## 5 Experiments and Results
### Impact of Parameters on Rule Extraction  
The outcome and the quality of rule extraction process depend on the distance \(d\) used to define the stations' neighbourhood, on the critical threshold \(cr\) used to mark a station as critical, and on \(minSupport\) and \(minConfidence\) thresholds. Whereas the first two parameters are intrinsic of the problem itself, defining the way the stations are critical and the operational range of the rebalancing operations, the latter two parameters are related to the association rule mining algorithm and define the minimum strength of the patterns to be mined. All rules extraction in this subsection are performed by considering Mondays of June using the per day partitioning approach. We set the neighbourhood radius empirically to 500 m and 1 km, considering it a reasonable distance for users to be travelled on foot/by bicycle in a reasonable amount of time in case the designated station for pick-up is empty or the one designated for drop is full. Moreover, by doing so, the operators in charge of the actual rebalance need a short amount of time to travel between bike stations to move bicycles. We analysed the performance of the framework with critical threshold \(cr\) values of 0.1, 0.2 and 0.3. The higher the \(cr\) value, the smaller the number of critical situations because, by increasing \(cr\), a higher absolute difference between the occupancy level of a station and the average occupancy rate of its neighbourhood is needed to highlight a critical status. Considering that almost all stations have a total number of slots in range 25-30, lower or higher threshold values are not meaningful for the detection of critical situations. Too many stations will become critical even if the difference in terms of number of used slots is only one or two slots if \(cr\) values lower than 0.1 are considered. Conversely, few or no critical situations would be identified if higher values of \(cr\) are considered. The impact of these two parameters can be evaluated using two metrics: number of critical stations identified per timestamp and number of rules extracted by the framework. Figure 4(a) represents on the horizontal axis the amount of critical stations and on the vertical axis the amount of timestamps having that number of critical stations. The two curves represent two different values of neighbourhood radius values \(d\) with fixed critical threshold \(cr\) of 0.2. As can be seen, increasing the radius shifts the curve to the right, with higher amount of critical stations. In fact, by increasing the radius \(d\) also the cardinality of the neighbourhood increases and consequently the probability that at least one station has significantly more or less used slots with respect to its neighbours. Figure 4(b) uses the same axis notation to show the impact of the critical threshold \(cr\) on the number of critical stations per timestamp. The results reported in Figure 4(b) were computed by setting \(d\) to 500 m. The chart shows the expected behaviour. Specifically, as the critical threshold \(cr\) shrinks, the amount of critical stations increases (stations that are critical with a threshold of 0.2 are also critical with a threshold of 0.1). \begin{table}
\begin{tabular}{l l l l} \hline \hline  & \(varianceThreshold=3\) & \(varianceThreshold=5\) & \(varianceThreshold=7\) \\ \hline \# unstable stations & 232 & 155 & 114 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Number of filtered unstable stations with variable \(varianceThreshold\) and fixed \(f\). A similar analysis can be performed on the number of discordant rules extracted by the framework by varying \(d\) and \(cr\). It is possible to notice in Figure 5(a) that, by increasing \(d\) from 500 m to 1 km, the number of extracted rules increases noticeably. A similar trend is observed independently from the applied partitioning strategy and also in Figure 5(b) as \(\sigma\) decreases. The aforementioned analyses about the number of extracted rules confirm the expected behaviour evidenced in Figure 4(a),b.