## 3 Results
### Probabilistic Neural Network Models  
The multivariate scaled t distribution was identified as the more suitable distribution to model multivariate distribution associated with potential input variables, to be used in modelling stock market return series of interest. This section illustrates the performance of proposed probabilistic neural network (PNN) model, after incorporating the identified multivariate distribution and comparison of performance with the existing PNN model. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Market**} & \multicolumn{2}{c}{**Maximum Function Value (f\_max)**} \\ \cline{2-3}  & **Local Optimization** & **Global Optimization** \\ \hline AORD & \(2.3623\times 0^{-7}\) & 22.13496 \\ GSPC & \(0.0386\) & 13.1762 \\ ASPI & \(6.2954\times 10^{-4}\) & 6.0352 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximum function values obtained from local and global optimization techniques when estimating parameters of multivariate distributions relating to three stock indices with their related inter-markets. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Market**} & **Number of Potential Inputs** & **Number of Parameters to Be Estimated** \\ \hline AORD & 9 & 91 \\ GSPC & 4 & 21 \\ ASPI & 13 & 183 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Number of parameters to be estimated with respect to multivariate distributions of stock market indices. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & & & \\ \cline{3-14}
**PNN Model** & **Market** & \begin{tabular}{c} **Proposed** \\ **Ensemble** \\ **Technique** \\ \end{tabular}  & \begin{tabular}{c} **Overall** \\ **Accuracy** \\ **(100\%)** \\ \end{tabular}  & \begin{tabular}{c} **Classes** \\ **(Clinical Loss)** \\ **(100\%)** \\ \end{tabular}  & \begin{tabular}{c} **Highly Important** \\ **(Clinical Loss)** \\ **(100\%)** \\ \end{tabular}  & \ \begin{tabular}{c} **Moderately** \\ **Important (Loss)** \\ **(100\%)** \\ \end{tabular}  & \begin{tabular}{c} **Less Important (No big** \\ **impact)** \\ \end{tabular}  \\ \cline{3-14}  & & & & **1** & **2** & **3** & **Syb** & **Syb** & **Syb** & **Syb** & **Syb** \\ \hline \multirow{4}{*}{Standard PrNN} & \multirow{4}{  
Table 5 reflects that the standard PNN model (without MCUB) is not capable of adequately classifying the direction of a stock market index. In addition, there is a higher variation of misclassification percentages among three classes. Therefore, to attain higher accuracy, an improvement to the standard probabilistic neural network model is essential. Further, Table 5 illustrates that the proposed PNN model (without MCUB) exhibits perceptible accuracy in classification. Both GSPC and ASPI indices exhibit similar overall accuracy while AORD index exhibits slightly lower overall accuracy than that of other two indices. Misclassification percentages of buy signal as sell signal and sell signal as buy signal, which lead to critical losses, are small values in the proposed PNN model while the moderately important losses due to classification of buy signal as hold signal and sell signal as hold signal are considerable in all three markets. Based on the results, a visible difference of misclassification percentages among the three classes can be identified. Table 6 consists of class imbalanced ratio (IR) of three markets, which is useful in the explanation of apparent variation among misclassification percentages of classes. Preceding table demonstrates that, a greater class imbalanced ratio presents in all three data sets under the study. Imbalance ratio of Class 2 dominates that of other classes in all three markets. A large variation of misclassification percentages can be observed among three classes in each market with reference to Table 5. This may be due to higher imbalance ratio among classes in all data sets. It is prominent that the misclassification percentage of Class 2 is very low compared to those of other two classes in data sets related to AORD, GSPC and ASPI indices.