JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2020, Vol. 55 ›› Issue (3): 58-69.doi: 10.6040/j.issn.1671-9352.1.2019.154

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Clustering method for multi-label symbolic value partition

Liu-ying WEN*(),Wei YUAN   

  1. School of Computer Science, Southwest Petroleum University, Chengdu 610500, Sichuan, China
  • Received:2019-10-31 Online:2020-03-20 Published:2020-03-27
  • Contact: Liu-ying WEN E-mail:wenliuying1983@163.com

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Abstract:

A clustering method for multi-label symbolic value partition (CMSVP) is proposed. First, the label ranking and K-means algorithms are employed to cluster the original label information. Then, an undirected weighted graph is constructed for each attribute. Each node represents an attribute value, and the weight of each edge represents the similarity between the nodes. Finally, random walks are performed on all undirected weighted graphs to obtain a clustering scheme of attribute values. The experiment is conducted on six multi-label data sets. The results show that the CMSVP algorithm can improve the classification performance of data while effectively compressing the data.

Key words: symbolic value partition, clustering, random walk, undirected weighted graph, multi-label

CLC Number: 

  • TP391

Table 1

Example of a multi-label decision system D"

U A L
a1 a2 l1 l2 l3
x1 0 6 1 0 1
x2 1 1 0 0 1
x3 5 3 0 1 0
x4 2 1 1 1 0
x5 1 2 1 0 1
x6 3 0 0 1 1
x7 4 5 1 1 1
x8 6 4 1 1 0

Table 2

New multi-label decision system DP"

U AP L
a1P a2P l1 l2 l3
x1 {0, 1, 4} {2, 5, 6} 1 0 1
x2 {0, 1, 4} {0, 1} 0 0 1
x3 {5, 6} {3, 4} 0 1 0
x4 {2, 3} {0, 1} 1 1 0
x5 {0, 1, 4} {2, 5, 6} 1 0 1
x6 {2, 3} {0, 1} 0 1 1
x7 {0, 1, 4} {2, 5, 6} 1 1 1
x8 {5, 6} {3, 4} 1 1 0

Table 3

Definition of symbols"

符号 定义说明
N U中实例的个数
q L中标签的个数
M A中属性的个数
CN 记录相同属性值的个数
Dai 关于属性ai的决策子表
V′i Dai中去重后的属性值域
Pj V′i中第j个属性值的概率
Gi 属性ai的无向加权图
vt Gi中的节点t
MGi 关于Gi的相关转移矩阵

Table 4

New decision system D′"

U A L′
a1 a2 {l1, l3} {l2}
x1 0 6 1 0
x2 1 1 0 0
x3 5 3 0 1
x4 2 1 0 1
x5 1 2 1 0
x6 3 0 0 1
x7 4 5 1 1
x8 6 4 0 1

Fig.1

Graph construction for attribute a1"

Fig.2

Clustering process of G1"

Table 5

Time complexity of CMSVP"

步骤描述 复杂度
标签聚类 O(qN)
构建无向加权图 O(qMN2)
随机游走聚类 O(RMN2)
总复杂度 O((q+R)MN2)

Table 6

Dataset information"

名称 领域 对象数目 属性数目 标签数目
Birds Audio 645 260 19
Cal500 Music 502 68 174
Emotions Music 593 72 6
Flags Images 194 19 7
Scene Image 2 407 294 6
Yeast Biology 2 417 103 14

Table 7

Algorithm performance"

数据集 Rank 压缩率
原始 处理后
Birds 3 279 1 045 0.319
Cal500 1 145 435 0.380
Emotions 1 806 275 0.152
Flags 104 48 0.462
Scene 5 785 981 0.170
Yeast 1 968 368 0.187

Table 8

Run time of each algorithm  ms"

数据集 IG-BR IG-LP RF-BR RF-LP CMSVP
Birds 2 191 2 057 19 176 3 741 66 151
Cal500 1 675 1 840 28 034 1 983 505 633
Emotions 473 456 1 696 764 10 746
Flags 241 301 333 334 5 133
Scene 5 793 5 669 91 033 20 388 259 470
Yeast 1 501 1 433 70 213 9 185 269 810

