Information Technology Reference
In-Depth Information
3
Principal component
analysis biplots
The general form of an asymmetric biplot was introduced in Chapter 2 as a multivariate
scatterplot. In this chapter we turn our attention to the first and, arguably, the simplest and
most popular form of asymmetric biplot, the principal component analysis (PCA) biplot.
Although only referred to as multivariate scatterplots, many of the figures in Chapter 2
are indeed PCA biplots. The PCA biplot is asymmetric because it represents the samples
and variables of
X
; a symmetric form that mainly represents covariance or correlation is
discussed in Chapter 10. In this chapter we first review algebraic and geometric properties
of PCA before discussing some examples of its biplot.
Our main instrument for constructing PCA biplots is our R function
PCAbipl
.We
discuss the capabilities of
PCAbipl
by introducing some interesting applications and
enhancements of more conventional PCA biplots. Readers are urged to explore the poten-
tial of
PCAbipl
using their own data.
3.1 An example: risk management
Van Blerk (2000) discuss an example where biplot methodology is applied in risk man-
agement. Risk management is critical for institutions such as banks, pension funds, asset
management and insurers to ensure that their liabilities can be met at all times. A popular
quantity for measuring exposure to market risk is the so-called value at risk (VAR). This
quantity is defined by Jorion (1997) as 'the expected maximum loss (or worst case) over
a given time interval at a given confidence level'.
In practice the calculation of VAR is complicated. The institution considered here uses
historical data and simulation to calculate the empirical probability distribution of the
return on a financial instrument. The empirical
π
th percentile is then used as an estimate
of the (100 -
π)
% VAR. Since risk management is concerned with expected losses the
