Page 191 - Computational Statistics Handbook with MATLAB

178 Computational Statistics Handbook with MATLAB

Zp2 = Z*proj2;
figure
plot(Zp1(:,1),Zp1(:,2),'k.'),title('Structure 1')
xlabel('\alpha^*'),ylabel('\beta^*')
figure
plot(Zp2(:,1),Zp2(:,2),'k.'),title('Structure 2')
xlabel('\alpha^*'),ylabel('\beta^*')
The results are shown in Figure 5.45 and Figure 5.46, where we see that pro-
jection pursuit did find two structures. The first structure has a projection
pursuit index of 2.67, and the second structure has an index equal to 0.572.

u
T
ou
T
Grand
Grand
Grand T To ou r r r
Grand
r
ou
The grand tour of Asimov [1985] is an interactive visualization technique that
enables the analyst to look for interesting structure embedded in multi-
dimensional data. The idea is to project the d-dimensional data to a plane and
to rotate the plane through all possible angles, searching for structure in the
data. As with projection pursuit, structure is defined as departure from nor-
mality, such as clusters, spirals, linear relationships, etc.
In this procedure, we first determine a plane, project the data onto it, and
then view it as a 2-D scatterplot. This process is repeated for a sequence of
planes. If the sequence of planes is smooth (in the sense that the orientation
of the plane changes slowly), then the result is a movie that shows the data
points moving in a continuous manner. Asimov [1985] describes two meth-
ods for conducting a grand tour, called the torus algorithm and the random
interpolation algorithm. Neither of these methods is ideal. With the torus
method we may end up spending too much time in certain regions, and it is
computationally intensive. The random interpolation method is better com-
putationally, but cannot be reversed easily (to recover the projection) unless
the set of random numbers used to generate the tour is retained. Thus, this
method requires a lot of computer storage. Because of these limitations, we
describe the pseudo grand tour described in Wegman and Shen [1993].
One of the important aspects of the torus grand tour is the need for a con-
tinuous space-filling path through the manifold of planes. This requirement
satisfies the condition that the tour will visit all possible orientations of the
projection plane. Here, we do not follow a space-filling curve, so this will be
called a pseudo grand tour. In spite of this, the pseudo grand tour has many
benefits:

• It can be calculated easily;
• It does not spend a lot of time in any one region;
• It still visits an ample set of orientations; and
• It is easily reversible.

© 2002 by Chapman & Hall/CRC