2.793

2018影响因子

(CJCR)

  • 中文核心
  • EI
  • 中国科技核心
  • Scopus
  • CSCD
  • 英国科学文摘

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

结构辨识和参数优化协同学习的概率TSK模糊系统

顾晓清 倪彤光 张聪 戴臣超 王洪元

顾晓清, 倪彤光, 张聪, 戴臣超, 王洪元.结构辨识和参数优化协同学习的概率TSK模糊系统.自动化学报, 2021, 47(2): 349-362 doi: 10.16383/j.aas.c180298
引用本文: 顾晓清, 倪彤光, 张聪, 戴臣超, 王洪元.结构辨识和参数优化协同学习的概率TSK模糊系统.自动化学报, 2021, 47(2): 349-362 doi: 10.16383/j.aas.c180298
Gu Xiao-Qing, Ni Tong-Guang, Zhang Cong, Dai Chen-Chao, Wang Hong-Yuan. Probabilistic TSK fuzzy system in the simultaneous learning of structure identiflcation and parameter optimization. Acta Automatica Sinica, 2021, 47(2): 349-362 doi: 10.16383/j.aas.c180298
Citation: Gu Xiao-Qing, Ni Tong-Guang, Zhang Cong, Dai Chen-Chao, Wang Hong-Yuan. Probabilistic TSK fuzzy system in the simultaneous learning of structure identiflcation and parameter optimization. Acta Automatica Sinica, 2021, 47(2): 349-362 doi: 10.16383/j.aas.c180298

结构辨识和参数优化协同学习的概率TSK模糊系统

doi: 10.16383/j.aas.c180298
基金项目: 

国家自然科学基金 61806026

国家自然科学基金 61572085

江苏省自然科学基金 BK20180956

江苏省教育科学"十三五"规划2018年度课题 Ba/2018/01/41

详细信息
    作者简介:

    顾晓清  常州大学信息科学与工程学院副教授. 2017年获江南大学博士学位.主要研究方向为模式识别和机器学习.E-mail:czxqgu@163.com

    张聪  常州大学信息科学与工程学院硕士研究生.主要研究方向为模糊理论.E-mail:18000052@smail.cczu.edu.cn

    戴臣超  常州大学信息科学与工程学院硕士研究生.主要研究方向为机器学习.E-mail:17000138@smail.cczu.edu.cn

    王洪元  常州大学信息科学与工程学院教授. 2004年获南京理工大学博士学位.主要研究方向为人工智能.E-mail:hywang@cczu.edu.cn

    通讯作者:

    倪彤光  常州大学信息科学与工程学院副教授. 2014年获江南大学博士学位.主要研究方向为人工智能.本文通信作者. E-mail:hbxtntg-12@163.com

  • 本文责任编委 许斌

Probabilistic TSK Fuzzy System in the Simultaneous Learning of Structure Identiflcation and Parameter Optimization

Funds: 

National Natural Science Foundation of China 61806026

National Natural Science Foundation of China 61572085

National Natural Science Foundation of Jiangsu Province BK20180956

Topics of the 13th Five-Year Plan for Educational Science in Jiangsu Province in 2018 Ba/2018/01/41

More Information
    Author Bio:

    GU Xiao-Qing   Associate professor at the School of Information Science and Engineering, Changzhou University. She received her Ph. D. degree from Jiangnan University in 2017. Her research interest covers pattern recognition and machine learning

    ZHANG Cong   Master student at the School of Information Science and Engineering, Changzhou University. His main research interest is fuzzy theory

    DAI Chen-Chao   Master student at the School of Information Science and Engineering, Changzhou University. His main research interest is machine learning

    WANG Hong-Yuan   Professor at the School of Information Science and Engineering, Changzhou University. He received his Ph. D. degree from Nanjing University of Science and Technology in 2004. His main research interest is artiflcial intelligence

    Corresponding author: NI Tong-Guang   Associated professor at the School of Information Science and Engineering, Changzhou University. He received his Ph. D. degree from Jiangnan University in 2014. His main research interest is artiflcial intelligence. Corresponding author of this paper.E-mail:hbxtntg-12@163.com
  • Recommended by Associate Editor XU Bin
  • 摘要: 传统Takagi-Sugeno-Kang (TSK)模糊系统的结构辨识和参数优化往往分阶段进行, 同时模糊规则数需要预先设定, 因此TSK模糊系统的逼近性能和解释性往往不理想.针对此问题, 提出了一种结构辨识和参数优化协同学习的概率TSK模糊系统(Probabilistic TSK fuzzy system, PTSK).首先, PTSK使用概率模型表示模糊回归系统, 将结构辨识和参数优化作为一个整体来考虑.其次, PTSK不借助于专家经验, 使用粒子滤波方法对规则数和前后件参数协同学习, 得到系统全部参数的最优解.实验结果表明, PTSK具有良好的逼近性能, 同时能获得较少的模糊规则数.
    Recommended by Associate Editor XU Bin
    1)  本文责任编委 许斌
  • 图  1  PTSK结构辨识和参数优化的协同学习示意图

