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基于K互近邻与核密度估计的DPC算法

周玉 夏浩 刘虹瑜 白磊

周玉,夏浩,刘虹瑜,等. 基于K互近邻与核密度估计的DPC算法[J]. 北京航空航天大学学报,2025,51(6):1978-1990 doi: 10.13700/j.bh.1001-5965.2023.0342
引用本文: 周玉,夏浩,刘虹瑜,等. 基于K互近邻与核密度估计的DPC算法[J]. 北京航空航天大学学报,2025,51(6):1978-1990 doi: 10.13700/j.bh.1001-5965.2023.0342
ZHOU Y,XIA H,LIU H Y,et al. DPC algorithm based on K-reciprocal neighbors and kernel density estimation[J]. Journal of Beijing University of Aeronautics and Astronautics,2025,51(6):1978-1990 (in Chinese) doi: 10.13700/j.bh.1001-5965.2023.0342
Citation: ZHOU Y,XIA H,LIU H Y,et al. DPC algorithm based on K-reciprocal neighbors and kernel density estimation[J]. Journal of Beijing University of Aeronautics and Astronautics,2025,51(6):1978-1990 (in Chinese) doi: 10.13700/j.bh.1001-5965.2023.0342

基于K互近邻与核密度估计的DPC算法

doi: 10.13700/j.bh.1001-5965.2023.0342
基金项目: 

国家自然科学基金(U1504622, 31671580); 河南省青年骨干教师培养计划(2018GGJS079)

详细信息
    通讯作者:

    E-mail:zhouyu_beijing@126.com

  • 中图分类号: TP311

DPC algorithm based on K-reciprocal neighbors and kernel density estimation

Funds: 

National Natural Science Foundation of China (U1504622,31671580); Henan Province Young Backbone Teachers Training Program (2018GGJS079)

More Information
  • 摘要:

    快速搜索和发现密度峰值聚类(DPC) 算法是一种基于密度的聚类算法。该算法不需要迭代和过多的设定参数,但由于计算局部密度时没有考虑数据的局部结构,导致无法识别簇密度小的聚类中心。针对此问题,提出基于K互近邻(KN)和核密度估计(KDE)的DPC (KKDPC)算法。通过K近邻和核密度估计方法得到数据点的K互近邻数量和局部核密度;将K互近邻数量与局部核密度进行加和获得新的局部密度;根据数据点的局部密度得到相对距离,并通过构建决策图选取聚类中心及分配非中心点。利用人工数据集和真实数据集进行实验,并与DPC、基于密度的噪声空间聚类应用(DBSCAN)、K-means、模糊C均值聚类算法(FCM)、基于K近邻的DPC(DPC-KNN)、近邻优化DPC(DPC-NNO)、基于模糊加权共享邻居的DPC(DPC-FWSN)算法进行对比。通过计算调整互信息(AMI)、调整兰德指数(ARI)、归一化互信息(NMI)来验证KKDPC算法的性能。实验结果表明:KKDPC算法能更加准确地识别聚类中心,有效地提高聚类精度。

     

