Despite the widespread utilization of Gaussian process models for versatile nonparametric modeling, they exhibit limitations in effectively capturing abrupt changes in function smoothness and accommodating relationships with heteroscedastic errors. Addressing these shortcomings, the heteroscedastic Gaussian process (HeGP) regression seeks to introduce flexibility by acknowledging the variability of residual variances across covariates in the regression model. In this work, we extend the HeGP concept, expanding its scope beyond regression tasks to encompass classification and state-space models. To achieve this, we propose a novel framework where the Gaussian process is coupled with a covariate-induced precision matrix process, adopting a mixture formulation. This approach enables the modeling of heteroscedastic covariance functions across covariates. To mitigate the computational challenges posed by sampling, we employ variational inference to approximate the posterior and facilitate posterior predictive modeling. Additionally, our training process leverages an EM algorithm featuring closed-form M-step updates to efficiently evaluate the heteroscedastic covariance function. A notable feature of our model is its consistent performance on multivariate responses, accommodating various types (continuous or categorical) seamlessly. Through a combination of simulations and real-world applications in climatology, we illustrate the model's prowess and advantages. By overcoming the limitations of traditional Gaussian process models, our proposed framework offers a robust and versatile tool for a wide array of applications.

翻译：尽管高斯过程模型在灵活的非参数建模中得到广泛应用，但它们在有效捕捉函数平滑度的突变以及处理异方差误差关系方面存在局限性。为解决这些不足，异方差高斯过程回归通过在回归模型中考虑残差方差随协变量的变化性来引入灵活性。本文扩展了异方差高斯过程的概念，将其应用范围从回归任务拓展至分类和状态空间模型。为此，我们提出了一种新颖框架，该框架将高斯过程与协变量诱导的精度矩阵过程耦合，并采用混合模型形式。该方法能够对跨协变量的异方差协方差函数进行建模。为缓解采样带来的计算挑战，我们采用变分推断近似后验分布并促进后验预测建模。此外，我们的训练过程利用一种包含闭式M步更新的期望最大化算法，以高效评估异方差协方差函数。该模型的一个显著特点是在多变量响应上表现出一致性能，可无缝适应连续或分类等不同类型。通过气候学中的模拟实验和实际应用案例，我们展示了该模型的优越能力和优势。通过克服传统高斯过程模型的局限性，我们提出的框架为广泛的应用领域提供了稳健且通用的工具。