# Machine Learning for Evaluating Hospital Mobility: An Italian Case Study
## 4 Experimental Results and Discussions
### Feature Contributions to the Hospital Mobility Model  
The analysis of feature importance, as depicted in Figure 8 and quantitatively detailed in Table 5, reveals a complex interaction between structural, operational, and systemic factors that influence the model's classification performance. SHAP values (SHapley Additive exPlanations) quantify the absolute average impact of each feature on the magnitude of the model's output, which predicts the distance patients travel to reach a hospital. Each feature's MEAN and STD scores provide further quantification of their respective influences. For example, a higher standard deviation indicates greater variability in a feature's influence across different model iterations, pointing to potential instabilities in its impact. The bar chart in Figure 8 illustrates the reduction in model accuracy, measured by the area under the curve (AUC), when each feature is individually removed. The graphical representation was facilitated by the Features Importance widget, displayed in the green box in Figure 2. The length of the bars indicates the extent to which the omission of a specific feature, such as DEATHS or PHYSICIANS, reduces the AUC, thereby underscoring its importance in the predictive model. A longer bar represents a greater decrease in AUC, highlighting the significant role of the feature in determining patient hospital mobility. The evaluation of the importance of various medical and structural features through SHAP values (SHapley Additive exPlanations) illustrates how each factor influences the distance patients are willing to travel to access a hospital. The variable \"Deaths\" (DEATHS), with an average impact value of 0.128 and a standard deviation of 0.012, emerges as the most significant; it indicates that high mortality rates tend to dissuade patients from choosing a hospital, reflecting concerns about the quality of care provided. The \(\lambda_{i}\) values in the table represent the normalized coefficients ranging from 0 to 1, whose total sum is 1, of the mean impact scores of the features. They indicate the proportion of contribution of each feature to the model output, taking into account the relative importance of each feature compared to others in influencing the model's predictions. The \(\lambda_{i}\) values are calculated using the following formula:  
\[\lambda_{i}=\frac{MEAN_{I}}{\sum_{j=1}^{n}MEAN_{J}}x\frac{1}{\sum_{j=1}^{n}MEAN _{J}} \tag{12}\]  
where \(\lambda_{i}\) is the normalized value of feature \(i\); \(MEAN_{i}\) is the mean impact score of feature \(I;j\) is an index representing each feature in the calculation of \(\lambda_{i}\), varying from 1 to \(nn\), the total number of features considered. Similarly, the presence of an adequate number of doctors (PHYSICIANS), which has an average impact of 0.042 and a standard deviation of 0.014, is perceived positively and encourages patients to opt for specific healthcare facilities, associating a higher number of doctors with better quality of care. Likewise, the use of interventions (INTERVENTIONS) records an average impact of 0.012 and a notable standard deviation of 0.08, highlighting how the volume of surgical interventions can variably influence perceptions of the hospital's capacity and specialization. Readmissions (READMISSIONS), with an average impact of 0.010 and a standard deviation of 0.007, can be seen as an indicator of deficiencies in the quality of care, pushing patients to consider more reliable alternatives. Moreover, the number of available beds (BEDS), with an average value of 0.023 and a standard deviation of 0.008, suggests that a greater capacity to accommodate can make a hospital more attractive, especially in areas with limited healthcare infrastructure. Hospital staff (HOSPITAL STAFF), with an average impact of 0.011 and a standard deviation of 0.009, demonstrates that a large and qualified workforce is essential to attract patients, viewed as a sign of service efficiency and quality.