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              基于多目標PSO混合優化的虛擬樣本生成

              王丹丹 湯健 夏恒 喬俊飛

              王丹丹, 湯健, 夏恒, 喬俊飛. 基于多目標PSO混合優化的虛擬樣本生成. 自動化學報, 2024, 50(4): 790?811 doi: 10.16383/j.aas.c211091
              引用本文: 王丹丹, 湯健, 夏恒, 喬俊飛. 基于多目標PSO混合優化的虛擬樣本生成. 自動化學報, 2024, 50(4): 790?811 doi: 10.16383/j.aas.c211091
              Wang Dan-Dan, Tang Jian, Xia Heng, Qiao Jun-Fei. Virtual sample generation method based on hybrid optimization with multi-objective PSO. Acta Automatica Sinica, 2024, 50(4): 790?811 doi: 10.16383/j.aas.c211091
              Citation: Wang Dan-Dan, Tang Jian, Xia Heng, Qiao Jun-Fei. Virtual sample generation method based on hybrid optimization with multi-objective PSO. Acta Automatica Sinica, 2024, 50(4): 790?811 doi: 10.16383/j.aas.c211091

              基于多目標PSO混合優化的虛擬樣本生成

              doi: 10.16383/j.aas.c211091
              基金項目: 國家自然科學基金(62073006, 62173120, 62021003), 北京市自然科學基金資助項目(4212032, 4192009), 科技創新2030 —— “新一代人工智能”重大項目(2021ZD0112301, 2021ZD0112302)資助
              詳細信息
                作者簡介:

                王丹丹:北京工業大學信息學部碩士研究生. 主要研究方向為基于虛擬樣本生成的小樣本數據建模. E-mail: wangdandan@emails.bjut.edu.cn

                湯?。罕本┕I大學信息學部教授. 主要研究方向為小樣本數據建模, 城市固廢處理過程智能控制. 本文通信作者. E-mail: freeflytang@bjut.edu.cn

                夏恒:北京工業大學信息學部博士研究生. 主要研究方向為小樣本數據建模和城市固廢焚燒過程二噁英排放預測. E-mail: xiaheng@emails.bjut.edu.cn

                喬俊飛:北京工業大學信息學部教授. 主要研究方向為污水處理過程智能控制, 神經網絡結構設計與優化. E-mail: junfeiq@bjut.edu.cn

              Virtual Sample Generation Method Based on Hybrid Optimization With Multi-objective PSO

              Funds: Supported by National Natural Science Foundation of China (62073006, 62173120, 62021003), Beijing Natural Science Foundation (4212032, 4192009), and National Key Research and Development Program of China (2021ZD0112301, 2021ZD0112302)
              More Information
                Author Bio:

                WANG Dan-Dan Master student at the Faculty of Information Technology, Beijing University of Technology. Her main research interest is small sample data modeling based on virtual sample generation

                TANG Jian Professor at the Faculty of Information Technology, Beijing University of Technology. His research interest covers small sample data modeling and intelligent control of municipal solid waste treatment process. Corresponding author of this paper

                XIA Heng Ph.D. candidate at the Faculty of Information Technology, Beijing University of Technology. His research interest covers small sample data modeling and dioxin emission prediction of the municipal solid waste incineration process

                QIAO Jun-Fei Professor at the Faculty of Information Technology, Beijing University of Technology. His research interest covers intelligent control of waste water treatment process and structure design and optimization of neural networks

              • 摘要: 受限于檢測技術難度、高時間與經濟成本等原因, 難測參數的軟測量模型建模樣本存在數量少、分布稀疏與不平衡等問題, 嚴重制約了數據驅動模型的泛化性能. 針對以上問題, 提出一種基于多目標粒子群優化(Multi-objective particle swarm optimization, MOPSO)混合優化的虛擬樣本生成(Virtual sample generation, VSG)方法. 首先, 設計綜合學習粒子群優化算法的種群表征機制, 使其能夠同時編碼用于連續變量和離散變量; 然后, 定義具有多階段多目標特性的綜合學習粒子群優化算法適應度函數, 使其能夠在確保模型泛化性能的同時最小化虛擬樣本數量; 最后, 提出面向虛擬樣本生成的多目標混合優化任務以改進綜合學習粒子群優化算法, 使其能夠適應虛擬樣本優選過程的變維特性并提高收斂速度. 同時, 首次借鑒度量學習提出用于評價虛擬樣本質量的綜合評價指標和分布相似指標. 利用基準數據集和真實工業數據集驗證了所提方法的有效性和優越性.
              • 圖  1  虛擬樣本與真實樣本間的關系

