Advances in credit scoring: combining performance and interpretation in kernel discriminant analysis

Abstract

Due to the recent financial turmoil, a discussion in the banking sector about how to accomplish long term success, and how to follow an exhaustive and powerful strategy in credit scoring is being raised up. Recently, the significant theoretical advances in machine learning algorithms have pushed the application of kernel-based classifiers, producing very effective results. Unfortunately, such tools have an inability to provide an explanation, or comprehensible justification, for the solutions they supply. In this paper, we propose a new strategy to model credit scoring data, which exploits, indirectly, the classification power of the kernel machines into an operative field. A reconstruction process of the kernel classifier is performed via linear regression, if all predictors are numerical, or via a general linear model, if some or all predictors are categorical. The loss of performance, due to such approximation, is balanced by better interpretability for the end user, which is able to order, understand and to rank the influence of each category of the variables set in the prediction. An Italian bank case study has been illustrated and discussed; empirical results reveal a promising performance of the introduced strategy.

Appendix: Results of characterization of classification

Table 7 Categories characterizing the group of the bad instances classified as good

Table 8 Categories characterizing the group of the bad instances classified as bad

Table 9 Categories characterizing the group of the good instances classified as bad

Table 10 Categories characterizing the group of the good instances classified as good

Characterization of the test partition has been carried out by finding a ranking among all the characterizing variables of a group by means of probabilistic criterion: value-test (Lebart et al. 1984). More specifically, absolute values of such test are the simple measures of similarities among groups and variables. Therefore, a category of a variable can be considered characteristic of a group if its presence is significantly higher respect to what we expected, given its presence in the sample. The value-test is distributed as an hypergeometric but can be easily approximated to a standardized normal applying the Laplace–Gauss approximation. Thus, in formula is:

$$\begin{aligned} t_{q}(N)=\frac{N-E(N)}{s_{q}(N)} \end{aligned}$$

(12)

where $N \sim Hyp(n,n_{\nu }, n_{q})$, $E(N)=n_{q}\frac{n_{\nu }}{n}$ and $s^{2}_{q}=n_{q} \frac{n-n_{q}}{n-1} \frac{n_{\nu }}{n} (1-\frac{n_{\nu }}{n})$, $n_{q}$ is the number of instances sampled without replacement belonging to qth group and $n_{\nu }$ is the number of instances with $\nu$th category.

Tables 7, 8, 9 and 10 aid the interpretation of the test classification obtained by means of the reconstructed Cauchy kernel discriminant. The first column of each table collects the characteristic categories, the second shows the percentages of instances with $\nu$th category in the group q ($n_{\nu q}/n_{q}$, where $n_{\nu q}$ is the number of instances with $\nu$th category among those belonging to the class q), the third, the percentages of instances with $\nu$th category in the test set ($n_{\nu }/n$), the forth column shows the percentages of qth group with $\nu$th category ($n_{\nu q}/n_{\nu }$), the fifth and the sixth columns collect the value-test and probability values respectively. Such measures synthesize the homogeneity and the selectivity of the partition.