Monoplots - Understanding Biplots - page 434

forms of factor analysis, which replace the diagonals by nonunit communalities to give
a matrix of rank r . However, a simple alternative way forward was proposed by Hills
(1969) who suggested that the correlation matrix be analysed by MDS and, in particular,
by PCO. This is equivalent to replacing the correlation matrix R by a squared-distance
matrix 11 - R . Any MDS method fits only the off-diagonal values, so approximating the
zero diagonal is not an issue. In accordance with the standard PCO procedure (Chapter 5),
we require the spectral decomposition of

1
2

( I − 11 / p )( 11 − R )( I − 11 / p )

−

(10.3)

1
2

( I − 11 / p R ( I − 11 / p )

which is the same as . The distances approximated are given by

d ii = ( 1 − r ii ).

Now, the origin is at the centroid of the p points which, rather than unit distance, has
squared distances from the centroid given by

1
2

( 1 + 1 R1 / p 2 ) 1 − 1 R / p .

Thus, the angle properties do not apply but the distance properties do and with better
approximation.
Rather than operate on R we may operate on R ∗ R , the matrix whose entries are
squared correlations. Both R and R ∗ R are positive semi-definite so PCO remains avail-
able. This gives distances defined by

d ii = ( 1 − r ii ).

The obvious difference is that only the absolute values of the correlations, not their signs,
are approximated.
Even though we are basing these approximations on PCO, neither nonlinear biplots
nor the regression method described above may be used to upgrade these monoplots to
biplots. This is because the monoplot already gives information on the variables. It is
likely that a method might be found for adding the n units, but we have not investigated
this possibility further.
The correlation monoplot resulting from a PCO of (10.3) for the flotation data is given
in the top panel of Figure 10.8, while the bottom panel is the corresponding monoplot
when the R in (10.3) is replaced by R ∗ R . The distances between the points in these
monoplots provide good approximations to the intercorrelations given in Table 10.2.
Figure 10.8 is a result of the call

MonoPlot.cor2(X = Flotation.data[,-1], offset.m = rep(0.4, 6),
pos.m = rep(2,6), rotate.degrees = -17)