                Fig.  1  Relationship between virtual samples and real samples

                圖  2  基于MOPSO混合優化的VSG策略

                Fig.  2  VSG based on hybrid optimization with MOPSO

                圖  3  基于混合優化策略的粒子設計

                Fig.  3  Particle design based on hybrid optimization strategy

                圖  4  非支配解的Pareto前沿

                Fig.  4  Pareto front of non-dominant solutions

                圖  5  非支配解的建模性能指標對比

                Fig.  5  Comparison of modeling performance indexes of non-dominant solutions

                圖  7  非支配解的分布相似度對比

                Fig.  7  Comparison of distribution similarity of non-dominant solutions

                圖  6  非支配解的綜合評價指標對比

                Fig.  6  Comparison of comprehensive evaluation indexes of non-dominant solutions

                圖  8  基準數據預測輸出對比

                Fig.  8  Comparison of prediction output for benchmark data

                圖  9  MSWI過程工藝流程圖

                Fig.  9  Flow chart of MSWI process

                圖  10  非支配解的Pareto前沿 —— DXN排放濃度

                Fig.  10  Pareto front of non-dominated solutions ——DXN emission concentration

                圖  11  非支配解的建模性能和綜合評價指標對比

                Fig.  11  Comparison of modeling performance indexes and comprehensive evaluation indexes of non-dominant solutions

