Tensor extrapolation: Forecasting large-scale relational data

Abstract

Tensor extrapolation attempts to forecast relational time series data using multi-linear algebra. It proceeds as follows. Multi-way data are arranged in the form of tensors, ie multi-dimensional arrays. Tensor decompositions are then used to retrieve periodic patterns in the data. Afterwards, these patterns serve as input for time series methods. However, previous approaches to tensor extrapolation are limited to preselected time series approaches and binary data. To permit automatic forecasting, the paper at hand connects state-of-the-art tensor decompositions with a general class of state-space time series models. Moreover, it highlights the need for data preprocessing in settings with real-valued data. In doing so, it enables data-driven model selection and estimation in large-scale forecasting problems. Numerical experiments show the effectiveness of the proposed method in identifying relevant underlying patterns and demonstrate its superiority over established extrapolation methods in terms of forecast accuracy.

Keywords:

• Forecast accuracy
• multi-linear algebra
• relational data
• tensor decomposition
• time series analysis

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 Using the Frobenius norm assumes that the random variation in the tensor data follows a Gaussian distribution. For alternative models, compare Chi and Kolda (2012) and Papalexakis et al. (2016).

2 Our approach to tensor extrapolation performs “local” optimisations based on the periodic patterns in the “time” component matrix. The results are consistent with those of “global” optimisations that resort to the tensor entries associated with the single periodic patterns. The proof is straightforward and, hence, left to the reader.

3 Wang et al. (2018) apply tensor extrapolation to binary data as well as to real-valued data. With regard to the latter, the authors report disappointing results. This could be due to the fact that the time series involved are of different scale. We return to this point in Section 3.

4 Hoff (2011) examines yearly trade data in a tensor-based framework. He detects correlations among the index sets of the data, but does not use this information for prediction.