## 3 Implementations and Results  
### Forecasting of the Stock and Price (S&P) Index Using the FLR-FSWGARCH Model  
Daily data from the S&P index covering a period from 2 January 2015 to 10 September 2015 was obtained from Yahoo Finance. Figure 3 plots the daily S&P used in this study. The \(x\)-axis is the day numbered from 1 to 252, while the \(y\)-axis is the S&P. The trend is apparent from the plot, indicating the necessity of a suitable model for nonlinear time-series forecasting, such as the FLR-FSWGARCH model. Next, the minimum and maximum observations of the S&P indexes were 1867.61 and 2130.82, respectively. The data was fuzzified using IFNNs, as shown in Figure 4.  
A \(12\times 1\) matrix of the current year \(Y\) and \(12\times 1\) matrices of 12 sliding windows (\(SW\)) were formed from the data and tabulated, as shown in Table 1.  
Figure 3: The time-series plot of the Stock and Price (S&P) dataset. After partitioning the sliding windows, the differences between the current values and previous values were calculated to find the Euclidean distance of each window. The size of the forecasting was \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Current Year, \(Y\)**} & \multicolumn{4}{c}{**Previous Year, \(PY\)**} \\ \hline
**Day** & **S\&P** & **TFN** & **Day** & **S\&P** & **TFN** \\ \hline
20-Dec-15 & 2043.41 & (2000,2050,2100) & 8-Dec-14 & 2088.87 & (2050,2100,2150) \\
21-Dec-15 & 2073.07 & (2000,2050,2100) & 9-Dec-14 & 2090.11 & (2050,2100,2150) \\
22-Dec-15 & 2041.89 & (2000,2050,2100) & 10-Dec-14 & 2080.41 & (2050,2100,2150) \\
23-Dec-15 & 2005.55 & (1950,2000,2050) & 11-Dec-14 & 2102.63 & (2050,2100,2150) \\
24-Dec-15 & 2021.15 & (1950,2000,2050) & 12-Dec-14 & 2079.51 & (2050,2100,2150) \\
25-Dec-15 & 2038.97 & (2000,2050,2100) & 13-Dec-14 & 2049.62 & (2000,2050,2100) \\
26-Dec-15 & 2064.29 & (2000,2050,2100) & 14-Dec-14 & 2091.69 & (2050,2100,2150) \\
27-Dec-15 & 2060.99 & (2000,2050,2100) & 15-Dec-14 & 2077.07 & (2050,2100,2150) \\
28-Dec-15 & 2056.50 & (2000,2050,2100) & 16-Dec-14 & 2063.59 & (2000,2050,2100) \\
29-Dec-15 & 2078.36 & (2050,2100,2150) & 17-Dec-14 & 2047.62 & (2000,2050,2100) \\
30-Dec-15 & 2063.36 & (2000,2050,2100) & 18-Dec-14 & 2052.23 & (2000,2050,2100) \\
31-Dec-15 & 2043.94 & (2000,2050,2100) & 19-Dec-14 & 2012.37 & (1950,2000,2050) \\ \hline  & & & 20-Dec-14 & 2021.94 & (1950,2000,2050) \\  & & & 21-Dec-14 & 2043.41 & (2000,2050,2100) \\  & & & 22-Dec-14 & 2073.07 & (2000,2050,2100) \\  & & & 23-Dec-14 & 2041.89 & (2000,2050,2100) \\  & & & 24-Dec-14 & 2005.55 & (1950,2000,2050) \\  & & & 25-Dec-14 & 2021.15 & (1950,2000,2050) \\  & & & 26-Dec-14 & 2038.97 & (2000,2050,2100) \\  & & & 27-Dec-14 & 2064.29 & (2000,2050,2100) \\  & & & 28-Dec-14 & 2060.99 & (2000,2050,2100) \\  & & & 29-Dec-14 & 2056.50 & (2000,2050,2100) \\
30-Dec-14 & 2078.36 & (2050,2100,2150) & \(SW_{2}\) \\
31-Dec-14 & 2063.36 & (2000,2050,2100) & \(SW_{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Matrix of the current year, \(Y\) and the previous year of S&P data, \(PY\). Figure 4: The triangular fuzzy numbers of the S&P dataset. determined from the selected window that produced the minimum mean Euclidean distance. An example of a calculation using matrices \(Y\) and \(SW_{1}\) to produce the Euclidean distance is shown below. 1. The difference between the current and previous values (only displays the calculation of the first row). \[Y_{1}-SW_{1,1}=(2000-2050,2050-2000,2100-1950)=(-50,50,150).\]
2. The defuzzification: \[Edi_{1,1}=\frac{-50+2(50)+150}{4}=50.\]
3. The mean Euclidean distance: \[Edi_{i}=\sqrt{\sum\limits_{j=1}^{12} \begin{array}{c}\left(50\right)^{2}+ \left(0\right)^{2}+\left(0\right)^{2}+\left(-50\right)^{2}+\left(0\right)^{2} +\left(50\right)^{2}+\left(0\right)^{2}+\left(0\right)^{2}+\left(0\right)^{2} +\left(50\right)^{2}\\ +\left(-50\right)^{2}+\left(0\right)^{2}\end{array}