# Enhancing Disease Diagnosis: Biomedical Applications of Surface-Enhanced Raman Scattering
## 4 Multivariate Chemometrics  
The SERS spectra obtained from biological samples are multivariate in nature. That is to say, they consist of thousands of complex and combination vibrational modes which are difficult to interpret  
Figure 1: Surface-enhanced Raman scattering (SERS) approaches: a. label-based (extrinsic) and b. label-free (intrinsic) SERS. Reproduced with permission from [PERSON] et al. [54]. Copyright © 2015, Springer Nature. by simple stare & compare. The fundamental tenet of multivariate analysis (MVA) is to simplify multivariate data systematically to a small number of variables whilst preserving maximum variance within a data matrix to guide scientific reasoning. This is accomplished through unsupervised and/or supervised MVA (machine learning) modelling. Unsupervised MVA explore natural variance within a data matrix using spectral variables in vertical columns (the \(X\)-data) as the only input data measured from objects in horizontal rows (the \(Y\)-data) of a data matrix. Principal components analysis (PCA) is a widely used traditional, unsupervised approach to reduce data dimensionality, classify spectra into specific groups and to identify outliers [55]. In principle, PCA decomposes multivariate data into scores (clusters) and associated spectral loadings. The scores plots consist of uncorrelated orthogonal hyperplanes called principal components (PCs), which display the differences, similarities and the total explained variance in the dataset; e.g., cancerous _vs._ noncancerous conditions. PC1 is extracted from the input \(X\)-data to account for the largest variance, whilst PC2, PC3,... PC_n_ (where \(n\) is an integer), explain the remaining natural variance in decreasing order. The PC loadings spectra are plotted to highlight the most important input spectral variables responsible for the clustering pattern observed on a PCA scores plot. In the case of disease diagnosis, the variables on a loadings plot often denote intensity ratio differences and/or spectral band shifts between diseased and healthy groups. Since PCA does not rely on prior knowledge of investigated samples, it is _exploratory_ by nature which implies that the algorithm can discover novel hidden biological patterns within samples under review, and is very useful for outlier detection [56]. Other unsupervised models are dendrograms, self-organising maps, autoassociative neural network, etc. By contrast, supervised models are calibrated with a known response variable(s) (the \(Y\)-data) to act as building blocks for supervised algorithms. Discriminant analysis and partial least-squares regression (PLSR) are commonly used supervised models for classification and quantitative predictions of SERS spectral data [57]. Discriminant function analysis (DFA) combines PCs extracted from the \(X\)-data and _a priori_ knowledge: e.g., sample objects (classes) in the \(Y\)-data to minimise within-class and maximise between-class variance for classification purposes. On the other hand, PLSR has proved to be powerful and useful in _quantitative_ analysis of biomarkers by SERS, which employs correlated variables in both \(X\)- and \(Y\)-data to build a linear relationship. For PLSR modelling, the sample spectral dataset is first divided into two parts: the training and test sets. The first subset of sample spectra are used to calibrate or train a PLSR model followed by quantitative prediction of the remaining unseen samples in the test set [58]. A PLSR model of good quality or performance will aim to generate a correlation coefficient (\(R^{2}\) or \(Q^{2}\) for trained and tested sets, respectively), which is close to one and low root mean squared error (RMSE). Partial least-squares-discriminant analysis (PLS-DA), a variant of PLSR, is also a very powerful multivariate model particularly applied when the \(Y\)-data are categorical by nature [59].