## Classification
### Discriminant Techniques  
Discriminant approaches are probably the most classical solutions for dealing with classification-related problems [97]. As their name suggests, they make it possible to highlight differences between samples belonging to distinct classes. Their basic principle is simple: they divide the multivariate space of the registered variables into several subregions equal to the number of categories of objects considered (say \(Z\)) and they assign each one of such objects to a certain class if the point corresponding to its measurement vector falls within the boundaries associated with that class. Subsequently, owing to these characteristics:  
1. discriminant classification models need to be calibrated on training sets composed by specimens belonging to all the categories under study;
2. every analyzed sample is always assigned to one and only one of these categories;
3. samples coming from other classes (not considered in the study) will always be (erroneously) recognized as members of one and only one of these categories. For all these reasons, discriminant techniques can be particularly useful, e.g., when the amount of expected classes is limited and/or if it can be reasonably supposed that all the samples to be assessed are drawn uniquely from them. The mother of all discriminant classification strategies is undoubtedly Linear Discriminant Analysis (LDA) [100]. In the simplest case encompassing two categories of objects, LDA estimates a direction of maximum separation between classes (i.e., a so-called _canonical variate_) as:  
\[\mathbf{w}^{\mathrm{T}}=(\mathbf{m}_{1}-\mathbf{m}_{2})^{\mathrm{T}}\mathbf{S }^{-1} \tag{24}\]  
with \(\mathbf{m}_{1}\) and \(\mathbf{m}_{2}\) (both of dimensions \(J\times 1\)) being the vectors of variable means (_centroids_) for class #1 and class #2, respectively, and \(\mathbf{S}\) (\(J\times J\)) an estimate of the class covariance matrix (which, here, is assumed to be the same for both categories (this assumption is relaxed in the most direct extension of LDA known as Quadratic Discriminant Analysis (QDA) [101; 102])). \(\mathbf{m}_{1}\), \(\mathbf{m}_{2}\) and \(\mathbf{S}\) are computed from training samples. A class delimiter, \(w_{0}\), is then calculated as:  
\[w_{0}=\mathbf{w}^{\mathrm{T}}\frac{(\mathbf{m}_{1}+\mathbf{m}_{2})}{2} \tag{25}\]  
and the classification rule established such that if the projection coordinate of each observation vector, say \(\mathbf{x}_{n}\) (\(J\times 1\)), along \(\mathbf{w}\) (\(t_{n}=\mathbf{w}^{\mathrm{T}}\mathbf{x}_{n}\)) is found to be lower than \(w_{0}\), the corresponding object is predicted as belonging to class #1 and _vice versa_. The rationale behind LDA can also be interpreted by assuming that data within each category are Gaussianly distributed so as to calculate the so-called _posterior_ probability of \(\mathbf{x}_{n}\) belonging to the \(z\)-th class, \(p(z|\mathbf{x}_{n})\), as:  
\[p(z|\mathbf{x}_{n})=\frac{p_{0}(z)}{(2\pi)^{\frac{l}{2}}|\mathbf{S}|}e^{-\frac {1}{2}(\mathbf{x}_{n}-\mathbf{m}_{z})^{\mathrm{T}}\mathbf{S}^{-1}(\mathbf{x} _{n}-\mathbf{m}_{z})} \tag{26}\]  
with \(p_{0}(z)\) equal to the probability of observing an individual from the same category before carrying out any measurement (_prior_ probability). Accordingly, the final classification rule can be reformulated by stating that a sample should be assigned to the class it has the highest posterior probability of proceeding from. Even though LDA can be in principle extended to scenarios involving a larger number of categories, it requires the inversion of the covariance matrix \(\mathbf{S}\), which can be singular or nearly singular in the presence of highly collinear (inter-correlated) descriptors (as for spectral profiles) [51].