  • 图 1  二维数据

    Figure 1.  Two-dimensional data

    图 2  决策图

    Figure 2.  Decision graph

    图 3  Data数据集

    Figure 3.  Data dataset

    图 4  Jain数据集

    Figure 4.  Jain dataset

    图 5  Data 数据集在不同dc值下的决策图

    Figure 5.  Decision graphs of Data dataset under different values of dc

    图 6  Jain数据集在不同dc值下的决策图

    Figure 6.  Decision graphs of Jain dataset under different values of dc

    图 7  Data数据集聚类结果

    Figure 7.  Clustering results of Data dataset

    图 8  Jain数据集聚类结果

    Figure 8.  Clustering results of Jain dataset

    图 9  Data数据集聚类中心分析

    Figure 9.  Analysis of cluster centers in Data dataset

    图 10  Jain数据集聚类中心分析

    Figure 10.  Analysis of cluster centers in Jain dataset

    图 11  A3数据集[27]聚类结果

    Figure 11.  A3 dataset[27] clustering results

    图 12  Aggregation数据集[28]聚类结果

    Figure 12.  Aggregation dataset[28] clustering results

    图 13  Flame数据集[29]聚类结果

    Figure 13.  Flame dataset[29] clustering results

    图 14  Jain数据集[30]聚类结果

    Figure 14.  Jain dataset[30] clustering results

    图 15  Line blobs数据集[31]聚类结果

    Figure 15.  Line blobs dataset[31] clustering results

    图 16  Spiral数据集[32]聚类结果

    Figure 16.  Spiral dataset[32] clustering results

    图 17  Olivetti Face数据集[34]聚类结果

    Figure 17.  Olivetti Face dataset[34]clustering results

    表  1  人工数据集

    Table  1.   Artificial datasets

    数据集数据量维度类簇个数
    A3[27]7500250
    Aggregation[28]78827
    Flame[29]21922
    Jain[30]37322
    Line blobs[31]26623
    Spiral[32]31223
    下载: 导出CSV

    表  2  人工数据集实验结果

    Table  2.   Experimental results on artificial datasets

    算法 AMI
    A3 Aggregation Flame Jain Line blobs Spiral
    KKDPC 0.977 5 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
    DPC[16] 0.976 7 0.992 2 1.000 0 0.709 2 0.779 9 1.000 0
    DBSCAN[12] 0.867 4 0.984 1 0.938 6 1.000 0 1.000 0 1.000 0
    K-means[8] 0.963 9 0.835 7 0.550 6 0.491 6 0.590 5 −0.004 7
    FCM[24] 0.937 3 0.805 0 0.589 7 0.515 2 0.586 6 −0.004 7
    DPC-KNN[17] 0.977 4 0.992 2 1.000 0 0.631 9 0.779 4 1.000 0
    DPC-NNO[22] 0.979 4 0.990 5 1.000 0 1.000 0 1.000 0 1.000 0
    DPC-FWSN[23] 0.977 3 0.995 5 1.000 0 1.000 0 1.000 0 1.000 0
    算法 NMI
    A3 Aggregation Flame Jain Line blobs Spiral
    KKDPC 0.965 9 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
    DPC[16] 0.962 9 0.995 6 1.000 0 0.822 4 0.721 0 1.000 0
    DBSCAN[12] 0.685 9 0.991 1 0.986 6 1.000 0 1.000 0 1.000 0
    K-means[8] 0.921 4 0.765 3 0.565 8 0.576 7 0.500 8 −0.005 0
    FCM[24] 0.859 3 0.731 8 0.622 4 0.585 3 0.494 2 −0.005 0
    DPC-KNN[17] 0.965 6 0.995 6 1.000 0 0.757 7 0.717 9 1.000 0
    DPC-NNO[22] 0.966 5 0.994 9 1.000 0 1.000 0 1.000 0 1.000 0
    DPC-FWSN[23] 0.963 9 0.997 8 1.000 0 1.000 0 1.000 0 1.000 0
    算法 NMI
    A3 Aggregation Flame Jain Line blobs Spiral
    KKDPC 0.978 6 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
    DPC[16] 0.977 8 0.992 4 1.000 0 0.742 9 0.788 5 1.000 0
    DBSCAN[12] 0.920 4 0.986 0 0.969 0 1.000 0 1.000 0 1.000 0
    K-means[8] 0.968 6 0.883 6 0.565 2 0.528 8 0.595 8 0.001 1
    FCM[24] 0.945 7 0.840 1 0.604 8 0.554 7 0.591 5 0.000 4
    DPC-KNN[17] 0.978 7 0.992 4 1.000 0 0.676 2 0.787 1 1.000 0
    DPC-NNO[22] 0.980 3 0.991 5 1.000 0 1.000 0 1.000 0 1.000 0
    DPC-FWSN[23] 0.978 4 0.995 8 1.000 0 1.000 0 1.000 0 1.000 0
    下载: 导出CSV

    表  3  UCI数据集

    Table  3.   UCI datasets

    UCI数据集 数据量 维度 类簇个数
    Blood 748 4 2
    Breast 277 9 2
    Diabetes 768 8 2
    Ecoli 336 7 8
    Heart 270 13 2
    Ionosphere 351 34 2
    Iris 150 4 3
    Pima 768 8 2
    Seeds 210 7 3
    Thyroid 215 5 3
    Vehicle 846 18 4
    WDBC 569 30 2
    Wine 178 13 3
    Zoo 101 16 7
    Banknote 1372 4 2
    Waveform 5000 21 3
    MFCCs 7195 22 10
    Pendigits 10992 16 10
    下载: 导出CSV

