## 2 Materials and Methods
### Classification  
SVM, logistic regression (LR), k-nearest neighbors (kNN), naive Bayes (NB), decision tree (DT), random forest (RF), bagging, AdaBoost and neural network (NN) classifiers are used to differentiate malignant lung nodules from benign ones using breath data. * Support vector machineSVM [38] is a marginal predictor that finds the best hyperplane in the feature vector space, thereby establishing a boundary that maximizes the margin between data samples in distinct categories, which results in strong generalization capabilities. In this study, a hyperplane linear classifier is used with the support vectors closest to the decision boundary being used. SVM with N support vectors \(v_{1},v_{2},\,\ldots,\,v_{n}\) and weights \(w_{1},w_{2},\,\ldots,\,w_{n}\) was used to estimate the classification given by: \[SVM=\sum_{i=1}^{n}w_{i}(v_{i},\,\,x)+b\] (5) where a feature vector is represented by \(x\), while a bias is represented by \(b\). * Logistic regression In LR [39], the predictor variable is generated by a linear combination of the input variables. The values of this predictor variable are converted into probabilities using a logistic function. It uses the logit function to calculate the chance of an event occurring: \[logit=\log(\frac{p}{1-p})\] (6) where \(p\) is the probability of the event occurring. Maximum likelihood is used to estimate the significance of input characteristics, and the coefficients of the model give this information. * kNN classifier The k-nearest neighbors algorithm [39] is a non-parametric classification and regression. The input in the feature space is made up of the \(k\) closest training examples. A class membership is the outcome of kNN. The object is assigned to the most frequent class among its k nearest neighbors after a majority of its neighbors vote to categorize it. If \(k=1\), the item is simply assigned to the class of the item's closest neighbor. In the proposed system, we used \(k=7\) so that the class with the majority vote for the closest seven training samples would be selected as the classification result. * Naive Bayes The Bayes rule is used to calculate NB, which assumes that features (variables) are independent of one another given the class. For a training sample \(s\) with \(m\) VOCs values levels \(\{v_{1},v_{2},\,\ldots,\,v_{m}\}\) for the \(m\) features, the posterior probability that \(s\) belongs to a class \(h_{k}\) is \[p(h_{k}\mid s)\propto\prod_{i\in s}p(v_{i}|h_{k})\] (7) where \(p(v_{i}\mid h_{k})\) are conditional tables (or conditional densities) that are constructed from training instances [38]. * Decision tree (DT) The generation of a decision tree is a very efficient approach for constructing classifiers from data. The most commonly used logic technique is the tree representation. A typical decision-tree learning system uses a top-down approach to find a solution in a specific section of the search space. It splits the working area into subparts and uses the Gini Index, Gain Ratio or Information Gain to verify the purity of a class division. These metrics are utilized in the decision tree that forms the classification and regression tree (CART), C4.5 and ID3, respectively. In this study, CART is used [39]. * Random forest (RF) Random forest [39] is an ensemble learning system for classification, regression and other tasks that works by creating a large number of decision trees during training and generating a class that is the mode of the individual tree classes (classification) or a mean/average predictor (regression). Random forest is a method of averaging a large number of deep decision trees that have been trained to minimise variation across various parts of the same training set. This comes at the cost of a small increase in bias and some lack of interpretability, but it usually results in a significant gain in the efficiency of the final model. In this study, RF is built using 1000 CART decision trees.