    Fig.  1  The diagram of simultaneous learning of structure identiflcation and parameter optimization in PTSK

    图  2  PTSK在mexihat数据集上的实验结果

    Fig.  2  Experimental results of PTSK on mexihat dataset

    图  3  PTSK在mexihat数据集上得到的模糊集示意图

    Fig.  3  Fuzzy sets obtained by PTSK on mexihat dataset

    图  4  PTSK某次运行的收敛曲线

    Fig.  4  Convergence curves of PTSK at a certain simulation

    图  5  8种算法的Friedman检验结果

    Fig.  5  Friedman results of eight algorithms

    表  1  聚类法TSK模糊系统中常用的模糊规则前件/后件参数学习方法

    Table  1  The common learning methods for the antecedent/consequent parameters in the clustering based TSK fuzzy system

    前件参数学习
    FCM模糊聚类[5] 优点 获得的空间划分具有模糊性, 算法实现简单
    缺点 聚类数需要预先设定
    Gustafson-Kessel[13]和Gath-Geva聚类[14] 优点 均使用诱导矩阵识别数据集的结构
    缺点 矩阵计算量较大
    One-pass聚类[15] 优点 数据集只需要遍历一遍即可完成空间划分, 适用于增量或在线学习模式
    缺点 对凸型数据分布识别较差; 且与聚类数有关的参数需要预先设定
    后件参数学习
    最小二乘法[3, 8] 优点 最常用; 显式地得到后件参数的解析解, 计算简单
    缺点 对噪声数据敏感
    支持向量回归机[15] 优点 保证参数的全局最优解, 逼近性能较强
    缺点 二次规划问题求解的计算量较大
    进化计算[4] 优点 模拟自然随机优化算法, 不依赖对象的数学模型
    缺点 遗传编码的选择较难解决, 时间复杂度较高
    反向传播算法[16] 优点 在小数据集上很快达到局部最优解
    缺点 收敛较慢, 不适用于大规模数据集
    下载: 导出CSV

    表  2  数据集基本信息

    Table  2  Basic information of datasets

    数据集 规模 维数
    abalone 4 177 8
    anacalt 4 052 7
    autompg6 392 5
    autompg8 392 7
    bodyfat 252 14
    compactiv 8 192 21
    concrete 1 030 8
    dee 365 6
    delta-ail 7 129 5
    delta-elv 9 517 6
    diabert 43 2
    elevators 16 599 18
    friedman 1 200 5
    gc-s 56 000 6
    gc-x 56 000 6
    gc-p 56 000 6
    housing 506 13
    mexihat 2 500 2
    mg 1 385 6
    mortgage 1 049 15
    plastic 1 650 2
    pole 14998 26
    puma32h 8 192 32
    quake 2 178 3
    stock 950 9
    treasury 1 049 15
    wankara 1 609 9
    wizmir 1 461 9
    下载: 导出CSV

    表  3  算法参数设置

    Table  3  Parameter setting

    算法 参数设置
    L2-TSK-FS 模糊规则数$K\in \{{{2}^{2}}, {{3}^{2}}, \cdots, {{11}^{2}}\}$, 尺度参数$h\in \{{{0.2}^{2}}, {{0.4}^{2}}, \cdots, {{2}^{2}}\}$, 模糊指数$m=2$, 正则化参数$C\in\{{{2}^{-4}}, {{2}^{0}}, \cdots, {{2}^{7}}\}$.
    TSK-IRL-R 适应性函数最小匹配度= 1.5, 种群数= 61, 交叉概率= 0.1, 种群比例=0.2.
    MOGUL-R 正类样本的匹配度参数= 0.05, 负类样本的允许比例参数= 1.5, 适应性函数最小匹配度= 0.1, 种群数= 15.
    B-ZTSK-FS 模糊规则数$K\in \{{{2}^{2}}, {{3}^{2}}, \cdots, {{11}^{2}}\}$, 尺度参数$h\in \{{{0.2}^{2}}, {{0.4}^{2}}, \cdots, {{2}^{2}}\}$, 模糊指数$m=2$, 狄利克雷参数= 1 000.
    WM 标签数= 5.
    ENSEMBLE 隐层数= 2, 隐结点数= 15, 学习系数= 0.15, 动量系数= 0.1, 集成方法类型: BEM, 网络数= 10.
    PSVR 正则化参数$C\in \{{{10}^{-3}}, {{10}^{-2}}, \cdots, {{10}^{3}}\}$, 高斯核核参$\sigma \in\{{{10}^{-3}}, {{10}^{-2}}, \cdots, {{10}^{3}}\}$.
    PTSK 模糊指数$m=2$, 最大迭代次数${10}^{3}$, 阈值$\varepsilon =10^{-3}$, 收敛阈值$miter=50$, 稀疏参数$\beta \in \{1, 2, \cdots, 8\}$, 粒子数$P=10$.
    下载: 导出CSV

