Title: Combinatorial regression model in abstract simplicial complexes
Authors: Andrej Srakar - University of Ljubljana (Slovenia) [presenting]
Abstract: In the regression analysis of market share data, four main parametric type models are prevalent: multinomial logistic regression, attraction models of various types, Dirichlet covariance models, and compositional regression. We extend this arsenal of possibilities with a fifth type, a completely novel regression perspective labelled combinatorial regression, based on combining $n$-tuplets of sampling units into groups and treating them as compositions. The novel perspective, estimated using Bradley-Terry based maximum likelihood approach, allows an extensive number of perspectives in the analysis of, for example, triplets, quadruplets or quintuplets or using as a measure of disparity between the units (to construct regressors) different divergence measures. It also allows applications to very small datasets as the number of units in the new model can be expressed in terms of factorial products of units of the original sample. We provide the analysis of the new approach for triplets and quadruplets using Jensen-Shannon and generalized Jensen-Shannon divergence measures and provide the Gaussian asymptotic limits of the approach with exploring its properties also in a Monte Carlo simulation study. In a short application, we analyze sessile hard-substrate marine organisms image data from Italian coast areas, which allows exploring the new approach in relative abundance data setting.