SIMPLICIAL INFERENCE 9
There are a number of useful and equivalent ways
[A5,
Chapter 4] in which to
summarize such a
sufficient
set of second-order moment characteristics: the
log
mtio covariance matrix
(6.7)
~(x)
=
[ai;]
=
cov{log(xdxn),log(x;fxn)],
using only the final component
xn
as the common ratio divisor; the
centered log
mtio covariance matrix
(6.8)
r(x)
=
[cov{log(xdg(x)}, log{x;/g(x)}];
or the variation matrix
(6.9)
with elements involving only pairs of components. These three dispersion char-
acteristics are equivalent: each can be derived from any other by simple matrix
operations
[A5,
Chapter 4]. Note that
Tis
symmetric, has zero diagonal elements,
and cannot be expressed as the standard covariance matrix of some vector, but has
the advantage of considering components only two at a time. We shall see later in
Section 8 that it has an important restricted negative definite property.
A first reaction to this variation matrix characterization is often surprise be-
cause it is defined in terms of variances only. The simplest statistical analogue is
in the use of a completely randomized block design in, say, an industrial experi-
ment From such a situation, information about
var(yi - Y;)
for all
i,
j is a sufficient
description of the variability for purposes of inference.
Note also that if
x
and y are independent compositions then, for any form
V
ofthe dispersion characteristic
V(x
o
y)
=
V(x)
+
V(y),
meeting, in particular, the
perturbation criterion (c), and
V(a·x)
=
a2 V(x),
meeting the power transformation
criterion (d).
Relative variances such as var{log(xdx;)} provide some compensation for de-
privation of the meaningless product-moment correlation interpretations. For ex-
ample,
Tij
=
0 means a perfect relationship between
Xi
and
x;
in the sense that
the ratio
xdx;
is constant, replacing the unusable idea of perfect positive corre-
lation between
Xi
and
x;
by one of perfect proportionality. Again, the larger the
value of
Tij
the more the departure from proportionality with
Tij
=
oo replacing
the unusable idea of zero correlation or independence between
Xi
and
x;.
Note
that if we are really interested in hypotheses of independence these are most ap-
propriately expressed in terms of independence of subcompositions. For example
independence of the (1,2,3)- and ( 4,5)-subcompositions would be reflected in the
following statements:
(6.10)
In the study of unconstrained variability in RD it is often convenient to have
available a measure of total variability, for example in principal component analysis
and in biplots. For such a sample space the trace of the covariance matrix is
the appropriate measure. Here we might consider trace{r(x)}, the trace of the
symmetric centered log ratio covariance matrix r(x). Equally, we might argue on
common sense grounds that the sum of all the possible relative variances in
T(x),
namely Lij var{log(xdx;)}, would be equally good. These two measures indeed
differ only by a constant factor and so we can define totvar(x), a measure of total
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