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value µ we must have e k ( M 1
) J )( JM 1 e k ) = µ and therefore
τ = µ
e k ( M ) 1 JM 1 e k .
This shows that the position of the marker for predicting the value µ on the k th variable
is given by
µ
e k ( M ) 1 JM 1 e k
e k ( M 1
) J .
4.5 Adding new points and variables to a CVA biplot
4.5.1 Adding new sample points
Similarly to PCA biplots, the vector-sum method may be used for adding new points to
a CVA biplot. The process is illustrated in Figure 4.10. First we construct a CVA biplot
with interpolation axes with the following function call:
CVAbipl(Ocotea.data[,3:5], X.new.samples = NULL,
G = indmat(Ocotea.data[,2]), ax.type = "interpolative",
alpha = 0.95, colours = c("red","blue","green"),
n.int = c(10,10,10), pch.means = c(15,16,17),
pch.samples = 0:2)
The function vectorsum.interp is then called and the values Ve sD = 134, Ve s L =
375 and FibL = 1170 selected on the respective interpolation axes to obtain the position
of the specimen of unknown origin as indicated by the solid red circle in Figure 4.10.
Note that we have increased the default n.int settings to obtain axis calibrations that
allow accurate determination of the input values. As our preferred alternative we can
use the incorporated algebraic calculation of (4.9) in the CVAbipl function with the
following function call:
CVAbipl(Ocotea.data[,3:5], X.new.samples = matrix(c(134,375,1170),
nrow = 1), G = indmat(Ocotea.data[,2]), ax.type = "predictive",
alpha = 0.95, colours = c("red","blue","green"), pch.means =
c(15,16,17), pch.samples = 0:2, pch.new.col = "black",
pch.new = 8)
The latter call results in a CVA biplot exactly like Figure 4.4 but without the classification
regions. We draw the reader's attention to the different directions of the interpolative axes
of Figure 4.10 in comparison to the directions of the corresponding predictive axes of
Figure 4.4.
4.5.2 Adding new variables
We saw in Section 3.5 how to use the regression method to add a new variable to a
PCA biplot. The same method can also be used for adding a new variable to a CVA
biplot. Let x : n × 1 denote a vector of sample values for a new variable. We assume
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