    表  4  8 种算法在 28 个数据集上的 MSE (标准差) 比较

    Table  4  MSE (Standard deviation) comparison of 8 algorithms on 28 datasets

    数据集 ENSEMBLE-R PSVR WM-R MOGUL-R TSK-IRL-R L2-TSK-FS B-ZTSK-FS PTSK
    mexihat 0.0205 0.0222 0.0219 0.0207 0.0231 0.0226 0.0239 $ \bf{0.0204}$
    1.004$\times {{10}^{-3}}$ 1.023$\times {{10}^{-3}}$ 1.011$\times {{10}^{-3}}$ 1.025$\times {{10}^{-3}}$ 1.017$\times {{10}^{-3}}$ 1.003$\times {{10}^{-3}}$ 1.020$\times {{10}^{-3}}$ 1.001$\times {{10}^{-3}}$
    abalone 4.1738 5.0107 6.6225 4.7323 5.8258 5.1987 4.1186 $ \bf{ 4.0810}$
    0.383 0.376 0.301 0.273 0.326 0.298 0.213 0.196
    anacalt 0.0531 0.0478 0.0595 0.0521 0.0829 0.0542 0.0496 $ \bf{0.0435 }$
    1.452$\times {{10}^{-3}}$ 1.023$\times {{10}^{-3}}$ 4.520$\times {{10}^{-3}}$ 1.102$\times {{10}^{-3}}$ 4.239$\times {{10}^{-3}}$ 2.358$\times {{10}^{-3}}$ 2.102$\times {{10}^{-3}}$ 1.404$\times {{10}^{-3}}$
    autompg6 11.8797 11.8827 13.3320 14.1473 $\bf{ 8.2574}$ 13.6705 12.3298 11.0676
    4.831 2.681 1.821 8.553 1.325 1.964 1.759 1.553
    autompg8 8.6182 8.2351 8.7713 9.6884 $ \bf{ 6.6799}$ 9.9872 8.8537 8.2122
    1.131 1.756 1.098 1.205 0.975 1.210 1.026 1.003
    bodyfat 5.1200$\times {{10}^{-4}}$ $ \bf{2.0201}\times{{10}^{-4}}$ 5.8424$\times {{10}^{-4}}$ 9.9749$\times{{10}^{-4}}$ 4.4523$\times {{10}^{-4}}$ 5.2704$\times{{10}^{-4}}$ 5.0125$\times {{10}^{-4}}$ 3.4402$\times{{10}^{-4}}$
    1.711$\times {{10}^{-5}}$ 2.310$\times {{10}^{-5}}$ 6.418$\times {{10}^{-5}}$ 6.528$\times {{10}^{-5}}$ 3.245$\times {{10}^{-5}}$ 4.718$\times {{10}^{-5}}$ 3.610$\times {{10}^{-5}}$ 2.036$\times {{10}^{-5}}$
    compactiv $ \bf{ 34.8932}$ 37.6408 35.7480 37.1190 37.4625 39.6401 38.3607 36.8515
    4.544 5.008 4.646 5.121 4.673 4.786 4.145 4.006
    concrete 52.6524 $ \bf{ 50.5003}$ 56.2245 55.1906 58.2542 58.8612 55.8678 50.5285
    3.535 4.388 6.541 3.325 3.875 3.764 3.757 3.400
    dee 0.2524 0.2627 $ \bf{0.2296 }$ 0.3034 0.6920 0.2956 0.2765 $ \bf{0.2294 }$
    0.024 0.032 0.086 0.184 0.085 0.030 0.027 0.021
    delta-elv 2.7764$\times {{10}^{-6}}$ 2.2719$\times{{10}^{-6}}$ 3.3007$\times {{10}^{-6}}$ 3.5924$\times{{10}^{-6}}$ $ \bf{ 1.9863\times {{10}^{-6}}}$ 4.3942$\times{{10}^{-6}}$ 3.5075$\times{{10}^{-6}}$ 2.1432$\times{{10}^{-6}}$
    6.892$\times {{10}^{-7}}$ 5.231$\times {{10}^{-7}}$ 3.390$\times {{10}^{-7}}$ 6.743$\times {{10}^{-7}}$ 4.721$\times {{10}^{-7}}$ 5.432$\times {{10}^{-7}}$ 5.121$\times {{10}^{-7}}$ 3.976$\times {{10}^{-7}}$
    delta-ail 3.5813$\times {{10}^{-8}}$ 3.5842$\times{{10}^{-8}}$ 5.7242$\times {{10}^{-8}}$ 3.6223$\times{{10}^{-8}}$ 2.9804$\times {{10}^{-8}}$ 3.7946$\times{{10}^{-8}}$ 5.1313$\times {{10}^{-8}}$ $ \bf{ 2.7236\times{{10}^{-8}}}$
    2.017$\times {{10}^{-9}}$ 4.654$\times {{10}^{-9}}$ 8.523$\times {{10}^{-9}}$ 7.487$\times {{10}^{-9}}$ 6.754$\times {{10}^{-9}}$ 5.987$\times {{10}^{-9}}$ 6.000$\times {{10}^{-9}}$ 3.003$\times {{10}^{-9}}$
    diabert 0.4932 0.5785 0.7266 0.6225 0.9208 1.1231 0.7334 $ \bf{0.4883 }$
    0.259 0.236 0.699 0.354 0.435 0.500 0.465 0.398
    elevator 2.5947$\times{{10}^{-4}}$ 2.5398$\times{{10}^{-4}}$ $ \bf{2.3237\times {{10}^{-4}} }$ 5.6023$\times{{10}^{-4}}$ 5.7413$\times{{10}^{-4}}$ 7.6498$\times{{10}^{-4}}$ 6.5429$\times {{10}^{-4}}$ 5.5531$\times{{10}^{-4}}$
