## 2 Materials and Methods
### Dimensional Reduction Techniques  
The first step of our analysis consists in the dimensional reduction. A wide variety of methods has been proposed to highlight the degrees of freedom of a data set endowed with most of the informative content. Such kinds of techniques are generally named embedding or dimensional reduction techniques; they are usually intended to project data into a subspace with a lower dimensionality to ease the data exploration, for example with the support of visual inspection or emphasize the informative content by improving the signal-to-noise-ratio. Several techniques have been proposed in the last two decades. In particular, we consider, here, the adaptive dimension reduction (ADR) approach [13]. ADR is a methodology iteratively combining the calculation of principal components and the application of k-means clustering. The motivation underlying the choice is based on a specific advantage of ADR over other effective possibilities: the absence of any parameter ruling the number of neighbors defining a local patch with respect to other popular choices such as Laplacian eigenmap [16] and distinguishing variance embedding [17; 18]. Besides, ADR presents two important characteristics:  
* on the one hand, the application of a principal component analysis (PCA) takes into account the information contained in the whole data set;
* on the other hand, the definition of clusters during each iteration rules further projections in the subspace spanned by principal components of the covariance matrix of cluster centers coordinates, weighted by the cardinality of each cluster, whose separation is improved. Thus, this embedding technique perfectly matches the purposes of this work. Besides, to evaluate the robustness of the analysis, we considered other embedding techniques and compared their results with those obtained by ADR. ADR adopts PCA as a first step, followed by \(k\)-means clustering for the patients' sample embedded in the space with a reduced number of dimensions. In the latter each patient is associated with a position vector \(\mathbf{y}_{i}\) with \(1\leq i\leq M\) and included in a cluster \(C_{l}\) with \(1\leq l\leq k\), whose centroid reads  
\[\mathbf{\mu}_{l}=\frac{1}{|C_{l}|}\ \sum_{i\in C_{l}}\mathbf{y}_{i}\]  
where \(|C_{l}|\) stands for the number of patients in the cluster or equivalently its cardinality. This partitioning procedure is ruled by the cost function. \[E=\sum_{l=1}^{k}|C_{l}|\ \sum_{i\in C_{l}}d(y_{i},\mathbf{\mu}_{l})^{2}\]  
which is the sum of the squared Euclidean distances between the objects \(y_{i}\) and their centroids. It has to be minimized with respect to centroids coordinates, while the number of clusters represents a fixed parameter [22]. Once cluster memberships in the embedded space are defined, they are exploited in the original feature space, where the associated centroids coordinates define a new covariance matrix, whose PCA represents the starting of the next iterative embedding. The algorithm pursues this flowchart until it reaches a convergence to a stable configuration of embedded patients. Such iterations evaluate patients' similarities defining clusters in the reduced dimensions space. This partition is then induced in the original space with full dimensions to define a new embedding, where another clustering is implemented until memberships are not modified in successive steps. For further comparison and to assess the robustness of ADR results we considered embedding techniques which for their popularity can be considered the state of the art: PCA, Laplacian eigenmap and distinguishing variance embedding (DVE). PCA represents a minimal implementation of an embedding map. The term minimal is referred to the number of operations required by the procedure: considering the diagonalization of a covariance matrix, followed by a projection onto the subspace spanned by the eigenvectors endowed with the largest eigenvalues, PCA allows the representation of high dimensional data in suitable low dimensional spaces.