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摘要: 模糊認知圖(Fuzzy cognitive map, FCM)是建立在認知圖和模糊集理論上的一類代表性的軟計算理論, 兼具神經網絡和模糊決策兩者的優勢, 已成功地應用于復雜系統建模和時間序列分析等眾多領域. 學習權重矩陣是基于模糊認知圖建模的首要任務, 是模糊認知圖研究領域的焦點. 針對這一核心問題, 首先, 全面綜述模糊認知圖的基本理論框架, 系統地總結近年來模糊認知圖的拓展模型. 其次, 歸納、總結和分析模糊認知圖學習算法的最新研究進展, 對學習算法進行重新定義和劃分, 深度闡述各類學習算法的時間復雜度和優缺點. 然后, 對比分析各類學習算法在不同科學領域的應用特點以及現有的模糊認知圖建模軟件工具. 最后, 討論學習算法未來潛在的研究方向和發展趨勢.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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圖 4 專家知識驅動的學習算法的基本流程
Fig. 4 The basic process of expert knowledge-driven learning algorithms
表 1 拓展認知圖模型對比
Table 1 Comparison of extension cognitive map models
類別 名稱 特點 優點 缺點 應用領域 基于不同
模糊理論
的拓展
認知圖RBFCM[28] 引入模糊進位累加器計算因果權重 涵蓋多種概念關系并具有
多功能性和簡單性建模要求高 決策支持 FGCM[29] 引入灰色數衡量因果強度 建模概念之間的不確定信息 推理復雜 可靠性工程 IFCM[30] 利用直覺模糊集建模因果關系 衡量了因果關系中的不確定性 推理復雜 時間序列預測 IVFCM[31] 利用區間值描述因果關系的強度 考慮了非結構化環境相關的不確定性 依賴專家知識 決策支持, 時間序列
預測模型ECM[32] 在因果推理中融入了證據理論 既能表示不確定性又能進行知識融合 依賴專家知識,
推理復雜決策支持 RCN[33] 利用粗糙集表示因果關系 解決了不確定情況下的決策問題 依賴專家知識 決策支持軟件,
可靠性評估FRCN[34] 利用模糊粗糙結構構建神經網絡 建模了因果關系的不確定性 推理復雜 決策支持 zT2FSs-FCM[35] 引入二型模糊集建模節點間的
因果權重捕獲了概念間的不確定性關系 推理復雜 系統評估 面向動態
系統建模
的拓展
認知圖DCN[37] 考慮了因果關系的時變性 結構上具有更高的可擴展性和靈活性 依賴拉普拉斯框架,
建模復雜度高決策支持 DRFCM[39] 推理過程中引入非線性動態函數 能夠捕獲動態因果關系,
具有自適應性建模要求高 風險評估, 決策支持 FTCM[40] 考慮了因果關系強度和時間滯后性 能夠隨時間推移分析系統的動態行為 建模復雜 時間序列預測 E-FCM[41] 采用檢查機制模擬動態因果關系 能夠自我進化適應不斷發展的行為 計算耗時 動態場景建模 HFCM[42] 考慮了復雜系統建模過程中的
多階動態性準確地描述了系統行為 隨著階數增加,
計算復雜度增加時間序列預測 TAFCM[43] 引入了定時自動機理論建模
系統的時間粒度推理過程具有動態性和自適應性 建模要求高 人類情緒建模 DFCM[44] 嵌入在深度神經網絡的框架中 構建可解釋預測器, 挖掘隱藏的
因果關系訓練耗時, 容易面臨
“數據饑餓”問題時間序列預測 AFCM[45] 構建基于趨勢的信息粒引入
自適應更新機制自適應權重長期預測 計算耗時 時間序列預測 
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表 2 基于學習范式的模糊認知圖學習算法分類
Table 2 Classification of fuzzy cognitive map learning algorithms based on the learning paradigm
類別 學習方法 時間復雜度 優點 缺點 作者 發表年份 專家知識
驅動的方法DHL[47] ${\rm{O}}(N^2)$ 簡單, 易操作 只考慮了當前的一對概念 Dickerson等 1994 BDA[48] ${\rm{O}}(N^2)$ 考慮多個概念的影響 只適用于二進制計算 Huerga 2002 AHL[49] ${\rm{O}}(N^2)$ 考慮了所有概念的影響 訓練耗時 Papageorgiou等 2004 NHL[50] ${\rm{O}}(N^2)$ 保留了原始的圖結構, 具有合理的物理解釋性 依賴專家標準 Papageorgiou等 2003 INHL[51] ${\rm{O}}(N^2)$ 避免陷入局部最小值 需要先驗知識 Li等 2004 DDNHL[52] ${\rm{O}}(N^2)$ 數據驅動 依賴專家知識 Stach 等 2008 帶終端約束的
