# Disease Severity Index in Parkinson's Disease Based on Self-Organizing Maps
## 2 Materials and Methods
### Selection of the Data Processing Methodology  
The Self-Organizing Maps (SOMs) method was selected for this study, which consists of an unsupervised Artificial Neural Network (ANN) composed of uni- or two-dimensional neurons. These neurons can adapt their synaptic weights through a competitive learning process, forming prototypes or clusters with similar data [14]. In the context of ANN learning, data are grouped into various categories. The winning neuron is determined based on the assigned input values, which are calculated using the Euclidean distance metric. This distance represents the shortest straight-line path between two points. Once the winning neuron is identified, a neighborhood is defined around it, allowing the winning neuron to interact with neighboring neurons and reinforce synaptic connections, establishing a cooperative network. By reinforcing these synaptic cooperations, the response of the winning neuron improves whenever the network receives patterns like those it was trained on [19]. The Self-Organizing Map (SOM) technique groups similar events together. In this study, the elements represent PD-related questions directed at patients diagnosed with the disease or those suspected of having it. SOM functions by transforming high-dimensional information into a two-dimensional representation, creating a mapping from \(R^{n}\) to \(R^{2}\). Initially, a two-dimensional grid of neurons (\(m\times n\)) was developed. These neurons are activated by the inputs of the neural network and can undergo movement within the grid, causing them to become proximate to each other during training. Each neuron is associated with coordinates in the space provided by the weight vector (\(W\)). To initialize the weights of each neuron, we assign values corresponding to the severity levels generated by the responses to each question in the database. After the creation and initialization of the grid, an event from the database is applied to all neurons. Subsequently, the Euclidean distance between the event and each weight (neuron) in the grid is \begin{table}
\begin{tabular}{c c c} \hline \hline
**Groups** & **Answers** & **Codification** \\ \hline Group 1 & 0—No & 0 \\  & 1—Yes & 1 \\ \hline \multirow{4}{*}{Group 2} & 0—No & \multirow{4}{*}{0} \\  & 1—Slight & \\  & 2—Smooth & \\ \cline{1-1} \cline{2-3}  & 3—Moderate & \\  & 4—Severe & \\ \hline \multirow{4}{*}{Group 3} & 0—No & \multirow{4}{*}{0} \\  & 1—Slight & \\ \cline{1-1} \cline{2-3}  & 2—Moderate & \\ \cline{1-1} \cline{2-3}  & 3—Serious & 1 \\ \cline{1-1} \cline{2-3}  & 4—Extremely serious & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coding of the questions. calculated [20]. For illustration, Figure 1 presents an example of a two-dimensional map of neurons. The winning neuron serves as the center of the neighborhood function radius, which updates the vectors (\(W\)) throughout the training of the ANN. As SOM activates different areas of the map corresponding to similar input patterns, it produces a non-linear map based on the provided input information, generating a two-dimensional representation of the achieved neighborhood [21]. For this study, a map size of 8 \(\times\) 6, comprising a total of 48 neurons, was defined. Subsequently, the map was associated with the \(K\)-means algorithm to form prototype vectors for the input space and obtain the desired classes [22]. The unified distance matrix (U-Matrix) generated by the SOM calculates the distance between the weights (\(W_{\text{j}}\)) of neuron j and those of neighboring neurons. The result is a visual representation where rhombuses indicate the calculated distances. This proximity is then evaluated on a color scale, where dark colors (black) indicate proximity (i.e., many inputs activated these neurons), and light colors (yellowish) represent greater distance (i.e., few inputs activated these neurons). Figure 2a displays the U-Matrix visualization. Following the U-Matrix analysis, the K-means method was employed to create the data classes, as depicted in Figure 2b.