On the application of principal component analysis and factor analysis in electron spectroscopy: the limit of detectability of factors in a set of noisy spectra

Application of PCA and FA in electron spectroscopy. II. The minimum energy shift of factors that can be resolved in a set of noisy spectra

“…By visual inspection of loadings on the second PCs in both figures, one can see that IND tends to overlook quite a considerable part of the real structure of the set of spectra. For this reason we proposed in our previous paper 10 another criterion based on the look of the loadings on the second PC. Now we can state, however, that an alternative to that criterion is to use PCA and IND for a properly selected sub-range of the spectra.…”

“…In order to find the answer to the problem of minimum energy shift between factors that can be detected at a given noise level of the recorded spectra, we exploited the approach presented in our previous paper. 10 The next section briefly presents this approach to the problem and introduces a function, related to shapes of factors, that will play a key role in further discussion of the problem. The subsequent section deals with calculations of this function performed for several variants of assumed shapes of the factors present in spectra.…”

“…Calculations underlying PCA of a data matrix are based on linear algebra, however some aspects of the structure of D can be clarified easily in terms of geometry. 12,13 The geometrical approach presented in our previous paper 10 enabled us to derive a relationship between properties of the factors, composition of the analysed set of spectra and signal-to-noise ratio (SNR) of the spectra. This equation can be adapted easily to the present problem.…”

“…The reason is that it relies on the idea of comparison of the dimension of the swarm of r points in c-dimensional space related to the structure of the data with the dimension related to the noise. 10 Noise addition enlarges the dimensions of the swarm in all directions and the structure of the data begins to be hidden in the noise when structure-related and noise-related dimensions start to be comparable. However, the term 'comparable' has rather a fuzzy meaning and therefore a precise value of Á is difficult to define theoretically.…”

Biomolecular Crystallization in Silico as a Source of the Colour Noise

“…Account must be taken of such parameters as: hydrophobicity and hydrophilicity of the surface, the #H to #P ratio for the given morphology, the presence of the "sliding" pathways. It is possible that the use of Principal Component Analysis (PCA) [28,29], successfully used for the analysis of multidimensional data, will allow us to find some important correlations.…”