Hierarchical Time Series Forecasting with Operator-Valued Kernels

Hierarchical forecasting exploits aggregation structures that expose multiple views of the same underlying spatiotemporal forecasting problem, while ensuring coherence among forecasts. Individual series provide fine-grained information, while progressively coarser aggregates reveal temporal and spatial patterns at different scales. In the associated deep learning architectures, the hierarchy determines which aggregate series are available and how they are processed together with individual series (e.g., through reconciliation or message-passing operators). Different hierarchies thus provide distinct learning signals and inductive biases for forecasting the same collection of time series. Nonetheless, existing architectures are typically trained on a single, often fixed, structure. In this work, we propose a graph deep learning framework that leverages variations across hierarchies as multi-task regularization. We show that jointly learning across these structures can reduce over-specialization to any single hierarchy and improve forecasting performance. Indeed, training across hierarchies provides an inductive bias toward representations that are useful across different structures, and as such, generalize better. Making forecasts coherent with the aggregation constraints for each sampled hierarchy is computationally expensive: to complement the framework, we propose a novel operator-based on a top-down approach-that produces coherent forecasts with linear time and space complexity w.r.t. the hierarchy size. Experiments show that our approach improves forecasting accuracy over training on a single structure and is competitive with current state-of-the-art methods.

In many real-world applications, such as retail sales, energy usage, and supply chain planning, forecasting is performed across hierarchical structures. These structures often represent aggregations (e.g., products to categories to regions), where forecasts must not only be accurate but also coherent, meaning that lower-level predictions sum correctly to higher-level forecasts. Traditional statistical methods, such as Bottom-Up and MinT, enforce coherence through post-processing but fail to model complex nonlinear temporal dependencies and covariate interactions. We propose Hierarchical Temporal Fusion (HTF), a novel extension of the Temporal Fusion Transformer (TFT) that integrates structured hierarchical embeddings with a coherence-aware loss function to ensure consistent forecasts across all levels of a hierarchy. Rather than applying reconciliation after forecasting, HTF embeds coherence directly into the training objective. The coherence loss penalizes the difference between aggregated child forecasts and their corresponding parent forecasts during training, enabling the model to learn both temporal dynamics and structural consistency simultaneously. We evaluate HTF on two publicly available benchmark datasets: the M5 Walmart forecasting dataset and a publicly available hierarchical energy consumption dataset. Results demonstrate that HTF substantially reduces forecast incoherence while improving forecasting accuracy compared with classical reconciliation methods and deep learning baselines. In addition, attention visualization and embedding analysis provide insight into how temporal and structural information contribute to hierarchical forecasting performance.

Multi-scale modeling has become an effective approach for long-term time series forecasting, capturing temporal patterns that range from fine-grained local dynamics to coarse global trends. Representations across these temporal scales are inherently hierarchical, with coarser scales abstracting and aggregating information from finer ones. While existing approaches readily exchange information across these scales, the hierarchy itself is typically left as an emergent byproduct of such interactions rather than captured as a geometric structure in its own right. In this paper, we introduce HypLTSF, a framework that endows the multi-scale hierarchy with a concrete geometric form by embedding scale-wise representations into the Poincaré ball, whose exponentially expanding volume naturally accommodates hierarchical structures. To align this geometry with the temporal hierarchy, HypLTSF imposes two constraints: (1) a radial constraint that orders embeddings by their level of abstraction, and (2) an angular constraint that groups fine-scale patterns sharing a common coarser-scale ancestor. Extensive experiments on long-term time series forecasting benchmarks show that HypLTSF achieves state-of-the-art performance, suggesting that explicitly modeling the multi-scale hierarchy as a geometric structure is effective for forecasting.