TCP-ARMA: A Tensor-Variate Time Series Forecasting Method

Analysis of complex data structures in the form of matrix or tensor format data has gained immense popularity in diverse fields. However, forecasting time series based on high-order historical tensor data presents significant challenges due to the huge number of parameters derived by the high-dimensional nature of these data. Traditional time series models, designed for scalar or vector data, are insufficient for handling such data, necessitating the development of novel techniques to tackle these challenges. To address this issue, we propose a Tensor-variate method with Compressed Parameters in Auto-Regressive Moving Average (TCP-ARMA) model for time series forecasting, which integrates a smoothed mean and a tensor-variate autoregressive moving average (ARMA) model with a parameter reduction technique. The proposed method captures the global trend within each dimension of tensors as well as the time-dimension by a tensor-based smoothed mean. The high-order parameters, commonly with tremendous elements, are compressed into a series of factor matrices, significantly reducing computational difficulty and complexity. To solve the optimization problem efficiently and avoid the computational challenge of inverting large matrices, we have designed an algorithm named BCD-PALM that combines block coordinate descent (BCD) with proximal alternating linearized minimization (PALM). We have employed a real-world case study to validate our proposed approach, and the results demonstrate its effectiveness in addressing the challenges associated with high-dimensional tensor data. Note to Practitioners—In response to the challenges associated with capturing the evolution within high-order tensor time series data, we develop a tensor-variate time series forecasting method that incorporates a smoothed mean and a tensor-variate autoregressive moving average (ARMA) model with parameter reduction. To effectively implement this method, there are three key considerations to bear in mind. Firstly, it is crucial to ensure that sufficient historical data is available for the model training process to be completed successfully. Secondly, while we have chosen the B-spline as the smoothing method for capturing the smoothed mean, it is only one among various smoothing methods available. Depending on the specific context or scenario, alternative smoothing techniques may be more suitable. Lastly, it is essential to carefully determine the CP rank in the decomposition process, taking into account the actual compression requirements of the data being analyzed. By considering these factors, our proposed method can be tailored and optimized to address the unique challenges posed by high-order tensor time series data.