Journal of Data Science

Abstract: In compositional data, an observation is a vector with non-negative components which sum to a constant, typically 1. Data of this type arise in many areas, such as geology, archaeology, biology, economics and political science among others. The goal of this paper is to extend the taxicab metric and a newly suggested metric for com-positional data by employing a power transformation. Both metrics are to be used in the k-nearest neighbours algorithm regardless of the presence of zeros. Examples with real data are exhibited.

Compositional data consist of known compositions vectors whose components are positive and defined in the interval (0,1) representing proportions or fractions of a “whole”. The sum of these components must be equal to one. Compositional data is present in different knowledge areas, as in geology, economy, medicine among many others. In this paper, we propose a new statistical tool for volleyball data, i.e., we introduce a Bayesian anal- ysis for compositional regression applying additive log-ratio (ALR) trans- formation and assuming uncorrelated and correlated errors. The Bayesian inference procedure based on Markov Chain Monte Carlo Methods (MCMC). The methodology is applied on an artificial and a real data set of volleyball.

Compositional data are positive multivariate data, constrained to lie within the simplex space. Regression analysis of such data has been studied and many regression models have been proposed, but most of them not allowing for zero values. Secondly, the case of compositional data being in the predictor variables side has gained little research interest. Surprisingly enough, the case of both the response and predictor variables being compositional data has not been widely studied. This paper suggests a solution for this last problem. Principal components regression using the 𝛼 -transformation and Kulback-Leibler divergence are the key elements of the proposed approach. An advantage of this approach is that zero values are allowed, in both the response and the predictor variables side. Simulation studies and examples with real data illustrate the performance of our algorithm.