## 2 Materials and Methods
### Instruments and Statistical Analysis
PLS-DA is a linear classification method combining the properties of PLS regression with the discrimination power of a classification technique. PLS-DA is based on the PLS regression algorithm (PLS1 when dealing with one dependent Y variable and PLS2 in the presence of several dependent Y variables), which searches for latent variables with a maximum covariance with the Y-variables. The main advantage of PLS-DA is that the relevant sources of data variability are modelled by the so-called Latent Variables (LVs), which are a linear combination of the original variables, and, consequently, it allows graphical visualization and understanding of the different data patterns and relations by LV scores and loadings. Loadings are the coefficients of variables, in the linear combinations, which determine the LVs and therefore they can be interpreted as the influence of each variable on each LV, while scores represent the coordinates of samples in the LV projection hyperspace. In detail, for the calibration dataset, samples containing asbestos fibers (i.e., crocidolite chrysotile and amosite) from the certified samples were chosen. Organic, aggregates, mortars bricks and glass fibers samples were also chosen so that to generate four classes characterized by a comparable amount of pixels. Through an exploratory analysis in PCA the best preprocessing procedures, according with those more utilized to handle NIR spectra [46; 47; 48], allowing the separation between the classes have been chosen. Finally four PLS-DA models have been created to perform discriminations:  
* PLDA1: AGGREGATES MORTAR and BRICK + ACM from FIBER GLASS + ORGANIC
* PLSDA2: FIBER GLASS from ORGANIC  
Figure 2: Hypercube image as resulting after mosaicking and samples individuals splitting in two data sets one to be utilized for calibration and the other for their validation (**a**). Four classes have been defined inside the data set (**b**). The class asbestos is referred to ACM individuals. * PLSDA3: ACM with a dimension of about 300 \(\upmu\)m from GGREGATES MORTAR and BRICK
* PLDA4: ACM WITH A DIMENSION below 300 \(\upmu\)m from AGGRAGATES MORTAR and BRICK  
The 4 models have been hierarchized in order to analyze the entire mosaic in a single model. Classification models were then evaluated using the following parameters: Sensitivity and Specificity in calibration (Cal) and cross-validated (CV):  
\[\text{Sensitivity: TP/(TP + FN)} \tag{1}\] \[\text{Specificity: TN/(TN + FP)} \tag{2}\]  
being TP the true positive and FN the false negatives. The best models are obtained when similar values are obtained for Sensitivity and Specificity in Cal and CV, thus demonstrating the robustness of the developed model (32). Cross validation is a very useful tool serving two critical functions in chemometric: (i) it enables an assessment of the optimal complexity of a model and (ii) it allows an estimation of the performance of a model when it is applied to unknown data. For a given data set, cross validation involves a series of experiments, hereby called sub-validation experiments, involving the removal of a subset of objects from a dataset (the test set), the construction of a model using the remaining objects in the dataset (the model building set), and the subsequent application of the resulting model to the removed objects. This way, each sub-validation experiment involves testing a model with objects that were not used to build the model. The PLS-DA models created to discriminate the different classes, were subsequently hierarchized. Adopting a hierarchical model, objects are preliminarily divided into subsets and are then subdivided into further subsets, until each subset contains a single object [11]. During each step, the object that is the most different, in respect of the others, is selected and isolated, and a comparison is carried out based, for each object, on a new PLS-DA classification model [12].