Convex clustering method for compositional data modeling

Compositional data refer to a vector with parts that are positive and subject to a constant-sum constraint. Examples of compositional data in the real world include a vector with each entry representing the weight of a stock in an investment portfolio, or the relative concentration of air pollutants in the environment. In this study, we developed a Convex Clustering approach for grouping Compositional data. Convex clustering is desirable because it provides a global optimal solution given its convex relaxations of hierarchical clustering. However, when directly applied to compositions, the clustering result offers little interpretability because it ignores the unit-sum constraint of compositional data. In this study, we discuss the clustering of compositional variables in the Aitchison framework with an isometric log-ratio (ilr) transformation. The objective optimization function is formulated as a combination of a L₂-norm loss term and a L₁-norm regularization term and is then efficiently solved using the alternating direction method of multipliers. Based on the numerical simulation results, the accuracy of clustering ilr-transformed data is higher than the accuracy of directly clustering untransformed compositional data. To demonstrate its practical use in real applications, the proposed method is also tested on several real-world datasets.