Robust aggregation of compositional and interval-valued data: The mode on the unit simplex

Abstract

We consider calculation of the mode of compositional data, the data related to each other through a linear constraint. Compositional data arises in various extensions of the fuzzy sets theory (type-2 interval-valued, intuitionistic, hesitant fuzzy sets), biomedicine (relative abundance, genome sequencing, activity recognition), and data analytics (various wealth indices, interval-valued observations, traffic congestion, etc.). Mode is a pre-aggregation function in the case of single variable, used as a classical estimator robust to outliers, but its multivariate extensions face the challenges of high computational complexity and potential oversmoothing. In this work we present several novel techniques for mode estimation on the unit k-simplex representing compositional, interval-valued, and general vector-valued data. We highlight the re-weighted k-nearest neighbours algorithm based on the Choquet integral with respect to a 2-additive fuzzy measure, compare its performance against other approaches based on spatial partitioning, and illustrate its applications to aggregation of real-world interval-valued data sets.