Monoplots - Understanding Biplots - page 431

show the corresponding monoplots of the covariance matrices of the unscaled data and
the scaled data, respectively. We note that the latter monoplot is a simple form of a
correlation monoplot (see Section 10.2.2). The monoplot in the top panel of Figure 10.4
results from the function call

MonoPlot.cov(X = Flotation.data[,-1], samples.plot = FALSE,
as.axes = TRUE, scaled.mat = FALSE, offset = c(0,0.3,0.2,0.1))

Setting argument samples.plot = TRUE gives the joint plot in the bottom panel.
Figure 10.5 is the result of similar function calls with argument scaled.mat = TRUE .

10.2.2 Correlation monoplot

When X has been normalized to unit sums of squares, X X becomes a correlation matrix,
R . Then the squared distance between two points (variables) i and i has elements

d ii =

2 ( 1 − r ii ) (10.1)

and so gives a direct measure of correlation that is approximated in r dimensions. In an
exact representation each of the p plotted points of the map will be at unit distance from
the origin. With a unit correlation the i th and i th points coincide. A correlation of

1
generates points furthest apart. Further, (10.1) implies that if θ ii is the angle subtended
at the origin then cos( θ ii ) = r ii . By plotting a unit circle (see Figure 10.6) we can see
the degree of approximation of each variable. This is not a geometrical representation
of the adequacy criterion discussed in Section 3.3, but shows how well each variable
approximates the unit correlation of exact representations, given algebraically by the
square root of diag ( V

−

2 JV ) .
Figure 10.6 is a result of the function call

MonoPlot.cor(X = Flotation.data[,-1],
offset = c(-0.1, 0.25, 0.1, 0.1),
print.ax.approx = TRUE, offset.m = rep(-0.2, 6))

We note that the relative positions of the red solid circles in Figure 10.6 are similar
to those in Figure 10.5. By considering the correlation matrix in Table 10.2, it can
be concluded that the monoplot in Figure 10.6 approximates the intercorrelations quite
well: in the case of the variable with the highest degree of approximation, BMCFD
(0.98), for example, its position is almost orthogonal to those of RMOF and EXFFBB ,
indicating almost zero correlations, while its position relative to those of BMCFF , RMP
and BMSWA shows the increasingly negative correlations with these variables.

10.2.3 Coefficient of variation monoplots

The coefficient of variation, CV, of a variable is its standard deviation divided by its
mean. The CV should not be used with interval scales for then the mean depends additively