加权K近邻和多簇合并的密度峰值聚类算法

doi:10.3778/j.issn.1673-9418.2102021

计算机科学与探索 ›› 2022, Vol. 16 ›› Issue (9): 2163-2176.DOI: 10.3778/j.issn.1673-9418.2102021

• 理论与算法 • 上一篇

加权K近邻和多簇合并的密度峰值聚类算法

陈磊, 吴润秀(), 李沛武, 赵嘉

南昌工程学院信息工程学院,南昌 330099

收稿日期:2021-02-05 修回日期:2021-04-02 出版日期:2022-09-01 发布日期:2021-04-19
通讯作者: + E-mail: wurunxiu@tom.com
作者简介:陈磊（1997—）,男,硕士研究生,主要研究方向为数据挖掘。
吴润秀（1971—）,女,教授,硕士生导师,主要研究方向为群智能算法及应用。
李沛武（1963—）,男,博士,教授,硕士生导师,主要研究方向为计算机安全、大数据分析。
赵嘉（1981—）,男,博士,教授,硕士生导师,主要研究方向为数据挖掘、群智能算法。
基金资助:
江西省教育厅科技项目(GJJ180940);国家自然科学基金(52069014);国家自然科学基金(51669014);江西省杰出青年基金(2018ACB21029)

Weighted K-nearest Neighbors and Multi-cluster Merge Density Peaks Clustering Algorithm

CHEN Lei, WU Runxiu(), LI Peiwu, ZHAO Jia

School of Information Engineering, Nanchang Institute of Technology, Nanchang 330099, China

Received:2021-02-05 Revised:2021-04-02 Online:2022-09-01 Published:2021-04-19
About author:CHEN Lei, born in 1997, M.S. candidate. His research interest is data mining.
WU Runxiu, born in 1971, professor, M.S. supervisor. Her research interests include swarm intelligence algorithm and application.
LI Peiwu, born in 1963, Ph.D., professor, M.S. supervisor. His research interests include computer security and big data analysis.
ZHAO Jia, born in 1981, Ph.D., professor, M.S. supervisor. His research interests include data mining and swarm intelligence algorithm.
Supported by:
Science and Technology Project of Jiangxi Province Department of Education(GJJ180940);National Natural Science Foundation of China(52069014);National Natural Science Foundation of China(51669014);Science Fund for Distinguished Young Scholars of Jiangxi Province(2018ACB21029)

摘要/Abstract

摘要：

密度峰值聚类（DPC）算法是一种基于密度的聚类算法。该算法原理简单、运行高效,可以找到任意非球形类簇。但是该算法存在一些缺陷：首先,该算法局部密度定义的度量准则不统一且两者的聚类结果存在较大差异;其次,该算法的分配策略易产生分配连带错误,即一旦某一个样本分配错误,会导致后续一连串的样本分配错误。为解决这些问题,提出了一种加权$K$近邻和多簇合并的密度峰值聚类算法（WKMM-DPC）。该算法结合加权$K$近邻的思想,引入样本的权重系数,重新定义样本的局部密度,使局部密度更加依赖于K近邻内样本的位置,且统一了密度定义的度量准则;定义了类簇间的相似度,并据此度量准则进行多簇合并,以避免分配剩余样本时的分配连带错误。在人工和UCI数据集上的实验表明,该算法的聚类效果优于FKNN-DPC、DPCSA、FNDPC、DPC和DBSCAN算法。

关键词: 聚类, 局部密度, 密度峰值, K近邻（KNN）, 多簇合并

Abstract:

Density peaks clustering (DPC) algorithm is a clustering algorithm based on density. The algorithm is simple in principle and efficient in operation, and can find any non-spherical class clusters. However, there are some defects in the algorithm. Firstly, the measurement criteria defined by the local density are not uniform and there are great differences in the clustering results. Secondly, the allocation strategy is prone to allocation errors, that is once a sample is incorrectly allocated, a series of subsequent samples will be incorrectly allocated too. In order to solve these problems, this paper proposes a weighted K-nearest neighbors and multi-cluster merge density peaks clustering (WKMM-DPC) algorithm. Combined with the idea of weighted K-nearest neighbors, the local density of the sample is redefined by introducing the weight coefficient of the sample, which makes the local density more dependent on the position of the sample in the K-nearest neighbors, and unifies the measurement criteria of density definition. The similarity between clusters is defined, and the clusters are merged according to the metric to avoid the joint error in the allocation of remaining samples. Experiments on artificial and UCI datasets show that the clustering performance of the proposed algorithm is better than that of FKNN-DPC, DPCSA, FNDPC, DPC and DBSCAN algorithms.