Table 9

Time complexity of each algorithm"

算法 时间复杂度
RF-BR O(qN+qMN+q2M2)
RF-LP O(qN+qMN+q2M2)
IG-BR O(qN+M2+qMN)
IG-LP O(qN+M2+qMN)
CMSVP O((q+R)MN2)

Table 10

Performance comparison under RAkEL classifier"

Hamming Loss(↓)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 0.051 7±0.007 0 0.036 4±0.009 6 0.050 3±0.005 6 0.052 6±0.006 1 0.050 8±0.006 2 0.049 5±0.005 9
Cal500 0.140 2±0.002 8 0.099 1±0.002 7 0.139 0±0.002 5 0.138 4±0.002 4 0.139 2±0.002 9 0.138 5±0.002 1
Emotions 0.238 4±0.015 3 0.172 1±0.026 6 0.271 8±0.031 8 0.260 0±0.034 0 0.248 5±0.034 7 0.239 3±0.036 7
Flags 0.271 9±0.032 5 0.217 6±0.053 7 0.257 9±0.024 4 0.260 8±0.022 8 0.265 2±0.027 7 0.265 2±0.027 7
Scene 0.137 3±0.009 5 0.146 5±0.011 5 0.158 8±0.007 4 0.158 1±0.007 1 0.159 5±0.009 3 0.159 2±0.008 3
Yeast 0.244 7±0.011 4 0.172 2±0.009 3 0.224 8±0.010 9 0.223 4±0.011 6 0.225 1±0.012 0 0.225 2±0.011 8
Ranking Loss(↓)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 0.231 1±0.027 8 0.112 5±0.030 8 0.215 7±0.020 1 0.244 3±0.026 6 0.239 5±0.025 4 0.212 1±0.019 7
Cal500 0.422 8±0.010 1 0.434 1±0.007 0 0.413 2±0.009 6 0.417 4±0.008 9 0.395 5±0.007 4 0.413 9±0.009 6
Emotions 0.227 2±0.026 5 0.100 8±0.027 5 0.288 3±0.043 6 0.284 0±0.049 3 0.262 4±0.038 9 0.261 9±0.043 0
Flags 0.240 4±0.055 1 0.233 7±0.075 8 0.273 3±0.048 4 0.271 4±0.050 1 0.276 9±0.049 8 0.276 9±0.049 8
Scene 0.172 2±0.014 9 0.155 7±0.026 8 0.229 3±0.025 0 0.233 0±0.019 2 0.232 1±0.014 6 0.232 2±0.016 0
Yeast 0.241 7±0.015 7 0.185 4±0.015 1 0.271 8±0.014 0 0.278 8±0.015 2 0.262 3±0.017 6 0.262 3±0.019 5
One Error(↓)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 0.820 3±0.044 8 0.877 5±0.052 5 0.773 6±0.043 6 0.835 5±0.045 3 0.818 4±0.045 0 0.765 8±0.061 7
Cal500 0.266 7±0.059 7 0.591 2±0.086 4 0.205 0±0.067 8 0.207 1±0.072 7 0.258 8±0.056 9 0.219 1±0.077 8
Emotions 0.377 8±0.060 7 0.495 9±0.049 3 0.438 6±0.063 2 0.438 4±0.063 6 0.396 4±0.063 2 0.386 3±0.072 4
Flags 0.257 1±0.098 5 0.412 6±0.058 1 0.292 9±0.095 2 0.277 6±0.104 8 0.283 2±0.083 8 0.283 2±0.083 8
Scene 0.395 9±0.033 1 0.719 2±0.032 7 0.491 9±0.036 0 0.491 5±0.026 7 0.489 8±0.026 3 0.489 4±0.025 9
Yeast 0.295 0±0.043 1 0.304 5±0.027 2 0.260 2±0.032 4 0.256 9±0.027 7 0.260 2±0.026 8 0.257 3±0.027 1
Coverage(↓)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 5.421 4±0.502 9 1.618 1±0.387 7 5.313 2±0.408 9 5.684 0±0.443 3 5.670 4±0.467 4 5.286 0±0.419 6
Cal500 168.929 2±0.555 4 108.805 6±1.413 3 168.893 1±0.535 3 168.872 0±0.741 1 168.843 5±0.676 0 168.798 3±0.743 8
Emotions 2.104 6±0.120 0 0.460 8±0.117 1 2.401 5±0.219 7 2.401 7±0.277 3 2.299 9±0.208 2 2.299 9±0.221 7
Flags 3.921 3±0.497 1 1.744 2±0.423 8 4.093 2±0.423 8 4.103 4±0.427 9 4.112 4±0.433 1 4.112 4±0.433 1
Scene 0.953 4±0.064 2 0.492 7±0.090 2 1.235 9±0.118 0 1.257 5±0.094 0 1.253 8±0.077 0 1.255 0±0.079 3
Yeast 7.989 9±0.300 9 3.022 1±0.173 1 8.561 3±0.263 2 8.670 0±0.310 3 8.398 4±0.349 9 8.349 6±0.403 4
Average Precision(↑)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 0.356 7±0.050 8 0.464 7±0.112 8 0.409 9±0.041 3 0.324 9±0.048 9 0.344 4±0.044 6 0.414 6±0.057 1
Cal500 0.363 8±0.011 2 0.193 8±0.011 3 0.376 8±0.011 6 0.374 0±0.011 3 0.385 2±0.011 4 0.375 1±0.011 3
Emotions 0.740 2±0.027 6 0.838 4±0.042 0 0.690 0±0.038 8 0.693 4±0.041 3 0.718 3±0.040 4 0.722 4±0.042 7
Flags 0.803 2±0.041 2 0.759 6±0.059 7 0.786 3±0.038 7 0.788 4±0.042 3 0.783 5±0.038 8 0.783 5±0.038 8
Scene 0.748 2±0.018 0 0.707 0±0.037 4 0.684 0±0.025 8 0.681 1±0.019 6 0.682 2±0.016 9 0.682 5±0.017 0
Yeast 0.691 5±0.022 0 0.728 1±0.016 7 0.693 1±0.019 2 0.689 8±0.018 8 0.698 3±0.016 9 0.697 7±0.018 8

