Transform-Based Multilinear Dynamical System for Tensor Time Series Analysis

Weijun Lu, Xiao-Yang Liu, Qingwei Wu, Yue Sun, Anwar Walid

Keywords: Tensor time series, transform, spatiotemporal correlation, frequency, multilinear structure

TL;DR: We propose a novel multilinear dynamical system (MLDS) in a transform domain to capture spatiotemporal-frequency correlations in tensor time series.

Abstract: We propose a novel multilinear dynamical system (MLDS) in a transform domain, named $\mathcal{L}$-MLDS, to model tensor time series. With transformations applied to a tensor data, the latent multidimensional correlations among the frontal slices are built, and thus resulting in the computational independence in the transform domain. This allows the exact separability of the multidimensional problem into multiple smaller LDS problems. To estimate the system parameters, we utilize the expectation-maximization (EM) algorithm to determine the parameters of each LDS. Further, $\mathcal{L}$-MLDS significantly reduces the model parameters and allows parallel processing. Our general $\mathcal{L}$-MLDS model is implemented based on discrete Fourier transform, discrete cosine transform and discrete wavelet transform, respectively. Due to the nonlinearity of these transformations, $\mathcal{L}$-MLDS is able to capture the nonlinear correlations within the data while the MLDS \cite{rogers2013multilinear} assumes multi-way linear correlations. On four real datasets, the proposed $\mathcal{L}$-MLDS achieves much higher prediction accuracy than the state-of-the-art MLDS and LDS with an equal number of parameters under different noise models. In particular, the relative errors are reduced by $50\% \sim 99\%$. Simultaneously, $\mathcal{L}$-MLDS achieves an exponential improvement in the model's training time than MLDS.