# Data Reduction in the String Space for Efficient kNN Classification Through Space Partitioning
## 5 Results
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Classification Performance (\%)**} & \multicolumn{3}{c}{**Set Size (\%)**} \\ \cline{2-7}
**Dataset** & \multicolumn{2}{c}{**String-Based**} & \multicolumn{2}{c}{**Feature-Based**} & \multicolumn{2}{c}{**String-Based**} & \multicolumn{2}{c}{**Feature-Based**} \\ \cline{2-7}  & **ALL** & **RHC** & **ALL** & **RHC** & **ALL** & **RHC** & **ALL** & **RHC** \\ \hline NIST & **89.2** & **84.2** & 82.5 & 77.2 & **100** & **22.6** & 100 & 27.2 \\ \hline MNIST & **94.1** & **90.7** & 92.4 & **88.6** & **100** & **15.0** & 100 & **10.3** \\ \hline USPS & **90.2** & 85.1 & 89.5 & **84.7** & **100** & 20.6 & 100 & **12.8** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results in terms of \(\text{F}_{1}\) as classification rate and resulting set size for the different corpora considered. ALL and RHC stand for the classification results obtained with the initial set and the reduced one, respectively. Set size percentages are obtained referring to the ALL case for each corpus and data representation. Standard deviation values are not reported as they are lower or equal than 0.01. Bold values highlight the non-dominated elements for each dataset. since the non-reduced feature-based elements suffer a certain performance drop due to the mapping process which is not paired with any reduction in the set size. In spite of this, the cases based on RHC in statistical representation improve the reduction figures with respect to the structural space, hence obtaining non-dominated results for both MNIST and USPS feature-based corpora. Oppositely, the RHC method on string space has obtained better size-reduction level in NIST dataset, turning it into another non-dominated result. Note that, depending on the corpus considered, the RHC-based non-dominating solutions which define the frontier are obtained from either the string space or the feature one, if not both. Concretely, the non-dominated solution in the NIST dataset is depicted by the case of the RHC in the string space whereas for the USPS corpus it is the RHC in the statistical one. The only case in which the RHC contributes to define the frontier in both spaces is with the MNIST dataset. As a last analysis on the results, Table 3 presents the figures obtained in terms of the estimated profit per prototype metric. As the values retrieved are generally quite low, for an easier comparison all results have been scaled up two orders of magnitude. The first point that can be observed is that all reduced cases achieve better profit figures than the non-reduced ones. This fact depicts that, while both set size and classification rate decrease with the reduction process, the accuracy drop is less accused than the size one, which is the desired behavior. Note that, while the best profit figures for both the MNIST and USPS datasets are obtained in the feature space, in the case of the NIST corpus this optimum is obtained in the string encoding. This fact of obtaining better results directly in the structural space without mapping the data onto a statistical space justifies the research efforts towards the development of reduction strategies in the native structural representation. Finally, in order to statistically assess the figures obtained, a significance analysis has been performed on these results. More precisely we have considered the Wilcoxon rank-sum test [31] for comparing the results obtained before and after applying the RHC reduction algorithm for each representation space separately with a significance level of \(p\)-value < 0.05. As a figure of merit we have considered the estimated profit per prototype since it properly summarizes both performance and reduction capabilities of the strategy considered in one single value. Furthermore, since the idea is comparing the non-reduced and reduced scenarios, we are not particularizing in any of the corpora. Thus, the individual results obtained for each fold, corpus, and reduction scenario constitute each of the samples of the distributions to compare.