    表  4  UCI数据集实验结果

    Table  4.   Experimental results on UCI datasets

    UCI数据集 AMI
    KKDPC DPC[16] DBSCAN[12] K-means[8] FCM[24] DPC-KNN[17] DPC-NNO[22] DPC-FWSN[23]
    Blood 0.059 1 0.088 5 0.008 4 −0.000 8 −0.000 7 0.010 1 0.043 7 0.050 3
    Breast 0.074 1 −0.001 4 0.001 7 0.074 1 −0.001 4 0.067 2 0.063 4 0.069 6
    Diabetes 0.045 0 0.034 1 0.077 6 0.051 2 0.062 5 0.001 7 0.044 0 0.002 2
    Ecoli 0.679 8 0.489 0 0.530 5 0.585 5 0.465 3 0.614 6 0.676 2 0.647 7
    Heart 0.313 3 0.229 7 0.085 0 0.283 2 0.255 2 0.245 3 0.279 3 0.263 2
    Ionosphere 0.321 3 0.087 6 0.212 3 0.129 4 0.124 6 0.150 4 0.366 7 0.381 1
    Iris 0.862 3 0.862 3 0.604 4 0.733 1 0.737 2 0.862 3 0.883 1 0.883 1
    Pima 0.045 0 0.034 1 0.077 6 0.051 2 0.077 1 0.001 7 0.044 0 0.002 2
    Seeds 0.723 3 0.717 2 0.541 4 0.661 5 0.678 5 0.706 3 0.773 8 0.785 5
    Thyroid 0.513 3 0.354 4 0.550 1 0.516 8 0.589 7 0.194 5 0.668 5 0.412 3
    Vehicle 0.236 5 0.164 7 0.163 6 0.158 4 0.094 9 0.173 6 0.193 7 0.181 4
    WDBC 0.779 2 0.026 7 0.374 9 0.611 0 0.607 6 0.506 9 0.735 5 0.707 6
    Wine 0.778 3 0.706 5 0.548 4 0.873 5 0.828 1 0.739 1 0.888 6 0.888 6
    Zoo 0.899 4 0.889 6 0.820 6 0.858 1 0.740 9 0.747 1 0.848 7 0.726 2
    Banknote 0.755 3 0.931 7 0.762 4 0.016 8 0.032 9 0.923 5 0.937 7 0.935 9
    Waveform 0.367 9 0.240 5 0.012 7 0.364 2 0.330 2 0.356 8 0.362 9 0.355 0
    MFCCs 0.796 1 0.639 0 0.726 6 0.754 1 0.461 6 0.745 1 0.764 1 0.723 3
    Pendigits 0.733 2 0.732 1 0.517 5 0.702 9 0.618 1 0.769 6 0.808 2 0.782 1
    UCI数据集 ARI
    KKDPC DPC[16] DBSCAN[12] K-means[8] FCM[24] DPC-KNN[17] DPC-NNO[22] DPC-FWSN[23]
    Blood 0.111 0 0.196 9 0.032 8 −0.006 2 −0.007 2 0.031 1 0.026 2 0.065 2
    Breast 0.162 2 −0.003 1 0.015 9 0.157 0 −0.003 1 0.152 8 0.128 4 0.162 2
    Diabetes 0.107 2 0.078 0 0.148 1 0.104 0 0.106 9 0.014 3 0.099 5 0.013 1
    Ecoli 0.760 1 0.561 8 0.649 1 0.537 3 0.368 9 0.734 7 0.734 6 0.741 0
    Heart 0.403 6 0.305 6 0.018 0 0.366 6 0.331 4 0.322 7 0.366 3 0.348 5
    Ionosphere 0.435 0 0.131 7 0.222 7 0.177 6 0.172 7 0.235 7 0.484 0 0.491 5
    Iris 0.885 7 0.885 7 0.639 7 0.761 3 0.728 7 0.885 7 0.903 8 0.903 8
    Pima 0.107 2 0.078 0 0.148 1 0.104 0 0.117 6 0.014 3 0.099 5 0.013 1
    Seeds 0.764 1 0.734 1 0.544 9 0.693 4 0.714 9 0.741 9 0.824 6 0.836 9
    Thyroid 0.599 9 0.425 8 0.793 2 0.638 2 0.692 7 0.273 1 0.771 2 0.455 0
    Vehicle 0.157 8 0.127 3 0.068 2 0.109 2 0.074 5 0.109 3 0.155 2 0.136 9
    WDBC 0.870 2 0.004 8 0.494 1 0.730 2 0.730 5 0.577 3 0.830 6 0.811 3
    Wine 0.786 9 0.672 4 0.529 2 0.899 2 0.849 8 0.726 9 0.914 9 0.914 9
    Zoo 0.957 0 0.957 0 0.807 4 0.894 2 0.644 4 0.622 4 0.926 0 0.770 6
    Banknote 0.837 9 0.962 4 0.826 6 0.022 3 0.045 2 0.956 7 0.962 4 0.965 3
    Waveform 0.316 1 0.226 7 0.003 6 0.253 6 0.243 6 0.305 7 0.297 7 0.303 2
    MFCCs 0.905 3 0.729 2 0.855 4 0.895 6 0.208 0 0.841 1 0.884 2 0.813 3
