## 5 Experimental Part
### Results

Table 1: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Company**} & \multirow{2}{*}{**Model Structure**} & \multicolumn{4}{c}{**Type of ANN-Based Model**} \\ \cline{3-6}  & & **FA-ANN** & **TA-ANN** & **FA-TA-ANN** & **ANN** \\ \hline \multirow{10}{*}{VISTULA} & ANN structure & MLP(10-15-1) & MLP(13-27-1) & MLP(13-14-1) & MLP(15-22-1) \\ \cline{2-6}  & Transfer function & sigmoidal & sigmoidal & sigmoidal & sigmoidal \\ \cline{2-6}  & MSE & 0.00003 & 0.00013 & 0.00004 & 0.00056 \\ \cline{2-6}  & & & & \(\text{FRAMA}_{10C}(b)\) & \\  & & & & \(\text{EMA}_{30C}(b)\) & \(\text{EMA}_{30C}(b)\) & \\  & & & & \(\text{SL}_{3\text{-MACD}}(b)\) & \(\text{EMA}_{30C}(b)\) & \\  & & & & \(\text{LWMA}_{10C}(b)\) & \(\text{CFL}_{10C}(b)\) & \\  & & & & \(\text{FRAMA}_{10C}(b)\) & \(\text{HULL}_{20C}(b)\) & \(\text{MACD}_{2}\),FRAMA(b) & \\  & & & & \(\text{CCI}(b)\) & \(\text{LWMA}_{50C}(b)\) & \\  & & & & \(\text{ROC}(b)\) & \(\text{EMA}_{30C}(b)\) & \(\text{CCI}(b-1)\) \\  & & & & \(\text{ROC}(b)\) & \(\text{EMA}_{30  
A comparison of the results shows that CLOSE value predictions using the hybrid ANN-based approach with technical and fractal analyses are more stable over a long period of time than those using the ANN model. The best prediction accuracy was obtained for the FA-ANN and FA-TA-ANN models. The most important prediction parameter for investors is the highest absolute error. For the purposes of comparison, the following error measures were used:  
1. The highest prediction error per month \(E_{max}\)--i.e., the highest difference between the real CLOSE value and the value predicted by the ANN per month: \[E_{max}=max\lfloor C(k)-\hat{C}(k)\rfloor,\text{ for }k\;=\;1,\;2,\ldots,\;N.\] (22)
2. Arithmetical mean of the month \(E_{max}\) values per tested period of time (one year): \[\overline{E_{max}}=\frac{1}{N}{\sum_{i=1}^{N}E_{max_{i}}}.\] (23)  
These error measures were used in the assessment and validation of the tested models. In order to assess the precision and stability of the models, the maximum absolute errors \(E_{max}\) of one day-ahead predictions were calculated from Equation (22) for each month in the test period. The results obtained for ZYWIEC SA in the period of one year for the ANN-based model and three hybrid models are given in Figure 6. Analogical results obtained for BPH Bank SA are shown in Figure 7 and for BUDIMEX SA in Figure 8.  
The results shown in Figures 6-8 imply that the hybrid approach using technical analysis with fractal indicators combined with the ANN (FA-ANN) ensured the best accuracy in terms for short-term predictions of CLOSE values. The worst results were obtained with the purely ANN-based approach. Similar results were obtained for other tested companies. A comparison of the arithmetical means \(\overline{E_{\text{max}}}\) of monthly \(E_{max}\) errors (absolute and relative) in the tested period of one year (2010) for the five selected companies and for four kinds of ANN-based models is presented in Table 2. These results confirm that the best accuracy was achieved by the hybrid model combining TA, FA, and ANN and the worst using the purely ANN-based approach. Figure 5: Short-term forecast of ANN-based models and the real CLOSE value for BUDIMEX SA in 2010. Figure 6: Comparison of \(E_{\max}\) values per month between all ANN-based models in a period of one year (2010) for ZYWIEC SA.