Page 194 - Computational Statistics Handbook with MATLAB

Chapter 5: Exploratory Data Analysis 181

rotates. We believe that the extra dimension conveyed by the speed is difficult
to understand unless the analyst has experience looking at grand tour mov-
ies.
In order to implement the pseudo grand tour, we need a way of obtaining
the projection vectors α αα α t() and β ββ β t() . First we consider the data vector x. If d
is odd, then we augment each data point with a zero, to get an even number
of elements. In this case,

x = ( x … x 0, , , ); for d odd.
1 d
This will not affect the projection. So, without loss of generality, we present
the method with the understanding that d is even. We take the vector α αα α t() to
be

⁄
,
,
,
α α α α t() = 2 d sin( ω t cos, ω t … sin ω d 2 t cos ω d 2 t) , (5.24)
⁄
⁄
1
1
and the vector β ββ β t() as
,
,
⁄
,
β β β β t() = 2 d cos( ω 1 t –, sin ω 1 t … cos ω d 2⁄ t – sin ω d 2⁄ t) . (5.25)
We choose ω i and ω j such that the ratio ω ω⁄ j is irrational for every i and
i
j. Additionally, we must choose these such that no ω ω⁄ is a rational multi-
i j
ple of any other ratio. It is also recommended that the time step ∆t be a small
positive irrational number. One way to obtain irrational values for ω is to let
i
ω = P , where P is the i-th prime number.
i i i
The steps for implementing the 2-D pseudo grand tour are given here. The
details on how to implement this in MATLAB are given in Example 5.28.
PROCEDURE - PSEUDO GRAND TOUR
to an irrational number.
1. Set each ω i
2. Find vectors α αα α t() and β ββ β t() using Equations 5.24 and 5.25.
3. Project the data onto the plane spanned by these vectors using
Equations 5.23 and 5.24.
α α α α t() β β β β t()
4. Display the projected points, z i and z i , in a 2-D scatterplot.
5. Using ∆t irrational, increment the time, and repeat steps 2
through 4.

Before we illustrate this in an example, we note that once we stop the tour at
an interesting projection, we can easily recover the projection by knowing the
time step.

© 2002 by Chapman & Hall/CRC