Preserving Curve Regression Procedure Based on Bilateral Kernel Estimation

摘要: 跳跃曲线回归在某些实际问题中非常重要，但是由于数据采集过程中受到噪声的干扰，传统的曲线回归方法对于跳跃位置的保持效果欠佳。本文基于分段双边核估计提出了一个保持跳跃的曲线拟合过程。该过程不仅在x方向上用核函数进行加权，亦在y方向上使用核函数。在目标点的邻域内，分别在左半邻域、右半邻域和全邻域对函数进行估计，并计算加权残差平方和以进行比较，选取较小的加权残差平方和的一个作为目标点的最终估计，以此来达到无需提前探测跳跃点的同时保持曲线跳跃特性的目的。数值模拟和实际数据分析表明该方法的可行性和有效性。

Abstract: It is well known that curve regression is very important in many applications. However, since data collection procedures are disturbed by errors, traditional curve regression methods cannot play well in jump points. This paper proposes a jump-preserving curve fitting procedure, which is based on bilateral kernel estimation. Kernel functions are not only added to x-axis, but also added to y-axis. Then, we estimate given points from left side, right side and whole neighborhood. Weighted residual sums of squares are calculated to compare. The estimate with smaller weighted residual sums of squares is selected as the final estimate of the given point, so that we can achieve jump- preserving while not to detect jump points at first. Numerical simulation and real data analysis demonstrate the feasibility and efficiency of this method.