## 2 Multiple-Criteria Decision-Making
### The TOPSIS Method
Finally, the SabiSPQ database consists of 944 instances and 16 explanatory variables that describe firms whose accounts are established in the Spanish Mercantile Registry [39]. This constitutes a fully balanced data set with 472 healthy companies and 472 companies that failed during the period 2000-2003. \begin{table}
\begin{tabular}{l l} \hline \hline
**Model** & **Parameters** \\ \hline Bnet & Initial count for estimating the conditional probability tables of the Bayes network = 0.5; \\  & Naive Bayes network used as the initial structure; K2 hill climbing algorithm for structure \\  & learning; Bayesian Dirichlet score to evaluate the structure learned \\ MLP & Broyden–Fletcher-Goldfarb–Shanno optimization algorithm; Sigmoid transfer function; \\  & Learning rate = 0.3; Momentum = 0.2; Maximum number of training epochs = 500; \\  & Neurons in the hidden layer = 2 \\ RBF & Normalized Gaussian RBF; Center vectors of the functions determined using K-means clustering \\ logR & Multinomial logistic regression; Quasi-Newton optimization method; Ridge value in the log-likelihood = \(1.0\times 10^{-8}\) \\ SVM & Linear kernel; Soft margin constant = 1.0: Tolerance = 0.001; Round-off error \(\epsilon\) = \(1.0\times 10^{-12}\); Sequential minimal optimization algorithm \\
1 NN & Euclidean distance \\ RIPPER & Number of folds = 3 (one fold is used for pruning, the rest for growing the rules); \\  & Minimum total weight of the instances in a rule = 2.0; Number of optimization runs = 2 \\ C4.5 & Number of folds = 3 (one fold is used for pruning, the rest for growing the tree); Minimum number of instances per leaf = 2; Error-based pruning; Pruning confidence factor = 0.25 \\ randF & Number of trees = 100; Number of randomly chosen attributes at each node = log\({}_{2}(D)+1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter values of the classifiers. ### Performance Assessment Measures

Standard performance assessment measures for credit risk prediction include accuracy, area under the ROC curve, Kolmogorov-Smirnov statistic, geometric mean of accuracies, root mean squared error, Gini coefficient, and F-measure [38; 40; 41], among many others. For a problem with two classes, as is the case of the data set used in our experiments, most of these measures are easily obtained from a \((2\times 2)\) confusion matrix as that shown in Table 4, where each entry represents the amount of correct (true-positive, true-negative) or wrong (false-positive, false-negative) decisions (classifications or predictions). Numerous prediction systems typically employ the accuracy (Acc) rate to assess the performance of the classifiers, thus describing the proportion of correct classifications on a given data set. Nevertheless, practical and theoretical evidences demonstrate that the accuracy can be heavily biased regarding imbalance in class distribution and proportions of correct and incorrect classifications. As financial data are commonly strongly skewed, the area under the ROC curve (AUC) has been proposed as a suitable measure without regard to class distribution or misclassification costs [17; 42]. For all practical purposes, the AUC for a two-class problem can be calculated as the arithmetic average of sensitivity (or true-positive rate, TP-rate) and specificity (or true-negative rate, TN-rate) [43]:

\[AUC=\frac{sensitivity+specificity}{2}, \tag{2}\]

where the sensitivity is the proportion of non-defaulters correctly classified, and the specificity denotes the proportion of defaulters classified as defaulters. Other powerful measures based on simple indices are the geometric mean of accuracies (G-mean) and the F-measure. The geometric mean attempts to maximize the accuracy on each individual class while keeping a small difference between sensitivity and specificity. This metric penalizes those classifiers that yield large differences between true-positive and true-negative rates.