## 4 Results
### Analysis of Results Using a Generalized F-Score  
The F-score is a commonly-used assessment to determine a classification algorithm's accuracy in a way that takes into account both precision and accuracy. The traditional \(F_{1}\) score is calculated as the harmonic mean of the precision and recall values and is bounded between zero and one. \(F_{1}\) can be calculated as follows:  
\[F_{1}=2\cdot\frac{Precision\cdot Recall}{Precision+Recall}=\frac{2\cdot TP}{2 \cdot TP+FP+FN} \tag{7}\]  
where \(TP\) represents the number of true positive outcomes, \(FP\) the number of false positive outcomes and \(FN\) the number of false negative outcomes. A more generalized version of the F-score can be calculated by introducing a variable \(\beta\) that allows more emphasis to be placed on the precision or recall component. This generalized F-score, \(F_{\beta}\), is calculated as:  
\[F_{\beta}=(1+\beta^{2})\cdot\frac{Precision\cdot Recall}{\beta^{2}\cdot Precision +Recall} \tag{8}\]  
where a larger value of \(\beta\) weights recall more highly than precision and a smaller value of \(\beta\) emphasizes precision at the expense of recall. Table 12 shows values of \(F_{0.1}\), \(F_{1}\) and \(F_{10}\) calculated using the precision and recall values from Table 8. The \(F_{0.1}\) represents a measure in which precision is 10-times more important than recall. The \(F_{10}\) measure weights precision and recall in a way that makes recall 10-times more important that precision. Finally, \(F_{1}\) is the balanced weighting of the precision and recall values. Based on these three measures across the various datasets, we can see that, in general, the nearest neighbor classifier can produce satisfactory results if recall is preferred over precision, but it does not do so consistently. In particular, the Las Vegas and Natanz datasets very rarely have values greater than 0.5. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & \multicolumn{3}{c}{**Columbia**} & \multicolumn{3}{c}{**Las Vegas**} & \multicolumn{3}{c}{**Natanz**} \\ \cline{2-10}
**Dataset** & \(F_{0.1}\) & \(F_{1}\) & \(F_{10}\) & \(F_{0.1}\) & \(F_{1}\) & \(F_{10}\) & \(F_{0.1}\) & \(F_{1}\) & \(F_{10}\) \\ \hline A1 & 0.28 & 0.44 & 0.97 & 0.21 & 0.34 & 0.96 & 0.05 & 0.10 & 0.85 \\ A2 & 0.27 & 0.42 & 0.90 & 0.15 & 0.25 & 0.85 & 0.03 & 0.05 & 0.28 \\ A3 & 0.27 & 0.40 & 0.75 & 0.13 & 0.22 & 0.59 & 0.01 & 0.02 & 0.05 \\ B1 & 0.39 & 0.36 & 0.34 & 0.34 & 0.32 & 0.29 & 0.13 & 0.15 & 0.17 \\ B2 & 0.53 & 0.43 & 0.36 & 0.31 & 0.40 & 0.58 & 0.09 & 0.05 & 0.04 \\ B3 & 0.51 & 0.39 & 0.32 & 0.35 & 0.32 & 0.29 & n/a & n/a & n/a \\ C1 & 0.51 & 0.43 & 0.37 & 0.55 & 0.26 & 0.17 & 0.22 & 0.09 & 0.06 \\ C2 & 0.47 & 0.44 & 0.41 & 0.53 & 0.40 & 0.32 & 0.21 & 0.09 & 0.06 \\ C3 & 0.50 & 0.48 & 0.46 & 0.67 & 0.59 & 0.54 & 0.19 & 0.08 & 0.05 \\ D1 & 0.33 & 0.45 & 0.70 & 0.46 & 0.28 & 0.20 & 0.06 & 0.10 & 0.28 \\ D2 & 0.37 & 0.46 & 0.62 & 0.51 & 0.39 & 0.32 & 0.08 & 0.10 & 0.15 \\ D3 & 0.42 & 0.49 & 0.59 & 0.46 & 0.37 & 0.30 & 0.09 & 0.10 & 0.13 \\ E1 & 0.33 & 0.44 & 0.70 & 0.35 & 0.34 & 0.33 & 0.02 & 0.04 & 0.15 \\ E2 & 0.34 & 0.45 & 0.64 & 0.31 & 0.29 & 0.28 & 0.02 & 0.03 & 0.10 \\ E3 & 0.36 & 0.46 & 0.62 & 0.26 & 0.27 & 0.27 & 0.02 & 0.03 & 0.07 \\ F1 & 0.54 & 0.49 & 0.45 & 0.57 & 0.34 & 0.25 & 0.22 & 0.13 & 0.09 \\ F2 & 0.53 & 0.50 & 0.47 & 0.52 & 0.38 & 0.30 & 0.22 & 0.10 & 0.06 \\ F3 & 0.50 & 0.48 & 0.47 & 0.47 & 0.37 & 0.31 & 0.15 & 0.07 & 0.04 \\ G1 & 0.52 & 0.44 & 0.39 & 0.39 & 0.35 & 0.32 & 0.11 & 0.06 & 0.04 \\ G2 & 0.52 & 0.47 & 0.42 & 0.29 & 0.25 & 0.22 & 0.11 & 0.06 & 0.04 \\ G3 & 0.51 & 0.47 & 0.43 & 0.24 & 0.23 & 0.22 & 0.09 & 0.04 & 0.03 \\ \hline \end{tabular}
\end{table} 
Table 12: Analysis of nearest neighbor change prediction results using a generalized F-Score.