Fig. 3

For synthetic datasets, it is possible to evaluate the proximity between the actually extracted principal components and the “true” ones obtained in the noise-free case. This is characterized by the angular difference between the actual and noise-free eigenspectra. a shows an ideal case of the perfect PCA decomposition when each eigenspectrum coincides with its true reference and has a zero projection to the other eigenspectra in the PCA basis. Then, the proximity diagram represents the sequence of bars of height 1 with the colors fitting the right component index: red for the 1st component, orange for the 2nd one, pink for the 3rd one and so on. However, the observed eigenspectra always deviate from the true ones in realistic cases. This looks like a redistribution of each colored bar over many of components in proximity diagrams: (b) represents such a diagram for the unweighted decomposition of the noisy synthetic dataset, while (c) and (d) show those for the weighted decomposition without and with filtering pre-treatment, respectively. Only the treatment variant (d) delivers the satisfactory proximity between the observed and true principal components