NHL算法[53]${\rm{O}}(N^2)$ 提高結果的可行性 需要先驗知識 陳寧等 2016 FBN[54] ${\rm{O}}(N^2)$ 利用模糊因果規則推理 性能受激活參數的影響 Carvalho等 2007 基于bagging增強的NHL算法[55] ${\rm{O}}(N^2)$ 泛化性能較好 依賴專家知識 Papageorgiou等 2012 自動學習
算法GA[56] ${\rm{O}}(N^2)$ 數據驅動 受限于二進制編碼 Mateou等 2005 RCGA[57] ${\rm{O}}(N^2)$ 數據驅動, 實數編碼 參數尋優耗時 Stach 等 2005 PSO[58?59] ${\rm{O}}(N!)$ 數據驅動, 元啟發式算法 依賴專家知識 Parsopoulos 等Oikonomou 等 2003
2013SOMA[60] ${\rm{O}}(N^2)$ 數據驅動 計算耗時 Va??ák 2010 ACO[61] ${\rm{O}}(N^2)$ 概率型算法魯棒性強 計算耗時, 容易早熟收斂 Chen等 2012 ABC[62] ${\rm{O}}(N^2)$ 數據驅動 參數尋優耗時 Yesil等 2013 ICA[63] ${\rm{O}}(N^3)$ 數據驅動 計算復雜, 耗時 Ahmadi 等 2015 DE[64] ${\rm{O}}(N^2)$ 容易理解, 計算簡單 易局部收斂 Juszczuk等 2009 SA[65?66] ${\rm{O}}(N^2)$ 計算簡單 參數尋優耗時 Ghazanfari 等
Alizadeh等2007
2009BB-BC[67] ${\rm{O}}(N^2)$ 算法簡單, 泛化能力較好 不適用于解決高維問題 Yesil等 2010 CA[68] ${\rm{O}}(N^2)$ 全局搜索與局部搜索結合 參數尋優耗時, 對問題的依賴性強 Ahmadi等 2014 基于互信息的
模因算法[70]${\rm{O}}(N^2)$ 適用于大規模圖學習 無法在搜索過程中
關注圖的稀疏性Zou等 2018 MARO[71] ${\rm{O}}(N^2)$ 只需調用一次目標
函數, 無需設置參數計算復雜, 易陷入局部最優 Salmeron等 2019 分解RCGA[72?73] ${\rm{O}}(N^2)$ 分解并行計算 計算復雜 Chen等, Stach等 2015, 2010 D&C RCGA[74] ${\rm{O}}(N^2)$ 可并行計算并具有可擴展性 隨著圖的大小和處理器數量增加, 算法性能下降 Stach等 2007 dMAGA[75] ${\rm{O}}(N^2)$ 適用于大規模圖學習
具有魯棒性受 FCM 節點的取值范圍限制, 需在算法執行前進行數據歸一化 Liu等 2015 MA-NN[76] ${\rm{O}}(N^2)$ 分布式計算框架適用于
大規模網絡重建受FCM節點的取值范圍限制, 需在算法執行前進行數據歸一化 Chi等 2019 MOEA[77, 79?80] ${\rm{O}}(N^2)$ 多目標進化考慮了圖的稀疏性 不適用于大規模圖學習 Liu等, Poczeta 等,
Chi等2019, 2018, 2016 IMFPSO[78] ${\rm{O}}(N!)$ 優化過程考慮了知識遷移 算法易早熟, 過早收斂 Liang等 2022 SRCGA[15] ${\rm{O}}(N^2)$ 考慮了圖的稀疏性 不適用于處理大規模數據 Stach等 2012 MMMA[17] ${\rm{O}}(N^2)$ 多圖優化知識轉移 有可能發生負信息遷移, 導致收斂速度緩慢 Shen等 2020 CS[81] ${\rm{O}}(N^3)$ 適用于大規模稀疏圖學習 參數尋優耗時 Wu等 2017 內點法[82] ${\rm{O}}(N^4)$ 精度高, 可擴展性好 對初值敏感, 難以處理
不等式約束問題Lu等 2020 約束優化[83] ${\rm{O}}(N^3)$ 考慮了矩陣分布具有抗噪能力 僅適用于有監督學習 Feng等 2021 近似梯度下降[84] ${\rm{O}}(N^3)$ 適用于解決大規模數據問題 對初始點敏感, 可能
陷入局部最優Ding等 2021 Moore-Penrose逆[85] ${\rm{O}}(N^3)$ 參數較少 計算復雜 Vanhoenshoven等 2020 Lasso回歸[86] ${\rm{O}}(N^3)$ 考慮了圖的稀疏性, 適用于
大規模圖學習可能出現過擬合 Wu等 2016 嶺回歸[87] ${\rm{O}}(N^3)$ 泛化性能較好, 適用于
大規模圖學習對特征的縮放敏感 Yang等 2018 彈性網絡回歸[88] ${\rm{O}}(N^3)$ 增加了L1 和L2 正則化, 適用于大規模圖學習 參數調節困難 Shen等 2020 支持向量回歸[89] ${\rm{O}}(N^4)$ 適用于高維非線性數據 對缺失數據敏感 Gao等 2020 貝葉斯嶺回歸[90] ${\rm{O}}(N^4)$ 簡單、模型適應性較強 對模型的假設較多依賴先驗分布 Liu等 2020 FTRL[91] ${\rm{O}}(N^3)$ 在線學習 計算、推理過程復雜 Wu 等 2021 半自動學習算法 DE+NHL[92] ${\rm{O}}(N^2)$ 進化過程中保留了圖的物理意義 依賴專家知識 Papageorgiou等 2005 RCGA+NHL[93] ${\rm{O}}(N^2)$ 利用了遺傳算法的全局優化能力 受限于專家經驗 Zhu等 2008 PSO+NHL[94] ${\rm{O}}(N!)$ 避免人為因素產生的訓練誤差 受限于專家經驗 Yazdi等 2008 EGDA+NHL[95] ${\rm{O}}(N^2)$ 全局搜索, 參數少 受限于專家經驗 Ren 2012 DDNHL+GA[96] ${\rm{O}}(N^3)$ 數據驅動分類推理能力強 受限于專家經驗 Natarajan等 2016 RCGA+DE+
