Jump Gaussian Process Model for Estimating Piecewise Continuous Regression Functions
Jump Gaussian Process Model for Estimating Piecewise Continuous Regression Functions
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发表时间：
2022
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通讯作者：
Chiwoo Park
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作者：
Chiwoo Park
本文提出了一种估计分段连续回归函数的高斯过程模型。在回归分析的许多科学和工程应用中，底层回归函数通常是分段连续的，因为数据遵循不同输入区域的不同连续回归模型，区域间具有不连续性。然而，许多传统的GP回归方法不是为分段回归分析而设计的。存在使用显式域划分的分段GP模型和在划分的区域上构成独立的GP模型。它们不够灵活，无法对真实的数据集进行建模，其中数据域由复杂且弯曲的跳跃边界划分。我们提出了一种新的GP建模方法来估计未知的分段连续回归函数。新的GP模型寻求在每个测试位置处的未知回归函数的局部GP估计，使用邻近测试位置的局部数据。考虑到局部数据可能来自不同区域，该方法通过局部数据划分函数将局部数据划分为若干块。它只使用可能来自同一地区的本地数据作为回归估计的测试位置。由于我们不知道哪些局部数据点来自相关区域，我们提出了一种数据驱动的方法，通过局部分区函数来分割和子集局部数据。我们讨论了局部数据划分函数的几种建模选择，包括局部线性函数和局部多项式函数。我们还研究了一个优化问题，共同优化的分区函数和其他协方差参数使用似然最大化标准。各种模拟实验和真实的数据研究表明，使用所提出的方法比传统的GP和分段GP建模方法的几个优点。
This paper presents a Gaussian process (GP) model for estimating piecewise continuous regression functions. In many scientific and engineering applications of regression analysis, the underlying regression functions are often piecewise continuous in that data follow different continuous regression models for different input regions with discontinuities across regions. However, many conventional GP regression approaches are not designed for piece-wise regression analysis. There are piecewise GP models to use explicit domain partitioning and pose independent GP models over partitioned regions. They are not flexible enough to model real datasets where data domains are divided by complex and curvy jump boundaries. We propose a new GP modeling approach to estimate an unknown piecewise continuous regression function. The new GP model seeks a local GP estimate of an unknown regression function at each test location, using local data neighboring the test location. Considering the possibilities of the local data being from different regions, the proposed approach partitions the local data into pieces by a local data partitioning function. It uses only the local data likely from the same region as the test location for the regression estimate. Since we do not know which local data points come from the relevant region, we propose a data-driven approach to split and subset local data by a local partitioning function. We discuss several modeling choices of the local data partitioning function, including a locally linear function and a locally polynomial function. We also investigate an optimization problem to jointly optimize the partitioning function and other covariance parameters using a likelihood maximization criterion. Several advantages of using the proposed approach over the conventional GP and piecewise GP modeling approaches are shown by various simulated experiments and real data studies.