Abstract The modeling and prediction of functions that can exhibit non-stationarity characteristics is important in many applications; for example, this is often the case for simulator output. One approach to predict a function with unknown stationarity properties is to model it as a draw from a flexible stochastic process that can produce stationary or non-stationary realizations. One such model is the composite Gaussian process (CGP) which expresses the large-scale (global) trends of the output and the small-scale (local) adjustments to the global trend as independent Gaussian processes; an extension of the CGP model can produce realizations with non-constant variance by allowing the variance of the local process to vary over the input space. A new, Bayesian extension of a global-trend plus local-trend model is proposed that also allows measurement errors. In contrast to the original CGP model, the new Bayesian CGP model introduces a weight function to allow the total process variability to be apportioned between the large- and small-scale processes. The proposed prior distributions ensure that the fitted global mean is smoother than the local deviations, a feature built into the CGP model. The log of the process variance for the Bayesian CGP is modeled as a Gaussian process to provide a flexible mechanism for handling variance functions that vary across the input space. A Markov chain Monte Carlo algorithm is proposed that provides posterior estimates of the parameters for the Bayesian CGP. It also yields predictions of the output and quantifies uncertainty about the predictions. The method is illustrated using both analytic and real-data examples.

中文翻译：

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使用贝叶斯复合高斯过程预测非平稳响应函数

摘要 具有非平稳性的函数的建模和预测在许多应用中都很重要。例如，模拟器输出通常就是这种情况。预测具有未知平稳特性的函数的一种方法是将其建模为来自可以产生平稳或非平稳实现的灵活随机过程的抽取。一种这样的模型是复合高斯过程 (CGP)，它将输出的大尺度（全局）趋势和对全局趋势的小尺度（局部）调整表示为独立的高斯过程；通过允许局部过程的方差在输入空间上变化，CGP 模型的扩展可以产生具有非恒定方差的实现。一个新的，提出了全局趋势加局部趋势模型的贝叶斯扩展，它也允许测量误差。与原始 CGP 模型相比，新的贝叶斯 CGP 模型引入了一个权重函数，允许在大尺度和小尺度过程之间分配总过程可变性。提议的先验分布确保拟合的全局平均值比局部偏差更平滑，这是 CGP 模型中内置的一个特征。贝叶斯 CGP 的过程方差的对数被建模为高斯过程，以提供一种灵活的机制来处理跨输入空间变化的方差函数。提出了马尔可夫链蒙特卡罗算法，该算法为贝叶斯 CGP 提供参数的后验估计。它还产生对输出的预测并量化预测的不确定性。