A variational inference-based framework for training a multi-output Gaussian process latent variable model, specifically tailored to the tails-up spatio-temporal stream network, is developed. Training, given a censored observational data set subject to missing values, proceeds by maximising a secondary variational lower bound on the model log marginal likelihood using gradient-based optimisation. Consequently, the theoretical development for a new family of tails-up spatio-temporal stream network models is introduced which rely on the sparse Gaussian process inducing variable framework, the Bayesian Gaussian process latent variable model, and local variational methods. These spatio-temporal models use stream distance instead of Euclidean distance and capture spatial and temporal dependencies using auto/cross-correlation and process convolution, respectively, which allows for the development of valid separable spatio-temporal stream network-based covariance functions. Results from the simulation-based case studies indicate that the proposed framework performs well when considering benchmark comparisons and several performance metrics.

翻译：本文开发了一种基于变分推理的多输出高斯过程潜变量模型训练框架，该框架专为尾上时空流网络设计。对于存在缺失值的删失观测数据集，训练过程通过梯度优化最大化模型对数边际似然的次级变分下界。由此，本文引入了一类新型尾上时空流网络模型的理论发展，该模型基于稀疏高斯过程诱导变量框架、贝叶斯高斯过程潜变量模型及局部变分方法。这些时空模型采用流距离替代欧氏距离，分别通过自/互相关函数与过程卷积捕获空间依赖性和时间依赖性，从而构建有效的可分离时空流网络协方差函数。基于仿真的案例研究结果表明，所提出的框架在基准对比与多项性能指标上均表现优异。