Agglomerative hierarchical clustering algorithm based on hesitant fuzzy set

doi:10.11772/j.issn.1001-9081.2023010094

Journal of Computer Applications ›› 2023, Vol. 43 ›› Issue (12): 3755-3763.DOI: 10.11772/j.issn.1001-9081.2023010094

• Data science and technology • Previous Articles Next Articles

Agglomerative hierarchical clustering algorithm based on hesitant fuzzy set

Wenquan LI(), Yimin MAO, Xindong PENG

School of Information Engineering，Shaoguan University，Shaoguan Guangdong 512005，China

Received:2023-02-07 Revised:2023-05-05 Accepted:2023-05-08 Online:2023-06-06 Published:2023-12-10
Contact: Wenquan LI
About author:MAO Yimin， born in 1970， Ph. D.， professor. Her research interests include data mining， big data security.
PENG Xindong， born in 1990， Ph. D.， associate professor. His research interests include fuzzy mathematics， artificial intelligence.
Supported by:
National Natural Science Foundation of China(62006155);Scientific Research Project of Department of Education of Guangdong Province(2022ZDJS048);Characteristic Innovation Project in Ordinary Universities in Guangdong Province(2023KTSCX137)

基于犹豫模糊集的凝聚式层次聚类算法

李文全(), 毛伊敏, 彭新东

韶关学院信息工程学院，广东韶关 512005

通讯作者: 李文全
作者简介:李文全（1980—），男，江西龙南人，副教授，硕士，主要研究方向：数据挖掘、模糊数学；Email：78192128@qq.com
毛伊敏（1970—），女，新疆伊犁人，教授，博士，主要研究方向：数据挖掘、大数据安全
彭新东（1990—），男，江西九江人，副教授，博士，主要研究方向：模糊数学、人工智能。
基金资助:
国家自然科学基金资助项目(62006155);广东省教育厅科研项目(2022ZDJS048);广东省普通高校特色创新类项目(2023KTSCX137)

Abstract

Abstract:

Aiming at the problems of information distortion， poor objectivity of attribute weights， and high time complexity in hesitant fuzzy clustering analysis， an Agglomerative Hierarchical Clustering algorithm based on Hesitant Fuzzy set （AHCHF） was proposed. Firstly， the average value of hesitancy fuzzy elements was used to expand the data object with small hesitation. Secondly， the weights of data object before and after expansion were calculated by using the original information entropy and internal maximum difference， and the comprehensive attribute weight was determined according to the minimum discrimination information between the two weight vectors. Finally， with the goal of making the sum of weighted distances smaller， a center point construction method with constant hesitation was given. Experimental results on specific examples and synthetic datasets show that compared with the classic Hesitant Fuzzy Hierarchical Clustering algorithm （HFHC） and the recent Fuzzy Hierarchical Clustering Algorithm （FHCA）， the proposed AHCHF increases the mean Silhouette Coefficient （SC） by 23.99% and 9.28% respectively， and shortens the running time by 27.18% and 6.40% averagely and respectively， proving that the proposed algorithm can effectively solve the problems of information distortion and poor objectivity of attribute weights， and improve the clustering effect and performance well.

Key words: hesitant fuzzy set, clustering analysis, hesitation, data mining, fuzzy entropy

摘要：

针对犹豫模糊聚类分析存在信息失真、属性权重客观性差、时间复杂度高的问题，提出一种基于犹豫模糊集的凝聚式层次聚类算法（AHCHF）。首先，采用犹豫模糊元的平均值扩充犹豫度小的数据对象；其次，利用原始信息熵和内部最大差异计算数据对象扩充前后的权重，并根据两个权重向量之间的最小鉴别信息确定属性的综合权重；最后，以加权距离和更小为目标，给出犹豫度恒定的中心点构造方法。在具体实例和人造数据集上进行的实验结果表明，相较于经典的犹豫模糊层次聚类算法（HFHC）和较新的模糊层次聚类算法（FHCA），AHCHF的轮廓系数（SC）均值分别提高了23.99%和9.28%，运行时间分别平均减少了27.18%和6.40%。以上结果验证了所提算法可以有效解决信息失真、属性权重客观性差的问题，并较好地提升聚类效果和聚类性能。

关键词: 犹豫模糊集, 聚类分析, 犹豫度, 数据挖掘, 模糊熵

CLC Number:

TP391.7

Wenquan LI, Yimin MAO, Xindong PENG. Agglomerative hierarchical clustering algorithm based on hesitant fuzzy set[J]. Journal of Computer Applications, 2023, 43(12): 3755-3763.