    2.646$\times {{10}^{-5}}$ 1.658$\times {{10}^{-5}}$ 1.765$\times {{10}^{-5}}$ 2.991$\times {{10}^{-5}}$ 3.102$\times {{10}^{-5}}$ 3.832$\times {{10}^{-5}}$ 3.801$\times {{10}^{-5}}$ 3.124$\times {{10}^{-5}}$
    friedman 2.7852 2.2769 3.1595 2.1445 3.0082 3.0603 2.4206 $ \bf{ 2.1408}$
    0.352 0.615 0.371 0.201 0.308 0.312 0.311 0.353
    gc-s 0.5734 0.6024 0.4216 0.5015 0.2601 0.3267 0.2602 $\bf{ 0.2304}$
    0.064 0.078 0.060 0.078 0.012 0.038 0.014 0.010
    gc-x 4.6492$\times {{10}^{-3}}$ 4.9121$\times {{10}^{-3}}$ 4.8530$\times {{10}^{-3}}$ 4.8000$\times {{10}^{-3}}$ 3.5912$\times {{10}^{-3}}$ 3.8955$\times {{10}^{-3}}$ 3.4279$\times {{10}^{-3}}$ $ \bf{ 3.2687\times{{10}^{-4}}}$
    3.042$\times {{10}^{-5}}$ 3.550$\times {{10}^{-5}}$ 3.706$\times {{10}^{-5}}$ 3.743$\times {{10}^{-5}}$ 3.001$\times {{10}^{-5}}$ 3.330$\times {{10}^{-5}}$ 3.328$\times {{10}^{-5}}$ 2.004$\times {{10}^{-5}}$
    gc-p 0.0826 0.0980 0.0900 0.0900 0.0807 0.0856 0.0754 $\bf{0.0717 }$
    2.998$\times {{10}^{-3}}$ 3.005$\times {{10}^{-3}}$ 3.071$\times {{10}^{-3}}$ 3.026$\times {{10}^{-3}}$ 3.053$\times {{10}^{-3}}$ 3.251$\times {{10}^{-3}}$ 3.117$\times {{10}^{-3}}$ 3.010$\times {{10}^{-3}}$
    housing $ \bf{ 29.6062}$ 33.7403 34.8514 30.4763 34.9782 37.5164 34.0108 33.0672
    6.899 7.041 6.948 6.389 5.839 5.214 5.317 5.215
    mg 0.0203 0.0214 0.0179 0.0163 0.0166 0.0188 0.0176 $\bf{ 0.0157}$
    1.873$\times {{10}^{-3}}$ 1.431$\times {{10}^{-3}}$ 1.351$\times {{10}^{-3}}$ 1.572$\times {{10}^{-3}}$ 1.313$\times {{10}^{-3}}$ 1.082$\times {{10}^{-3}}$ 1.277$\times {{10}^{-3}}$ 1.139$\times {{10}^{-3}}$
    mortgage 0.0843 0.0448 0.0925 0.6160 0.0881 0.0589 0.0409 $ \bf{ 0.0407}$
    3.985$\times {{10}^{-3}}$ 2.751$\times {{10}^{-3}}$ 3.618$\times {{10}^{-3}}$ 1.916$\times {{10}^{-2}}$ 2.936$\times {{10}^{-3}}$ 2.643$\times {{10}^{-3}}$ 2.603$\times {{10}^{-3}}$ 2.517$\times {{10}^{-3}}$
    plastic 2.6657 2.3495 2.3646 2.3735 2.8642 2.9098 2.8477 $ \bf{ 2.2153}$
    0.401 0.098 0.446 0.110 0.100 0.218 0.231 0.200
    pole 206.5032 203.5998 229.1911 207.0983 233.7895 225.5987 216.9002 $ \bf{ 200.9751}$
    4.167 5.531 6.258 4.003 5.638 4.980 4.678 4.236
    puma 6.2417$\times {{10}^{-3}}$ $ \bf{ 2.3405\times{{10}^{-3}}}$ 4.6104$\times {{10}^{-3}}$ 5.8963$\times{{10}^{-3}}$ 4.3104$\times {{10}^{-3}}$ 4.7403$\times{{10}^{-3}}$ 4.7612$\times {{10}^{-3}}$ 3.7208$\times{{10}^{-3}}$
    7.835$\times {{10}^{-6}}$ 7.230$\times {{10}^{-6}}$ 7.200$\times {{10}^{-6}}$ 7.737$\times {{10}^{-6}}$ 6.875$\times {{10}^{-6}}$ 6.943$\times {{10}^{-6}}$ 7.032$\times {{10}^{-6}}$ 7.053$\times {{10}^{-6}}$
    quake 0.0591 $ \bf{ 0.0350}$ 0.0538 0.0371 0.0461 0.0570 0.0532 0.0356
    4.752$\times {{10}^{-3}}$ 1.995$\times {{10}^{-3}}$ 6.863$\times {{10}^{-3}}$ 2.700$\times {{10}^{-3}}$ 3.286$\times {{10}^{-3}}$ 3.274$\times {{10}^{-3}}$ 2.863$\times {{10}^{-3}}$ 2.965$\times {{10}^{-3}}$
    stock 1.5008 $ \bf{ 1.1681}$ 1.3863 1.6803 1.5430 1.6518 1.7626 1.1938
    0.321 0.477 0.459 0.287 0.300 0.348 0.372 0.265
    treasury 0.4287 0.4562 0.4199 0.5553 0.5421 0.4198 0.4568 $ \bf{ 0.4124}$
    0.102 0.021 0.122 0.099 0.078 0.063 0.067 0.066
    wankara 1.6955 2.0040 2.6569 2.8312 2.7434 1.8889 1.7313 $ \bf{1.6096 }$
    0.409 0.143 0.588 0.164 0.296 0.302 0.228 0.199
    wizmir 2.1475 1.6275 2.2321 2.2404 2.3343 1.9649 1.9772 $\bf{ 1.5245}$
    0.467 0.146 0.502 0.161 0.278 0.222 0.234 0.211
    下载: 导出CSV

