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.
Modelling the dynamics of urban venues is a challenging task as it is multifaceted in nature. Demand is a function of many complex and nonlinear features such as neighborhood composition, real-time events, and seasonality. Recent advances in Graph Convolutional Networks (GCNs) have had promising results as they build a graphical representation of a system and harness the potential of deep learning architectures. However, there has been limited work using GCNs in a temporal setting to model dynamic dependencies of the network. Further, within the context of urban environments, there has been no prior work using dynamic GCNs to support venue demand analysis and prediction. In this paper, we propose a novel deep learning framework which aims to better model the popularity and growth of urban venues. Using a longitudinal dataset from location technology platform Foursquare, we model individual venues and venue types across London and Paris. First, representing cities as connected networks of venues, we quantify their structure and note a strong community structure in these retail networks, an observation that highlights the interplay of cooperative and competitive forces that emerge in local ecosystems of retail businesses. Next, we present our deep learning architecture which integrates both spatial and topological features into a temporal model which predicts the demand of a venue at the subsequent time-step. Our experiments demonstrate that our model can learn spatio-temporal trends of venue demand and consistently outperform baseline models. Relative to state-of-the-art deep learning models, our model reduces the RSME by ~ 28% in London and ~ 13% in Paris. Our approach highlights the power of complex network measures and GCNs in building prediction models for urban environments. The model could have numerous applications within the retail sector to better model venue demand and growth.