李文全, 毛伊敏, 彭新东. 基于犹豫模糊集的凝聚式层次聚类算法[J]. 《计算机应用》唯一官方网站, 2023, 43(12): 3755-3763.

Figures/Tables 13

Tab.1 Hesitant fuzzy object sets

对象集	$h (o 1)$	$h (o 2)$	$h (o 3)$
$r 1$	｛0.2，0.3｝	｛0.3，0.4，0.5｝	｛0.5｝
$r 2$	｛0.7，0.8，0.9｝	｛0.5，0.8｝	｛0.6，0.7，0.8｝
$r 3$	｛0.4，0.5｝	｛0.4，0.5，0.6｝	｛0.5，0.6，0.7｝
$r 4$	｛0.3，0.4，0.5｝	｛0.5，0.8｝	｛0.2，0.5｝
$r 5$	｛0.7｝	｛0.7，0.8｝	｛0.7，0.8，0.9｝
$r 6$	｛0.4，0.5｝	｛0.3，0.6｝	｛0.3，0.4｝
$r 7$	｛0.8｝	｛0.7，0.9｝	｛0.7，0.8｝

Tab.1 Hesitant fuzzy object sets

对象集	$h (o 1)$	$h (o 2)$	$h (o 3)$
$r 1$	｛0.2，0.3｝	｛0.3，0.4，0.5｝	｛0.5｝
$r 2$	｛0.7，0.8，0.9｝	｛0.5，0.8｝	｛0.6，0.7，0.8｝
$r 3$	｛0.4，0.5｝	｛0.4，0.5，0.6｝	｛0.5，0.6，0.7｝
$r 4$	｛0.3，0.4，0.5｝	｛0.5，0.8｝	｛0.2，0.5｝
$r 5$	｛0.7｝	｛0.7，0.8｝	｛0.7，0.8，0.9｝
$r 6$	｛0.4，0.5｝	｛0.3，0.6｝	｛0.3，0.4｝
$r 7$	｛0.8｝	｛0.7，0.9｝	｛0.7，0.8｝

Tab.2 Expanded hesitant fuzzy object sets

对象集	$h (o 1)$	$h (o 2)$	$h (o 3)$
$r 1$	｛0.20，0.25，0.30｝	｛0.30，0.40，0.50｝	｛0.50，0.50，0.50｝
$r 2$	｛0.70，0.80，0.90｝	｛0.50，0.65，0.80｝	｛0.60，0.70，0.80｝
$r 3$	｛0.40，0.45，0.50｝	｛0.40，0.50，0.60｝	｛0.50，0.60，0.70｝
$r 4$	｛0.30，0.40，0.50｝	｛0.50，0.65，0.80｝	｛0.20，0.35，0.50｝
$r 5$	｛0.70，0.70，0.70｝	｛0.70，0.75，0.80｝	｛0.70，0.80，0.90｝
$r 6$	｛0.40，0.45，0.50｝	｛0.30，0.45，0.60｝	｛0.30，0.35，0.40｝
$r 7$	｛0.80，0.80，0.80｝	｛0.70，0.80，0.90｝	｛0.70，0.75，0.80｝