    表  5  8种算法在28个数据集上的平均训练时间(s)的比较

    Table  5  Comparison of the average training time (s) of 8 algorithms on 28 datasets

    数据集 ENSEMBLE-R PSVR WM-R MOGUL-R TSK-IRL-R L2-TSK-FS B-ZTSK-FS PTSK
    mexihat 18.82 5.03 20.36 50.14 160.23 $ \bf{ 4.25 }$ 90.23 70.18
    abalone 21.31 $ \bf{ 3.00 }$ 28.22 5 238.68 7 192.52 3.17 51.89 40.35
    anacalt 36.94 $ \bf{ 2.31 }$ 27.58 1 006.54 60.48 2.45 22.31 18.96
    autompg6 8.38 $ \bf{ 1.37 }$ 7.14 320.26 70.69 1.89 8.15 6.38
    autompg8 8.55 $ \bf{ 1.55 }$ 8.01 309.65 392.45 1.81 20.64 9.10
    bodyfat 7.10 0.65 7.71 420.68 856.98 $ \bf{ 0. 52 }$ 6.69 5.98
    compactiv 44.88 $ \bf{ 28.38 }$ 98.39 4 124.81 4 792.17 30.34 729.93 640.63
    concrete 30.95 $ \bf{ 5.01 }$ 41.30 1 100.41 8 649.22 5.33 72.66 41.80
    dee 10.43 $ \bf{0.36 }$ 9.31 30.82 89.47 0.93 9.39 5.61
    delta-elv 29.47 $ \bf{ 3.56 }$ 35.72 965.03 7536.12 3.88 79.38 51.69
    delta-ail 30.10 $ \bf{ 5.78 }$ 27.26 1 167.60 2 546.60 6.06 109.63 69.61
    diabert 7.81 0.45 3.64 24.10 16.935 $ \bf{ 0.43 }$ 1.96 1.74
    elevator 186.64 $ \bf{ 30.07 }$ 181.18 3 508.54 3 286.65 100.64 828.80 559.20
    friedman 53.26 $ \bf{ 2.81 }$ 47.96 124.85 1 003.47 2.95 68.036 31.47
    gc-s 245.27 79.34 268.96 3 976.33 4 024.20 $ \bf{ 79.11}$ 964.45 520.76
    gc-x 252.85 78.02 270.82 4 034.87 4 035.65 $ \bf{ 78.00}$ 968.02 518.32
    gc-p 248.38 80.13 268.40 3 956.67 4 020.55 $ \bf{ 80.05 }$ 966.27 523.44
    housing 20.37 $ \bf{ 3.87 }$ 13.86 398.50 445.20 4.12 75.82 47.83
    mg 32.54 $ \bf{ 3.25}$ 23.54 786.92 765.32 3.56 60.43 23.67
    mortgage 20.15 $ \bf{ 3.88 }$ 22.16 1 039.27 1238.48 4.02 68.26 57.20
    plastic 22.74 $ \bf{ 4.79 }$ 23.58 305.20 268.09 5.07 120.3 78.64
    pole 96.36 $ \bf{ 40.86 }$ 720.44 6 290.63 8 533.63 42.75 1075.56 713.70
    puma 40.58 $ \bf{ 30.05 }$ 130.83 4 903.38 4 893.30 32.70 1073.12 796.57
    quake 28.57 $ \bf{2.54 }$ 19.80 360.25 400.39 2.97 63.92 18.48
    stock 19.43 $ \bf{ 3.50 }$ 25.32 1 490.34 2 175.56 3.89 78.46 22.46
    treasury 19.45 $ \bf{ 5.83 }$ 18.21 1 202.40 2 543.82 6.02 117.20 78.30
    wankara 28.76 $ \bf{ 4.38 }$ 27.33 2 009.35 2 464.50 4.85 164.64 79.43
    wizmir 19.29 $ \bf{ 4.02}$ 16.36 1 344.52 2 032.38 4.21 100.28 70.35
    下载: 导出CSV