                表  1  本文采用符號的含義

                Table  1  The meaning of the symbols used in this article

                序號 符號 含義
                1 ${\rho _i}$ 全局最優粒子選擇指標
                2 ${\rho _j}$ 虛擬樣本綜合評價指標
                3 $\eta $ 數據分布相似度
                4 ${{\boldsymbol{F}}}\left( {{\boldsymbol{z}}} \right)$ 多目標優化問題的目標函數集
                5 ${ {\boldsymbol{z} } }, {\boldsymbol{z}}_n^p\left( {t + 1} \right)$ 優化問題決策變量(粒子的位置矢量), 表示第$t + 1$次迭代時, 粒子$p$的第$n$維位置值
                6 ${ {\boldsymbol{v} } }, {\boldsymbol{v} }_n^p( {t + 1} )$ 粒子的速度矢量, 表示第$t + 1$次迭代時, 粒子$p$的第$n$維速度值
                7 ${w_{{{\rm{inertia}}}}}$ 粒子速度更新的慣性權重
                8 ${{{\boldsymbol{d}}}^p}\left( {t + 1} \right)$ 第$t + 1$次迭代時, 粒子$p$的個體最優
                9 $E_n^p$ 粒子$p$的第$n$維的學習樣例值
                10 ${N_{{{\rm{refresh}}}}}$ 個體最優未更新閾值, 用于控制學習樣例的更新
                11 $P_c^p$ 粒子$p$的學習概率, 用于控制學習樣例的更新概率
                12 $ran{k^p}$ 粒子$p$個體最優的適應度在種群中排名
                13 $K$ RF模型中決策樹數量
                14 ${L_F}$ RF模型中切分特征數
                15 ${\theta _{{{\rm{leaf}}}}}$ RF模型中決策樹的葉節點包含樣本數量的閾值
                16 $F_{_{{{\rm{sel}}}}}^q$ RF模型中決策樹的節點$q$最佳切分特征
                17 ${s^q}$ RF模型中決策樹的節點$q$最佳分裂點取值
                18 $f_{_{{{\rm{tree}}}}}^k\left( \cdot \right)$ RF模型中第$k$個決策樹模型
                19 $f_{_{{{\rm{RF}}}}}^{}\left( \cdot \right)$ RF 模型
                20 ${{\boldsymbol{z}}}_{_{{{\rm{para}}}}}^{}$ 指導候選虛擬樣本生成的參數決策變量
                21 ${{\boldsymbol{z}}}_{_{{{\rm{vss}}}}}^{}$ 篩選候選虛擬樣本選擇決策變量
                22 $\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{train}}}}} $ 原始小樣本訓練集
                23 ${ { {\boldsymbol{x} } }_{ { {\rm{vsg\text{-}min} } } }}, { { {\boldsymbol{x} } }_{ { {\rm{vsg\text{-}max} } } }}$ 采用改進MTD進行擴展后的輸入擴展域的上限和下限
                24 ${y_{ { {\rm{vsg\text{-}min} } } }}, {y_{ { {\rm{vsg\text{-}max} } } }}$ 采用改進MTD進行擴展后的輸出擴展域的上限和下限
                25 $\mathop { {\boldsymbol{X} } }\nolimits_{_{ { {\rm{vs\text{-}g} } } }}$ 混合插值生成的虛擬樣本輸入
                26 $\mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{equal}}}}} , \mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{rand}}}}} $ 等間隔插值、隨機插值生成的虛擬樣本輸入
                27 $\mathop { {\boldsymbol{y} } }\nolimits_{_{ { {\rm{vs\text{-}g1} } } } } , \mathop { {\boldsymbol{y} } }\nolimits_{_{ { {\rm{vs\text{-}g2} } } } }$ 基于虛擬樣本輸入, 結合RF、RWNN映射模型生成的虛擬樣本輸出
                28 ${ {\boldsymbol{R} } }_{ { {\rm{vs\text{-}g1} } } }^p, { {\boldsymbol{R} } }_{ { {\rm{vs\text{-}g2} } } }^p$ 基于虛擬樣本輸入, 結合RF、RWNN 映射模型生成的虛擬樣本
                29 $\mathop { {\boldsymbol{R} } }\nolimits_{_{ { {\rm{vs\text{-}g} } } }}$ 生成的混合虛擬樣本
                30 $\mathop { {\boldsymbol{R} } }\nolimits_{_{ { {\rm{vs\text{-}d} } } } }$ 對$\mathop { {\boldsymbol{R} } }\nolimits_{_{ { {\rm{vs\text{-}g} } } }}$進行刪減后的候選虛擬樣本
                31 $\mathop { {\boldsymbol{R} } }\nolimits_{_{ { {\rm{vs\text{-}s} } } }}$ 對候選虛擬樣本進行選擇后獲得的虛擬樣本
                32 $\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{valid}}}}} $ 原始小樣本驗證集
                33 $\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{vs}}}}} $ 最優虛擬樣本
                34 ${f_{{{\rm{num}}}}}({{\boldsymbol{z}}})$ 多目標優化問題的目標之一, 篩選后的虛擬樣本數量
                35 ${f_{{{\rm{mod}}}}}({{\boldsymbol{z}}})$ 多目標優化問題的目標之一, 篩選后的虛擬樣本與原始訓練集構建RF模型的性能指標
                36 $z_{{\rm{MTD}}}$ 粒子的參數決策變量之一, 對應基于MTD方法的擴展率${\gamma _{{{\rm{extend}}}}}$
                37 $z_{{{\rm{RF}}}}^{{\rm{1}}}$ 粒子的參數決策變量之一, 對應RF映射模型的切分特征數${L_F}$
                38 $z_{{{\rm{RF}}}}^{{\rm{2}}}$ 粒子的參數決策變量之一, 對應RF映射模型中決策樹的中葉節點包含樣本數量的閾值${\theta _{{{\rm{leaf}}}}}$
                39 ${z_{{{\rm{RWNN}}}}}$ 粒子的參數決策變量之一, 對應RWNN映射模型的隱含層神經元數量$I$
                40 ${\gamma _{{{\rm{extend}}}}}$ 基于MTD方法的擴展率
                41 $I$ RWNN映射模型的隱含層神經元數量
                42 $\mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{train}}}}} $ 原始小樣本訓練集輸入
                43 ${{{\boldsymbol{y}}}_{{{\rm{train}}}}}$ 原始小樣本訓練集輸出
                44 ${y_{{{\rm{ave}}}}}$ ${{{\boldsymbol{y}}}_{{{\rm{train}}}}}$的均值
                45 ${{{\boldsymbol{y}}}_{{{\rm{high}}}}}, {{{\boldsymbol{y}}}_{{{\rm{low}}}}}$ $\mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{train}}}}} $中大于/小于${y_{{{\rm{ave}}}}}$的輸出集合
                46 ${y_{{{\rm{max}}}}}, {y_{{{\rm{min}}}}}$ ${{{\boldsymbol{y}}}_{{{\rm{train}}}}}$中最大值、最小值
                47 ${y_{ { {\rm{H\text{-}ave} } } } }, {y_{ { {\rm{L\text{-}ave} } } } }$ ${{{\boldsymbol{y}}}_{{{\rm{high}}}}}, {{{\boldsymbol{y}}}_{{{\rm{low}}}}}$的均值
                48 $\mathop N\nolimits_{_{{{\rm{equal}}}}} , \mathop N\nolimits_{_{{{\rm{rand}}}}} $ 等間隔插值、隨機插值倍數
                49 ${{\boldsymbol{W}}}, {{\boldsymbol}}$ RWNN模型輸入層與隱含層間神經元的連接權重與偏置
                50 ${{\boldsymbol{H}}}_{}^{_{{{\rm{ori}}}}}$ RWNN模型隱含層輸出矩陣
                51 ${{\boldsymbol{\beta}}}$ RWNN模型隱含層與輸出層神經元的連接權重
                52 $\mathop N\nolimits_{_{ { {\rm{vs\text{-}g} } } } } \mathop {, N}\nolimits_{{ { {\rm{vs\text{-}d} } } } } , \mathop N\nolimits_{_{ { {\rm{vs\text{-}s} } } } }$ 生成、候選、選擇后虛擬樣本的數量
                53 ${\theta _{{{\rm{select}}}}}$ 虛擬樣本的選擇閾值
                54 ${{{\boldsymbol{\tilde z}}}_{{{\rm{vss}}}}}$ 對${{{\boldsymbol{z}}}_{{{\rm{vss}}}}}$進行變維度處理后獲得
                55 $F$ 使用虛擬樣本集$\mathop { {\boldsymbol{R} } }\nolimits_{_{ { {\rm{vs\text{-}s} } } }}$的建模性能指標
                56 $\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{mix}}}}} $ 原始訓練集$\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{train}}}}} $與$\mathop { {\boldsymbol{R} } }\nolimits_{_{ { {\rm{vs\text{-}s} } } }}$的混合樣本集
                57 ${P_{{{\rm{num}}}}}$ 種群中粒子數量
                58 ${N_{{{\rm{iter}}}}}$ 種群迭代次數
                59 ${{\boldsymbol{A}}}$ 種群的外部檔案, 保存非支配解
                下載: 導出CSV

