Amalgamations are valid in compositional data analysis, can be used in agglomerative clustering, and their logratios have an inverse transformation

Abstract

Amalgamations (i.e. summing) of parts can be included as new parts in compositional data analysis, and logratios can then be formed using these amalgamations as well as any of the individual parts themselves. In the first contribution of this paper, a comparison is made of the performance of different logratio transformations in explaining the structure of a geochemical data set ​− ​some of the transformations include amalgamations. The second contribution shows how amalgamations suggest a natural way of clustering compositional data, leading to a new clustering algorithm for compositional data. The third contribution deals with the inverse transformation where amalgamations are involved in the logratios. In the case of a linearly independent set of logratios of parts and amalgamated parts, consisting of one less logratio than the number of compositional parts and where each part is present in at least one logratio, it is possible to back-transform this set of logratios to the original parts. The solution is defined by a set of linear equations. A special case is a set of linearly independent pairwise logratios of parts, which are also invertible back to the original parts.

Keywords

• Amalgamation
• Balance
• Clustering
• Compositional data
• Inverse transformation
• Logratio transformation