    表  6  6种TSK模糊系统在28个数据集上的平均模糊规则数比较

    Table  6  Comparison of the average number of fuzzy rules of six TSK fuzzy systems on 28 datasets

    数据集 WM-R MOGUL-R TSK-IRL-R L2-TSK-FS B-ZTSK-FS PTSK
    mexihat 8.8 10.4 28.2 6.8 6.6 $ \bf{4.2 }$
    abalone 217.6 114.4 6434.0 16.0 9.0 $ \bf{6.0 }$
    anacalt 124.6 313.6 185.4 16.0 25.0 $ \bf{4.8 }$
    autompg6 117.0 81.2 786.0 36.0 36.0 $ \bf{18.2 }$
    autompg8 182.0 380.0 2658.0 36.0 25.0 $ \bf{6.6 }$
    bodyfat 190.6 101.2 1715.2 9.0 6.0 $ \bf{ 3.2}$
    compactiv 1 599.6 536.2 2097.0 25.0 16.0 $ \bf{11.0 }$
    concrete 309.2 360.4 1 497.2 49.0 36.0 $ \bf{ 8.6}$
    dee 161.4 112.2 3051.4 36.0 36.0 $ \bf{ 34.6}$
    delta-elv 708.8 220.6 6 510.0 36.0 25.0 $ \bf{5.8 }$
    delta-ail 241.8 104.6 1 476.6 25.0 36.0 $ \bf{8.8 }$
    diabert 16.4 32.8 22.8 25.0 16.0 $ \bf{ 8.4}$
    elevator 4286.7 801.0 191 25.0 25.0 $ \bf{ 23.8}$
    friedman 767.8 432.2 3 043.2 25.0 16.0 $ \bf{6.6 }$
    gc-s 62.8 40.2 226.2 16.0 9.0 $ \bf{6.4 }$
    gc-x 60.8 43.2 220.8 16.0 9.0 $ \bf{ 6.2}$
    gc-p 61.4 40.0 218.8 16.0 9.0 $ \bf{ 6.4}$
    housing 291.2 288.4 2 673.0 49.0 49.0 $ \bf{ 40.0}$
    mg 240.0 175.0 3 887.0 9.0 9.0 $ \bf{5.4 }$
    mortgage 198.2 62.8 122.6 25.0 16.0 $ \bf{ 14.4}$
    plastic 14.8 97.4 87.8 49.0 25.0 $ \bf{ 21.4}$
    pole 3 228.8 100.2 1 775.0 36.0 36.0 $ \bf{ 23.4}$
    puma 6 553.4 188.0 3 221.0 81.0 64.0 $ \bf{60.8 }$
    quake 54.2 173.4 985.8 36.0 $ \bf{25.0 }$ 29.0
    stock 264.8 80.6 578 36.0 36.0 $ \bf{8.6 }$
    treasury 197.2 63.6 70.0 49.0 36.0 $ \bf{35.6 }$
    wankara 458.6 127.8 836.0 25.0 25.0 $ \bf{ 22.0}$
    wizmir 413.8 119.3 189.4 36.0 $ \bf{ 25.0}$ $ \bf{ 25.0}$
    下载: 导出CSV

    表  7  mexihat, elevators, bodyfat 和 wizmir 数据集上 β 参数敏感性实验

    Table  7  Sensitivity experiments of parameter β on mexihat, elevators, bodyfat and wizmir datasets

