## 4 Discussion  
The main objective of this study is to improve the standard probabilistic neural network algorithm by introducing an appropriate multivariate distribution for non-Gaussian input variables and incorporating a feasible technique to handle multi-class imbalanced problem. This objective was further elaborated into finding the most suitable marginal distributions of the identified input variables, estimating parameters of multivariate distributions associated with these input variables and introducing a proper method to handle multi-class imbalanced problem. Comparisons of proposed and existing methods were carried out to assess the performance of proposed methods. Preliminary analysis identified nine variables, lag 4 (i.e., return series four days back) of AORD index, lag 1 of oil index, lag 1 of gold index, lag 14 of cocoa index, lag 1 of GSFC index, lag 0 of exchange rate of Australian dollar to United States dollar (AUD-USD), lag 1 of AUD-USD, lag 8 of AUD-USD and lag 12 of AUD-USD exchange rates; four variables, lag 5 of GSFC index, lag 12 of oil index, lag 13 of gold index, lag 6 of cocoa index, and thirteen variables, lag 1 of ASPI index, lag 3 to lag 5 of ASPI index, lag 1 of oil index, lag 5 of oil index, lag 13 of oil index, lag 14 of oil index, lag 9 of gold index, lag 1 of GSFC index, lag 2 of GSFC index, lag 13 of GSFC index as potential inputs to PNN models with respect to three stock market indices AORD, GSFC and ASPI, respectively. Different marginal distributions, normal distribution, lognormal distribution, inverse Gaussian distribution, levy distribution and Pareto distribution, were used to model return series of stock market indices and other financial markets under study, and inadequacies of all above mentioned distributions to model return series of considered financial indices were identified. Finally, scaled t distribution was identified as a more suitable distribution to model return series of financial indices and respective parameters are present in Table 2. The shape parameter \(\
u\) of the scaled t distribution can capture the fat tails of the fitted distributions. Smaller \(\
u\) parameter estimates between 2 to 5 for all indices in Table 2 indicate that the fat tails of the distributions are well captured by the fitted distributions. All estimated location parameters are small and around zero, which implies that the identified scaled t distributions are symmetric. It can be said that the fitted marginal distributions for return series of considered markets are suitable based on the estimated parameter values. Adequacy of fitted distributions was assessed using probability-probability plots (p-p plots) and cumulative plots. All results confirm that the fitted scaled t distributions with respective parameters are suitable to model respective financial return series. Therefore, the scaled t distribution can be considered as a more suitable distribution to model return series of AORD, GSFC, ASPI, Amex oil, Amex gold, world cocoa indices and AUD-USD exchange rate. Identifying most suitable multivariate distribution for input variables is a difficult task as the parameter estimation is associated with heavy mathematical calculations. Three parameter estimation techniques, batch approximation algorithm, local optimization method with matlab \"fmincon\" function and global optimization method with DSO algorithm, were considered in parameter estimation of multivariate distributions, and results show that the global optimization provides better results than the other two techniques. The number of parameters to be estimated associated with each multivariate distribution varies with the number of potential input variables associated with them or the dimension of the multivariate distribution. Number of potential input variables\(\backslash\)dimensions of considered three multivariate distributions of AORD, GSFC and ASPI and their related other markets are nine, four and thirteen, respectively.