Graphics Reference
In-Depth Information
Nowadays many graphical tools permit one to enhance basic information with
many additional elements that transform a
-D representation into a higher-dimen-
sional representation.
Wainer and Velleman (
) noted that some concepts of geometry are inconsis-
tently imported in the statistical literature. However, in spite of differences in defi-
nitions and notations, statisticians and mathematicians use visualization techniques
for the same goal: to reveal (hidden) relationships in the data structure (also known
as fitting in statistics).
In the next subsection we separately introduce the notions of Cartesian space,
distance, and metric space. hese are the foundations of principal coordinates and
dendrograms. hen, in the following sections, we present innovative visualization
techniques that can be realized by combining the old and the new.
Distance and Metric Space
4.2.1
he method statisticians use to collect multivariate data is the data matrix.An
X
n,p
data matrix represents a set of n multidimensional observations, described by a set
of p variables. From a geometric point of view, the generic row vector
x
i
(where
i
,...,n)representsapointintheR
p
Cartesian space. Of course the alterna-
tive representation consists in representing p points (variable) in the n-dimensional
space.
According to Euclidean geometry, Cartesian space refers to a couple of ordered
and oriented orthogonal axes, which admits a definition of a unit measure. In the
year
B.C. approximately, the Greek mathematician Euclid formalized the math-
ematical knowledge of his time. His book Elements is considered the second most
popular book in history, ater the Holy Bible. he greatness of his contribution is
demonstrated by the fact that most of our current knowledge in geometry is in so-
called Euclidean geometry.
However, we must wait for Descartes for the formalization of the isomorphism
between algebraic and geometric structures. In fact, what we call a Cartesian space
was first introduced by Newton several years later.
he interest and the long story behind the definition of Cartesian space demon-
strates the importance of such a mathematical concept.
Given a set Ω, any function d
=
+
Ω
Ω
R
satisfying the following three prop-
erties is a distance:
a) d
(
X
i
,
X
j
)=
X
i
=
X
j
b) d
(
X
i
,
X
j
)=
d
(
X
j
,
X
i
)
(symmetry)
c) d
(
X
i
,
X
j
)
d
(
X
i
,
X
h
)+
d
(
X
h
,
X
j
)
,
where
X
i
,
X
h
,
X
j
Ω(triangular inequality)
We will call Ω a metric space if its elements define a distance.
Looking at the properties of distance, we can easily understand the reasons that
induced statisticians to turn to indexes having the properties of distance. It is impor-
tant to note that the most commonly used methods satisfy special cases of minimal
distance.