Key words: clustering, local density, density peaks, K-nearest neighbors (KNN), multi-cluster merge

中图分类号:

TP301.6

陈磊, 吴润秀, 李沛武, 赵嘉. 加权K近邻和多簇合并的密度峰值聚类算法[J]. 计算机科学与探索, 2022, 16(9): 2163-2176.

CHEN Lei, WU Runxiu, LI Peiwu, ZHAO Jia. Weighted K-nearest Neighbors and Multi-cluster Merge Density Peaks Clustering Algorithm[J]. Journal of Frontiers of Computer Science and Technology, 2022, 16(9): 2163-2176.

图/表 18

参考文献 32

[1]	SRIVASTAVA A N. Data mining: concepts, models, methods, and algorithms[J]. Journal of Computing and Information Science and Engineering, 2005, 5(4): 394-395. DOI URL
[2]	JAIN A K. Data clustering: 50 years beyond K-means[J]. Pattern Recognition Letters, 2010, 31(8): 651-666. DOI URL
[3]	ZHANG T, RAMAKRISHNAN R, LIVNY M. BIRCH: a new data clustering algorithm and its applications[J]. Data Mining and Knowledge Discovery, 1997, 1(2): 141-182. DOI URL
[4]	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 2nd International Conference on Knowledge Discovery and Data Mining, Portland, 1996. Menlo Park: AAAI, 1996: 226-231.
[5]	WEI W, YANG J, MUNTZ R R. STING: a statistical informa-tion grid approach to spatial data mining[C]// Proceedings of the 23rd International Conference on Very Large Data Bases, Los Angeles, Aug 25-29, 1997. San Francisco: Morgan Kaufmann, 1997: 186-195.
[6]	XU X, DING S, DU M, et al. DPCG: an efficient density peaks clustering algorithm based on grid[J]. International Journal of Machine Learning and Cybernetics, 2018, 9(5): 743-754. DOI URL
[7]	RODRIGUEZ A, LAIO A. Clustering by fast search and find of density peaks[J]. Science, 2014, 344(6191): 1492-1496. DOI URL
[8]	高诗莹, 周晓锋, 李帅. 基于密度比例的密度峰值聚类算法[J]. 计算机工程与应用, 2017, 53(16): 10-17. DOI
	GAO S Y, ZHOU X F, LI S. Clustering by fast search and find of density peaks based on density-raito[J]. Computer Engineering and Applications, 2017, 53(16): 10-17.
[9]	ZHAO J, TANG J J, SHI A Y, et al. Improved density peaks clustering based on firefly algorithm[J]. International Journal of Bio-Inspired Computation, 2020, 15(1): 24-42. DOI URL
[10]	SUN L, LIU R N, XU J C, et al. An adaptive density peaks clustering method with fisher linear discriminant[J]. IEEE Access, 2019, 7: 72936-72955. DOI URL
[11]	纪霞, 姚晟, 赵鹏. 相对邻域与剪枝策略优化的密度峰值聚类算法[J]. 自动化学报, 2020, 46(3): 562-575.
	JI X, YAO S, ZHAO P. Relative neighborhood and pruning strategy optimized density peaks clustering algorithm[J]. Acta Automatica Sinica, 2020, 46(3): 562-575.
[12]	薛小娜, 高淑萍, 彭弘铭, 等. 结合K近邻的改进密度峰值聚类算法[J]. 计算机工程与应用, 2018, 54(7): 36-43. DOI
	XUE X N, GAO S P, PENG H M, et al. Improved density peaks clustering algorithm combining K-nearest neighbors[J]. Computer Engineering and Applications, 2018, 54(7): 36-43.
[13]	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 URL
[14]	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 URL
[15]	贾露, 张德生, 吕端端. 物理学优化的密度峰值聚类算法[J]. 计算机工程与应用, 2020, 56(13): 47-53. DOI
	JIA L, ZHANG D S, LV D D. Optimized density peak clustering algorithm in physics[J]. Computer Engineering and Applications, 2020, 56(13): 47-53. DOI