Tab.2 Expanded hesitant fuzzy object sets

对象集	$h (o 1)$	$h (o 2)$	$h (o 3)$
$r 1$	｛0.20，0.25，0.30｝	｛0.30，0.40，0.50｝	｛0.50，0.50，0.50｝
$r 2$	｛0.70，0.80，0.90｝	｛0.50，0.65，0.80｝	｛0.60，0.70，0.80｝
$r 3$	｛0.40，0.45，0.50｝	｛0.40，0.50，0.60｝	｛0.50，0.60，0.70｝
$r 4$	｛0.30，0.40，0.50｝	｛0.50，0.65，0.80｝	｛0.20，0.35，0.50｝
$r 5$	｛0.70，0.70，0.70｝	｛0.70，0.75，0.80｝	｛0.70，0.80，0.90｝
$r 6$	｛0.40，0.45，0.50｝	｛0.30，0.45，0.60｝	｛0.30，0.35，0.40｝
$r 7$	｛0.80，0.80，0.80｝	｛0.70，0.80，0.90｝	｛0.70，0.75，0.80｝

Tab.3 Attribute weights calculated by different methods

权重	$h (o 1)$	$h (o 2)$	$h (o 3)$
原始权重 $w j ¯$	0.412 12	0.267 31	0.320 57
扩充后权重 $w j ̲$	0.383 02	0.277 68	0.339 30
综合权重 $w j$	0.397 48	0.272 56	0.329 95

Tab.3 Attribute weights calculated by different methods

权重	$h (o 1)$	$h (o 2)$	$h (o 3)$
原始权重 $w j ¯$	0.412 12	0.267 31	0.320 57
扩充后权重 $w j ̲$	0.383 02	0.277 68	0.339 30
综合权重 $w j$	0.397 48	0.272 56	0.329 95

Tab.4 Distance among hesitation fuzzy objects

簇	$r 1$	$r 2$	$r 3$	$r 4$	$r 5$	$r 6$	$r 7$
$r 1$	0.000 00	0.392 16	0.155 32	0.198 67	0.383 26	0.158 14	0.430 91
$r 2$		0.000 00	0.243 39	0.323 37	0.120 28	0.318 16	0.103 01
$r 3$			0.000 00	0.171 51	0.237 05	0.149 36	0.286 11
$r 4$				0.000 00	0.332 16	0.121 48	0.357 53
$r 5$					0.000 00	0.345 28	0.080 53
$r 6$						0.000 00	0.368 77
$r 7$							0.000 00

Tab.4 Distance among hesitation fuzzy objects

簇	$r 1$	$r 2$	$r 3$	$r 4$	$r 5$	$r 6$	$r 7$
$r 1$	0.000 00	0.392 16	0.155 32	0.198 67	0.383 26	0.158 14	0.430 91
$r 2$		0.000 00	0.243 39	0.323 37	0.120 28	0.318 16	0.103 01
$r 3$			0.000 00	0.171 51	0.237 05	0.149 36	0.286 11
$r 4$				0.000 00	0.332 16	0.121 48	0.357 53
$r 5$					0.000 00	0.345 28	0.080 53
$r 6$						0.000 00	0.368 77
$r 7$							0.000 00

Tab.5 Comparison of clustering results by different algorithms

簇数	AHCHF	HFHC	FHCA
7	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$
6	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$
5	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4, r 6}, {r 5, r 7}$	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$
4	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 2}, {r 3, r 4, r 6}, {r 5, r 7}$	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$
3	${r 1}, {r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 3}, {r 1, r 4, r 6}, {r 2, r 5, r 7}$
2	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$
1	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$

Tab.5 Comparison of clustering results by different algorithms

簇数	AHCHF	HFHC	FHCA
7	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$
6	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$
5	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4, r 6}, {r 5, r 7}$	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$
4	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 2}, {r 3, r 4, r 6}, {r 5, r 7}$	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$
3	${r 1}, {r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 3}, {r 1, r 4, r 6}, {r 2, r 5, r 7}$
2	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$
1	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$

Tab.6 Mean silhouette coefficients of different algorithms

算法	SC均值	算法	SC均值
AHCHF	0.710 75	FHCA	0.650 38
HFHC	0.573 23

Tab.7 Comparison of clustering results by proposed algorithm under different expansion methods

