# Investigation of Critical Gap for Pedestrian Crossing Using Fuzzy Logic System
## 6 Discussion of Results  
Fuzzy logic analysis was adopted to predict critical gaps for pedestrians' crossing using a fuzzy logic system. The fuzzy inputs (the lengths and number of rejected gaps and lengths of accepted gaps) were built each using three parameters (low, medium and high) in order to assess and calculate the critical gaps for each individual in the set. Triangular membership functions for each input were calibrated for each decision maker. A dataset of 500 observations that was collected in 2017 using two GoPro cameras in Edinburgh (UK) was used to assess pedestrian critical gaps. The site was located on a two-way midblock crossing location.  
The results are encouraging and show a number of positive contributions. Firstly, the analysis presented in this paper shows that it is evidently possible to use fuzzy logic analysis to assess and predict pedestrians' crossing behavior. This is a good outcome as it may offer opportunities to enhance understanding of pedestrians' crossing behavior using other tools than those conventionally used.  
Critical gap values obtained using the proposed fuzzy logic system range from 0.987 s to 11.00 s with an optimal value of 6.0 s. For the accepted gaps, the fuzzy logic system predicts an optimum value of 1.92 s while the range is 0.214 s to 7.5 s. Number of rejected gaps ranges from 0.892-4.0 with an optimum value of 1.93. Finally, length of rejected gap is ranging from 0.144 to 3.4 s with an optimum value of 0.554s. These values are within reasonable limits with the input values. The outcomes look reasonable and are in agreement with findings from previous studies that applied conventional analytical methods. For example, critical gap values were estimated in the range from 3-8 s in [2]. Ref [39] also calibrated values of critical gap ranges from 5.75 s and 6.25 s using conventional statistical methods. Accordingly, the predicted critical gap value can be endorsed as reasonable. It should  
Figure 8: Results from the fuzzy system.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variables/Values** & **Accepted Gap** & \
\begin{tabular}{c} **Number of** \\ **Rejected Gaps** \\ \end{tabular} &
\begin{tabular}{c} **Length of** \\ **Rejected Gap** \\ \end{tabular} & **Critical Gap** \\ \hline Minimum value & 0.214 s & 0.892 & 0.144 s & 0.987 s \\ Maximum value & 7.5 s & 4.0 & 3.5 s & 11.0 s \\ Optimum value & 1.92 s & 1.93 & 0.554 s & 6.0 s \\ \hline \hline \end{tabular}
\end{table}
Table 3: Output results from the proposed fuzzy system.  
be noted here that no further validation of these values has been carried out and, therefore, this is suggested for further research. In addition, it is possible for future research to assume different membership function forms of fuzzy logic systems in the analysis, as well as including more variability by including pedestrians' characteristics and waiting times. Further statistical analysis can also be adopted. For example, in the estimation toolbox included in MATLAB, the real-time data was matched with the adopted fuzzy model; other mathematical justifications can be used, such as sum-of-squared-error minimization or any other suitable rule. This could also be suggested for further research.