梯度下降[97]${\rm{O}}(N^2)$ 全局搜索 參數尋優耗時 Madeiro 等 2012 注: 時間復雜度為該算法更新一次FCM權重矩陣所需時間開銷, 未考慮數據量大小及最大迭代次數. N表示節點個數.
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表 3 大規模模糊認知圖學習算法分類
Table 3 Large-scale fuzzy cognitive map learning algorithm classification
類別 方法 轉換函數 最大FCM規模 發表年份 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 基于維度縮減的方法 MIMA[70] sigmoid 500 2018 文獻[100] sigmoid/tanh 25 2015 文獻[101] sigmoid 10 2018 基于分解的方法 CS[81] sigmoid 1 000 2017 MMMA[17] sigmoid/tanh 600 2020 內點法[82] sigmoid/tanh 200 2020 約束優化[83] sigmoid/tanh 200 2021 近似梯度下降[84] sigmoid 200 2021 Lasso回歸[86] sigmoid 500 2016 彈性網絡回歸[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 模糊認知圖學習算法的應用文獻總結
Table 4 Literature review on the application of fuzzy cognitive map learning algorithms
類別 應用領域 文獻 專家知識驅動的方法 模式分類 [55] 前列腺癌診斷 [105] 公司信用風險評估 [106] 自閉癥預測 [107] 結構損傷檢測 [108] 帕金森病預測 [110] 事故成因預測 [111] 裂紋嚴重程度分級 [112] 乳腺癌風險評估 [113] 自動學習算法 基因調控網絡重建 [17, 75–77, 82, 84, 86] 多變量時間序列預測 [44?45, 78, 85, 116–119] 單變量時間序列預測 [83, 87, 89?91, 120?127, 140–143] 情景意識評估 [114] 病情趨勢預測 [134] 前列腺癌預測 [135] 日需水量預測 [136] 電器能耗預測 [137] RFID物流操作評估 [138] 分類 [5, 128–133, 140] 半自動學習算法 醫學診斷 [139] 甘蔗產量預測 [96] 決策支持 [92] 化學控制 [94] 太陽能發電 [97]
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表 5 模糊認知圖建模工具對比
Table 5 Comparison of fuzzy cognitive map modeling tools
工具名稱 受眾定位 適用場景 應用形式 學習算法數量 圖形頁面 年份 FCM Modeler[144] 學術研究 靜態建模, 群體決策 Java Applet 1 √ 1997 FCMappers.net[145] 學術研究 網絡分析, 系統建模 網站 — — 2009 FCM Tool[146] 商業產品, 學術研究 決策支持, 系統建模 軟件 1 √ 2011 FCM Designer[147] 學術研究 系統建模 Java Applet — √ 2010 FCM Designer Version 2.0[148] 學術研究 醫學診斷, 推薦系統建模 Java Applet — √ 2016 Mental Modeler[149] 商業產品, 學術研究 群體決策, 系統建模 Web 頁面 — √ 2013 JFCM[150] 教學工具, 學術研究 系統建模 Java開源庫 — — 2014 ISEMK[152] 商業產品, 學術研究 決策支持, 時間序列預測 — 6 √ 2015 FCM Expert[154] 學術研究 決策支持, 系統建模 Java軟件 4 √ 2017 FCMpy[158] 學術研究 系統建模 開源Python模塊 5 √ 2022 注: “—”表示“無”或者未查詢到.
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