簇数	平均值扩充	乐观扩充	悲观扩充
7	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$
6	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$
5	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$
4	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$
3	${r 1}, {r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3}, {r 4, r 6}, {r 2, r 5, r 7}$
2	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$
1	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$

Tab.7 Comparison of clustering results by proposed algorithm under different expansion methods

簇数	平均值扩充	乐观扩充	悲观扩充
7	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$
6	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$
5	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$
4	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$
3	${r 1}, {r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3}, {r 4, r 6}, {r 2, r 5, r 7}$
2	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$
1	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$

Tab.8 Mean silhouette coefficients of proposed algorithm under different expansion methods

扩充方法	SC均值	扩充方法	SC均值
平均值扩充	0.710 75	悲观扩充	0.707 47
乐观扩充	0.700 85

Tab.9 Comparison of center distance of different algorithms

算法	$D i s (d i)$	算法	$D i s (d i)$
AHCHF	0.188 92	FHCA	0.202 34
HFHC	0.291 66

Tab.9 Comparison of center distance of different algorithms

算法	$D i s (d i)$	算法	$D i s (d i)$
AHCHF	0.188 92	FHCA	0.202 34
HFHC	0.291 66

Tab.10 Comparison of clustering results under different weights

簇数	综合权重（0.397 48，0.272 56，0.329 95）	主观权重1 （0.333 33，0.333 33，0.333 33）	主观权重2 （0.6，0.2，0.2）	主观权重3 （0.1，0.45，0.45）
7	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$
6	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$
5	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$
4	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$
3	${r 1}, {r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 4}, {r 1, r 3, r 6}, {r 2, r 5, r 7}$
2	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$
1	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$

Tab.10 Comparison of clustering results under different weights

簇数	综合权重（0.397 48，0.272 56，0.329 95）	主观权重1 （0.333 33，0.333 33，0.333 33）	主观权重2 （0.6，0.2，0.2）	主观权重3 （0.1，0.45，0.45）
7	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 5}, {r 6}, {r 7}$
6	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$	${r 1}, {r 2}, {r 3}, {r 4}, {r 6}, {r 5, r 7}$
5	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$
4	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3}, {r 4}, {r 6}, {r 2, r 5, r 7}$
3	${r 1}, {r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3}, {r 4, r 6}, {r 2, r 5, r 7}$	${r 1}, {r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 4}, {r 1, r 3, r 6}, {r 2, r 5, r 7}$
2	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$	${r 1, r 3, r 4, r 6}, {r 2, r 5, r 7}$
1	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$	${r 1, r 2, r 3, r 4, r 5, r 6, r 7}$

Tab.11 Mean silhouette coefficients under different weights

权重	SC均值	权重	SC均值
综合权重	0.710 75	主观权重2	0.712 35
主观权重1	0.703 85	主观权重3	0.677 48

Tab.12 Comparison of clustering time among different algorithms

样本数	不同算法的聚类时间/ms
样本数	AHCHF	HFHC	FHCA
10	8.687	11.143	9.065
20	17.401	22.810	18.416
30	26.486	35.456	28.230
40	34.935	50.096	37.821
50	45.242	69.168	49.443

Tab.13 Comparison of spatial complexity among different algorithms

算法	不同情況下空间复杂度
算法	最好情况	最坏情况
HFHC	$O (n)$	$O (犹豫度 n)$
FHCA	$O (n)$	$O (犹豫度 × n)$
AHCHF	$O (n)$	$O (犹豫度 × n)$

Tab.13 Comparison of spatial complexity among different algorithms

算法	不同情況下空间复杂度
算法	最好情况	最坏情况
HFHC	$O (n)$	$O (犹豫度 n)$
FHCA	$O (n)$	$O (犹豫度 × n)$
AHCHF	$O (n)$	$O (犹豫度 × n)$

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Agglomerative hierarchical clustering algorithm based on hesitant fuzzy set

基于犹豫模糊集的凝聚式层次聚类算法

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