# Estimating the Tour Length for the Close Enough Traveling Salesman Problem
## 3 Regression Results
### Best Subset Model Selection  
Although the regression model shown with all 14 variables performed well in estimating the SZVNS tour lengths, two of the variables were not significant at the 10% level and one variable was significant only at the 10% level. We now examine whether we can remove some variables to create a parsimonious model without compromising much on the model fit. We create models with 1 to 14 independent variables in the following way. We start by constructing models with only one independent variable and identifying the 1-variable model that produced the largest adjusted R\({}^{2}\) value. Then we find the 2-variable model that produced the largest adjusted R\({}^{2}\) value, and so on until we consider the 14-variable model. In Table 3, we show the best subset models based on the adjusted R\({}^{2}\) value. For example, out of all possible models with two independent variables, the model containing the variables SumMaxP and AvgR produced the largest adjusted R\({}^{2}\) value. Following this process, we create 14 models that contain 1 to 14 variables. In Table 4, we show the adjusted R\({}^{2}\), Mallows's C\({}_{p}\), and BIC values of the best subset models from Table 3.  
Over these 14 models, we select the model that has the largest adjusted R\({}^{2}\) value. This is the model with 13 variables and an adjusted R\({}^{2}\) value of 0.921. There are other models with the same adjusted R\({}^{2}\) value when rounded to three decimal places. We show the coefficients of the first model in Table 5 under the column labeled \"Best adjusted R\({}^{2}\)\". We note that the adjusted R\({}^{2}\) value of 0.921 indicates a very good model fit. The variable SZ is not in this model and the variable n is not significant at the 10% level. The remaining 12 variables are significant at various levels. In Figure 2b, 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 \(-0.192\%\) which is close to zero. The MAPE value indicates that the tour-length estimates from this regression model differ by an average of 3.983% from the SZVNS tour lengths. \begin{table}
\begin{tabular}{c c} \hline \hline
**Coefficient** & **Mean Values** \\ \hline Intercept (\(\beta_{1}\)) & 15.231 *** \\ n (\(\beta_{2}\)) & 0.225 \\ A (\(\beta_{3}\)) & 0.048 *** \\ MinP (\(\beta_{4}\)) & \(-0.654\) *** \\ MaxP (\(\beta_{5}\)) & 0.224 ** \\ VarP (\(\beta_{6}\)) & 0.106 * \\ SumMinP (\(\beta_{7}\)) & 0.361 *** \\ SumMaxP (\(\beta_{8}\)) & 0.036 ** \\ MinM (\(\beta_{9}\)) & 0.459 *** \\ MaxM (\(\beta_{10}\)) & \(-0.418\) ** \\ SumM (\(\beta_{11}\)) & \(-0.067\) ** \\ VarM (\(\beta_{12}\)) & 0.683 *** \\ VarX\(\times\)VarY (\(\beta_{13}\)) & 0.014 *** \\ AvgR (\(\beta_{14}\)) & \(-9.092\) *** \\ SZ (\(\beta_{15}\)) & \(-0.026\) \\ \hline Adjusted R\({}^{2}\) & 0.921 \\ MPE & \(-0.192\%\) \\ MAPE & 3.984\% \\ \hline \hline \end{tabular} \({}^{*}\)\(p<0.1\); \({}^{**}\)\(p<0.05\); \({}^{***}\)\(p<0.01\); \({}^{***}\)\(p<0.001\).  
\end{table} 
Table 2: Regression results. Next, from the 14 models in Table 3, we select the model that has its Mallows's \(C_{p}\) value closest to the number of independent variables in the model. This is the model with 10 independent variables and a Mallows's \(C_{p}\) value of 13.0. We show the coefficients of the second model in Table 5 under the column labeled \"Best Mallows's \(C_{p}\)\". We note that the adjusted R\({}^{2}\) value of 0.921 indicates a very good model fit. The four variables n, VarP, SumM, and SZ are not in this model. The remaining 10 variables are significant at various levels. In Figure 2c, 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 \(-0.194\%\) which is close to zero.