# Estimating the Tour Length for the Close Enough Traveling Salesman Problem
## 3 Regression Results
### Best Subset Model Selection
The MAPE value indicates that the tour-length estimates from this regression model differ by an average of 3.995% from the SZVNS tour lengths. Finally, from the 14 models in Table 3, we select the model that has the smallest BIC value. This is the model with eight independent variables and a BIC value of \(-1921\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Best Subset Model** & **Adjusted R\({}^{2}\)** & **Mallows’s \(C_{p}\)** & **BIC** \\ \hline
1-variable & 0.598 & 3189.7 & \(-698\) \\
2-variable & 0.787 & 1320.5 & \(-1189\) \\
3-variable & 0.876 & 447.1 & \(-1605\) \\
4-variable & 0.899 & 218.1 & \(-1762\) \\
5-variable & 0.911 & 107.7 & \(-1850\) \\
6-variable & 0.918 & 42.1 & \(-1906\) \\
7-variable & 0.919 & 30.2 & \(-1912\) \\
8-variable & 0.920 & 16.9 & \(-1921\) \\
9-variable & 0.921 & 15.0 & \(-1918\) \\
10-variable & 0.921 & 13.0 & \(-1916\) \\
11-variable & 0.921 & 14.5 & \(-1911\) \\
12-variable & 0.921 & 15.7 & \(-1906\) \\
13-variable & 0.921 & 16.9 & \(-1901\) \\
14-variable & 0.921 & 18.0 & \(-1895\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Adjusted R\({}^{2}\), Mallows’s \(C_{p}\), and BIC values of the best subset models from Table 3.  
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Variable**} & \multicolumn{11}{c}{\multirow{2}{*}{**2**}} & \multicolumn{11}{c}{\multirow{2}{*}{**3**}} & \multicolumn{11}{c}{\multirow{2}{*}{**4**}} & \multicolumn{11}{c}{\multirow{2}{*}{**5**}} & \multicolumn{1}{c}{\multirow{2}{*}{**6**}} & \multicolumn{1}{c}{\multirow{2}{*}{**7**}} & \multicolumn{1}{c}{\multirow{2}{*}{**8**}} & \multicolumn{1}{c}{\multirow{2}{*}{**9**}} & \multicolumn{1}{c}{\multirow{2}{*}{**10**}} & \multicolumn{1}{c}{\multirow{2}{*}{**11**}} & \multicolumn{1}{c}{\multirow{2}{*}{**12**}} & \multicolumn{1}{c}{\multirow{2}{*}{**13**}} & \multicolumn{1}{c}{\multirow{2}{*}{**14**}} \\ \cline{2-2} \cline{11-16}  & **1** & & & & & & & & & & & & & & \\ \hline n & & & & & & & & & & & & & & \\ A & & & \(*\  
We show the coefficients of the third model in Table 5 under the column labeled \"Best BIC\". We note that the adjusted R\({}^{2}\) value of 0.920 indicates a very good model fit. The six variables n, MaxP, VarP, MaxM, SumM, and SZ are not in this model. The remaining eight variables are significant at the 0.1% level. In Figure 2d, we give the histogram of Studentized residuals for this model. This histogram shows that there is almost an equal distribution of instances with positive and negative residuals. This is also indicated by the MPE value of \(-0.202\%\) which is close to zero. The MAPE value indicates that the tour-length estimates from this regression model differ by an average of 4.008% from the SZVNS tour lengths. All three best subset models based on adjusted R\({}^{2}\), [PERSON]'s \(C_{p}\), and BIC performed well. In fact, their adjusted R\({}^{2}\) values are nearly the same (about 0.921), as are their MAPE values (about 4%). We recommend the model with the least number of independent variables (Best BIC model with eight variables) to estimate the SZVNS tour lengths for CETSP instances with random node locations. The coefficients of all eight variables in the Best BIC model are highly significant (\(p<0.001\)).