## 2 Description of the Method
### Clustering of Data  
Artificial neural networks (ANNs) are mathematical tools based on the so-called \"connectionist paradigm\" [47,48], which stores information by means of the connections of a structure rather than through a set of memory cells. ANNs are made of networks of elementary processors linked together by weighted connections. This is, in fact, the typical paradigm adopted by the human brain, where the knowledge is built by processing data through the synapse network rather than by simply storing information in neurons. Although much less complex than the human brain, ANNs derive their name just from the paradigm similarity with it. Among the brain processes that ANNs can imitate, the most appealing is that of \"learning\" from experience, i.e., building functional relations on the basis of sets of learning examples. A wide variety of techniques may be adopted to train ANNs, all of them being based on two essential elements: a training set and a training rule. The training rule establishes how the weights of the synapses should be modified to adapt the network to the training set. Since this rule mostly does not lead explicitly to a complete learning, iterative algorithms are usually needed. In general, ANNs can be based on supervised [49] or on unsupervised learning [50,51]. In the supervised learning, the neural network is exploited to reproduce a functional connection between input and output variables, the values of which are known for all the learning examples. A target value must be set for each learning example, which is taken as the expected output. Because the discrepancy between calculated and expected output values depends on the connection weights, the learning strategy tries to modify such weights in order to minimize the error. Should the learning process be successfully performed, the network can generalize the acquired information, being able even to associate outputs to inputs not belonging to the training set. The supervised learning has been the most adopted, and it is responsible for the widespread success that ANNs had in a lot of scientific fields. However, neither target values nor strategies to minimize the errors are adopted by the human brain, the learning of which is in fact unsupervised. Inspired by the brain's learning strategy, the ANNs originally used unsupervised approaches, where the weights of the synapses continuously change in response to both external stresses (electrical impulses in the brain) and neuron replies. According to the information given by the inputs, the unsupervised learning can handle properties of a set of patterns and map its distribution in the input space. Analogies between different patterns can also be put in evidence, thus leading to a subdivision of a population into aggregates (clusters). The connection weights that are linked to the same neuron represent a prototype of the cluster, generally not coinciding with any example of the training set, even if it would have all the properties to be one of them. The learning process tries to make the prototype as similar as possible to the network input. One of the main activities of unsupervised ANNs is the clustering, which is usually based on a competitive learning (self-organized map, SOM) [51], where the different prototypes compete to win the learning step, i.e., the possibility to change their weights. The most known rule is \"the winner takes all\" [52], which results in an iterative process of changing the prototype to make it to be more like the input and to increase such a likeness as much as possible. Thus, the distribution can be divided into clusters, each cluster being defined by a prototype which is the winner of the points belonging to that cluster. At the end of the process, each prototype tends to converge to the barycenter of the relevant cluster. Once the positions of the prototypes are stabilized, i.e., once the weights have converged to a stable value, they are frozen, and the training phase is concluded.