A parsimonious multivariate poisson model for market basket analysis

Intensified research on multivariate Poisson models offers new opportunities for the analysis of purchase quantities in market basket data. The investigation of positive or negative correlations in quantity decisions among product categories facilitates a deeper understanding of consumer purchase behavior. The applied multivariate log-normal Poisson model introduces interdependencies between categories with multivariate normal-distributed latent effects by means of a covariance matrix. As the size of this covariance matrix depends on the number of categories in the model, its ation may become tedious. Furthermore, we assume that quantity decisions do not interact for all pairs of categories. That is why we propose to use covariance selection to derive a parsimonious representation of the correlation structure. For two market basket data sets, we show that the vast majority of off-diagonal elements in the covariance matrix are irrelevant. For a data set with product categories, the model with a partly restricted covariance matrix achieves a better fit to the holdout data than the model with full covariance matrix. For a data set with subcategories of the broader category beverage, the proposed model with restricted covariance outperforms the model with full covariance matrix even on the calibration data. We conclude that interactions of quantity decisions are overall the exception, even for complements-in-use.