Co-evolving time series appears in a multitude of applications such as environmental monitoring, financial analysis, and smart transportation. This paper aims to address the following challenges, including (C1) how to incorporate explicit relationship networks of the time series; (C2) how to model the implicit relationship of the temporal dynamics. We propose a novel model called Network of Tensor Time Series, which is comprised of two modules, including Tensor Graph Convolutional Network (TGCN) and Tensor Recurrent Neural Network (TRNN). TGCN tackles the first challenge by generalizing Graph Convolutional Network (GCN) for flat graphs to tensor graphs, which captures the synergy between multiple graphs associated with the tensors. TRNN leverages tensor decomposition to model the implicit relationships among co-evolving time series. The experimental results on five real-world datasets demonstrate the efficacy of the proposed method.

Real-world applications often involve domain-specific and task-based performance objectives that are not captured by the standard machine learning metrics, such as mean squared error, mean absolute error, and cross-entropy loss, but are critical for decision making. A key challenge for direct integration of more meaningful domain and task-based evaluation criteria into an end-to-end gradient-based training process is the fact that often such performance objectives are not necessarily differentiable and may even require additional decision-making optimization processing. We propose Task-Oriented Prediction Network (TOPNet), an end-to-end learning scheme that automatically integrates task-based evaluation criteria into the learning process via a task-oriented estimator and directly learns a model with respect to the task-based goal. A major benefit of the proposed TOPNet learning scheme lies in its capability of automatically integrating non-differentiable evaluation criteria. This makes it particularly suitable for diversified and customized task-based evaluation criteria in real-world prediction tasks. We validate the performance of TOPNet on two real-world financial prediction tasks, revenue surprise forecasting and credit risk modeling. The experimental results on multiple real-world data sets demonstrate that TOPNet significantly outperforms both traditional modeling with standard losses and modeling with differentiable (relaxed) surrogate losses.