# NMR in Metabolomics: From Conventional Statistics to Machine Learning and Neural Network Approaches
## 2 Conventional Approaches
### Unsupervised Methods
#### 2.1.1 Principal Component Analysis (PCA)  
Principal component analysis (PCA) is employed for lowering the dimensionality of high-dimensional datasets, preserving as much information as possible by means of a \"linear\" multivariate analysis [29; 30]. This approach employs a linear transformation to define a new smaller set of \"summary indices\"--or \"principal components\" (PCs)--that are more easily visualized and analyzed [31]. In this frame, principal components correspond to new variables obtained by the linear combination of the initial variables by solving an eigenvalue/eigenvector problem. The first principal component (PC1) represents the \"path\" along which the variance of the data is maximized. As happens for the first principal component, the second one (PC2) also defines the maximum variance in the database. Nevertheless, it is completely uncorrelated to the PC1 following a direction that is orthogonal to the first component path. This step reiterates based on the dimensionality of the system, where a next principal component is the direction orthogonal to the prior components with the most variance. If there are significant distinctions between the ranges of initial variables (those variables with smaller ranges will be dominated by those with larger ones), distorted results may occur. To avoid this kind of problem, it is required to perform a standardization operation before executing PCA that corresponds to a transformation of the data into comparable scales. This can be done by using different scaling transformations, such as autoscaling, the generalized logarithm transform or the Pareto scaling with the aim to enhance the importance of small NMR signals, whose variation is more affected by the noise [32]. One of the most used transformation is the mean centered autoscaling:  
\[\frac{value-mean}{st.deviation}, \tag{1}\]  
Furthermore, the computation of the covariance matrix is required to discard redundant information mainly due to the presence of any relationship between the initial variables of the data. The covariance matrix is symmetric (a \(n\ \times\ n\)) being composed by the covariances of all pairs of the considered n variables \((x_{1},...,x_{n})\):  
\[ \begin{bmatrix}Cov(x_{1},x_{1})&\cdots&Cov(x_{1},x_{n})\\ \vdots&\ddots&\vdots\\ Cov(x_{n},x_{1})&\cdots&Cov(x_{n},x_{n})\end{bmatrix}  \tag{2}\]  
In this frame, PCs can be obtained by finding the eigenvectors and eigenvalues from this covariance matrix. Figure 2 shows a graph with only three variables axes of the n-dimensional variables space. The red point in this figure represents the average point used to move the origin of the coordinate system by means of the mean-centering procedure in the standardization process. Once we define PC1 and PC2, as shown in Figure 2, they define a plane that allows inspecting the organization of the studied database. Further, the projection of the data with respect to the new variables (PCs) is called the score plot, and if the data are statistically different/similar, they can be regrouped and classified. PCA is usually applied in NMR-metabolomic studies because it simplifies the investigation of hundreds of thousands of chemical components in metabolomic database composed of several collected NMR spectra. In this way, each NMR spectrum is confined to a single point in the score plot in which similar spectra are regrouped, and differences on the PC axes shed light on experimental variations between the measurements [28; 33; 34; 35]. Figure 2: Example plot with 3 variable axes in a n-dimensional variable space. The principal components PC1 and PC2 are reported. However, it is noteworthy that PCA, like the other latent structure techniques, must be applied to matrices where the number of cases is greater than the number of variables [36]. The PCA technique can also be combined with other statistical approaches, including the analysis of variance (ANOVA) as reported by [PERSON] et al. [37] in their ANOVA-simultaneous component analysis (ASCA).