Abstract :
Due to propagation of errors in the normalization process the interpretation of the score plots and loading tables, as obtained from principal component analysis (PCA), may be misleading or even meaningless. In the usually applied normalization method the row sum is calculated for each case or sample. It turns out that error propagation occurs when the errors in this sum average out poorly. Normalization then proceeds by dividing each value by this sum and the result is therefore affected by the remaining error in the sum. A normalization method based on a logarithmic transformation (Ln method), which is widely used in geology, and a weighted normalization method with flexible weighting constants (weight method), are applied and minimize this remaining error. As a result, the introduction of spurious correlations due to error propagation is then significantly reduced. This is demonstrated by using a simulated data set with a heteroscedastic noise pattern to which the usual normalization method (norm) and the method commonly used in mass spectrometry (MS) are applied, as well as the weight and Ln methods. It turned out that the normal method and especially the MS method were highly unsatisfactory and introduced a significant amount of spurious correlations. After application of the weight and Ln methods to a real data set, which consisted of chromatograms with subtle differences, patterns could be identified whereas with the normal method randomlike plots were obtained, demonstrating their usefulness in these situations. However important they may be, these two normalization methods still close the data set, i.e., the row sum is subjected to the constraint of a constant sum, and, as a consequence, induce false correlations. The size and sign of these correlations are a function of the means and variances of the variables after normalization. A method to correct for the induced correlations due to closure is suggested.