    Datasets $\beta = 1$ $\beta = 2$ $\beta = 3$ $\beta = 4$ $\beta = 5$ $\beta = 6$ $\beta = 7$ $\beta = 8$
    mexihat MSE 0.0367 0.0304 0.0257 0.0222 0.0204 0.0213 0.0248 0.0248
    Rules 10.4 8.6 6.6 5.2 4.0 3.8 3.6 3.6
    elevators MSE 5.9476$\times {{10}^{-4}}$ 5.5531$\times{{10}^{-4}}$ 5.6307$\times {{10}^{-4}}$ 5.6948$\times {{10}^{-4}}$ 5.6281$\times {{10}^{-4}}$ 5.5845$\times {{10}^{-4}}$ 6.0934$\times {{10}^{-4}}$ 6.4256$\times {{10}^{-4}}$
    Rules 24.4 23.8 22.6 21.2 20.8 20.2 18.4 16.6
    bodyfat MSE 3.4876$\times {{10}^{-4}}$ 3.5512$\times{{10}^{-4}}$ 3.4402$\times {{10}^{-4}}$ 3.6802$\times{{10}^{-4}}$ 3.9823$\times {{10}^{-4}}$ 3.8963$\times{{10}^{-4}}$ 3.9027$\times {{10}^{-4}}$ 3.9216$\times {{10}^{-4}}$
    Rules 3.5 3.0 3.2 3.0 2.8 2.6 2.6 2.6
    wizmir MSE 1.7657 1.7606 1.6435 1.5245 1.5288 1.7578 1.9244 2.0864
    Rules 28.2 28.0 26.0 25.0 25.0 22.6 20.8 19.8
    下载: 导出CSV

    表  8  Holm post-hoc检验结果

    Table  8  Holm post-hoc results

    Algorithm $z$ $ p$ Holm = $a/i$ Hypothesis
    L2-TSK-FS 6.9284 0 7.143$\times {{10}^{-3}}$ Rejected
    WM-R 5.7555 0 8.333$\times {{10}^{-3}}$ Rejected
    MOGUL-R 5.5918 0 0.0100 Rejected
    TSK-IRL-R 5.4009 0 0.0125 Rejected
    B-ZTSK-FS 5.0190 1.0$\times {{10}^{-6}}$ 0.0167 Rejected
    ENSEMBLE-R 3.8461 1.2$\times {{10}^{-5}}$ 0.0250 Rejected
    PSVR 3.2460 0.00117 0.0500 Rejected
    下载: 导出CSV
  • [1] 张必山, 马忠军, 杨美香.既含有一般多个随机延迟以及多个测量丢失和随机控制丢失的鲁棒H∞模糊输出反馈控制.自动化学报, 2017, 43(9): 1656-1664 doi: 10.16383/j.aas.2017.e150082

    Zhang Bi-Shan, Ma Zhong-Jun, Yang Mei-Xiang. Robust H∞ fuzzy output-feedback control with both general multiple probabilistic delays and multiple missing measurements and random missing control. Acta Automatica Sinica, 2017, 43(9): 1656-1664 doi: 10.16383/j.aas.2017.e150082
    [2] 顾晓清, 蒋亦樟, 王士同.用于不平衡数据分类的0阶TSK型模糊系统.自动化学报, 2017, 43(10): 1773-1788 doi: 10.16383/j.aas.2017.c160200

    Gu Xiao-Qing, Jiang Yi-Zhang, Wang Shi-Tong. Zero-order TSK-type fuzzy system for imbalanced data classification. Acta Automatica Sinica, 2017, 43(10): 1773-1788 doi: 10.16383/j.aas.2017.c160200
    [3] Luo M, Sun F C, Liu H P. Joint block structure sparse representation for multi-input-multi-output (MIMO) T-S fuzzy system identification. IEEE Transactions on Fuzzy Systems, 2014, 22(6): 1387-1400 doi: 10.1109/TFUZZ.2013.2292973
    [4] Garcia A M, Carmona C J, Gonzalez P, Jesus M J. MOEA-EFEP: Multi-objective evolutionary algorithm for the extraction of fuzzy emerging patterns. IEEE Transactions on Fuzzy Systems, 2018, 3(9): 2861-2872
    [5] Deng Z H, Choi K S, Wang S T. Scalable TSK fuzzy modeling for very large datasets using minimal-enclosing-ball approximation. IEEE Transactions on Fuzzy Systems, 2011, 19(4): 210-226
    [6] Leski J M. SparseFIS: Data-driven learning of fuzzy systems with sparsity constraints. IEEE Transactions on Fuzzy Systems, 2010, 18(2): 396-411
    [7] 蒋亦樟, 邓赵红, 王士同. ML型迁移学习模糊系统.自动化学报, 2012, 38(9): 1393-1409 doi: 10.3724/SP.J.1004.2012.01393

