Gaussian process (GP) emulators have become essential tools for approximating complex simulators, significantly reducing computational demands in optimization, sensitivity analysis, and model calibration. While traditional GP emulators effectively model continuous and Gaussian-distributed simulator outputs with homogeneous variability, they typically struggle with discrete, heteroskedastic Gaussian, or non-Gaussian data, limiting their applicability to increasingly common stochastic simulators. In this work, we introduce a scalable Generalized Deep Gaussian Process (GDGP) emulation framework designed to accommodate simulators with heteroskedastic Gaussian outputs and a wide range of non-Gaussian response distributions, including Poisson, negative binomial, and categorical distributions. The GDGP framework leverages the expressiveness of DGPs and extends them to latent GP structures, enabling it to capture the complex, non-stationary behavior inherent in many simulators while also modeling non-Gaussian simulator outputs. We make GDGP scalable by incorporating the Vecchia approximation for settings with a large number of input locations, while also developing efficient inference procedures for handling large numbers of replicates. In particular, we present methodological developments that further enhance the computation of the approach for heteroskedastic Gaussian responses. We demonstrate through a series of synthetic and empirical examples that these extensions deliver the practical application of GDGP emulators and a unified methodology capable of addressing diverse modeling challenges. The proposed GDGP framework is implemented in the open-source R package dgpsi.

翻译：高斯过程(GP)仿真器已成为近似复杂模拟器的重要工具，显著降低了优化、灵敏度分析和模型校准中的计算需求。传统GP仿真器能够有效模拟具有均匀变异性的连续高斯分布模拟器输出，但通常难以处理离散型、异方差高斯型或非高斯型数据，这限制了其在日益普遍的随机模拟器中的应用。本文提出一种可扩展的通用深度高斯过程(GDGP)仿真框架，旨在适应具有异方差高斯输出的模拟器以及包括泊松分布、负二项分布和类别分布在内的广泛非高斯响应分布。GDGP框架利用深度高斯过程的表达能力，并将其扩展至潜高斯结构，从而既能捕捉许多模拟器固有的复杂非平稳行为，又能建模非高斯模拟器输出。通过引入用于处理大量输入位置的Vecchia近似，我们实现了GDGP的可扩展性，同时开发了用于处理大量重复样本的高效推理程序。特别是，我们提出了针对异方差高斯响应的方法论改进，进一步增强了该方法的计算效率。通过一系列合成和实证案例，我们证明这些扩展实现了GDGP仿真器的实际应用，并形成了一种能够应对多样化建模挑战的统一方法论。所提出的GDGP框架已在开源R包dgpsi中实现。