Sparse large-scale high-order fuzzy cognitive maps guided by spearman correlation coefficient

Time series prediction is one of the most important applications of Fuzzy Cognitive Maps (FCMs). In general, the state of FCMs in forecasting depends only on the state of the previous moment, but in fact it is also affected by the past state. Hence Higher-Order Fuzzy Cognitive Maps (HFCMs) are proposed based on FCMs considering historical information and have been widely used for time series forecasting. However, using HFCMs to deal with sparse and large-scale multivariate time series are still a challenge, while large-scale data makes it difficult to determine the causal relationship between nodes because of the increased number of nodes, so it is necessary to explore the relationship between nodes to guide large-scale HFCMs learning. Therefore, a sparse large-scale HFCMs learning algorithm guided by Spearman correlation coefficient, called SG-HFCM, is proposed in the paper. The SG-HFCM model is specified as follows: First, the solving of HFCMs model is transform into a regression model and an adaptive loss function is utilized to enhance the robustness of the model. Second, $l_1$-norm is used to improve the sparsity of the weight matrix. Third, in order to more accurately characterize the correlation relationship between variables, the Spearman correlation coefficients is added as a regular term to guide the learning of weight matrices. When calculating the Spearman correlation coefficient, through splitting domain interval method, we can better understand the characteristics of the data, and get better correlation in different small intervals, and more accurately characterize the relationship between the variables in order to guide the weight matrix. In addition, the Alternating Direction Multiplication Method and quadratic programming method are used to solve the algorithms to get better solutions for the SG-HFCM, where the quadratic programming can well ensure that the range of the weights and obtaining the optimal solution. Finally, by comparing with five algorithms, the SG-HFCM model showed an average improvement of 11.93% in prediction accuracy for GRNs, indicating that our proposed model has good predictive performance.