# The Modeling of Time Series Based on Least Square Fuzzy Cognitive Map
## 5 Results
### The Influence of the Parameters of the Proposed Model  
The number of the concepts is highly sensitive to the performance of FCM models [35]. The parameters \(c\) and \(\lambda\) are particular discussed in the experiments. Optimal values of the parameters are established by inspecting the predictive error (RMSE) of the training set. The values of \(c\) range from 2, 3,..., 20, while several representative values of \(\lambda\) are selected. Figure 6 plots line chart groups of RMSE of corresponding different parameter values. Examination of the data shown in Figure 6 leads us to the conclusion that the value of RMSE becomes substantially lower when increasing the number of concepts; however, the value becomes slightly higher or even higher still when the number of concepts exceeds a certain number. In other words, the predictive accuracy does not continuous increase with \begin{table}
\begin{tabular}{c c c} \hline \hline
**NO.** & **Time Series** & **Size** \\ \hline (a) & Mean daily flow, Oldman River near Brocket, Jan 01, 1988 to Dec 31, 1991 & 1461 \\ (b) & Monthly Boston armed robberies Jan.1966-Oct.1975 & 118 \\ (c) & Nigeria power consumption & 123 \\ (d) & Annual water use in New York city, liters per capita per day, 1898-1968 & 71 \\ (e) & Annual sheep population (1000s) in England and Wales 1867–1939 & 73 \\ (f) & Births per 10,000 of 23-year-old women, U.S., 1917–1975 & 59 \\ (g) & Daily open prices of the S\&P 500 stock index, May 16, 2017 to May 15, 2020 & 756 \\ (h) & Daily close prices of the Dow Jones industrial index, May 1, 2019 to April 29, 2020 & 252 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental data. Figure 4: The strategy of concept adjustment. the increasing number of \(c\). For example, regarding the Oldman time series, the optimal value of \(c\) is 10. Whereas for the Annual water time series, once the number of \(c\) moves past eight, there will be no substantial improvement of the reconstruction error, or it will become even worse. The topology of the FCM is more complex with the growth of the number of concepts. Therefore, the optimal value of \(c\) is selected under comprehensive consideration of the topological complexity of the FCM and the predictive accuracy. Figure 5: The experimental data. Different from the number of \(c\) which directly affects the precision of the least square fuzzy cognitive map (LSFCM) model, the values of \(\lambda\) do not have an apparent effect for the precision of the LSFCM model, but a large one for the weight matrix. The weight matrix varies linearly inversely with the shape parameter of sigmoid function \(\lambda\). The results of the numerical experiments illustrate that various values of \(\lambda\) represent different \(W^{*}\), but models with same performance are obtained. For the numerical example of the Births time series (f), the changes \(W\) with \(\lambda\) are as follows. Figure 6: \(Cont\). Letting \(\lambda=1\), we can calculate the weight matrix,  
\[W^{1}=\left[ \begin{array}{cccccc}5.0174&-8.6692&-7.2166&-5.1458&-8.3879&-7.8711 \\ -9.1504&4.0895&-7.9115&-8.6434&-4.0537&-7.1306\\ -5.7369&-6.1585&3.5081&-3.8613&-5.6261&-4.4168\\ -3.3991&-6.2223&-2.6043&1.4834&-5.8400&-5.0940\\ -7.1786&-3.1379&-5.1858&-6.4852&-0.3340&-2.2041\\ -6.1414&-4.5083&-4.1489&-5.3633&-2.6671&2.4939\end{array} \right].\]  
Then, letting \(\lambda=\max\lvert W^{1}\rvert=9.1504\), we have the weight matrix conforming to the definition  
\[W^{9.1504}=\left[ \begin{array}{cccccc}0.5483&-0.9474&-0.7887&-0.5624&-0.9167& -0.8602\\ -1.0000&0.4469&-0.8646&-0.9446&-0.4430&-0.7793\\ -0.6270&-0.6730&0.3834&-0.4220&-0.6148&-0.4827\\ -0.3715&-0.6800&-0.2846&0.1621&-0.6382&-0.5567\\ -0.7845&-0.3429&-0.5667&-0.7087&-0.0365&-0.2409\\ -0.6712&-0.4927&-0.4534&-0.5861&-0.2915&0.2725\end{array} \right].\]  
If setting \(\lambda=10\), we can also calculate the weight matrix  
\[W^{10}=\left[