# Effect of Formation Size on Flocking Formation Performance for the Goal Reach Problem
## 3 Results
### Experimental Results  
Each experiment was simulated 20 times (for 400 experiments in total) using NetLogo, and the obtained performance measures for each simulation were recorded. We report the average values obtained for each of the performance metrics over the twenty runs. Performance metric data for all simulations were analyzed using SPSS software [36]. Our aim is to explore the existence of an association between the change in flock size (as an independent variable) and each of the performance measures (as dependent variables). Figure 4: Arena specifications ratio. The circle on the left displays an enlarged patch. We use regression curves to approximate the function or curve that has the best fit to a number of observed data points in our experiments [37; 38; 39]. * The correlation coefficient, \(r\in[-1,1]\), describes the relationship and measures the strength of the association between two variables, and is defined in Equation (20) [40], where \(m\) is the sample size (i.e., the number of experiments), \(x\) is the value of the independent variable (i.e., the flock size), and \(y\) is the value of the dependent variable (i.e., the performance metric). * The coefficient of determination, \(R^{2}\in[0,1]\), explains the variations in the performance measure accounted for by flock size. It is obtained by squaring the \(r\) value and is normally expressed as a percentage in the interpretation of results. An \(R^{2}\) close to 0 reflects a model that does not explain the change in the performance measure around its mean, while an \(R^{2}\) close to 1 reflects a model that explains all the variation of the performance measure around its mean due to change in the flock size. * The \(p\)-value tests the null hypothesis, which states that there is no association between changes in the flock size and shifts in the performance measure. If the \(p\)-value is less than the significance level \(\alpha\), then it indicates that the data from the experiments provides enough evidence to reject the null hypothesis. * The distribution of data points in the scatter plots shown in Figures 5 and 6 are obtained from the experiments and are estimated for each of the six functions generated by the SPSS statistical software (linear, inverse, logarithmic, quadratic, cubic, and exponential). The \(R^{2}\) for each of these functions allows us to determine the association between the flock size and each of the performance measures. \[r=\frac{m\sum xy-\sum x\sum y}{\sqrt{\left[m\sum x^{2}-\left(\sum x\right)^{2} \right]}\sqrt{m\sum y^{2}-\left(\sum y\right)^{2}}}, \tag{20}\]  
The remainder of this section is organized as follows: Section 3.2.1 reports the results of experiments excluding obstacles. Next, Section 3.2.2 reports the results of experiments including obstacles. All of the experiments used the settings and parameters described in Table 2.  
Figure 5: \(Cont.\)  
Figure 5: Relationship between the flock size and each of the performance metrics in an environment without obstacles. (**a**) Flock size versus the goal localization time. (**b**) Flock size versus the distance to the center of mass. (**c**) Flock size versus the average number of collisions per member. (**d**) Flock size versus the average number of applied separation rules per member. (**e**) Flock size versus the average number of applied alignment rules per member. (**f**) Flock size versus the average number of applied cohesion rules per member. (**g**) Flock size versus the average time of task completion. (**h**) Flock size versus the average speed matching time. (**i**) Flock size versus the average goal localization window. Figure 6: Relationship between the flock size and each of the performance metrics in an environment with obstacles. (**a**) Flock size versus the goal localization time. (**b**) Flock size versus the distance to the  
center of mass. (**c**) Flock size versus the average number of collisions per member. (**d**) Flock size versus the average number of applied separation rules per member.