A Measure of Difference for Compositional Data Based on Measures of Divergence

For the application of many statistical methods it is necessary to establish the measure of di erence to be used This measure has to be de ned in accordance with the nature of the data In this study we propose a measure of di erence when the data set is compositional We analyze its properties and we present examples to illustrate its performance INTRODUCTION It is well known that the usual dissimilarities and distances are inadequate to measure the di erence between two compositonal data see for further details In Aitchison proposes that a suitable measure of di erence de ned on the simplex S should verify two essential requirements perturbation invariance and subcompositional dominance One of the most widely used measures of divergence between two multinomial prob ability distributions is the Kullback Leibler information number The purpose of this paper is to propose a measure of di erence between two compositional data based on the Kullback Leibler divergence In the next section we de ne the new measure we analyze its propierties and we show that the compositional requirements are veri ed Then we expose an interpretation of the measure and present an example to illustrate its performance MEASURE OF DIFFERENCE BETWEEN TWO COMPOSITIONS In Aitchison proposes that any scalar measure of di erence between two compositions x x S can be expressed in terms of the ratios of the components More accurately a suitable measure of di erence should be a function of the compositions x x and x x where simbolizes the perturbation operation introduced in We call these compositions as the perturbation di erences between x and x Note that to the special case x x we obtain the perturbation di erence x x e where e D D D is the center of the simplex S It is well knwon that a suitable measure is the distance dA squared called Aitchison distance d A x x D d E clr x clr x where dE represents the Euclidean distance and clr the clr transformation see for more details Because the distance dA is perturbation invariant we can express as function of the perturbation di erences d A x x d A e x x d A e x x D d E clr e clr x x d E clr e clr x x On the other hand we can express the Kullback Leibler information number see between two compositonal data x x S as the expression