INDCLUS for spatial proximity data

A suitable extension of the INDCLUS model is proposed for clustering spatial units in three-way proximity data taking into account the spatial nature of the units. Specifically, our concern is three-way two-mode data consisting of square symmetric matrices S_h (h = 1,...,H) of pairwise proximities of a set of I spatial units coming from H domains. INDCLUS searches for a covering of the units, which is common to all the H domains, and a set of weights and an additive constant, which are different for each domain. The model is fitted by solving a least-squares optimization problem. In order to identify a covering of spatial units accounting for and taking advantage of the spatial nature of the units themselves, a penalty term based on a suitable spatial contiguity matrix of size I is added to the objective function. Furthermore, a tuning coefficient allows to balance the identification of both a common classification of units for all domains and approximately spatial homogeneous clusters. An Alternating Least-Squares algorithm is provided to solve the penalized problem. The proposed method has been applied to the subset of BES indicators included in the Economic and Financial Document (DEF), submitted annually to the Government and approved by Parliament.