                表  2  基準數據集劃分

                Table  2  Benchmark data set partitioning

                數據集 特征數 訓練集 驗證集 測試集 數據集編號
                數量 $\eta $ 數量 $\eta $ 數量 $\eta $
                混凝土抗壓強度 8 20 0.3327 20 0.3598 100 0.1255 A1
                40 0.2444 40 0.2628 A2
                60 0.1853 60 0.2070 A3
                超導臨界溫度 81 20 0.3351 20 0.3388 100 0.1538 B1
                40 0.2309 40 0.2423 B2
                60 0.1949 60 0.1966 B3
                下載: 導出CSV

                表  3  基準數據基于多目標PSO混合優化的VSG參數設定

                Table  3  Parameter setting of VSG based on hybrid optimization with multi-objective PSO for benchmark data

                數據集 ${P_{{{\rm{num}}}}}$ ${N_{{{\rm{iter}}}}}$ ${N_{{{\rm{refresh}}}}}$ $K$ ${z_{{{\rm{MTD}}}}}$ $z_{{{\rm{RF}}}}^{{\rm{1}}}$ $z_{{{\rm{RF}}}}^{{\rm{2}}}$ ${z_{{{\rm{RWNN}}}}}$
                混凝土抗壓強度 30 30 3 30 (0, 1) (1, 6) (2, 10) (3, 20)
                超導臨界溫度 30 30 3 50 (0, 1) (1, 30) (2, 10) (3, 20)
                下載: 導出CSV

