It must be clear from the above aspects of interpretation that the fundamental
elements of a compositional biplot are the links, not the rays as in the case of
variation diagrams for unconstrained multivariate data. The complete set of links,
by specifying all the relative variances, determines the compositional covariance
structure and provides direct information about subcompositional variability and
independence. It is also obvious that interpretation of the relative variation diagram
is concerned with its internal geometry and would, for example, be unaffected by
any rotation or indeed mirror-imaging of the diagram.
Another fundamental difference between the practice of biplots for uncon-
strained and compositional data is in the use of data scaling. For unconstrained
data, if there are substantial differences in the variances of the components, biplot
approximation may concentrate its effort on capturing the nature of the variability
of the most variable components and fail to provide any picture of the pattern of
variability within the less variable components. Since such differences in variances
may simply arise because of scales of measurement, a common technique in such
biplot applications is to apply some form of individual scaling to the components of
the unconstrained vectors prior to application of the singular value decomposition.
No such individual scaling is necessary for compositional data when the analysis in-
volves log ratio transformations. Indeed, since for any set of constants (
c1, ... ,
we have
log(ckxk/clxl)} =
cov{log(xdxj}, log(xk/x1)},
it is obvious that the covariance structure and therefore the compositional biplot
are unchanged by any differential scaling or perturbation of the compositions. Only
the centering process is affected by such differential scaling. Moreover any attempt
at differential scaling of the
log mtios
of the components would be equivalent to ap-
plying differential power transformations to the
of the compositions, a
distortion which would prevent any compositional interpretation from the resulting
For some applications of biplots to compositional data in a variety of geological
contexts, see [A9], and for applications in other disciplines and to extensions to
conditional biplots, see
12. Subcompositional analysis
A common problem in compositional data analysis appears to be marginal
analysis in the sense of locating subcompositions of greatest or of least variabil-
ity. For this purpose, the measure of total variation discussed in Section 6.2 pro-
vides for any subcomposition
of a full compositions
the estimate of the ratio
trace r(s)/tracer(x) as the proportion of the total variation explained by the sub-
composition. In such forms of analysis it should be noted that a
(1, ... ,
subcomposition is a set of
C - 1
particular log contrasts and so the variability
explained by a C-part subcomposition can also be compared with that achieved by
the first
C -
1 principal log contrasts.
Another interesting form of subcompositional analysis is where the composition
plays the role of regressor, for example in categorical regression, where we wish to
examine the extent to which, for example, type of rock depends on full major oxide
composition or some subcomposition. For binary regression a sensible approach
is to set the conditional model of type
say 0 and 1, for given composition
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