# Incorporating Grey Total Influence into Tolerance Rough Sets for Classification Problems
## 5 Computer Simulations
### Evaluating Classification Performance  
There is no optimal setting for parameter specifications for genetic algorithms, but we can refer to the principles introduced in [40; 41]. Parameter specifications were specified for all experiments as follows: \(\mathrm{n_{size}}=50\), \(\mathrm{n_{max}}=500\), \(\mathrm{n_{del}}=2\), \(\mathrm{Pr_{c}}=0.8\), and \(\mathrm{Pr_{m}}=0.01\). Five-fold cross-validation (5-CV) was considered for each classification method ten times by means of a distribution-balanced stratified CV (DBSCV) [42]. This study divided all patterns into five disjoint subsets of equal size such that four served as training patterns and one as test data. We iterated this procedure until each subset had been tested. The classification performance of the proposed GTI-TRSC was compared with that of several representative rough-set-based classification methods: a rule-based method with shortening optimization (RSES-O) using the Rough Set Exploration System (RSES) [3; 4; 5], a hierarchical version of the lattice machine (HLM) [43; 44], a hierarchical form of RSES-O (RSES-H) [44; 45], and the Rule Induction with Optimal Neighborhood Algorithm (RIONA) [46]. These classification methods are briefly introduced as follows:  
1. HLM: The lattice machine generates hypertuples as a model of the data. Some more general hypertuples can be used in the hierarchy that covers objects covered by the hypertuples. The covering hypertuples locate various levels of the hierarchy. 2. RSES-O: RSES-O is implemented in RSES. An optimal threshold for the positive region is used to shorten all decision rules with a minimal number of descriptors. 3. RSES-H: RSES-H can be obtained by constructing a hierarchy of rule-based classifiers. The levels of the hierarchy are defined by different levels of minimal rule shortening. A new pattern can be classified by a single hierarchy of the classifier. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Data** & **\# Patterns** & **\# Attributes** & **\# Classes** \\ \hline Australian approval & 690 & 14 & 2 \\ Glass & 214 & 9 & 6 \\ Hepatitis & 155 & 19 & 2 \\ Iris & 150 & 4 & 3 \\ Pima Indian diabetes & 768 & 8 & 2 \\ Sonar & 208 & 60 & 2 \\ Statlog Heart & 270 & 13 & 2 \\ Tic-Tac-Toe & 958 & 9 & 2 \\ Voting & 435 & 16 & 2 \\ Wine & 178 & 13 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Data used in computer simulations. * 4. RIONA: RIONA is also implemented in RSES. It uses the nearest neighbor method to induce distance-based rules. For a new pattern, the patterns most similar to it can vote for its decision, but patterns that do not match any rule are excluded from voting. The classification performance of the above methods, reported in [44], is summarized in Table 3.  
Variants of TRSC were also considered: TRSC with subset approximations (TRSC-SU), TRSC with concept approximations (TRSC-CO), flow-based TRSC (FTRSC) with subset approximations (FTRSC-SU), flow-based TRSC with concept approximations (FTRSC-CO) [22], Grey-tolerance-rough-set-based classifier (GTRSC) with subset approximations (GTRSC-SU), and GTRSC with concept approximations (GTRSC-CO) [19]. The GTRSC and FTRSC were chosen because it is interesting to investigate whether different measures of similarity or relationship can influence classification accuracy. To implement the basic differences between GTI-TRSC, GTRSC, and FTRSC, GTRSC and FTRSC are briefly introduced as follows:  
1. GTRSC: Instead of a simple distance measure used to evaluate the proximity of any two patterns, the GRG (grey relational grade) is used here to implement a relationship-based similarity measure that generates a tolerance class for each pattern. As mentioned above, only direct relationships were considered in the GTRSC. 2. FTRSC: The FTRSC uses preference information expressed by flows among patterns to measure similarity between patterns. The flow of each pattern is computed by the well-known preference ranking organization methods for enrichment evaluations (PROMETHEE) [47,48]. The results are summarized in Tables 3 and 4.