A comparison of three methods for principal component analysis of fuzzy interval data

Vertices Principal Component Analysis (V-PCA), and Centers Principal Component Analysis (C-PCA) generalize Principal Component Analysis (PCA) in order to summarize interval valued data. Neural Network Principal Component Analysis (NN-PCA) represents an extension of PCA for fuzzy interval data. However, also the first two methods can be used for analyzing fuzzy interval data, but they then ignore the spread information. In the literature, the V-PCA method is usually considered computationally cumbersome because it requires the transformation of the interval valued data matrix into a single valued data matrix the number of rows of which depends exponentially on the number of variables and linearly on the number of observation units. However, it has been shown that this problem can be overcome by considering the cross-products matrix which is easy to compute. A review of C-PCA and V-PCA (which hence also includes the computational short-cut to V-PCA) and NN-PCA is provided. Furthermore, a comparison is given of the three methods by means of a simulation study and by an application to an empirical data set. In the simulation study, fuzzy interval data are generated according to various models, and it is reported in which conditions each method performs best.