SIMPLICIAL INFERENCE 17

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, ... ,

c

D},

we have

cov{log(cixdcixi},

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

components

of the compositions, a

distortion which would prevent any compositional interpretation from the resulting

diagram.

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

[Al4].

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

s

of a full compositions

x

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, ... ,

C-

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

t,

say 0 and 1, for given composition

x

as