Introduction - Understanding Biplots - page 7

Asymmetric plots Symmetric plots

Euclidean embeddable
distance
Nonlinear biplots
Chapter 5

AoD biplots
Chapter 5

Mahalanobis distance
CVA biplots
Chapter 4 Pythagorean distance

PCA biplots
Chapter 3

Biadditive
biplots
Chapter 6

MDS biplots
Chapter 5

Generalized biplots
Chapter 9

CATPCA biplots
Chapter 8

Extended matching
coefficient

Chi-squared distance -

MCA biplots
Chapter 8

CA biplots
Chapter 7

Biplots

Figure 1.5 Summary of the different types of biplots discussed in subsequent chapters.

In a symmetric biplot, rows and columns have equal status and we aim to find two
sets of coordinates A and B , one for the rows and one for the columns respectively.
Now, the main interest is in the inner product AB and there is less interest in distance
interpretations. A popular version of correspondence analysis (CA) approximates chi-
squared distance, treating either the rows or columns as if they were 'variables' and thus
giving two asymmetric biplots, not linked by a useful inner product. This form of CA
is not a biplot and is sometimes referred to as a joint plot (see also Figure 10.4); other
forms of CA do treat X symmetrically.

1.3 Software

A library of functions has been developed in the R language (R Development Core
Team, 2009) and is available on the website www.wiley.com/go/biplots. Throughout this
topic reference will be made to the functions associated with the biplots being discussed.
Examples of the commands to reproduce the figures in this topic are given in the text.
Sections are also included with specific information about the core functions needed for
the different types of biplots.

1.4 Notation

Matrices are used extensively to enable the mathematically inclined reader to understand
the algebra behind the different biplots. Bold upper-case letters indicate matrices and