Numerous learning methods for fuzzy cognitive maps (FCMs), such as the Hebbian-based and the population-based learning methods, have been developed for modeling and simulating dynamic systems. However, these methods are faced with several obvious limitations. Most of these models are extremely time consuming when learning the large-scale FCMs with hundreds of nodes. Furthermore, the FCMs learned by those algorithms lack robustness when the experimental data contain noise. In addition, reasonable distribution of the weights is rarely considered in these algorithms, which could result in the reduction of the performance of the resulting FCM. In this article, a straightforward, rapid, and robust learning method is proposed to learn FCMs from noisy data, especially, to learn large-scale FCMs. The crux of the proposed algorithm is to equivalently transform the learning problem of FCMs to a classic-constrained convex optimization problem in which the least-squares term ensures the robustness of the well-learned FCM and the maximum entropy term regularizes the distribution of the weights of the well-learned FCM. A series of experiments covering two frequently used activation functions (the sigmoid and hyperbolic tangent functions) are performed on both synthetic datasets with noise and real-world datasets. The experimental results show that the proposed method is rapid and robust against data containing noise and that the well-learned weights have better distribution. In addition, the FCMs learned by the proposed method also exhibit superior performance in comparison with the existing methods.

In time-series forecasting, it is an important task to make an accurate and interpretable long-term prediction. In this article, we present a novel approach developed from the perspective of granular computing (GrC) to realize the long-term prediction of time series. The proposed method first employs a sliding window strategy to smooth on the raw time series. Subsequently, the smoothed time series is transformed into the corresponding granular time series that is depicted by evolving shape with the aid of the clustering algorithm based on the dynamic time warping (DTW) distance. Finally, a Takagi–Sugeno (TS) architecture-like granular model (GrM) is formed by deriving the relations implying in the granular time series and offers the granular output in the numeric vector format. The GrM adopts the pattern-to-pattern inference mechanism to realize the long-term prediction of time series at the vector level. Experiments on several datasets demonstrate that the proposed method not only has the ability to circumvent the cumulative error but also makes the resulting GrM equip better interpretability.