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摘要: 模糊认知图(Fuzzy cognitive map, FCM)是建立在认知图和模糊集理论上的一类代表性的软计算理论, 兼具神经网络和模糊决策两者的优势, 已成功地应用于复杂系统建模和时间序列分析等众多领域. 学习权重矩阵是基于模糊认知图建模的首要任务, 是模糊认知图研究领域的焦点. 针对这一核心问题, 首先, 全面综述模糊认知图的基本理论框架, 系统地总结近年来模糊认知图的拓展模型. 其次, 归纳、总结和分析模糊认知图学习算法的最新研究进展, 对学习算法进行重新定义和划分, 深度阐述各类学习算法的时间复杂度和优缺点. 然后, 对比分析各类学习算法在不同科学领域的应用特点以及现有的模糊认知图建模软件工具. 最后, 讨论学习算法未来潜在的研究方向和发展趋势.Abstract: Fuzzy cognitive maps (FCM) are a representative soft computing theory based on cognitive maps and fuzzy set theory. They combine the advantages of both neural networks and fuzzy decision-making and have been successfully applied in many fields, including complex system modeling and time series analysis. Learning the weight matrix is the primary task of modeling based on fuzzy cognitive maps and is the focus of research in this field. To address this core issue, we first comprehensively review the basic theoretical framework of fuzzy cognitive maps and systematically summarize the extended models developed in recent years. Next, the most recent advancements in fuzzy cognitive map learning algorithms are reviewed, analyzed, and summarized. The algorithms are redefined and categorized, with a detailed exploration of their time complexity, strengths, and weaknesses. Additionally, the application properties of various learning algorithms in various scientific domains are also compared and analyzed in this research, along with the software tools that are now available for creating fuzzy cognitive maps. Finally, potential research directions and development trends for learning algorithms are discussed.
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图 4 专家知识驱动的学习算法的基本流程
Fig. 4 The basic process of expert knowledge-driven learning algorithms
表 1 拓展认知图模型对比
Table 1 Comparison of extension cognitive map models
类别 名称 特点 优点 缺点 应用领域 基于不同
模糊理论
的拓展
认知图RBFCM[28] 引入模糊进位累加器计算因果权重 涵盖多种概念关系并具有
多功能性和简单性建模要求高 决策支持 FGCM[29] 引入灰色数衡量因果强度 建模概念之间的不确定信息 推理复杂 可靠性工程 IFCM[30] 利用直觉模糊集建模因果关系 衡量了因果关系中的不确定性 推理复杂 时间序列预测 IVFCM[31] 利用区间值描述因果关系的强度 考虑了非结构化环境相关的不确定性 依赖专家知识 决策支持, 时间序列
