Predicting Multidimensional Data via Tensor Learning

The analysis of multidimensional data is becoming a more and more relevant topic in statistical and machine learning research. Given their complexity, such data objects are usually reshaped into matrices or vectors and then analysed. However, this methodology presents several drawbacks. First of all, it destroys the intrinsic interconnections among datapoints in the multidimensional space and, secondly, the number of parameters to be estimated in a model increases exponentially. We develop a model that overcomes such drawbacks. In particular, we proposed a parsimonious tensor regression based model that retains the intrinsic multidimensional structure of the dataset. Tucker structure is employed to achieve parsimony and a shrinkage penalization is introduced to deal with over-fitting and collinearity. An Alternating Least Squares (ALS) algorithm is developed to estimate the model parameters. A simulation exercise is produced to validate the model and its robustness. Finally, an empirical application to Foursquares spatio-temporal dataset and macroeconomic time series is also performed. Overall, the proposed model is able to outperform existing models present in forecasting literature.