# Time to Capture a Moving Target Travelling along a Circular Trajectory
## 3 Results
### Fit of the Fitts' Model for Moving Targets with Various Speeds  
It was evaluated whether the collected target selection times were well fitted to the model proposed in this study. The degree of model fit was determined by the degree of linear correlation between measured target selection times and indexes of difficulty. In regression, the \(R^{2}\) coefficient of determination is a statistical measure of how well the regression predictions approximate the real data points. In order to evaluate the degree of model fit, an optimal \(K\) value should be also derived. The optimal \(K\) value is an empirically determined constant for calibrating the unit of the moving speed of the target. Therefore, linear regression analysis between measured selection time and index of the difficulty was performed while changing \(K\). The \(K\) value in the linear regression equation that provides the largest \(R^{2}\) value was determined as an optimal \(K\) value. Table 1 indicates the change of Equation (4) according to \(K\) value change. If the value of \(K\) approaches negative infinity or approaches positive infinity, Equation (4) approximates the traditional Fitts' law. This is because the speed (\(V\)) of the moving target becomes meaningless when \(K\) is large. On the other hand, if the value of \(K\) is 0, the index of difficulty in Equation (4) becomes 0 and the movement time is equal to a. If the optimal \(K\) value was determined as any negative number, it means that the participants was feeling the target width wider than the actual target width. On the other hand, if the optimal \(K\) value was positive, it indicates that the participants was feeling the target width narrower than the actual target width, due to the speed of moving target. The optimal \(K\) value in the [PERSON]'s study was positive and the result of this study was also expected to be positive. Contrary to our expectations, the optimal \(K\) value that gave the best model fit was negative. A negative value of \(K\) means that the effective width of the target was larger than the actual target width. Table 2 shows the change of the \(R^{2}\) value of the proposed model \begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{1}{c}{\(\mathbf{K}\)} & \multicolumn{1}{c}{**Models**} \\ \hline \(K\) = Negative infinity & MT = a + b \(log_{2}(2A/W)\) \\ \hline Negative infinity \(<K<0\) & MT = a + b \(log_{2}(A/(W/2+V/K))\) \\ \hline \(K=0\) & MT = a \\ \hline \(0<K<\) Positive infinity & MT = a + b \(log_{2}(A/(W/2-V/K))\) \\ \hline \(K=\) Positive infinity & MT = a + b \(log_{2}(2A/W)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variation of Equation (4) according to \(K\) value change. Figure 5: Interaction effects between the speed of moving target and the target width on the target selection time. according to the change of the \(K\) value. When the \(K\) value was between \(-95\) and \(-100\), the \(R^{2}\) value was the largest (\(R^{2}=0.9173\)). In some cases, linear regression analysis did not work during data analysis. For example, if \(W=25\), \(A=150\), \(V=200\), and \(K=10\), the ID value could not be calculated. If the value of each parameter is substituted into ID = \(log_{2}\) (\(A/(W/2-V/K)\)), the ID becomes \(log_{2}(-20)\). \(X\) must be a positive real number in \"\(log_{2}\)\", which is the log function of x with respect to base b. Since the value of \(X\) is negative, calculation of ID is not possible. This case frequently occurred when the absolute value of \(K\) was small. These cases were excluded from the analysis results. Figure 6 shows three cases of the linear regression equations shown in Table 2. The linear regression equation on the left is a case where the \(K\) value is close to infinity, so the speed of the moving target does not affect the mouse movement time. The graph presented in the center is the case of the best model fit, where the coefficient of determination of the regression equation is the largest.