预测模型ECM[32] 在因果推理中融入了证据理论 既能表示不确定性又能进行知识融合 依赖专家知识,
推理复杂决策支持 RCN[33] 利用粗糙集表示因果关系 解决了不确定情况下的决策问题 依赖专家知识 决策支持软件,
可靠性评估FRCN[34] 利用模糊粗糙结构构建神经网络 建模了因果关系的不确定性 推理复杂 决策支持 zT2FSs-FCM[35] 引入二型模糊集建模节点间的
因果权重捕获了概念间的不确定性关系 推理复杂 系统评估 面向动态
系统建模
的拓展
认知图DCN[37] 考虑了因果关系的时变性 结构上具有更高的可扩展性和灵活性 依赖拉普拉斯框架,
建模复杂度高决策支持 DRFCM[39] 推理过程中引入非线性动态函数 能够捕获动态因果关系,
具有自适应性建模要求高 风险评估, 决策支持 FTCM[40] 考虑了因果关系强度和时间滞后性 能够随时间推移分析系统的动态行为 建模复杂 时间序列预测 E-FCM[41] 采用检查机制模拟动态因果关系 能够自我进化适应不断发展的行为 计算耗时 动态场景建模 HFCM[42] 考虑了复杂系统建模过程中的
多阶动态性准确地描述了系统行为 随着阶数增加,
计算复杂度增加时间序列预测 TAFCM[43] 引入了定时自动机理论建模
系统的时间粒度推理过程具有动态性和自适应性 建模要求高 人类情绪建模 DFCM[44] 嵌入在深度神经网络的框架中 构建可解释预测器, 挖掘隐藏的
因果关系训练耗时, 容易面临
“数据饥饿”问题时间序列预测 AFCM[45] 构建基于趋势的信息粒引入
自适应更新机制自适应权重长期预测 计算耗时 时间序列预测 
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表 2 基于学习范式的模糊认知图学习算法分类
Table 2 Classification of fuzzy cognitive map learning algorithms based on the learning paradigm
类别 学习方法 时间复杂度 优点 缺点 作者 发表年份 专家知识
驱动的方法DHL[47] ${\rm{O}}(N^2)$ 简单, 易操作 只考虑了当前的一对概念 Dickerson等 1994 BDA[48] ${\rm{O}}(N^2)$ 考虑多个概念的影响 只适用于二进制计算 Huerga 2002 AHL[49] ${\rm{O}}(N^2)$ 考虑了所有概念的影响 训练耗时 Papageorgiou等 2004 NHL[50] ${\rm{O}}(N^2)$ 保留了原始的图结构, 具有合理的物理解释性 依赖专家标准 Papageorgiou等 2003 INHL[51] ${\rm{O}}(N^2)$ 避免陷入局部最小值 需要先验知识 Li等 2004 DDNHL[52] ${\rm{O}}(N^2)$ 数据驱动 依赖专家知识 Stach 等 2008 带终端约束的
NHL算法[53]${\rm{O}}(N^2)$ 提高结果的可行性 需要先验知识 陈宁等 2016 FBN[54] ${\rm{O}}(N^2)$ 利用模糊因果规则推理 性能受激活参数的影响 Carvalho等 2007 基于bagging增强的NHL算法[55] ${\rm{O}}(N^2)$ 泛化性能较好 依赖专家知识 Papageorgiou等 2012 自动学习
算法GA[56] ${\rm{O}}(N^2)$ 数据驱动 受限于二进制编码 Mateou等 2005 RCGA[57] ${\rm{O}}(N^2)$ 数据驱动, 实数编码 参数寻优耗时 Stach 等 2005 PSO[58−59] ${\rm{O}}(N!)$ 数据驱动, 元启发式算法 依赖专家知识 Parsopoulos 等Oikonomou 等 2003
2013SOMA[60] ${\rm{O}}(N^2)$ 数据驱动 计算耗时 Vaščák 2010 ACO[61] ${\rm{O}}(N^2)$ 概率型算法鲁棒性强 计算耗时, 容易早熟收敛 Chen等 2012 ABC[62] ${\rm{O}}(N^2)$ 数据驱动 参数寻优耗时 Yesil等 2013 ICA[63] ${\rm{O}}(N^3)$ 数据驱动 计算复杂, 耗时 Ahmadi 等 2015 DE[64] ${\rm{O}}(N^2)$ 容易理解, 计算简单 易局部收敛 Juszczuk等 2009 SA[65−66] ${\rm{O}}(N^2)$ 计算简单 参数寻优耗时 Ghazanfari 等
