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

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

王丹丹^{1, 2, 3,},
湯健^{1, 2, 3,},
夏恒^{1, 2, 3,},
喬俊飛^{1, 2, 3,}

1.
北京工業大學信息學部北京 100124
2.
北京工業大學智慧環保北京實驗室北京 100124
3.
北京工業大學智能感知與自主控制教育部工程研究中心北京 100124

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

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出版歷程
- 收稿日期: 2021-11-18
- 錄用日期: 2022-05-11
- 網絡出版日期: 2022-09-13
- 刊出日期: 2024-04-26

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

WANG Dan-Dan^{1, 2, 3
,},
TANG Jian^{1, 2, 3
,},
XIA Heng^{1, 2, 3
,},
QIAO Jun-Fei^{1, 2, 3
,}

1.
Faculty of Information Technology, Beijing University of Technology, Beijing 100124
2.
Beijing Laboratory of Smart Environmental Protection, Beijing University of Technology, Beijing 100124
3.
Engineering Research Center of Intelligent Perception and Autonomous Control, Ministry of Education, Beijing University of Technology, Beijing 100124

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

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.
- 小樣本建模 /
- 虛擬樣本生成 /
- 混合優化 /
- 多目標粒子群優化 /
- 分布相似度
Abstract: Due to the difficulty of detection technology, and high time and economic cost, the modeling samples of soft-sensing model with difficult parameters have some problems, such as small numbers, sparse distribution, and imbalance, which seriously restrict the generalization performance of data-driven models. To solve the above problems, a virtual sample generation (VSG) method based on multi-objective particle swarm optimization (MOPSO) hybrid optimization is proposed. First, the population representation mechanism of the integrated learning particle swarm optimization algorithm is designed, so that it can simultaneously encode the continuous and the discrete variables. Then, the fitness function of the integrated learning particle swarm optimization algorithm with multi-stage and multi-objective characteristics is defined to minimize the number of virtual samples while ensuring the generalization performance of the model. Finally, a multi-objective hybrid optimization task is generated for virtual samples to improve the integrated learning particle swarm optimization algorithm, so that it can adapt to the variable dimension characteristics of the virtual sample optimization process and improve the convergence speed. At the same time, the comprehensive evaluation index and distribution similarity index are proposed for evaluating the quality of virtual samples by referring to metric learning for the first time. In this paper, two benchmark datasets and an actual industrial dataset are used to verify the effectiveness and superiority of the proposed method.
- Small sample modeling /
- virtual sample generation (VSG) /
- hybrid optimization /
- multi-objective particle swarm optimization (MOPSO) /
- distribution similarity

HTML全文

圖 1 虛擬樣本與真實樣本間的關系

Fig. 1 Relationship between virtual samples and real samples

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圖 2 基于MOPSO混合優化的VSG策略

Fig. 2 VSG based on hybrid optimization with MOPSO

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圖 3 基于混合優化策略的粒子設計

Fig. 3 Particle design based on hybrid optimization strategy

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圖 4 非支配解的Pareto前沿

Fig. 4 Pareto front of non-dominant solutions

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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

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圖 7 非支配解的分布相似度對比

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

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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

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圖 8 基準數據預測輸出對比

Fig. 8 Comparison of prediction output for benchmark data

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圖 9 MSWI過程工藝流程圖

Fig. 9 Flow chart of MSWI process

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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

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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

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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

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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
數據集	特征數	數量	$\eta $	數量	$\eta $	數量	$\eta $	數據集編號
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

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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)

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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

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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
A1	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
B1	最大值	9.0	209.0	209.0	209.0	209.0	2.0	2.0	208.0	185.0

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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 $
數據集	${\gamma _{{{\rm{extend}}}}}$	${L_F}$	${\theta _{{{\rm{leaf}}}}}$	$I$	虛擬樣本數量	平均${\rm{RMSE}}$	平均$\rho $	平均${\rm{RMSE}}$	平均$\rho $	混合樣本$\eta $
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

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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 $
數據集	方法	虛擬樣本數量	混合樣本$\eta $	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

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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)

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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

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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

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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 $
方法	虛擬樣本數量	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

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