High-dimensional Autoregressive Modeling for Time Series with Hierarchical Structures

2025-11-29

AI Summary

High-dimensional time series often exhibit hierarchical structures naturally represented as tensors, yet existing statistical methods struggle to model such structures effectively. This paper proposes a supervised tensor factorization framework that hierarchically extracts low-dimensional features along a hierarchy-aware mode ordering, balancing structural fidelity and modeling flexibility. We introduce a novel sparse selection mechanism for optional mode orderings and intermediate ranks, automatically identifying a parsimonious set of effective orderings to uncover latent hierarchical relationships and significantly enhance interpretability. The framework integrates sequential factor extraction, tensor decomposition, and data-driven hyperparameter selection, and establishes non-asymptotic error bounds, providing theoretical guarantees for both regression and autoregressive forecasting. Evaluated on personality panel data, our method achieves superior predictive accuracy over state-of-the-art baselines while yielding clear, interpretable hierarchical structures.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: Multimodal Perception & Sensor Fusion

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

High-dimensional time series often exhibit hierarchical structures represented by tensors, while statistical methodologies that can effectively exploit the structural information remain limited. We propose a supervised factor modeling framework that accommodates general hierarchical structures by extracting low-dimensional features sequentially in the mode orders that respect the hierarchical structure. Our method can select a small collection of such orders to allow for impurities in the hierarchical structures, yielding interpretable loading matrices that preserve the hierarchical relationships. A practical estimation procedure is proposed, with a hyperparameter selection scheme that identifies a parsimonious set of action orders and interim ranks, thereby revealing the possibly latent hierarchical structures. Theoretically, non-asymptotic error bounds are derived for the proposed estimators in both regression and autoregressive settings. An application to the IPIP-NEO-120 personality panel illustrates superior forecasting performance and clearer structural interpretation compared with existing methods based on tensor decompositions and hierarchical factor analysis.

Problem

• Models high-dimensional time series with hierarchical tensor structures
• Extracts low-dimensional features preserving hierarchical relationships interpretably
• Proposes estimation with hyperparameter selection for latent structure discovery

Innovation

• Supervised factor modeling for hierarchical tensor structures
• Sequential low-dimensional feature extraction respecting hierarchy
• Parsimonious hyperparameter selection revealing latent hierarchies