Nm. Faber et al., ASPECTS OF PSEUDORANK ESTIMATION METHODS BASED ON AN ESTIMATE OF THE SIZE OF THE MEASUREMENT ERROR, Analytica chimica acta, 296(1), 1994, pp. 1-20
The estimation of the pseudorank of a matrix, i.e., the rank of a matr
ix in the absence of measurement error, is a major problem in multivar
iate data analysis. In the practice of analytical chemistry it is ofte
n even the only problem. An important example is the determination, of
the purity of a chromatographic peak. In this paper we discuss three
pseudorank estimation methods that make use of prior knowledge about t
he size of the measurement error. The first method (Method A) is based
on the standard errors in the diagonal elements of the row-echelon fo
rm of the matrix, the second method (Method B) is based on the eigenva
lues of principal component analysis (PCA) and the third method (a t-t
est) is based on the singular values. Methods A and B are modification
s of methods that are well known in analytical chemistry. However, the
se methods cannot provide significance levels for the estimated pseudo
rank. This holds for the original methods as well as the present modif
ications. The main reason for introducing these modifications is that
in this way relationships are established between the t-test and metho
ds that are already known. The aspects that are covered in this paper
include the sampling distribution of the test statistic, the number of
degrees of freedom to be used in the test, the adequacy of theoretica
l predictions and the bias that results from random measurement noise.
The object of this paper is to demonstrate that using prior knowledge
about the size of the measurement error may yield powerful pseudorank
estimation methods. This is illustrated by comparing the significance
levels obtained by the t-test and Malinowski's F-test. The t-test yie
lds sharper significance levels for experimental data obtained from th
e literature as well as simulated data. This can be satisfactorily exp
lained by the larger number of degrees of freedom that is employed in
this test. The viability of the new t-test is supported by a thorough
evaluation of the test data by a large number of conventional methods.
As a remarkable by-product of the present investigation we find that
a plot of the singular values yields a promising graphical pseudorank
estimation method. Graphical methods have proved their use in the past
in the case that the size of the measurement error is unknown. This n
ew graphical method therefore provides a natural complement to the t-t
est.