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通用非参数建模中被广泛使用，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承认回归模型中残差方差随协变量变化的灵活性，引入了改进方法。在本研究中，我们将异方差高斯过程的概念扩展至回归任务之外，涵盖分类和状态空间模型。为此，我们提出了一种新颖框架，其中高斯过程与协变量诱导的精度矩阵过程相结合，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，我们采用变分推断来逼近后验分布并促进后验预测建模。此外，我们的训练过程利用了包含闭式M步更新的EM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一致，并能无缝适应连续或分类等多种类型。通过模拟实验和气候学领域的实际应用，我们展示了该模型的优越性能。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提供了稳健且通用的工具。