[16]	王芙银, 张德生, 张晓. 结合鲸鱼优化算法的自适应密度峰值聚类算法[J]. 计算机工程与应用, 2021, 57(3): 94-102. DOI
	WANG F Y, ZHANG D S, ZHANG X. Adaptive density peaks clustering algorithm combining with whale optimization algorithm[J]. Computer Engineering and Applications, 2021, 57(3): 94-102.
[17]	赵嘉, 姚占峰, 吕莉, 等. 基于相互邻近度的密度峰值聚类算法[J]. 控制与决策, 2021, 36(3): 543-552.
	ZHAO J, YAO Z F, LV L, et al. Density peaks clustering based on mutual neighbor degree[J]. Control and Decision, 2021, 36(3): 543-552.
[18]	ZHAO J, TANG J J, FAN T H, et al. Density peaks clustering based on circular partition and grid similarity[J]. Concurrency and Computation: Practice and Experience, 2020, 32(7): e5567.
[19]	CHEN H, YANG B, WANG G, et al. A novel bankruptcy prediction model based on an adaptive fuzzy-nearest neighbor method[J]. Knowledge-Based Systems, 2011, 24(8): 1348-1359. DOI URL
[20]	LV L, WANG J Y, WU R X, et al. Density peaks clustering based on geodetic distance and dynamic neighborhood[J]. International Journal of Bio-Inspired Computation, 2021, 17(1): 24-33. DOI URL
[21]	YU D H, LIU G J, GUO M Z, et al. Density peaks clustering based on weighted local density sequence and nearest neighbor assignment[J]. IEEE Access, 2019, 7: 34301-34317. DOI URL
[22]	DU M J, DING S F, XUE Y. A robust density peaks clustering algorithm using fuzzy neighborhood[J]. International Journal of Machine Learning and Cybernetics, 2018, 9(7): 1131-1140. DOI URL
[23]	DIAS D B, MADEO R C B, ROCHA T, et al. Hand movement recognition for Brazilian sign language: a study using distance-based neural networks[C]// Proceedings of the 2009 International Joint Conference on Neural Networks, Atlanta, Jun 14-19,2009. Washington: IEEE Computer Society, 2009: 697-704.
[24]	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 URL
[25]	JAIN A K, LAW M H C. Data clustering: user's dilemma[C]// LNCS 3776: Proceedings of the 1st International Conference on Pattern Recognition and Machine Intelligence, Koikata,Dec 20-22, 2005. Berlin, Heidelberg: Springer, 2005: 1-10.
[26]	CHANG H, YEUNG D Y. Robust path-based spectral clustering[J]. Pattern Recognition, 2008, 41(1): 191-203. DOI URL
[27]	VEENMAN C J, REINDERS M J T, BACKER E. A maximum variance cluster algorithm[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2002, 24(9): 1273-1280. DOI URL
[28]	FRNTI P, VIRMAJOKI O. Iterative shrinking method for clustering problems[J]. Pattern Recognition, 2006, 39(5): 761-775. DOI URL
[29]	CHARYTANOWICZ M, NIEWCZAS J, KULCZYCKI P, et al. Complete gradient clustering algorithm for features analysis of X-ray images[J]. Information Technologies in Biomedicine, 2010, 69: 15-24.
[30]	BLAKE C L, MERZ C J. UCI repository of machine learning database[EB/OL]. [2020-12-28]. http://archive.ics.uci.edu/ml/index.html.
[31]	WU C R, JIA L, YAO J, et al. Efficient clustering method based on density peaks with symmetric neighborhood relationship[J]. IEEE Access, 2019, 7: 60684-60696. DOI URL
[32]	BREIMAN L, FRIEDMAN J H, OLSHENR A, et al. Classificationand regression trees(CART)[J]. Biometrics, 1984, 40(3): 358.

