## 2 Theoretical Background  
Numerous papers and research focused on the prediction of financial time series of stock indices using type-1 fuzzy logic can be found in the literature. However, classical type-1 fuzzy sets are not able to fully capture the additional uncertainty that is a feature of stock markets. For this reason, in recent years, research has focused on the application of type-2 fuzzy logic in economic areas. The latest research in this area is described below. The aim was to select research studies, taking into account the membership functions used, but there are not many articles focused on this topic. The type of function is not specified, as the vast majority of researchers do not pay much attention to choosing the appropriate membership function, but the accuracy of fuzzy model outputs and researchers state the function they use in their fuzzy models. Authors in [16] use several stock indices to predict market trends: Bombay Stock Exchange; CNX Nifty; and S&P 500. The authors chose the type-1 fuzzy model, which integrates ANFIS, to predict them. The authors used fuzzy sets in the Gaussian fuzzy model. However, the authors point out that the integration of neural networks through the ANFIS model significantly reduces the interpretability of the created system. Researches [17; 18] apply triangular fuzzy functions to the moving average prediction and the obtained forecast is implemented on the share at Bombay Stock Exchange. Paper [19] develops a fuzzy trading system integrating technical indicators on the basis of which they predict the development of the stock index. [20] predict the stock price of the State Bank of India (SBI) and the Dow-Jones Industrial Average (DJIA) using a fuzzy model. As part of fuzzification, the authors use a triangular fuzzy function, which is very simple and widespread, as the authors note. Other paper [21] integrates technical indicators into the model in order to improve the portability of the Athens Stock Exchange stock price forecast. The output of the strategy are signals to buy or sell a specific share. Authors [22] use two US stock indices, the DOW30 and the NASDAQ100 modified ANFIS model to predict. According to their empirical results, the bell functions of membership outperform trapezoids in performance. The authors also draw attention to a fundamental finding: the numbers and types of fuzzy functions have a fundamental and substantial influence in the process of predicting for financial time series. Paper [23] proposes a type-2 fuzzy model for stock index forecasting: Taiwan Stock Exchange Capitalization Weighted Stock Index, the Dow Jones Industrial Average and the National Association of Securities Dealers AutomatedQuotation. As an input variable, they use a time series delayed by one day in the fuzzy model. [24] predict the closing prices of TAIEX and NASDAQ stock indices. They also note that membership functions are an important aspect of prediction. The authors use triangular MF, because according to their knowledge and experience they provide better performance than Gaussian. Authors [25] use T2 FLS to form a profit analysis decision support model. The authors apply type-2 fuzzy sets because type-1 is too noisy. [26] in their research suggest the creation of T2 FLS to improve and refine the effectiveness of the TAIEX and COVID-19 stock index prediction model. As is mentioned above, the stock market, and therefore the entire financial system, is characterized by a highly unstable environment that is constantly changing. This is also one of the reasons why it is not possible to choose a universal form of membership functions within decision-making models or stock market prediction (see Table 1). However, this is a crucial area, as incorrect description of input data through membership functions results in incorrect output. In this case, incorrect investment signals or the indication of false signals to buy or sell, which may ultimately lead to the loss of investors or the impairment of their funds.