ASPECTS OF PSEUDORANK ESTIMATION METHODS BASED ON AN ESTIMATE OF THE SIZE OF THE MEASUREMENT ERROR

The estimation of the pseudorank of a matrix, i.e., the rank of a matrix in the absence of measurement error, is a major problem in multivariate data analysis. In the practice of analytical chemistry it is often 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 the size of the measurement error. The first method (Method A) is based on the standard errors in the diagonal elements of the row-echelon form of the matrix, the second method (Method B) is based on the eigenvalues of principal component analysis (PCA) and the third method (a t-test) is based on the singular values. Methods A and B are modifications of methods that are well known in analytical chemistry. However, these methods cannot provide significance levels for the estimated pseudorank. This holds for the original methods as well as the present modifications. The main reason for introducing these modifications is that in this way relationships are established between the t-test and methods 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 theoretical 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 yields sharper significance levels for experimental data obtained from the literature as well as simulated data. This can be satisfactorily explained 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 new graphical method therefore provides a natural complement to the t-test.