**5. Results Analysis**

## *5.1. Taiwan Stock Exchange Capitalization Weighted Stock Index*

In general, TAIEX is a widely-used dataset in stock market forecasting. In order to facilitate comparison with other forecasting models, this paper also uses it as the main dataset to verify the model. Using non-stationary data can lead to spurious regressions, so we first performed a stationarity test based on the unit root test by software Eviews (Eviews10.0 Enterprise Edition, Microsoft, Redmond, WA, USA). It can be concluded that the first-order di fference of TAIEX 1997–2005 was stationary data, which indicates that the fluctuation data used in this study were stationary. Further, other datasets in this study were also stationary data.

The model in this paper was based on high order, and thus, di fferent orders may a ffect the accuracy of the prediction. Hence, the experimental analysis showed that when the order of fuzzy fluctuation information entropy was 9–11, the stability of the model was more ideal. Table 3 shows the experimental errors for di fferent years under di fferent orders.


**Table 3.** Comparing average RMSEs based on di fferent order fuzzy fluctuation time series from 1997–2005.

Not surprisingly, accurate fluctuation trend predictions are very important and needed. Therefore, the performance of di fferent methods must be compared and evaluated, thus verifying the superiority or deficiency of the model. In order to verify the e ffects of model prediction, this section focuses on comparing this model's experimental results with those from other models. Comparing the errors across model showed that the current model had certain advantages in prediction accuracy. Table 4 shows the prediction errors for the di fferent methods between 1997 and 2005. The NFM-IE hybrid model achieved better prediction accuracy compared to the traditional regression model, autoregressive model, neural network model, and fuzzy model (Table 4). In addition, NFM-IE exhibited better predictive power in some years compared to other hybrid models based on the fuzzy theory.


**Table 4.** Performance comparison of prediction RMSEs with other models. NFM-IE, neutrosophic forecasting model based on information entropy.

## *5.2. Forecasting Shanghai Stock Exchange Composite Index*

SHSECI is one of the most typical stock indices in China, with certain representativeness. We selected it as an experimental dataset to verify the model's applicability.

Recently, scholars have proposed more comprehensive models based on traditional prediction methods. For example, Guan et al. [39] proposed a two-actor autoregressive moving average model based on the fuzzy logical relationships (ARMA-FR). Guan et al. [40] proposed a model based on back propagation neural network and high-order fuzzy-fluctuation trends (BPNN-HFT). This section compares several typical prediction methods. The results indicated that the model can also e ffectively predict the stock index. Table 5 and Figure 3 show a comparison of the di fferent prediction methods.

**Table 5.** RMSEs of forecast errors for the Shanghai Stock Exchange Composite Index SHSECI from 2007–2015.


**Figure 3.** RMSEs of forecast errors for SHSECI from 2007–2015.

The comparison shows that NFM-IE outperformed other methods in predicting SHSECI from 2007–2015.

Comparing the average value of the SHSECI prediction error showed that NFM-IE had better prediction accuracy and stability compared to the neural network-based BPNN-HFT model and the statistical-based ARMA-FR model.

## *5.3. Forecasting Hong Kong-Hang Seng Index*

Finally, the Hong Kong-Hang Seng Index (HSI) was selected as the experimental dataset. Comparing several authoritative prediction methods, we can verify the universality of the model in other stock markets. Table 6 and Figure 4 show a comparison of the di fferent prediction methods from 1998–2012.



**Figure 4.** RMSEs of forecast errors for HSI from 1998–2012.

To further evaluate the validity of the proposed model, we used Friedman's test to perform a significance test based on the study of Demšar [44]. For reference, Friedman's test is a parametric statistical test that was proposed by Milton Friedman [45,46]. To further illustrate the significance of the model's predictions compared to other prediction methods, this section will use Friedman's test and the post-hoc test for significance analysis. In the Friedman test phase, SPSS was used for statistical testing, and the post-hoc test phase was based on manual calculations.

In the first stage, Friedman's test requires comparison of the average ranking of different algorithms *Rj* = 1*N<sup>i</sup> rji* , where, *rji* is the rank of the *j*-th of *k* algorithms on the *i*-th of *N* datasets. The ranking of each method was based on the analysis of HSI forecast results as shown in Table 7.

**Table 7.** The rank of the forecasting results of the HSI.


Through software analysis, we concluded that the method had the highest comprehensive ranking. In addition, according to the Chi-square distribution, there were significant differences between these methods.

$$CD = q\_a \sqrt{\frac{k(k+1)}{6N}}\tag{31}$$

In the second stage, in order to further compare the different methods, we used the Nemenyi test [47]. According to Equation (31), α = 0.05 and CD = 1.575. Upon further comparison, we found that the method proposed in this study had significant advantages over Yu (2005) [41], Wan (2017) [42], Ren (2016) [43], etc. Although it was not significant compared with Cheng's method (2018) [10], the NFM-IE had certain advantages from the perspective of error mean and average level.
