Brownian Integrated Covariance Functions for Gaussian Process Modeling: Sigmoidal Versus Localized Basis Functions
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高斯过程建模的布朗积分协方差：S 形与局部化基函数
- 研究问题
- 确定性计算机模拟的高斯过程预测中，局部基函数回归先验均值可能使曲面起伏，如何获得非局部且能表示光滑曲面的基函数？
- 方法与思路
- 作者向平稳协方差引入积分器，构造布朗积分协方差类，使预测基函数呈非局部 S 形，并与常规幂指数模型比较。
- 材料陈述的贡献
- 摘要称新协方差兼具非局部基函数与光滑响应建模能力，讨论其与其他协方差的基本差异，并提供实例比较。
- 阅读关注点
- 核对协方差构造的数学条件、光滑性与边界行为，以及各模型的参数估计和预测误差比较是否采用一致设置。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要，不能推断刊本变化。摘要未给出协方差公式、具体条件或误差数值，不能断言预测普遍更好。
原文摘要
Gaussian process modeling, or kriging, is a popular method for modeling data from deterministic computer simulations, and the most common choices of covariance function are Gaussian, power exponential, and Matérn. A characteristic of these covariance functions is that the basis functions associated with their corresponding response predictors are localized, in the sense that they decay to zero as the input location moves away from the simulated input sites. As a result, the predictors tend to revert to the prior mean, which can result in a bumpy fitted response surface. In contrast, a fractional Brownian field model results in a predictor with basis functions that are nonlocalized and more sigmoidal in shape, although it suffers from drawbacks such as inability to represent smooth response surfaces. We propose a class of Brownian integrated covariance functions obtained by incorporating an integrator (as in the white noise integral representation of a fractional Brownian field) into any stationary covariance function. Brownian integrated covariance models result in predictor basis functions that are nonlocalized and sigmoidal, but they are capable of modeling smooth response surfaces. We discuss fundamental differences between Brownian integrated and other covariance functions, and we illustrate by comparing Brownian integrated power exponential with regular power exponential kriging models in a number of examples. Supplementary materials for this article are available online.