                表  4  基準數據基于多目標PSO混合優化獲得的最優虛擬樣本

                Table  4  Optimal virtual samples obtained based on multi-objective PSO hybrid optimization for benchmark data

                數據集 ${\mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{vs}}}}} }$ ${y_{{{\rm{vs}}}}}$
                A1 396.50 117.40 0 176.40 11.42 876.70 796.90 60.23 58.83
                200.50 16.35 115.80 161.60 8.27 1071.70 809.90 17.23 29.23
                240.90 0 100.30 183.50 5.87 977.30 852.40 14.00 18.25
                272.40 56.58 0 199.00 0 965.00 786.90 37.38 12.62
                347.40 0 0 190.80 0 1116.40 718.20 15.08 3.42
                B1 5.69 95.64 60.78 69.89 36.85 1.48 1.41 182.20 26.79
                4.08 77.39 51.82 60.19 35.09 1.22 1.27 121.40 95.32
                4.00 76.44 50.35 59.37 34.71 1.20 1.29 121.30 80.12
                4.46 82.72 56.99 64.52 36.03 1.30 1.09 131.20 51.89
                3.54 83.97 60.06 66.37 43.11 1.07 0.97 99.90 6.38
                下載: 導出CSV

                表  5  基準數據原始樣本輸入/輸出范圍

                Table  5  Input/output range of original samples for benchmark data

                數據集 輸入 輸出
                A1 最小值 102.0 0 0 121.8 0 801.0 594.0 1.0 2.3
                最大值 540.0 359.4 200.1 247.0 32.2 1145.0 992.6 365.0 82.6
                B1 最小值 1.0 6.9 6.4 5.3 2.0 0 0 0 0
                最大值 9.0 209.0 209.0 209.0 209.0 2.0 2.0 208.0 185.0
                下載: 導出CSV

                表  6  基準數據基于多目標PSO混合優化的全局最優解的統計結果

                Table  6  Statistical results of global optimal solution based on hybrid optimization with multi-objective PSO for benchmark data

                數據集 超參數 虛擬樣本數量 驗證集 測試集 混合樣本$\eta $
                ${\gamma _{{{\rm{extend}}}}}$ ${L_F}$ ${\theta _{{{\rm{leaf}}}}}$ $I$ 平均${\rm{RMSE}}$ 平均$\rho $ 平均${\rm{RMSE}}$ 平均$\rho $
                A1 0.6033 3 9 18 82 10.36 0.026 11.59 0.012 0.2354
                A2 0.6245 6 5 19 128 10.03 0.012 10.73 0.003 0.2099
                A3 0.6528 6 9 20 150 10.40 0.006 10.28 0.002 0.2002
                B1 0.3951 5 5 16 20 16.44 0.300 19.07 0.169 0.2407
                B2 0.4892 8 6 14 69 20.14 0.019 17.86 0.051 0.2118
                B3 0.6775 19 6 15 70 19.57 0 18.05 0.023 0.2076
                下載: 導出CSV

                表  7  基準數據不同VSG方法的對比統計結果

                Table  7  Comparative statistical results of different VSG methods for benchmark data

                數據集 方法 虛擬樣本數量 混合樣本$\eta $ 測試${\rm{RMSE} }$ 測試$\rho $
                均值 方差 最優 均值$(\times{10^{ - 3} })$ 方差$(\times{10^{ - 4} })$ 最優$(\times{10^{ - 3} })$
                A1 N-VSG 219 0.2770 16.47 8.785 14.11 4.09 15.44 4.62
                M-VSG 238 0.3018 17.08 8.575 13.65 2.26 19.73 4.55
                PSO-VSG 55 0.4235 16.35 3.822 12.75 3.76 30.20 5.88
                MP-VSG 165 0.2641 14.03 4.525 12.93 6.04 9.93 7.19
                MoHo-VSG 82 0.2354 11.59 0.107 9.67 12.46 1.34 14.72
                B1 N-VSG 176 0.2945 24.38 10.541 21.96 13.87 17.96 14.25
                M-VSG 281 0.3100 25.33 12.786 20.12 12.63 56.11 14.12
                PSO-VSG 36 0.3317 26.11 17.710 20.38 1.69 71.20 8.23
                MP-VSG 134 0.2513 20.84 3.452 19.47 17.43 4.37 18.89
                MoHo-VSG 20 0.2076 18.05 0.062 17.84 169.26 1.57 178.69
                下載: 導出CSV

