Multidimensional scaling and nonlinear biplots - Understanding Biplots - page 225

Figure 5.17 Circle projection prediction markers are found by orthogonal projection of
the origin in R onto the intersection spaces L ∩ N .
the point β k (µ) where the circle intersects the biplot axis ensuring that P is orthogo-
nally projected onto the line O β k (µ) . The predicted value of the original variable Y for
point P is therefore the marker corresponding to the point β k (µ) on the biplot trajectory.
This process is illustrated in Figure 5.20. Once p prediction biplot trajectories are fitted
to the nonlinear biplot, the predictions for all p original variables are obtained by the
one circle with diameter OP where the circle intersects each of the biplot trajectories.
If a biplot trajectory intersects the circle more than once, we follow the convention
of selecting the intersection point closest to O. At first sight this may seem an elab-
orate process, but it may be helpful to note that when the biplot trajectories happen
to be linear, circle projection and normal projection are the same thing, as we saw
in Section 3.8.8.
Notice that once the prediction biplot trajectory is constructed in L , there is no
need to consider R or R + nor for the embedded representation of the original axis
in these spaces. All that is available to the user is the biplot space L which contains