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.
Previous Page Next Page