Bayesian tensor regression based on the Horseshoe prior and its application in air quality data

For ultra-high dimensional sparse linear regression scenarios where the predictors and responses are in tensor form, the problems regarding their parameter estimation and variable selection have become significant challenges. To effectively address this issue, we propose a tensor-on-tensor regression model based on the Horseshoe prior, which is embedded within a fully Bayesian framework. Under this framework, we first employ a low-rank canonical polyadic (CP) decomposition tool to reduce the dimensionality of the coefficient array for computational convenience. A hierarchical Horseshoe prior is then imposed to capture sparsity patterns in the regression coefficients and the Gibbs sampling technique is used to sample from the full conditional distribution of the parameters iteratively. Meanwhile, we apply the deviance information criterion (DIC) to solve the Bayesian model selection problem. Moreover, through numerical simulation studies, we confirm that the tensor regression model proposed in this paper can demonstrate satisfactory effectiveness. Finally, the model is applied to the air pollution dataset to investigate the primary pollution factors influencing air quality in the Beijing-Tianjin-Hebei Urban Agglomeration in China, and the results show that they are largely consistent with the actual situation.