Hi .JOHN AITCHISON Case markers and recovery of data. Such markers have the easily established property that the scalar or dot product Ocn·ji represents the departure of log(xdxi) for case Cn from the average of this log ratio over all the cases. Let P and Pn in Figure 1 denote the projections of the center 0, and the compositional marker en on the possibly extended link ji. Then Ocn · ji = ±IPPniUil, where the positive sign is taken if the directions of PPn and ji are the same, otherwise the negative sign is taken. A simple interpretation can be obtained as follows. Consider the extended line ji as divided into positive and negative parts by the point P, the positive part being in the direction of ji from P. If Pn falls on the positive (negative) side of this line then the log ratio of log(xni/Xnj) of the nth composition exceeds (falls short of) the average value of this log ratio over all cases and the further Pn is from P the greater is this exceedance (shortfall) if Pn coincides with P, then the compositional log ratio coincides with the average. In Figure 1, the nth composition clearly has a log ratio log( xnd Xnj) which falls short of the overall average of this log ratio. A FIGURE 1. Components of a compositional relative variation di- agram (a) and interpretation of markers in relation to log ratio log(xdxi) (b) . (a) 0 ray 1 vertex j (b) 0 similar form of interpretation can be obtained from the fact that Ocn · Oi represents the departure of the centered log ratio log{xni/g(xn)} of the nth composition from the average of this centered log ratio over all replicates.

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