Spectral loadings

In general, because the noise in the concentration data is independent from the spectral noise, each optimum factor, W, will lie at some angle to the plane that contains the spectral data. But we can find the projection of each W, onto the plane containing the spectral data. These projections are called the spectral factors, or spectral loadings. They are usually assigned to the variable named P. Each spectral factor P, is usually organized as a row vector. 

Fig. 36.4, Spectral loadings for the first four principal components.

Figure 12.20 PLS spectral loading vector ( eigenspectrum ) of rank 1.

Each successive PLS component approximates both the concentration and spectral data better. For each component, there will be a spectral scores vector t, spectral loadings vector p and concentration loadings scalar q. 

Different schemes have been developed to allow the validation of three-way models. One method used for PARAFAC models is split-half analysis, where the data set is split into two parts and individual modelling is performed on the two halves. If the two models show similar spectral loadings, the variation expressed in the two halves is comparable and it can be assumed that an appropriate number of components have been chosen. As for PCA, visual interpretability is also important and usually substantially easier for PARAFAC, because the components directly represent chemically meaningful phenomena. Other tools that may aid in deciding the correct number of components can be found in the literature.15... 

Fig. 7. (A) The spectral loadings calculated by PARAFAC and (B) diffusion attenuation profiles resulting from the PARAFAC model.

Fig. 8. On the left-hand side, the acquired pure component spectra as well as the residual spectra when comparing with the spectral loadings calculated by PARAFAC shown in Fig. 7.

The MALT data [Allosio et al. 1997] are analyzed by PARAFAC with orthogonality constraints on the batch and wavelength modes. Four components can be extracted and their spectral loadings may be interpreted as intermediates and reaction products of the malting by looking at the line plots. The ANOVA structure is incomplete and cannot be interpreted easily. The paper is unclear about why four components are chosen or how much of the total sum of squares they explain. 

In (Schueller 1981) and a more detailed explanation in (Clough and Penzien 1975) is given to the approach to compute wind induced structural reaction with the help of spectral load formulation. The used schema for the spectral wind induced reaction can be found in (Schueller 1981). With the help of the power spectra velocity function Sv (o) an the aero admittance functionT/a(force spectrum can be expressed... 

The variation spectra are often called eigenvectors (a.k.a., spectral loadings, loading vectors, principal components, or factors) for the methods used to calculate them. The scaling constants used to reconstruct the spectra are generally known as scores. 

The main difference between PLS and PCR is that the concentration information is included in the calculations during the spectral decomposition. This results in two sets of eigenvectors a set of spectral loadings (Bx) that represent the common variations in the spectral data and a set of... 

Normalize the spectral loading vector by the spectral scores Bx,- = BXi/(SiS ... 

Prediction of concentrations of constituents in an unknown mixture generally follows the procedure of PCR, except that it is a more iterative procedure. The spectral loadings from the decomposition step are used to calculate the scores from the unknown absorbance spectrum. The scores are used with the loading vectors for the constituents to calculate the unknown concentrations. Since both A and C are decomposed, the concentrations of the constituents also have loading vectors. As with PCR, an ILS basis is used. 

FIGURE 7 Excitation and emission spectral loadings obtained for PARAFAC models with different number of factors. Top Models with no constraints. Bottom Models with non-negativity constraints. 

---