# The Modeling of Time Series Based on Least Square Fuzzy Cognitive Map
## 4 Modeling Time Series Using LSFCM
### Refinements of LSFCM Model  
The concepts of the LSFCM are obtained by the fuzzy c-means algorithm; however the performance index is prediction error. The initial concepts do not necessarily lead to the minimum prediction error. Thus, the idea is to optimize the LSFCM model by adjusting the position of the concepts (cluster centers) for a better performance index. The partition matrix \(\mathbf{U}\) includes the fuzzy membership of each sample to the concepts. Supposing that the position of the concepts is relocated in the domain of definition, on this basis, there will be acting in response to cause a new partition matrix. The structure of the LSFCM will be relearned based on the new partition matrix and concepts, then that impacts directly on the prediction error. The procedure of migration of concepts is depicted in Figure 3. The initial/updated population of concepts is used to construct the LSFCM model. Then, the reconstruction errors are calculated and the optimal concept whose LSFCM model has minimum errors is selected. The best concept is output after the iteration, which is used to construct the optimal LSFCM model. The refinements of concepts will be detailed discussed in the following.  
How to change the position of concepts plays a pivotal role in the process. In this study, the stochastic seeking strategy is considered to adjust the coordinate of all the concepts. It can be believed that the clustering centers of the fuzzy c-means algorithm are already close to the best concepts of the FCM but still make further progress. The position of the clustering center is in the neighborhood of the optimum position which makes the LSFCM model show a better performance index. To refine the FCM model, the initial concepts obtained by fuzzy c-means are used as the starting point. To modify the concepts, a sub-defined interval is introduced for the coordinates of the concepts. The radius of the sub-defined interval is described as follows,  
\[r=\frac{\max(\text{data})-\min(\text{data})}{2c} \tag{23}\]  
Each concept moves to a new position from its current position in a particular range rather than the entire domain and the new position \(v_{i}^{\prime}\in[v_{i}-r,\,v_{i}+r]\). Hence, the efficiency of adjustment is higher. The probability of a new concept position in the domain is a normal distribution. In other words, the closer the position to the initial concept, the higher the probability selected. The strategy of concept adjustment is illustrated in Figure 4. The initial population of concepts is a normal distribution. \(\mathbf{\varphi}_{i}(\mathbf{v}_{i})\)\((i=1,2\cdots c)\) is the probability density function that the position in the domain selects as the new concept. Each concept can move to the best position around the initial position. Although the movement of each concept in each coordinate is random, the movement of all concepts has a preference or tendency. The final result is to minimize the performance index RMSE, viz. \(\frac{1}{N}\sum_{t=1}^{N}\|x_{t}-\hat{x}_{t}\|^{2}\rightarrow\min\).  
Figure 3: Migration of concepts.  
The positions of the concepts are adjusted by the evolutionary algorithm Particle swarm optimization (PSO). The fuzzy membership matrix \(\mathbf{U}\) is refreshed corresponding to the new concepts. Once the optimal concepts are obtained, the final LSFCM model (LSFCM-ref) is formed.