Principal Component Analysis of symmetric fuzzy data

Principal Component Analysis (PCA) is a well-known tool often used for the exploratory analysis of a numerical data set. Here an extension of classical PCA is proposed, which deals with fuzzy data (in short PCAF), where the elementary datum cannot be recognized exactly by a specific number but by a center, two spread measures and a membership function. Specifically, two different PCAF methods, associated with different hypotheses of interrelation between parts of the solution, are proposed. In the first method, called Centers-related Spread PCAF (CS-PCAF), the size of the spread measures depends on the size of the centers. In the second method, called Loadings-related Spread PCAF (LS-PCAF), the spreads are not related directly to the sizes of the centers, but indirectly, via the component loadings. To analyze how well PCAF works a simulation study was carried out. On the whole, the PCAF method performed better than or equally well as PCA, except in a few particular conditions. Finally, the application of PCAF to an empirical fuzzy data set is described.