SIMPLICIAL INFERENCE 15 (Figure 1) then converts the second-order approximation to Z given by the singular value decomposition into a graphical display. As for standard multivariate biplots the quality of the approximations is measured in terms of the proportion k~ +k~ k~ + ... +kh of the total variability of the compositional data set retained by Z 2, with k2 ka an obvious requirement. Figure 1 consists of an origin 0 which represents the center of the compositional data set, a vertex at position (k1vn,k2Vi2)/(N -1)! for each of the parts, labeled 1, ... ,D, and a case marker at position (N -1)!(u,.1,ur2) for each of theN cases, labeled c1 , ••• , CN. We term the join of 0 to a vertex i the my Oi and the join of two vertices i and j the link ij. These features constitute a biplot with the following main properties for the interpretation of the compositional variability. Links, mys and covariance structure. The links and rays provide information on the covariance structure of the compositional data set, in the sense that (11.1) (11.2) (11.3) cos(iOj) ~ corr[log{xi/g(x)},log{x /g(x)}]. It is tempting to imagine that (11.2) can be used to replace discredited corr(xi,x ) as a measure of the dependence between two components. Unfortunately, this measure does not have subcompositional coherence. A more useful result is the following. If links ij and kl intersect in M, then (11.4) cos(iMk) ~ corr[log(xdx ), log(xk/xl)], A particular case of this is when the two links are at right angles so that cos( iMk) ~ 0, implying that there is zero correlation of the two log ratios. This is useful in investigation of subcompositions for possible independence. Subcompositional analysis. The center 0 is the centroid (center of gravity) of the D vertices 1, ... , D. Since ratios are preserved under formation of subcom- positions it follows that the biplot for any subcomposition s is simply formed by selecting the vertices corresponding to the parts of the subcomposition and taking the center of the subcompositional biplot as the centroid of these vertices. Coincident vertices. If vertices i and j coincide or nearly so this means that var(log(xifxj) is zero or nearly so, so that the ratio Xi/x is constant or nearly so. Collinear vertices. If a subset of vertices, say 1, ... , C is collinear, then we know from our comment on subcompositional analysis that the associated subcomposition has a biplot that is one-dimensional, and then a technical argument leads us to the conclusion that the subcomposition has one-dimensional variability. Technically, this one-dimensionality is described by the constancy of ( C- 2) log contrasts of the components x 1 , ••• , xc. Inspection of these constant log contrasts may then give further insights into the nature of the compositional variability.
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