## 2 Materials and Methods
### Data Analysis
The choice of RFC was motivated by the purpose of employing a classifier belonging to traditional ML characterized by strong non-linearity. * Multi-layer perceptron (MLPC) : In this investigation, this simple DL algorithm consisted of a single hidden layer with 400 neurons and zero bias. The hidden layer was followed by an activation layer employing a ReLU activation function. We carried out the training either with the ADAM [28] or with the L-BFGS-B [29] solvers, with an upper limit of 600 iterations. In the following, we refer to the aforementioned DL models as MLPC(ADAM) and MLPC(L-BFGS-B), respectively. We chose MLPC as representative of DL classifiers to compare the related behavior with traditional ML classifiers. We assessed the performances of the ML models mentioned before through 5-fold cross-validation, in terms of sensitivity (\(S\)), specificity (\(SP\)), and accuracy (\(A\)), widely employed in medicine. In particular, in the following, we refer to \(S\), \(SP\), and \(A\) as quantities averaged over the values of the label \(y\) and over the folds of the cross-validation. One of the purposes of this work was the search for the Raman components associated with molecules relevant to the classification process. By definition, these molecules can be considered biomarkers of the diseases under investigation. In particular, the search for cancer biomarkers could provide interesting information about the biochemical processes behind the occurrence of such diseases. Furthermore, this analysis could help to reduce the amplitude of the spectral region needed to reach a reliable diagnosis. This perspective could have important consequences in terms of the future implementation of this technology in simple and low-cost, engineered devices. To this aim, we adopted the so-called Raman Band Identification (RBI), inspired by the so-called permutation feature importance (PFI) [24,30,31]. Consider an ML or DL model \(M\), previously trained on the dataset represented by the matrix \(X\). Let \(s\) be a generic score, assessing the performance of \(M\) on a test dataset \(X^{\prime}\). Suppose we calculate the parameter \(s_{j}\), representing the aforementioned score, on the dataset \(X^{\prime}\) after a permutation of the elements of the \(j\)-th column. The basic idea behind the definition of RBI is that, if \(s_{j}\sim s\), the spectral component corresponding to the \(j\)-th column of \(X^{\prime}\) can be considered irrelevant for the classification. The RBI associated with the \(j\)-th spectral component is defined as  
\[RBI_{j}=s-\frac{1}{K}\sum_{l}^{K}s_{l,j}. \tag{2}\]  
In this definition, \(X^{\prime}\) is subjected to \(K\) permutations of the \(j\)-th column, and \(s_{l,j}\) represents the score obtained at the \(l\)-th permutation. By definition, the larger \(RBI_{j}\), the larger the importance of the \(j\)-th spectral component. Despite RFC providing an intrinsic definition of feature importance, based on the ability of a single feature to increase the \"purity\" of the classification domains [32], the use of RBI allowed us to compare the results of different ML models directly and to reinforce our considerations about the relevant chemical compounds in the biochemical processes at the origin of the diseases under interest. Hence, we calculated the RBI associated with the prediction accuracy (\(s\equiv A\)), on an \(X^{\prime}\) dataset obtained by randomly selecting spectra from \(X\), with a proportion of 20% with respect to the total number of spectra. We chose this proportion in line with the 5-fold cross-validation adopted. In addition, we chose \(K=5\). All the calculations were carried out through a hybrid script, obtained by customizing an Orange Data Mining (version 3.37.0) script with Python (version 3.10.12) codes [30].