Fuzzy Cognitive Maps Learning And Its Application In Time Series Prediction

In the current era of artificial intelligence,intelligent system has been developed and applied to various fields resulting in better performance.Knowledge representation and reasoning are indispensable in the intelligent system.As a knowledge-based representation and realizing inference processes,fuzzy cognitive maps(FCMs)have been used in modeling and simulation of complex systems,decision making,time series modeling and prediction,etc.The fast,effective and robust learning of FCMs is the key for effective application.The main learning methods of FCMs are population-based algorithms,which use data-driven techniques.However,those learning algorithms also exhibit obvious limitations.Some of them either are extremely time-consuming with high computational overload.The FCMs learned by those algorithms lack robustness when the experimental data contain noise.Reasonable distribution of the weights is rarely considered in these algorithms.Fuzzy cognitive maps have been widely used in the field of time series prediction.However,these methods perform well in one-stepahead or shortterm prediction but poorly in terms of long-term prediction.In addition,the existing FCMs models usually increase the number of concept nodes for modeling complex time series systems.Consequently,the prediction model is too complex and the prediction performance is not significantly improved.To solve these problems,the data-driven modeling method is used based on the reasoning mechanism of FCMs to investigate the rapid and robust learning method.And then,the time series prediction models with FCMs are investigated.Detailed research work is as follows:(1)The existing FCMs learning methods are inefficient and rarely considered noisy data.In order to solve these problems,a straightforward,rapid,and robust learning method is proposed.The overall number of weights of the FCMs to be learned quadratically increases with the number of nodes.The learning of FCMs is extremely time-consuming with high computational overload.The crux of the proposed algorithm is to equivalently transform the learning problem of FCMs to a classic-constrained convex optimization problem.Then the interior-point method is used to solve this optimization problem,consequently the optimal weight matrix of fuzzy cognitive map is obtained.In the learning process,the least square method is used to estimate the parameters to ensure the robustness of the well-learned FCMs.In addition,the existing learning methods rarely consider the reasonable distribution of the weight values.The weights that are learned by these methods are unreasonably distributed in the universe of discourse,which could result in the reduction of the resulting FCMs.To reduce this limitation,the maximum entropy term is added into the convex optimization problem for regularizing the distribution of the weights of the well-learned FCMs,which can greatly restore the distribution of weights.(2)The existing FCMs time series methods perform poorly in terms of long-term prediction.To address this problem,a sound conceptual framework is proposed for long-term time series prediction,which melds FCMs,time series segmentation and fuzzy clustering.Time series is divided into suitable and internally homogenous segments for high-level knowledge representation.Afterwards,the segments are equalized lengths to the prediction horizon based on dynamic time warping.Subsequently,the modified fuzzy c-means based on dynamic time warping is utilized to fuzzify these segments.The data learned by FCMs is expanded from scalar to vector.In the FCMs model,each node of FCMs represents one latent variation modality of time series and the weights depict the causal relationship that exists among these modalities.This model has not only better numeric prediction,but also has better interpretability.With this regard,the design of long-term time series forecasting model used FCMs is interpretable and easily comprehended by humans.(3)Building a single FCMs model for complex time series system will lead to the model too complex and poor prediction performance.To address these problems,the multi-mode fuzzy cognitive maps models are proposed for modeling time series.It is difficult to reveal the multi-modal characteristics of time series when using a single model to simulate it since it contain a variety of variation modality.In time dimension,the proposed method exploits the bootstrap method to select multiple sub-sequences from the original time series.These resulting sub-sequences contain the diverse modality in the original time series.Then around these subsequences,the fuzzy cognitive map sub-models are constructed respectively.Finally,the formed sub-models are further merged by means of granular computing method.The resulting multi-modal model has not only better numerical prediction,but also interval prediction.In space dimension,fuzzy clustering is adopted to partition time series into several sub-sequences.Subsequently,these sub-sequences are used to construct FCMs models respectively.In addition,the influence of the number of nodes on the performance of the model is discussed.The results of this investigation show that the forecast accuracy cannot continuously improve with the increasing nodes of FCMs.On the contrary,the prediction precision of the model increases as the number of partitions increases.