## Numerical Analysis
### Data Profiling and Filtering  
In this paper, the control and experimental attribute groups were defined for modeling and comparison (Figure 1). Table 1 lists selected results of the invariant check on attribute cd_000, representing class c, operational identifier cd, and bin id 000. As shown, attribute cd_000 had the same value for the mean, median, and mode, as well as the minimum and maximum values, which implies all valid data were the same. Therefore, this attribute should be removed from the list of attributes for predictive modeling. Including invariant attributes would only increase the computational cost. Table 2 presents the overall IR check results based on the class label in the training and test sets. The calculated IRs of NEG to POS in the training and test sets were 59.0 and 41.7, respectively, which satisfies the definition of high class imbalance. The IR is also a quick way of determining whether the training and test sets are consistently distributed for the given attributes. First of all, the examples of the attributes are labeled as VALID if they comprise real values. If the examples are shown as blank cells, they will be labeled as N/A. It is assumed that if the IR of individual attributes falls within a specified range centered on the overall IR, the assumption of a consistent distribution between the training and test sets was valid. Furthermore, \(2\times 2\) cross-tabulation was conducted. The calculated Chi-square results would clarify if the missing data behavior of the attributes was homogenous between NEG and POS classes disregarding whether the examples were from the training or test set. After performing IR checks on all attributes, the training set IR values were ranged between 55 and 61, whereas the test set values ranged between 39 and 43, with a single exception for the attribute ac_000. The ac_000 data was from the class a, the operational identifier was ac, and the bin id was 000. As shown in Table 3, the calculated IRs for the training and test sets were 104.3 and 75.9, which were larger than the overall IRs of 59.0 and 41.7, respectively. The calculated Pearson Chi-square and maximum-likelihood Chi-square results (CLASS x ac_000) indicated statistical significance (\(p\)-values were both 0.0000). For these findings, it is recommended to remove attribute ac_000 from the list of predictive modeling attributes. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Attribute** & **Valid N** & **\#Missing** & **Mean** & **Median** & **Mode** & **Minimum** & **Maximum** \\ \hline cd\_000 & 59,234 & 676 (1.13\%) & 1,209,600 & 1,209,600 & 1,209,600 & 1,209,600 & 1,209,600 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Invariant attribute assessment of the selected attribute cd_000. \begin{table}
\begin{tabular}{c c c} \hline \hline
**CLASS** & **Training** & **Test** \\ \hline NEG & 59,000 & 15,625 \\ POS & 1000 & 375 \\ \hline IR = NEG/POS & 59.0 & 41.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of instances of each class in the training and test sets. For the APS dataset, [PERSON] and [PERSON] [PERSON] and [PERSON] (2014) recommended removing attributes with over 10% missing data; however, they do not explain why the criterion for filtering attributes was set to 10%. Following this concept, we created bins by the sorted missing percentages, namely, >30%, 20-30%, 10-20%, and 5-10%. Figure 2 shows the bar chart presenting the bins by different missing percentages. Table 4 shows the details for the contained attributes in each bin. As shown, the first three bins had the same attributes and sorting results for both the training and test sets. The fourth bin, 5-10%, had the same attributes but different sorting results. According to the results in Table 4, the cumulative missing percentages were further used to define the control (CTRL) and experimental groups (EXP1, EXP2, and EXP3). After removing the attributes