Alizadeh等2007
2009BB-BC[67] ${\rm{O}}(N^2)$ 算法简单, 泛化能力较好 不适用于解决高维问题 Yesil等 2010 CA[68] ${\rm{O}}(N^2)$ 全局搜索与局部搜索结合 参数寻优耗时, 对问题的依赖性强 Ahmadi等 2014 基于互信息的
模因算法[70]${\rm{O}}(N^2)$ 适用于大规模图学习 无法在搜索过程中
关注图的稀疏性Zou等 2018 MARO[71] ${\rm{O}}(N^2)$ 只需调用一次目标
函数, 无需设置参数计算复杂, 易陷入局部最优 Salmeron等 2019 分解RCGA[72−73] ${\rm{O}}(N^2)$ 分解并行计算 计算复杂 Chen等, Stach等 2015, 2010 D&C RCGA[74] ${\rm{O}}(N^2)$ 可并行计算并具有可扩展性 随着图的大小和处理器数量增加, 算法性能下降 Stach等 2007 dMAGA[75] ${\rm{O}}(N^2)$ 适用于大规模图学习
具有鲁棒性受 FCM 节点的取值范围限制, 需在算法执行前进行数据归一化 Liu等 2015 MA-NN[76] ${\rm{O}}(N^2)$ 分布式计算框架适用于
大规模网络重建受FCM节点的取值范围限制, 需在算法执行前进行数据归一化 Chi等 2019 MOEA[77, 79−80] ${\rm{O}}(N^2)$ 多目标进化考虑了图的稀疏性 不适用于大规模图学习 Liu等, Poczeta 等,
Chi等2019, 2018, 2016 IMFPSO[78] ${\rm{O}}(N!)$ 优化过程考虑了知识迁移 算法易早熟, 过早收敛 Liang等 2022 SRCGA[15] ${\rm{O}}(N^2)$ 考虑了图的稀疏性 不适用于处理大规模数据 Stach等 2012 MMMA[17] ${\rm{O}}(N^2)$ 多图优化知识转移 有可能发生负信息迁移, 导致收敛速度缓慢 Shen等 2020 CS[81] ${\rm{O}}(N^3)$ 适用于大规模稀疏图学习 参数寻优耗时 Wu等 2017 内点法[82] ${\rm{O}}(N^4)$ 精度高, 可扩展性好 对初值敏感, 难以处理
不等式约束问题Lu等 2020 约束优化[83] ${\rm{O}}(N^3)$ 考虑了矩阵分布具有抗噪能力 仅适用于有监督学习 Feng等 2021 近似梯度下降[84] ${\rm{O}}(N^3)$ 适用于解决大规模数据问题 对初始点敏感, 可能
陷入局部最优Ding等 2021 Moore-Penrose逆[85] ${\rm{O}}(N^3)$ 参数较少 计算复杂 Vanhoenshoven等 2020 Lasso回归[86] ${\rm{O}}(N^3)$ 考虑了图的稀疏性, 适用于
大规模图学习可能出现过拟合 Wu等 2016 岭回归[87] ${\rm{O}}(N^3)$ 泛化性能较好, 适用于
大规模图学习对特征的缩放敏感 Yang等 2018 弹性网络回归[88] ${\rm{O}}(N^3)$ 增加了L1 和L2 正则化, 适用于大规模图学习 参数调节困难 Shen等 2020 支持向量回归[89] ${\rm{O}}(N^4)$ 适用于高维非线性数据 对缺失数据敏感 Gao等 2020 贝叶斯岭回归[90] ${\rm{O}}(N^4)$ 简单、模型适应性较强 对模型的假设较多依赖先验分布 Liu等 2020 FTRL[91] ${\rm{O}}(N^3)$ 在线学习 计算、推理过程复杂 Wu 等 2021 半自动学习算法 DE+NHL[92] ${\rm{O}}(N^2)$ 进化过程中保留了图的物理意义 依赖专家知识 Papageorgiou等 2005 RCGA+NHL[93] ${\rm{O}}(N^2)$ 利用了遗传算法的全局优化能力 受限于专家经验 Zhu等 2008 PSO+NHL[94] ${\rm{O}}(N!)$ 避免人为因素产生的训练误差 受限于专家经验 Yazdi等 2008 EGDA+NHL[95] ${\rm{O}}(N^2)$ 全局搜索, 参数少 受限于专家经验 Ren 2012 DDNHL+GA[96] ${\rm{O}}(N^3)$ 数据驱动分类推理能力强 受限于专家经验 Natarajan等 2016 RCGA+DE+
梯度下降[97]${\rm{O}}(N^2)$ 全局搜索 参数寻优耗时 Madeiro 等 2012 注: 时间复杂度为该算法更新一次FCM权重矩阵所需时间开销, 未考虑数据量大小及最大迭代次数. N表示节点个数.
