Mathematical programming models for classification problems with applications to credit scoring
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Mathematical programming (MP) can be used for developing classification models for the two–group classification problem. An MP model can be used to generate a discriminant function that separates the observations in a training sample of known group membership into the specified groups optimally in terms of a group separation criterion. The simplest models for MP discriminant analysis are linear programming models in which the group separation measure is generally based on the deviations of misclassified observations from the discriminant function. MP discriminant analysis models have been tested extensively over the last 30 years in developing classifiers for the two–group classification problem. However, in the comparative studies that have included MP models for classifier development, the MP discriminant analysis models either lack appropriate normalisation constraints or they do not use the proper data transformation. In addition, these studies have generally been based on relatively small datasets. This thesis investigates the development of MP discriminant analysis models that incorporate appropriate normalisation constraints and data transformations. These MP models are tested on binary classification problems, with an emphasis on credit scoring problems, particularly application scoring, i.e. a two–group classification problem concerned with distinguishing between good and bad applicants for credit based on information from application forms and other relevant data. The performance of these MP models is compared with the performance of statistical techniques and machine learning methods and it is shown that MP discriminant analysis models can be useful tools for developing classifiers. Another topic covered in this thesis is feature selection. In order to make classification models easier to understand, it is desirable to develop parsimonious classification models with a limited number of features. Features should ideally be selected based on their impact on classification accuracy. Although MP discriminant analysis models can be extended for feature selection based on classification accuracy, there are computational difficulties in applying these models to large datasets. A new MP heuristic for selecting features is suggested based on a feature selection MP discriminant analysis model in which maximisation of classification accuracy is the objective. The results of the heuristic are promising in comparison with other feature selection methods. Classifiers should ideally be developed from datasets with approximately the same number of observations in each class, but in practice classifiers must often be developed from imbalanced datasets. New MP formulations are proposed to overcome the difficulties associated with generating discriminant functions from imbalanced datasets. These formulations are tested using datasets from financial institutions and the performance of the MP-generated classifiers is compared with classifiers generated by other methods. Finally, the ordinal classification problem is considered. MP methods for the ordinal classification problem are outlined and a new MP formulation is tested on a small dataset.