## 2 Data Exploration
SVD decomposes \(\mathbf{X}\) as:  
\[\mathbf{X}=\mathbf{U}\mathbf{D}\mathbf{V}^{\text{T}}+\mathbf{E} \tag{2}\]  
with the columns of \(\mathbf{U}\) (\(N\times A_{\text{PCA}}\)) and \(\mathbf{V}\) (\(J\times A_{\text{PCA}}\)) being the first \(A_{\text{PCA}}\) left and right singular vectors of \(\mathbf{X}\), respectively, and \(\mathbf{D}\) (\(A_{\text{PCA}}\times A_{\text{PCA}}\)) a square diagonal array whose diagonal elements correspond to its first \(A_{\text{PCA}}\) non-zero singular values. Therefore, it holds that \(\mathbf{T}=\mathbf{U}\mathbf{D}\) and \(\mathbf{P}=\mathbf{V}\). PCA shows the following property:  
\[\mathbf{P}^{\text{T}}\mathbf{P}=\mathbf{I} \tag{3}\]  
where \(\mathbf{I}\) is an identity matrix of dimensions \(A_{\text{PCA}}\times A_{\text{PCA}}\), which translates into the fact that the dimensions of its subspace are orthogonal and, thus, can be inspected assuming that the information they individually capture is mutually unrelated. The assessment and interpretation of a PCA model is generally performed by examining both scores and loadings profiles (i.e., the columns of \(\mathbf{T}\) and \(\mathbf{P}\), respectively). The former yield insights into existing relations among analyzed objects, while the latter enable the identification of correlation patterns within ensembles of multiple spectral variables. The basic idea, here, is to simultaneously analyze these profiles not only to possibly pinpoint the presence of distinctive clusters of samples, but also to recognize sets of spectral features shared within each one of these clusters and responsible for their differentiation. Nevertheless, as a consequence of the orthogonality imposed to the principal components, PCA scores and loadings seldom provide physico-chemically meaningful information, since they do not capture single phenomena or events affecting the registered measurements, but rather heterogeneous effects resulting from combinations of such phenomena or events. To overcome this limitation, alternative approaches like Independent Component Analysis (ICA [23; 24]) can be resorted to. Unlike PCA, ICA regards the rows of the data matrix \(\mathbf{X}\) as a collection of _observed signals_ that are mixtures of a certain number of common unknown _source signals_ or _independent components_ and decomposes it as:  
\[\mathbf{X}=\mathbf{A}\mathbf{R}^{\text{T}}+\mathbf{E}_{\text{ICA}} \tag{4}\]  
with \(\mathbf{A}\) (\(N\times A_{\text{ICA}}\)) being the so-called _proportion_ matrix, carrying the mixing coefficients associated with these independent components contained in every column of \(\mathbf{R}\) (\(J\times A_{\text{ICA}}\)), and \(\mathbf{E}_{\text{ICA}}\) (\(N\times J\)) denoting the ICA residual array. As for PCA, the interpretation of an ICA model rests on the simultaneous inspection of the retrieved independent components and their respective proportion profiles. Independent components are analogous to PCA loadings, but, differently from them, are not necessarily orthogonal, and therefore, can be easily related to individual phenomena impacting the recorded spectral variables (e.g., Rayleigh or Raman scattering) [25; 26; 27; 28]. Subsequently, ICA proportions can give a quantitative idea of how they influence the raw measured spectra. There exist several algorithms by which the ICA solution can be attained. Examples are Infomax [29], FastICA [30] and JADE (Joint Approximate Diagonalization of Eigenmatrices) [26; 31]. Their basic principle, though, is common: source signals are estimated by maximizing their statistical independence. In essence, they can be looked at as the outcome of a rotation of PCA loadings that enhances their aforementioned interpretability properties.