Nm. Faber et al., ASPECTS OF PSEUDORANK ESTIMATION METHODS BASED ON THE EIGENVALUES OF PRINCIPAL COMPONENT ANALYSIS OF RANDOM MATRICES, Chemometrics and intelligent laboratory systems, 25(2), 1994, pp. 203-226
Nowadays, analytical instruments that produce a data matrix for one ch
emical sample enjoy a widespread popularity. However, for a successful
analysis of these data an accurate estimate of the pseudorank of the
matrix is often a crucial prerequisite. A large number of methods for
estimating the pseudorank are based on the eigenvalues obtained from p
rincipal component analysis (PCA). In this paper methods are discussed
that exploit the essential similarity between the residuals of PCA of
the test data matrix and the elements of a random matrix. In the lite
rature of PCA these methods are commonly denoted as parallel analysis.
Attention is paid to several aspects that have to be considered when
applying such methods. For some of these aspects asymptotic results ca
n be found in the statistical literature. In this study Monte Carlo si
mulations are used to investigate the practical implications of these
theoretical results. It is shown that for sufficiently large matrices
the distribution of the measurement error does not significantly influ
ence the results. Down to a very small signal-to-noise ratio the ratio
of the number of rows and the number of columns constitutes the major
influence on the expected value of the eigenvalues associated with th
e residuals. The consequences are illustrated for two functions of the
eigenvalues, i.e. the logarithm of the eigenvalues and Malinowski's r
educed eigenvalues. Both methods are graphical and have been applied i
n the past with considerable success for a variety of data. Malinowski
's reduced eigenvalues are of special interest since they have been us
ed to construct an F-test. Finally, a modification is proposed for pse
udorank estimation methods that are based on the principle of parallel
analysis.