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.