## 4 Discussion
Class imbalance ratio (IR) plays a significant role in controlling forecasting ability of PNN models. Table 6 indicates that the IR of Class 1, Class 2 and Class 3 of AORD series are 0.2, 0.6 and 0.2, respectively. By comparing overall misclassification percentage and IR of each class of AORD index, it can be said that the standard PNN model is biased towards the majority class and forecasts most of the test set observations to the majority class. Similar behavior can be observed when comparing misclassification percentages and class imbalance ratios of classes in GSPC and ASPI indices. Results suggest that the performance of the existing probabilistic neural network model heavily depends on the class imbalanced ratio. Correspondingly, Table 5 indicates that the misclassification percentages among three classes of each index differ significantly when using the proposed PNN. Similar to the results of standard PNN, Class 2 exhibits a lower misclassification percentage than that of Class 1 and Class 3 when forecasting the direction of AORD, GSPC and ASPI indices using proposed PNN. Even though the proposed probabilistic neural network model outperforms the existing PNN model, it still suffers from the class imbalanced problem. Findings described above highlight the necessity of an appropriate technique to handle class imbalanced problem in data sets. A technique called multi-class undersampling based bagging (MCUB) is proposed to handle the multi-class imbalanced problem persisting in data sets considered in the study. Both standard PNN and proposed PNN were embedded into the proposed MCUB technique separately for further investigations. Overall accuracy of standard PNN model increases after introducing MCUB technique in forecasting the direction of GSPC and ASPI indices, and AORD index predictions exhibit a slight decrease in the overall performance with reference to Table 5. After ingraining the proposed PNN to the proposed MCUB technique, an increase in overall accuracy can be observed with respect to all indices. Increase of two highly important types of misclassification percentages can be observed after introducing MCUB to standard PNN whereas two types of moderately important misclassification percentages have decreased in all indices. The crucial misclassification types of buy-sell and sell-buy signals have reduced considerably by the introduced ensemble technique in most of the circumstances, and misclassification percentages of buy-hold and sell-hold signals have been reduced in all the cases simultaneously in proposed PNN model. The remarkable fact is that the misclassification percentages among Class 1, Class 2 and Class 3 in forecasting the direction of each stock index when using both standard and proposed PNN models are very much similar. Accordingly, it is indisputable that the proposed ensemble technique, multi-class undersampling based bagging (MCUB), is able to unravel the multi-class imbalanced problem residing in the data sets. Results in Table 5 prove that the proposed MCUB technique is suitable to handle the multi-class imbalanced problem existing in probabilistic neural networks when classifying data. Hence, it is sensible to compare performance of standard PNN versus proposed PNN with MCUB technique. By referring to Table 5, it can be said that the proposed PNN with multivariate scaled \(t\) distribution as the joint distribution of input variables exhibits higher accuracy than that of standard PNN with respect to all considered stock indices. The balance between misclassification percentages among three classes has been enhanced when classifying using the proposed PNN. Misclassification percentages of most important buy-sell and sell-buy signals leading to critical losses have been decreased immensely by the proposed PNN. Slight increase in misclassification percentages of moderately important two misclassification types can be observed.