## 4 Data and Methodology  
Stock market price change is a dynamic system that is constantly changing, and these sudden changes and movements of stock price movements doubles the complication of successful stock price prediction. In addition, stock markets are characterized by their highly non-linear nature that makes it difficult for investors to make quick decisions about the right investments. It is essential to develop an intelligent system for obtaining real-time price information, reducing one investor obsession and helping them maximize their profits. However, financial time series show relatively complicated and non-stationary behavior, with the variable not having a linear trend or clear tendency to move to a fixed value according to [34]. Mainly for the reasons described above, researchers are increasingly focusing on the use of alternative techniques that can be used to predict the development of financial time series. These techniques certainly include fuzzy logic, which is able to include the uncertainty, nonlinearity and noise that occur in financial time series. The Dow Jones Industrial Average (DJIA) is one of the best-known indicators of developments in the US stock market. The DJIA is named after its founder, [PERSON], and its business partner, [PERSON]. The index was first calculated on 26 May 1896. At that time, it contained 12 exclusively industrial companies focused on areas such as the railway industry, cotton, tobacco, sugar cane, natural gas, etc. Today, the DJIA consists of 30 American companies, many of these are the largest and the most traded (blue-chip companies). The historical development of the DJIA index for the observed period is shown in Figure 2.  
Table 2 shows the basic statistics of selected stock index. The average price of the DJIA index for the selected period from January 2015 to January 2020 was 21,736.94. The price of the index rose to a maximum value of 28,645.26 and fell to a minimum value of 15,660.8. The standard deviation is a simple measure of the volatility of stock indices. For the DJIA index, the price fluctuated 3718.25. Skewness is a measure of asymmetry, or, more precisely, a lack of symmetry. Based on the values, it can be stated that the stock index has a positive asymmetric distribution of price probabilities. Furthermore, Kurtosis values show that it does not have a value close to 3, which is a theoretical value for the Gaussian probability distribution. This suggests that none of the probability distributions of this time series appear to be normally distributed. To confirm this assumption, the Jarque-Bera test was conducted with a zero hypothesis that the respective probability distribution was Gaussian (chi-square with 2 degrees of freedom). The reported values led to the rejection of the null hypothesis for all stock indices. This lack of normality is in line with the well-known \"stylized facts\" of market returns, as pointed out in previous studies [35]. Figure 2: Historical development of the DJIA stock index. Daily stock index closing rates for the selected period January 2015 to January 2020 are used, a medium-term period of 5 years. It is also clear from the Figure 2 that this is a bullish trend or a rising trend without long-term bearish trends. The time period is deliberately ended in January 2020, before a sharp decline due to the COVID-19 pandemic situation. This is a sudden event that would significantly distort the output of the fuzzy model. Real data are processed through standard procedures applied in [36; 37]. This step is purposeful, as the value of the stock index shows large fluctuations, which would not have to provide relevant outputs without pre-processing. \[y_{t}=\frac{x_{t}-m}{M-m} \tag{17}\]  
where \(x_{t}\) is the closing daily price of the stock index at time \(t\), \(M=max\{x_{t}\}\) and \(m=min\{x_{t}\}\).