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表 3 大规模模糊认知图学习算法分类
Table 3 Large-scale fuzzy cognitive map learning algorithm classification
类别 方法 转换函数 最大FCM规模 发表年份 基于暴力求解的方法 D&C RCGA[73] sigmoid 40 2010 并行RCGA[74] sigmoid 80 2007 dMAGA[75] sigmoid 200 2015 MA-NN[76] sigmoid 100 2019 MOEA[80] sigmoid 40 2015 SRCGA[15] sigmoid 40 2012 基于维度缩减的方法 MIMA[70] sigmoid 500 2018 文献[100] sigmoid/tanh 25 2015 文献[101] sigmoid 10 2018 基于分解的方法 CS[81] sigmoid 1 000 2017 MMMA[17] sigmoid/tanh 600 2020 内点法[82] sigmoid/tanh 200 2020 约束优化[83] sigmoid/tanh 200 2021 近似梯度下降[84] sigmoid 200 2021 Lasso回归[86] sigmoid 500 2016 弹性网络回归[88] sigmoid 200 2020 HTMA-DRA[99] sigmoid 200 2022 dMAGA-FCM$_D$[102] sigmoid 500 2017 NMMMAGA[103] sigmoid 200 2019 Parallel FCM[104] sigmoid 1 000 2023
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表 4 模糊认知图学习算法的应用文献总结
Table 4 Literature review on the application of fuzzy cognitive map learning algorithms
类别 应用领域 文献 专家知识驱动的方法 模式分类 [55] 前列腺癌诊断 [105] 公司信用风险评估 [106] 自闭症预测 [107] 结构损伤检测 [108] 帕金森病预测 [110] 事故成因预测 [111] 裂纹严重程度分级 [112] 乳腺癌风险评估 [113] 自动学习算法 基因调控网络重建 [17, 75–77, 82, 84, 86] 多变量时间序列预测 [44−45, 78, 85, 116–119] 单变量时间序列预测 [83, 87, 89−91, 120−127, 140–143] 情景意识评估 [114] 病情趋势预测 [134] 前列腺癌预测 [135] 日需水量预测 [136] 电器能耗预测 [137] RFID物流操作评估 [138] 分类 [5, 128–133, 140] 半自动学习算法 医学诊断 [139] 甘蔗产量预测 [96] 决策支持 [92] 化学控制 [94] 太阳能发电 [97]
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表 5 模糊认知图建模工具对比
Table 5 Comparison of fuzzy cognitive map modeling tools
工具名称 受众定位 适用场景 应用形式 学习算法数量 图形页面 年份 FCM Modeler[144] 学术研究 静态建模, 群体决策 Java Applet 1 √ 1997 FCMappers.net[145] 学术研究 网络分析, 系统建模 网站 — — 2009 FCM Tool[146] 商业产品, 学术研究 决策支持, 系统建模 软件 1 √ 2011 FCM Designer[147] 学术研究 系统建模 Java Applet — √ 2010 FCM Designer Version 2.0[148] 学术研究 医学诊断, 推荐系统建模 Java Applet — √ 2016 Mental Modeler[149] 商业产品, 学术研究 群体决策, 系统建模 Web 页面 — √ 2013 JFCM[150] 教学工具, 学术研究 系统建模 Java开源库 — — 2014 ISEMK[152] 商业产品, 学术研究 决策支持, 时间序列预测 — 6 √ 2015 FCM Expert[154] 学术研究 决策支持, 系统建模 Java软件 4 √ 2017 FCMpy[158] 学术研究 系统建模 开源Python模块 5 √ 2022 注: “—”表示“无”或者未查询到.
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