Vector autoregressive time series model by tensor singular value decomposition

Abstract

The Vector Autoregressive model (VAR) is a statistical model used to describe the relationship of multiple variables as they change over time. The model is widely applied in predicting stocks, rates, and taxes. In this thesis, we will first introduce previous works about the VAR model, including Reduced Rank Regression (RRR) model and Multilinear Low-rank Regression (MLR) model. We then provide definitions of tensors, several tensor decomposition methods, and their calculation methods and algorithms, such as the Higher-Order Singular Value Decomposition (HOSVD) and tensor Singular Value Decomposition (t-SVD). The t-SVD is related to an invertible matrix L. We introduce Discrete Cosine Transform (DCT) and its properties and select the L as the DCT matrix. Based on this specific t-SVD, we propose the Low Tubal-rank Regression (LTR) model and LTR estimator. Further, we prove the asymptotic property of the LTR estimator and use Regularized Alternating Least Squares (RALS) method to compute it. We also prove the convergence result of the algorithm. In the experimental part, we introduce the video prediction problem. The VAR model is applied to predict future frames from known grayscale images and RGB images. We present the experimental process, and the visual and numerical results both demonstrate that the LTR method outperforms the MLR method.