# The Modeling of Time Series Based on Least Square Fuzzy Cognitive Map
###### Abstract  
A novel design method for time series modeling and prediction with fuzzy cognitive maps (FCM) is proposed in this paper. The developed model exploits the least square method to learn the weight matrix of FCM derived from the given historical data of time series. A fuzzy c-means clustering algorithm is used to construct the concepts of the FCM. Compared with the traditional FCM, the least square fuzzy cognitive map (LSFCM) is a direct solution procedure without iterative calculations. LSFCM model is a straightforward, robust and rapid learning method, owing to its reliable and efficient. In addition, the structure of the LSFCM can be further optimized with refinements the position of the concepts for the higher prediction precision, in which the evolutionary optimization algorithm is used to find the optimal concepts. Withal, we discussed in detail the number of concepts and the parameters of activation function on the impact of FCM models. The publicly available time series data sets with different statistical characteristics coming from different areas are applied to evaluate the proposed modeling approach. The obtained results clearly show the effectiveness of the approach. time series; least square method; fuzzy cognitive map; refinements of concepts +
Footnote †: journal: Applied Soft  
given time series data set [10; 11; 12]. The interrelationships between all concepts corresponding to weights are confirmed during the learning process. Hebbian-based [13; 14; 15] and population-based [16; 17; 18; 19] methods are the mainstream for small-scale FCM learning problems. Moreover, the population-based methods outperform Hebbian-based methods in terms of time series prediction [20]. For large-scale FCM learning problems, [PERSON] et al. [21] introduced Lasso regularization as the sparsity penalty term into the objective function to ensure the sparse structure of the resulting FCM. [PERSON] et al. [22; 23] transformed FCM learning into a convex optimization problem with constraints, and the maximum entropy terms were invoked to solve the optimization problem. The obvious problem in the above-mentioned FCM learning methods of time series is the need for iterative calculation and intensive computing to perform inference. These methods are time consuming. Time series modeling with a fuzzy cognitive map has been applied in a range of quite diverse fields. [PERSON] [24] proposed a method that combines FCM with granular to time series prediction realized both at the linguistic and numerical level. They take advantage of real-coded genetic algorithms to learn FCM. [PERSON] and [PERSON] [25; 26] used evolutionary-based and multi-step enhancement of the evolutionary algorithm to learn the FCM to cope with the forecasting of patient states in the case of pulmonary infections. [PERSON] [27] resorted to wavelet transform to decompose original non-stationary time series into multivariate time series, then the high-order FCMs were applied to model and predict multivariate time series. [PERSON] [28] proposed a high-order fuzzy cognitive map (HFCM) to model and predict time series. The structure of the HFCM generated works in an automatic fashion. [PERSON] [29] proposed a hybrid algorithm based on an FCM for fuzzy time series prediction, in which a fuzzy C-means clustering algorithm was used to construct the framework of the FCM and a genetic algorithm was applied to learn the weights of the FCM. [PERSON] [30] adopted simplified fuzzy cognitive maps to construct the framework of time series modeling and introduced the selection criteria of concepts. In order to achieve a reasonable balance between complexity and accuracy, some simplification strategies which a posteriori remove nodes and weights were presented. [PERSON] [31] proposed a dynamic optimization of the fuzzy cognitive maps for univariate time series forecasting. In this model, the concept of a sliding window was applied to train the predictive model.