数据集	数据来源	样本规模	数据维数	类簇个数
Aggregation	[23]	788	2	7
Flame	[24]	240	2	2
Jain	[25]	373	2	2
Pathbased	[26]	300	2	3
Spiral	[26]	312	2	3
R15	[27]	600	2	15
D31	[27]	3 100	2	31
S2	[28]	5 000	2	15

数据集	数据来源	样本规模	数据维数	类簇个数
Aggregation	[23]	788	2	7
Flame	[24]	240	2	2
Jain	[25]	373	2	2
Pathbased	[26]	300	2	3
Spiral	[26]	312	2	3
R15	[27]	600	2	15
D31	[27]	3 100	2	31
S2	[28]	5 000	2	15

数据集	数据来源	样本规模	数据维数	类簇个数
Seeds	[29]	210	7	3
Libras	[23]	360	90	15
Iris	[30]	150	4	3
Wine	[30]	178	13	3
Ecoli	[30]	336	8	8
Dermatology	[30]	366	33	6
Glass	[31]	214	9	6
Waveform	[32]	2 310	19	7

数据集	数据来源	样本规模	数据维数	类簇个数
Seeds	[29]	210	7	3
Libras	[23]	360	90	15
Iris	[30]	150	4	3
Wine	[30]	178	13	3
Ecoli	[30]	336	8	8
Dermatology	[30]	366	33	6
Glass	[31]	214	9	6
Waveform	[32]	2 310	19	7

Clustering algorithm	Aggregation				Spiral
Clustering algorithm	AMI	ARI	FMI	Arg-	AMI	ARI	FMI	Arg-
WKMM-DPC	1.000 0	1.000 0	1.000 0	14	1.000 0	1.000 0	1.000 0	9
FKNN-DPC	0.990 5	0.994 9	0.996 0	20	1.000 0	1.000 0	1.000 0	6
DPCSA	0.953 7	0.958 1	0.967 3	—	1.000 0	1.000 0	1.000 0	—
FNDPC	0.986 4	0.991 3	0.993 2	0.02	1.000 0	1.000 0	1.000 0	0.07
DPC	0.992 2	0.995 6	0.996 6	4.00	1.000 0	1.000 0	1.000 0	1.80
DBSCAN	0.968 1	0.977 9	0.982 7	0.04/6	1.000 0	1.000 0	1.000 0	0.04/2
Clustering algorithm	Flame				R15
Clustering algorithm	AMI	ARI	FMI	Arg-	AMI	ARI	FMI	Arg-
WKMM-DPC	1.000 0	1.000 0	1.000 0	18	0.994 2	0.992 8	0.993 2	17
FKNN-DPC	0.926 7	0.966 7	0.984 5	5	0.993 8	0.992 8	0.993 3	25
DPCSA	1.000 0	1.000 0	1.000 0	—	0.988 5	0.985 7	0.986 6	—
FNDPC	1.000 0	1.000 0	1.000 0	0.13	0.993 8	0.992 8	0.993 3	0.03
DPC	1.000 0	1.000 0	1.000 0	2.80	0.993 8	0.992 8	0.993 2	0.70
DBSCAN	0.866 5	0.938 8	0.971 2	0.09/8	0.983 2	0.975 8	0.979 9	0.04/12
Clustering algorithm	Jain				D31
Clustering algorithm	AMI	ARI	FMI	Arg-	AMI	ARI	FMI	Arg-
WKMM-DPC	1.000 0	1.000 0	1.000 0	16	0.957 1	0.915 9	0.919 1	36
FKNN-DPC	0.709 2	0.822 4	0.935 9	43	0.965 4	0.952 3	0.953 8	28
DPCSA	0.216 7	0.044 2	0.592 4	—	0.955 2	0.935 3	0.937 4	—
FNDPC	0.596 1	0.725 7	0.905 1	0.47	0.955 5	0.936 4	0.938 5	0.04
DPC	0.618 3	0.714 6	0.881 9	0.90	0.955 4	0.936 5	0.938 5	0.60
DBSCAN	0.928 1	0.975 8	0.990 6	0.08/2	0.903 2	0.809 5	0.816 3	0.04/47
Clustering algorithm	Pathbased				S2
Clustering algorithm	AMI	ARI	FMI	Arg-	AMI	ARI	FMI	Arg-
WKMM-DPC	0.751 0	0.715 0	0.811 1	18	0.944 5	0.935 6	0.939 9	25
FKNN-DPC	0.930 5	0.949 9	0.966 5	9	0.918 0	0.888 9	0.896 3	22
DPCSA	0.707 3	0.613 3	0.751 1	—	0.933 3	0.915 2	0.920 9	—
FNDPC	0.575 1	0.506 7	0.706 5	0.01	0.943 1	0.935 1	0.939 5	0.03
DPC	0.521 2	0.471 7	0.666 4	3.80	0.943 7	0.935 2	0.939 5	1.50
DBSCAN	0.872 1	0.901 1	0.934 0	0.08/10	0.878 1	0.751 0	0.776 7	0.04/30

