## 2 Materials and Methods
### Methods
#### 2.2.1 The Hierarchical PLS-DA Classification Model  
Partial Least Squares Discriminant Analysis (PLS-DA) is a supervised technique that combines the predictive features of PLS regression with the discriminative ability of a classification method. This technique is used to develop models capable of classifying samples into known categories, even when applied to previously unexamined data, such as an unknown image. PLS-DA works by projecting the original data into a new space that maximizes the distinction between classes, while simultaneously modeling the relationships between input variables (predictors) and their respective class labels [52; 53]. The PLS-DA model utilizes the NIPALS (Non-linear Iterative Partial Least Squares) algorithm, an iterative technique that allows for the calculation of principal components even when the dataset contains many missing variables or is large. This makes it particularly useful for analyzing high-dimensional datasets, as NIPALS reduces computational load and simplifies the handling of complex data. By combining the dimensionality reduction in PLS with its classification objective, PLS-DA efficiently handles complex datasets, focusing on class separation. A hierarchical model based on Partial Least Square Discriminant Analysis (PLS-DA) was built and applied to obtain the correct classification of the microplastic particles. The hierarchical model is based on the creation of different rules for the classification of different subsets selected from the samples under investigation until each subset contains a single type of object [54; 55]. In a hierarchical model, objects are initially split into subsets. Each subset is then further divided into smaller subsets, continuing this process until each subset contains only a single object [56]. The object that differs the most from the rest is chosen and isolated during each phase. Following that, a comparison is performed using a new PLS-DA classification model for every object. PLS-DA is utilized to categorize samples into predefined groups by creating discriminant functions based on input variables (wavelengths). These functions produce transformed values enabling more precise discrimination than any individual variable alone [52]. A discriminant function is then built using reference samples to be utilized to recognize and classify samples belonging to unknown sets. Once the model is obtained, it can be applied to an entire hypercube and for the classification of new hypercubes. In this study, the application of a hierarchical model enabled the identification of EPS polymer, alongside PS, PP and PE. This approach works by organizing the data into different levels or categories, making it easier to differentiate between polymers with similar characteristics [57]. Hierarchical modeling proved to be an effective strategy for addressing a multi-class problem, particularly when dealing with nested structures--where some classes (e.g., polymer types) are grouped within broader categories (e.g., levels of degradation or environmental conditions). This allows for more accurate predictions and classifications, as it accounts for both general patterns and specific nuances within the data. Indeed, sometimes classification problems require the problem to be broken down into subgroups with individual models handling the increased detail of the problem. In Figure 2 the developed dendrogram for the classification of the different polymers according to hierarchical modeling is shown. The small number of unidentified materials includes biological substances, such as algae, black plastics, and fragments that were too small or thin to be accurately categorized, likely due to their low reflectance properties. All samples defined as 'unidentifiable' were then analyzed using Raman spectroscopy and/or FTIR. Using Principal Component Analysis (PCA) as an exploratory analysis, the most effective preprocessing techniques that permit class separation have been identified.