## 5 Results and Evaluations  
As previously described, fuzzy logic has a number of advantages, in particular being able to work with vague terms, chaotic and dynamic environments and nonlinear behaviors that are typical of everyday use in stock market analysis. Specifically, this paper applies a higher level of fuzzy logic known as type-2 fuzzy logic, whose milked fuzzy functions are able to include additional uncertainty. The Mamdani fuzzy inference method has been widely applied in processing fuzzy rule-based systems. The Mamdani model is suitable if the modeled area is described using natural language. The input data is converted into fuzzy sets as part of the fuzzification process using mathematical functions. A total of nine models are created for each of the four shapes of the fuzzy sets: L--low, M--medium and H--high. Triangular, trapezoidal, Gaussian and bell member functions in the range [0; 1] were used to create fuzzy models and to examine the degrees of uncertainty in each of these MFs. A total of nine models were created for each MF, which differ in the magnitude of the uncertainty contained in the duplicate membership functions. Examined MF with varying levels of uncertainty are graphically illustrated in Appendix A. When compiling fuzzy sets, the parameters lower membership function scaling factor, specified as a positive scalar less than or equal to 1, are used. Next, the lag value is set. This delay defines the point at which the lower membership function value starts increasing from zero based on the value of the upper membership function. For example, \begin{table}
\begin{tabular}{c c} \hline \hline
**Statistics** & **DJIA Index** \\ \hline Mean & 21,736.94 \\ Standard deviation & 3718.25 \\ Minimum & 15,660.18 \\ Maximum & 28,645.26 \\ Skewness & 0.08 \\ Kurtosis & \(-1.52\) \\ Jarque-Bera test & 122.98 ** \\ \hline \hline \end{tabular}  
* indicates \(p\)-value lower than 0.01.  
\end{table} 
Table 2: Summary statistics of DJIA index for the observed period. a lag value of 0.1 indicates that the lower membership function becomes positive when the upper membership function has a membership value of 0.1. For demonstration in this example, several values indefinitely are used to evaluate the accuracy of the created models. Gradually, a total of nine models are created, which differ in the degree of uncertainty contained in fuzzy sets describing the input variables. Model 1 contains membership functions with lower scale 1 and lower lag 0, essentially classical fuzzy sets of type 1. Model 2 consists of membership functions that already contain another degree of uncertainty, which was not mentioned for model 1. Specifically, model 2 contains fuzzy sets with lower scale 0.9 and lower lag 0.1. The following models contain more and more uncertainty, as can be seen from the figures corresponding to the membership functions. An overview of the plotting of the investigated types of IT2 FL MF is shown in Appendix A, where triangular, trapezoidal, Gaussian and bell MFs are plotted. The figure also shows how the distance between the individual functions gradually increases with the addition of uncertainty. Thus, a total of nine models are constructed for each type of fuzzy membership function. The level of uncertainty with which the individual models differ is as follows:  
* Model (1): T1 FLS: LowerScale 1; LowerLag 0
* Model (2): T2 FLS: LowerScale 0.95; LowerLag 0.05
* Model (3): T2 FLS: LowerScale 0.90; LowerLag 0.10
* Model (4): T2 FLS: LowerScale 0.85; LowerLag 0.15
* Model (5): T2 FLS: LowerScale 0.80; LowerLag 0.20
* Model (6): T2 FLS: LowerScale 0.75; LowerLag 0.25
* Model (7): T2 FLS: LowerScale 0.70; LowerLag 0.30
* Model (8): T2 FLS: LowerScale 0.65; LowerLag 0.35
* Model (9): T2 FLS: LowerScale 0.60; LowerLag 0.40  
The knowledge base represents rules in the form \"IF-THEN\" and expresses expert knowledge about the relationship between the delayed values of the DJIA stock index and the current value of this index. There are 27 defined fuzzy rules based on a team of experts.