    Pendigits 0.579 0 0.635 0 0.598 0 0.623 3 0.529 1 0.623 8 0.700 9 0.6645
    UCI数据集 NMI
    KKDPC DPC[16] DBSCAN[12] K-means[8] FCM[24] DPC-KNN[17] DPC-NNO[22] DPC-FWSN[23]
    Blood 0.067 0 0.092 7 0.026 1 0.000 3 0.000 4 0.035 0 0.048 3 0.057 5
    Breast 0.107 4 0.001 3 0.017 2 0.078 4 0.001 3 0.071 0 0.107 4 0.096 2
    Diabetes 0.062 3 0.035 6 0.080 6 0.052 7 0.064 7 0.017 1 0.059 8 0.015 4
    Ecoli 0.739 4 0.576 1 0.588 9 0.665 3 0.562 0 0.707 7 0.690 3 0.709 1
    Heart 0.316 1 0.238 1 0.151 3 0.286 2 0.258 3 0.255 0 0.286 8 0.269 6
    Ionosphere 0.351 6 0.091 6 0.272 3 0.134 9 0.129 9 0.152 9 0.401 8 0.415 3
    Iris 0.864 2 0.864 2 0.676 2 0.741 9 0.743 3 0.864 2 0.885 1 0.885 1
    Pima 0.062 3 0.035 6 0.080 6 0.052 7 0.080 2 0.017 1 0.059 8 0.015 4
    Seeds 0.726 6 0.723 8 0.609 0 0.665 4 0.682 2 0.710 3 0.775 9 0.787 5
    Thyroid 0.583 3 0.482 5 0.648 7 0.602 7 0.665 7 0.333 3 0.691 1 0.502 1
    Vehicle 0.266 8 0.196 0 0.193 5 0.203 4 0.098 6 0.203 3 0.211 4 0.214 1
    WDBC 0.781 1 0.056 2 0.377 2 0.623 2 0.615 2 0.547 5 0.747 1 0.718 2
    Wine 0.783 8 0.710 4 0.591 8 0.878 2 0.833 6 0.743 5 0.892 6 0.892 6
    Zoo 0.916 9 0.911 9 0.856 8 0.889 3 0.815 8 0.804 8 0.899 8 0.795 0
    Banknote 0.755 3 0.931 7 0.762 4 0.017 4 0.032 9 0.923 5 0.931 7 0.935 9
    Waveform 0.367 9 0.240 5 0.344 9 0.364 2 0.330 2 0.356 8 0.362 9 0.355 0
    MFCCs 0.796 1 0.639 0 0.726 6 0.754 1 0.461 6 0.745 1 0.764 1 0.723 3
    Pendigits 0.773 9 0.754 4 0.699 0 0.706 5 0.639 3 0.796 7 0.824 9 0.812 6
    下载: 导出CSV
  • [1] ZUO W D, HOU X M. An improved probability propagation algorithm for density peak clustering based on natural nearest neighborhood[J]. Array, 2022, 15: 100232. doi: 10.1016/j.array.2022.100232
    [2] KHATER I M, NABI I R, HAMARNEH G. A review of super-resolution single-molecule localization microscopy cluster analysis and quantification methods[J]. Patterns, 2020, 1(3): 100038. doi: 10.1016/j.patter.2020.100038
    [3] GRIFFIÉ J, BURN G L, WILLIAMSON D J, et al. Dynamic Bayesian cluster analysis of live-cell single molecule localization microscopy datasets[J]. Small Methods, 2018, 2(9): 1800008. doi: 10.1002/smtd.201800008
    [4] FANG U, LI J X, LU X Q, et al. Robust image clustering via context-aware contrastive graph learning[J]. Pattern Recognition, 2023, 138: 109340. doi: 10.1016/j.patcog.2023.109340
    [5] GUAN J Y, LI S, HE X X, et al. Peak-graph-based fast density peak clustering for image segmentation[J]. IEEE Signal Processing Letters, 2021, 28: 897-901. doi: 10.1109/LSP.2021.3072794
    [6] HU N, TIAN Z H, LU H, et al. A multiple-kernel clustering based intrusion detection scheme for 5G and IoT networks[J]. International Journal of Machine Learning and Cybernetics, 2021, 12(11): 3129-3144. doi: 10.1007/s13042-020-01253-w
    [7] 张喜梅, 解滨, 徐童童, 等. 基于反向K近邻和密度峰值初始化的加权Kmeans聚类入侵检测算法[J]. 南京理工大学学报, 2023, 47(1): 56-65.