    Jiang Y Z, Deng Z H, Wang S T. Mamdani-Larsen type transfer learning fuzzy system. Acta Automatica Sinica, 2012, 38(9): 1393-1409 doi: 10.3724/SP.J.1004.2012.01393
    [8] Pal N R, Mudi R, Pal K, Rule extraction through exploratory data analysis for self-tuning fuzzy controllers. International Journal of Fuzzy Systems, 2004, 6(2): 71-80
    [9] Juang C F, Hsieh C D. A fuzzy system constructed by rule generation and iterative linear SVR for antecedent and consequent parameter optimization. IEEE Transactions on Fuzzy Systems, 2012, 20(2): 372-384 doi: 10.1109/TFUZZ.2011.2174997
    [10] Liu J F, Chung F L, Wang S T. Bayesian zero-order TSK fuzzy system modeling. Applied Soft Computing, 2017, 55(6): 253-264
    [11] Zadeh L A. Discussion: Probability theory and fuzzy logic are complementary rather than competitive. Technometrics, 1995, 37(3): 271-276 doi: 10.1080/00401706.1995.10484330
    [12] Hardy A. The Poisson processes in cluster analysis, classification and multivariate analysis for complex data structures: studies in classification, data analysis, and knowledge organization. Springer-Verlag Berlin Heidelberg. July 2011
    [13] Puri C, Kumar N. Type-2 projected Gustafson-Kessel clustering algorithm. International Journal of Computer Applications, 2017, 167(14): 1-6 doi: 10.5120/ijca2017914445
    [14] Agounad S, Aassif E H, Khandouch Y, Maze G, Décultot D. Characterization and prediction of the backscattered form function of an immersed cylindrical shell using hybrid fuzzy clustering and bio-inspired algorithms. Ultrasonics, 2018, 83(2): 222-235
    [15] Cheng W Y, Juang C F. A fuzzy model with online incremental SVM and margin-selective gradient descent learning for classification problems. IEEE Transactions on Fuzzy System, 2012, 20(2): 372-384 doi: 10.1109/TFUZZ.2011.2174997
    [16] Lin C J, Chen C H. A self-constructing compensatory neural fuzzy and its applications. Mathematical and Computer Modelling, 2000, 42(3): 339-351
    [17] Chao C, Zare A, Trinh H N, Omotara G O, Cobb J T, Lagaunne T A. Partial membership latent Dirichlet allocation for soft image segmentation. IEEE Transactions on Image Processing, 2017, 26(12): 5590-5602
    [18] Mitianoudis N. A generalized directional Laplacian distribution: Estimation, mixture models and audio source separation. IEEE Transactions on Audio Speech and Language Processing, 2012, 20(9): 2397-2408 doi: 10.1109/TASL.2012.2203804
    [19] Gu X Q, Chung F L, Ishibuchi H, Wang S T. Imbalanced TSK fuzzy classifier by cross-class Bayesian fuzzy clustering and imbalance learning. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2017, 47(8): 2005-2020 doi: 10.1109/TSMC.2016.2598270
    [20] Gu X Q, Wang S. Bayesian Takagi-Sugeno-Kang fuzzy model and its joint learning of structure identification and parameter estimation. IEEE Transactions on Industrial Informatics, 2018, 47(8): 5327-5337
    [21] Chopin N. A sequential particle filter method for static models. Biometrika, 2002, 89(3): 539-551 doi: 10.1093/biomet/89.3.539
    [22] Cheng W C. PSO algorithm particle filters for improving the performance of lane detection and tracking systems in difficult roads. Sensors, 2012, 12(12): 17168-17185 doi: 10.3390/s121217168
    [23] LIBSVM Datasets[Online], available: https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets, November 11, 2017
    [24] KEEL Software and KEEL Datasets[Online], available: http://sci2s.ugr.es/keel, November 11, 2017
    [25] Cordón O, Herrera F. A two-stage evolutionary process for designing TSK fuzzy rule-based systems. IEEE Trans. Systems, Man and Cybernetics, Part B: Cybernetics, 1999, 29(6): 703-715 doi: 10.1109/3477.809026
    [26] Alcalá R, Alcala-Fdez J, Casillas J, Cordón O, Herrera F. Local identification of prototypes for genetic learning of accurate TSK fuzzy rule-based systems. International Journal of Intelligent Systems, 2007, 22(9): 909-941 doi: 10.1002/int.20232
    [27] Wang L X, Mendel J M. Generating fuzzy rules by learning from examples. IEEE Transactions on Systems, Man and Cybernetics, 1992, 22(6): 1414-1427 doi: 10.1109/21.199466
    [28] Pedrajas N G, Osorio C G, Fyfe C. Nonlinear boosting projections for ensemble construction. Journal of Machine Learning Research, 2007, 8(1): 1-33
    [29] Peng X J, Xu D. Projection support vector regression algorithms for data regression. Knowledge-Based Systems, 2016, 112(11): 54-66
    [30] Demsar J. Statistical comparisons of classifiers over multiple data sets. Journal of Machine Learning Research, 2006, 7(1): 1-30
    [31] Holm S. A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 1979, 6(2): 65-70
    [32] Eichhorn H, Cano J L, McLean F, Anderl R. A comparative study of programming languages for next-generation astrodynamics systems. CEAS Space Journal, 2018, 10(3): 115-123
  • 加载中
图(5) / 表(8)
计量
  • 文章访问数:  118
  • HTML全文浏览量:  93
  • PDF下载量:  74
  • 被引次数: 0
出版历程
  • 收稿日期:  2018-05-10
  • 录用日期:  2018-12-12
  • 刊出日期:  2021-02-26

目录

    /

    返回文章
    返回