Decomposition mathematical definition

The definition (7.32) masks the local and non-local contributions from bodies to the flow. A more systematic approach to characterising the Eulerian mean velocity is to decompose the flow into (i) a far field flow contribution - far from each body but still within the group of bodies - and (ii) a near field flow contribution - local to each body. This concept, originally described qualitatively by Kowe et al. [353], is strictly valid for dilute arrays since it formally requires the bodies to be widely separated, so that there is a separation of lengthscales between the near and far field, scaling approximately as 0(a) and 0(LS) respectively. The decomposition is defined formally here for potential flows. The far field flow, u, is defined mathematically as the sum of the dipolar and source contributions from the bodies, by assuming the bodies shrink to zero, so that (from (7.31))... 

These four elementary spectra H (P) are not yet the elementary hydration spectra we are looking for, which correspond to four independent hydration mechanisms. The reason is that the definition of these four elementary spectra only rests on mathematical criteria, and mathematics tells us that instead of taking these four spectra, we may as well have taken any four independent linear combinations of them. We would then have obtained an equally good decomposition of all experimental hydration spectra, with the same statistical errors. 

PCR method is a two-step process, in which the projection stage is separated and independent from the regression one. As discussed in Section 3.3, this can lead to the drawback that the components that are extracted in the decomposition step, based only on the information about the X-matrix, can be poorly predictive for the Y-block. Starting from these considerations, another method was proposed, PLS regression [2,18,19], in which information in Y is actively used also for the definition of the latent variable space. Indeed, PLS looks for components which compromise between explaining the variation in the X-block and predicting the responses in Y. This corresponds to a bilinear model, which can be summarized mathematically as ... 

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