    ZHANG X M, XIE B, XU T T, et al. Intrusion detection algorithm based on weighted Kmeans clustering with reverse K-nearest neighbor and density peak initialization[J]. Journal of Nanjing University of Science and Technology, 2023, 47(1): 56-65(in Chinese).
    [8] MACQUEEN J. Some methods for classification and analysis of multivariate observations[J]. Berkeley Symposium on Mathematical Statistics and Probability, 1967, 1: 281-297.
    [9] PARK H S, JUN C H. A simple and fast algorithm for K-medoids clustering[J]. Expert Systems with Applications, 2009, 36(2): 3336-3341. doi: 10.1016/j.eswa.2008.01.039
    [10] GUHA S, RASTOGI R, SHIM K. Cure: an efficient clustering algorithm for large databases[J]. Information Systems, 2001, 26(1): 35-58. doi: 10.1016/S0306-4379(01)00008-4
    [11] ZHANG T, RAMAKRISHNAN R, LIVNY M. BIRCH: an efficient data clustering method for very large databases[J]. ACM SIGMOD Record, 1996, 25(2): 103-114. doi: 10.1145/235968.233324
    [12] ESTER M, KRIEGEL H P, SANDER J, et al. A density-based algorithm for discovering clusters in large spatial databases with noise[C]//Proceedings of the Second International Conference on Knowledge Discovery and Data Mining. Palo Alto: AAAI Press, 1996: 226-231.
    [13] ANKERST M, BREUNIG M M, KRIEGEL H P, et al. OPTICS: ordering points to identify the clustering structure[J]. ACM SIGMOD Record, 1999, 28(2): 49-60. doi: 10.1145/304181.304187
    [14] SHEIKHOLESLAMI G, CHATTERJEE S, ZHANG A D. WaveCluster: a wavelet-based clustering approach for spatial data in very large databases[J]. The VLDB Journal, 2000, 8(3): 289-304.
    [15] WANG W, YANG J, MUNTZ R R. STING: a statistical information grid approach to spatial data mining[C]//Proceedings of the 23rd International Conference on Very Large Data Bases. New York: ACM, 1997: 186-195.
    [16] RODRIGUEZ A, LAIO A. Clustering by fast search and find of density peaks[J]. Science, 2014, 344(6191): 1492-1496. doi: 10.1126/science.1242072
    [17] DU M J, DING S F, JIA H J. Study on density peaks clustering based on K-nearest neighbors and principal component analysis[J]. Knowledge-Based Systems, 2016, 99: 135-145. doi: 10.1016/j.knosys.2016.02.001
    [18] JIANG D, ZANG W K, SUN R, et al. Adaptive density peaks clustering based on K-nearest neighbor and gini coefficient[J]. IEEE Access, 2020, 8: 113900-113917. doi: 10.1109/ACCESS.2020.3003057
    [19] YUAN X N, YU H, LIANG J, et al. A novel density peaks clustering algorithm based on K nearest neighbors with adaptive merging strategy[J]. International Journal of Machine Learning and Cybernetics, 2021, 12(10): 2825-2841. doi: 10.1007/s13042-021-01369-7
    [20] XIE J Y, GAO H C, XIE W X, et al. Robust clustering by detecting density peaks and assigning points based on fuzzy weighted K-nearest neighbors[J]. Information Sciences, 2016, 354: 19-40. doi: 10.1016/j.ins.2016.03.011
    [21] LIU R, WANG H, YU X M. Shared-nearest-neighbor-based clustering by fast search and find of density peaks[J]. Information Sciences, 2018, 450: 200-226. doi: 10.1016/j.ins.2018.03.031
    [22] 陈蔚昌, 赵嘉, 肖人彬, 等. 面向密度分布不均数据的近邻优化密度峰值聚类算法[J]. 控制与决策, 2024, 39(3): 919-928.

