Multi-linear Tensor Autoregressive Models

Abstract

Contemporary time series analysis has seen more and more tensor type data, from many fields. For example, stocks can be grouped according to Size, Book-to-Market ratio, and Operating Profitability, leading to a 3-way tensor observation at each month. We propose an autoregressive model for the tensor-valued time series, with autoregressive terms depending on multi-linear coefficient matrices. Comparing with the traditional approach of vectoring the tensor observations and then applying the vector autoregressive model, the tensor autoregressive model preserves the tensor structure and admits corresponding interpretations. We introduce three estimators based on projection, least squares, and maximum likelihood. Our analysis considers both fixed dimensional and high dimensional settings. For the former we establish the central limit theorems of the estimators, and for the latter we focus on the convergence rates and the model selection. The performance of the model is demonstrated by simulated and real examples.

References in corpus (10)

• Tensor Decomposition for Signal Processing and Machine Learning
• High-dimensional regression with noisy and missing data: Provable guarantees with nonconvexity
• Covariance Estimation in High Dimensions via Kronecker Product Expansions
• A Direct Estimation of High Dimensional Stationary Vector Autoregressions
• Low Rank and Structured Modeling of High-dimensional Vector Autoregressions
• KoPA: Automated Kronecker Product Approximation
• Tensor Factor Model Estimation by Iterative Projection
• Factor Models for High-Dimensional Tensor Time Series
• High-Dimensional Low-Rank Tensor Autoregressive Time Series Modeling
• Hybrid Kronecker Product Decomposition and Approximation