加权K近邻和多簇合并的密度峰值聚类算法

Weighted K-nearest Neighbors and Multi-cluster Merge Density Peaks Clustering Algorithm

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图/表 18

参考文献 32

相关文章 15

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Metrics

算法	秩均值
算法	AMI	ARI	FMI
WKMM-DPC	5.13	4.56	4.50
FKNN-DPC	3.94	4.00	4.19
DPCSA	2.50	2.63	2.63
FNDPC	3.38	3.56	3.63
DPC	3.63	3.81	3.63
DBSCAN	2.44	2.44	2.44

算法	秩均值
算法	AMI	ARI	FMI
WKMM-DPC	4.81	5.06	5.19
FKNN-DPC	4.56	4.69	4.44
DPCSA	2.31	2.44	2.81
FNDPC	4.19	4.56	4.56
DPC	2.88	3.00	2.75
DBSCAN	2.25	1.25	1.25

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Clustering algorithm	Seeds				Ecoli
Clustering algorithm	AMI	ARI	FMI	Arg-	AMI	ARI	FMI	Arg-
WKMM-DPC	0.705 4	0.730 6	0.820 7	5	0.716 2	0.747 9	0.824 4	25
FKNN-DPC	0.775 7	0.802 4	0.868 2	9	0.587 8	0.589 4	0.702 7	2
DPCSA	0.660 9	0.687 3	0.791 8	—	0.440 6	0.459 3	0.646 7	—
FNDPC	0.713 6	0.754 5	0.836 1	0.07	0.483 3	0.561 8	0.717 8	0.35
DPC	0.729 8	0.767 0	0.844 4	0.70	0.497 8	0.446 5	0.577 5	0.40
DBSCAN	0.591 2	0.529 1	0.671 1	0.24/16	0.516 9	0.536 7	0.669 2	0.20/22
Clustering algorithm	Libras				Dermatology
Clustering algorithm	AMI	ARI	FMI	Arg-	AMI	ARI	FMI	Arg-
WKMM-DPC	0.676 7	0.411 4	0.458 6	8	0.902 3	0.841 6	0.880 8	61
FKNN-DPC	0.555 4	0.345 9	0.404 4	10	0.806 6	0.836 1	0.870 9	35
DPCSA	0.538 8	0.309 5	0.379 1	—	0.745 1	0.606 2	0.689 6	—
FNDPC	0.549 4	0.329 0	0.386 9	0.17	0.789 8	0.799 5	0.841 8	0.17
DPC	0.535 8	0.319 3	0.371 7	0.30	0.608 6	0.611 0	0.705 6	1.50
DBSCAN	0.591 2	0.196 5	0.257 0	0.90/2	0.625 0	0.415 2	0.538 5	0.99/3
Clustering algorithm	Iris				Glass
Clustering algorithm	AMI	ARI	FMI	Arg-	AMI	ARI	FMI	Arg-
WKMM-DPC	0.883 1	0.903 8	0.935 5	11	0.699 3	0.668 9	0.758 3	12
FKNN-DPC	0.883 1	0.903 8	0.935 5	22	0.489 6	0.510 6	0.654 1	9
DPCSA	0.883 1	0.903 8	0.935 5	—	0.237 5	0.198 6	0.545 6	—
FNDPC	0.883 1	0.903 8	0.935 5	0.11	0.563 5	0.576 4	0.687 7	0.09
DPC	0.724 7	0.703 7	0.803 2	0.20	0.556 5	0.533 5	0.655 9	0.90
DBSCAN	0.640 1	0.612 0	0.729 1	0.12/5	0.504 0	0.110 6	0.278 4	0.1/1
Clustering algorithm	Wine				Waveform
Clustering algorithm	AMI	ARI	FMI	Arg-	AMI	ARI	FMI	Arg-
WKMM-DPC	0.772 9	0.771 3	0.848 0	14	0.299 1	0.282 7	0.582 5	8
FKNN-DPC	0.848 1	0.883 9	0.922 9	8	0.323 9	0.267 1	0.524 4	2
DPCSA	0.748 0	0.741 4	0.828 3	—	0.251 0	0.223 6	0.532 7	—
FNDPC	0.789 8	0.802 5	0.868 6	0.26	0.329 3	0.283 0	0.544 2	0.34
DPC	0.706 5	0.672 4	0.783 5	2.00	0.326 1	0.269 8	0.529 2	0.10
DBSCAN	0.590 5	0.529 2	0.712 1	0.50/21	0.104 9	0.009 4	0.481 1	0.38/5