    CHEN W C, ZHAO J, XIAO R B, et al. Density peaks clustering algorithm with nearest neighbor optimization for data with uneven density distribution[J]. Control and Decision, 2024, 39(3): 919-928 (in Chinese).
    [23] ZHAO J, WANG G, PAN J S, et al. Density peaks clustering algorithm based on fuzzy and weighted shared neighbor for uneven density datasets[J]. Pattern Recognition, 2023, 139: 109406. doi: 10.1016/j.patcog.2023.109406
    [24] BEZDEK J C. Pattern recognition with fuzzy objective function algorithms[M]. New York : Plenum Press, 1981.
    [25] DING S F, LI C, XU X, et al. A sampling-based density peaks clustering algorithm for large-scale data[J]. Pattern Recognition, 2023, 136: 109238. doi: 10.1016/j.patcog.2022.109238
    [26] CHENG D D, ZHANG S L, HUANG J L. Dense members of local cores-based density peaks clustering algorithm[J]. Knowledge-Based Systems, 2020, 193: 105454. doi: 10.1016/j.knosys.2019.105454
    [27] KARKKAINEN I, FRANTI P. Dynamic local search algorithm for the clustering problem[M]. Joensuu: University of Joensuu, 2002.
    [28] GIONIS A, MANNILA H, TSAPARAS P. Clustering aggregation[J]. ACM Transactions on Knowledge Discovery from Data, 2007, 1(1): 4. doi: 10.1145/1217299.1217303
    [29] FU L M, MEDICO E. FLAME, a novel fuzzy clustering method for the analysis of DNA microarray data[J]. BMC Bioinformatics, 2007, 8(1): 3. doi: 10.1186/1471-2105-8-3
    [30] JAIN A K, LAW M H C. Data clustering: a user’s dilemma[M]// Pattern Recognition and Machine Intelligence. Berlin: Springer, 2005: 1-10.
    [31] XU X, DING S F, WANG L J, et al. A robust density peaks clustering algorithm with density-sensitive similarity[J]. Knowledge-Based Systems, 2020, 200: 106028. doi: 10.1016/j.knosys.2020.106028
    [32] CHANG H, YEUNG D Y. Robust path-based spectral clustering[J]. Pattern Recognition, 2008, 41(1): 191-203. doi: 10.1016/j.patcog.2007.04.010
    [33] BACHE K, LICHMAN M. UCI machine learning repository[EB/OL]. (2020-12-04)[2023-02-01]. http://archive.ics.uci.edu/ml.
    [34] SAMARIA F S, HARTER A C. Parameterisation of a stochastic model for human face identification[C]// Proceedings of The IEEE Workshop on Applications of Computer Vision. Piscataway: IEEE Press, 1994: 138-142.
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出版历程
  • 收稿日期:  2023-06-12
  • 录用日期:  2023-10-13
  • 网络出版日期:  2023-10-20
  • 整期出版日期:  2025-06-30

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