证明了模糊灰色认知图在双曲正切和Sigmoid激活下的收敛性。
On the Convergence of Sigmoid and tanh Fuzzy General Grey Cognitive Maps
- 基于不动点定理与不等式，推导出收敛的充分条件。
- 验证了新理论涵盖传统模型的收敛性，具普适性。
- 为控制、预测等系统的建模提供理论支撑。
模糊一般灰色认知图（FGGCM）和模糊灰色认知图（FGCM）是处理不确定性问题的模糊认知图（FCM）扩展。FGGCM可处理具有多个区间的广义灰色数，使模型更适应不确定情境。尽管已有大量文献讨论了FCM与FGCM的收敛性，但对FGGCM的收敛性研究仍不充分。本文填补该空白：首先定义并证明了广义灰色数空间及其向量空间的度量性质，利用闵可夫斯基不等式证明其完备性；在此基础上，结合Banach不动点定理、Browder-Gohde-Kirk不动点定理、拉格朗日中值定理及柯西不等式，推导出当使用tanh与sigmoid函数作为激活函数时，FGGCM收敛到唯一不动点的充分条件，并分别给出核矩阵与灰色度的收敛条件。最后，基于网络体验与土木工程数据集，构建了采用广义灰色数权重的FGGCM模型，对比传统模型的收敛定理，验证了所提方法的有效性。结果表明，传统FCM的收敛定理是本文理论的特例。本研究对指导FGGCM学习算法设计、实现特定不动点建模具有重要意义，为控制、预测与决策支持系统中的应用奠定理论基础。
原文摘要 · Abstract (English)
Fuzzy General Grey Cognitive Map (FGGCM) and Fuzzy Grey Cognitive Map (FGCM) are extensions of Fuzzy Cognitive Map (FCM) in terms of uncertainty. FGGCM allows for the processing of general grey number with multiple intervals, enabling FCM to better address uncertain situations. Although the convergence of FCM and FGCM has been discussed in many literature, the convergence of FGGCM has not been thoroughly explored. This paper aims to fill this research gap. First, metrics for the general grey number space and its vector space is given and proved using the Minkowski inequality. By utilizing the characteristic that Cauchy sequences are convergent sequences, the completeness of these two space is demonstrated. On this premise, utilizing Banach fixed point theorem and Browder-Gohde-Kirk fixed point theorem, combined with Lagrange's mean value theorem and Cauchy's inequality, deduces the sufficient conditions for FGGCM to converge to a unique fixed point when using tanh and sigmoid functions as activation functions. The sufficient conditions for the kernels and greyness of FGGCM to converge to a unique fixed point are also provided separately. Finally, based on Web Experience and Civil engineering FCM, designed corresponding FGGCM with sigmoid and tanh as activation functions by modifying the weights to general grey numbers. By comparing with the convergence theorems of FCM and FGCM, the effectiveness of the theorems proposed in this paper was verified. It was also demonstrated that the convergence theorems of FCM are special cases of the theorems proposed in this paper. The study for convergence of FGGCM is of great significance for guiding the learning algorithm of FGGCM, which is needed for designing FGGCM with specific fixed points, lays a solid theoretical foundation for the application of FGGCM in fields such as control, prediction, and decision support systems.
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