                表  8  DXN數據基于多目標PSO混合優化的VSG算法參數設定

                Table  8  Parameter setting of VSG algorithm based on multi-objective PSO hybrid optimization for DXN data

                參數 ${P_{{{\rm{num}}}}}$ ${N_{{{\rm{iter}}}}}$ ${N_{{{\rm{refresh}}}}}$ $K$ ${z_{{{\rm{MTD}}}}}$ $z_{{{\rm{RF}}}}^{{\rm{1}}}$ $z_{{{\rm{RF}}}}^{{\rm{2}}}$ ${z_{{{\rm{RWNN}}}}}$
                數據 30 30 3 50 (0, 1) (1, 35) (2, 10) (3, 20)
                下載: 導出CSV

                表  9  DXN數據基于多目標PSO混合優化獲得的最優虛擬樣本

                Table  9  Optimal virtual samples obtained based on multi-objective PSO hybrid optimization for DXN data

                ${\mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{vs}}}}} }$ ${y_{{{\rm{vs}}}}}$
                4.366 1.54 68.78 27.31 241.4 3.96 334.7 0.0289
                4.206 0 68.94 28.15 222.5 3.77 306.8 0.0458
                4.449 7.69 72.48 30.23 222.8 3.98 315.8 0.0685
                4.432 10.00 71.83 30.00 225.9 3.99 319.5 0.0163
                4.461 17.69 74.65 30.77 228.5 3.99 321.8 0.0029
                下載: 導出CSV

                表  10  DXN數據面向VSG的多目標PSO混合優化全局最優解

                Table  10  DXN data for VSG-oriented multi-objective PSO hybrid optimization global optimal solution

                性能指標 最優解
                超參數${\gamma _{{{\rm{extend}}}}}$ 0.1206
                超參數${L_F}$ 2
                超參數${\theta _{{{\rm{leaf}}}}}$ 5
                超參數$I$ 15
                虛擬樣本數量 40
                驗證集的平均${\rm{RMSE}}$ 0.0231
                驗證集的平均$\rho $ 4.41 ×${10^{ - 5}}$
                測試集的平均${\rm{RMSE}}$ 0.0238
                測試集的平均$\rho $ 3.18 ×${10^{ - 5}}$
                驗證集, 小樣本建模的${\rm{RMSE}}$ 0.0259
                測試集, 小樣本建模的${\rm{RMSE}}$ 0.0251
                下載: 導出CSV

                表  11  DXN數據的不同VSG方法對比統計結果

                Table  11  Comparative statistical results of different VSG methods based on DXN dataset

                方法 虛擬樣本
                數量
                測試集的${\rm{RMSE} }$ 測試集的$\rho $
                均值 方差$(\times {10^{ - 4} })$ 最優 均值$(\times {10^{ - 5} })$ 方差 最優$(\times{10^{ - 5} })$
                N-VSG 129 0.0406 0.695 0.0262 0.19 1.94 ×${10^{ - 5}}$ 0.36
                M-VSG 116 0.0403 1.331 0.0231 0.26 8.83 ×${10^{ - 5}}$ 0.53
                PSO-VSG 27 0.0328 0.519 0.0245 0.56 8.44 ×${10^{ - 5}}$ 1.02
                MP-VSG 68 0.0377 1.208 0.0218 1.04 5.16 ×${10^{ - 7}}$ 1.78
                MoHo-VSG 40 0.0231 0.691 0.0220 3.18 4.47 ×${10^{ - 9}}$ 3.45
                下載: 導出CSV
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