Tensor Autoregression: A Multidimensional Time Series Model

What is the multidimensional time series? How to characterize these time series with tensor autoregression?

In the past decades, we witnessed great development of time series models, especially on the univariate time series and multivariate time series. One most classical time series model would be the time series autoregression, including univariate autoregression and vector autoregression. However, almost all of these models are not well-suited to the multidimensional time series. In this story, we introduce a tensor autoregression for modeling multidimensional time series. Let’s get started!

Time Series Models

Time series analysis has great significance in many scientific fields and industrial applications. In the real world, a large variety of time series is univariate or multivariate. Both univariate and multivariate time series are well studied in the past decades and there are many mature methods that have been developed on such kind of data, supporting both analysis and forecasting. However, multidimensional time series are not well explored, but of great significance in the real-world applications.

In practice, multidimensional time series have a more complicated representation than multivariate time series. For instance, multivariate time series usually consist of vector yt of length N (i.e., N variables), at time t, while multidimensional…