Clustering algorithm	Seeds				Ecoli
Clustering algorithm	AMI	ARI	FMI	Arg-	AMI	ARI	FMI	Arg-
WKMM-DPC	0.705 4	0.730 6	0.820 7	5	0.716 2	0.747 9	0.824 4	25
FKNN-DPC	0.775 7	0.802 4	0.868 2	9	0.587 8	0.589 4	0.702 7	2
DPCSA	0.660 9	0.687 3	0.791 8	—	0.440 6	0.459 3	0.646 7	—
FNDPC	0.713 6	0.754 5	0.836 1	0.07	0.483 3	0.561 8	0.717 8	0.35
DPC	0.729 8	0.767 0	0.844 4	0.70	0.497 8	0.446 5	0.577 5	0.40
DBSCAN	0.591 2	0.529 1	0.671 1	0.24/16	0.516 9	0.536 7	0.669 2	0.20/22
Clustering algorithm	Libras				Dermatology
Clustering algorithm	AMI	ARI	FMI	Arg-	AMI	ARI	FMI	Arg-
WKMM-DPC	0.676 7	0.411 4	0.458 6	8	0.902 3	0.841 6	0.880 8	61
FKNN-DPC	0.555 4	0.345 9	0.404 4	10	0.806 6	0.836 1	0.870 9	35
DPCSA	0.538 8	0.309 5	0.379 1	—	0.745 1	0.606 2	0.689 6	—
FNDPC	0.549 4	0.329 0	0.386 9	0.17	0.789 8	0.799 5	0.841 8	0.17
DPC	0.535 8	0.319 3	0.371 7	0.30	0.608 6	0.611 0	0.705 6	1.50
DBSCAN	0.591 2	0.196 5	0.257 0	0.90/2	0.625 0	0.415 2	0.538 5	0.99/3
Clustering algorithm	Iris				Glass
Clustering algorithm	AMI	ARI	FMI	Arg-	AMI	ARI	FMI	Arg-
WKMM-DPC	0.883 1	0.903 8	0.935 5	11	0.699 3	0.668 9	0.758 3	12
FKNN-DPC	0.883 1	0.903 8	0.935 5	22	0.489 6	0.510 6	0.654 1	9
DPCSA	0.883 1	0.903 8	0.935 5	—	0.237 5	0.198 6	0.545 6	—
FNDPC	0.883 1	0.903 8	0.935 5	0.11	0.563 5	0.576 4	0.687 7	0.09
DPC	0.724 7	0.703 7	0.803 2	0.20	0.556 5	0.533 5	0.655 9	0.90
DBSCAN	0.640 1	0.612 0	0.729 1	0.12/5	0.504 0	0.110 6	0.278 4	0.1/1
Clustering algorithm	Wine				Waveform
Clustering algorithm	AMI	ARI	FMI	Arg-	AMI	ARI	FMI	Arg-
WKMM-DPC	0.772 9	0.771 3	0.848 0	14	0.299 1	0.282 7	0.582 5	8
FKNN-DPC	0.848 1	0.883 9	0.922 9	8	0.323 9	0.267 1	0.524 4	2
DPCSA	0.748 0	0.741 4	0.828 3	—	0.251 0	0.223 6	0.532 7	—
FNDPC	0.789 8	0.802 5	0.868 6	0.26	0.329 3	0.283 0	0.544 2	0.34
DPC	0.706 5	0.672 4	0.783 5	2.00	0.326 1	0.269 8	0.529 2	0.10
DBSCAN	0.590 5	0.529 2	0.712 1	0.50/21	0.104 9	0.009 4	0.481 1	0.38/5