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, in this paper, we propose a parsimonious tensor regression 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. To estimate the model parameters, an Alternating Least Squares algorithm is developed. In order to validate the model performance and robustness, a simulation exercise is produced. Moreover, we perform an empirical analysis that highlight the forecasting power of the model with respect to benchmark models. This is achieved by implementing an autoregressive specification on the Foursquares spatio-temporal dataset together with a macroeconomic panel dataset. Overall, the proposed model is able to outperform benchmark models present in the forecasting literature.

多维数据分析正在成为统计和机器学习研究中越来越重要的话题。考虑到它们的复杂性，通常将此类数据对象重塑为矩阵或向量，然后进行分析。但是，这种方法存在一些缺点。首先，它破坏了多维空间中数据点之间的固有互连，其次，要在模型中估计的参数数量呈指数增长。我们开发了克服此类缺陷的模型。特别是，在本文中，我们提出了一个简约的张量回归模型，该模型保留了数据集的固有多维结构。采用塔克（Tucker）结构来达到简约性，并引入收缩惩罚来处理过度拟合和共线性。要估算模型参数，开发了交替最小二乘算法。为了验证模型的性能和鲁棒性，进行了模拟练习。此外，我们进行了一项经验分析，突出了该模型相对于基准模型的预测能力。这是通过在Foursquares时空数据集和宏观经济面板数据集上实现自回归规范来实现的。总体而言，所提出的模型能够胜过预测文献中存在的基准模型。这是通过在Foursquares时空数据集和宏观经济面板数据集上实现自回归规范来实现的。总体而言，所提出的模型能够胜过预测文献中存在的基准模型。这是通过在Foursquares时空数据集和宏观经济面板数据集上实现自回归规范来实现的。总体而言，所提出的模型能够胜过预测文献中存在的基准模型。