Table 11

Performance comparison under MLkNN classifier"

Hamming Loss(↓)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 0.053 8±0.005 5 0.033 9±0.006 6 0.050 9±0.006 8 0.051 2±0.006 0 0.049 4±0.005 8 0.050 2±0.006 9
Cal500 0.138 4±0.003 1 0.097 9±0.002 9 0.138 3±0.002 8 0.138 6±0.002 6 0.138 8±0.002 8 0.138 4±0.003 0
Emotions 0.199 0±0.016 0 0.131 6±0.016 5 0.233 5±0.025 8 0.240 6±0.023 5 0.245 4±0.020 8 0.239 0±0.018 8
Flags 0.284 5±0.039 4 0.203 4±0.049 9 0.269 2±0.041 5 0.286 2±0.030 1 0.274 9±0.029 2 0.274 9±0.029 2
Scene 0.101 4±0.004 4 0.113 9±0.010 2 0.150 3±0.005 7 0.147 3±0.005 6 0.148 3±0.004 5 0.146 8±0.005 8
Yeast 0.207 9±0.013 6 0.151 1±0.010 5 0.217 6±0.012 1 0.221 8±0.008 3 0.216 3±0.011 8 0.213 5±0.011 5
Ranking Loss(↓)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 0.117 6±0.023 7 0.060 7±0.017 2 0.082 3±0.019 8 0.083 3±0.020 0 0.084 7±0.022 5 0.084 1±0.021 4
Cal500 0.184 8±0.007 7 0.205 4±0.009 6 0.184 0±0.006 6 0.184 9±0.007 2 0.183 9±0.006 8 0.185 1±0.007 3
Emotions 0.165 3±0.024 4 0.056 3±0.017 4 0.213 5±0.026 0 0.224 5±0.031 2 0.224 3±0.017 4 0.225 0±0.021 5
Flags 0.214 5±0.044 5 0.228 9±0.078 9 0.198 5±0.036 9 0.206 3±0.028 7 0.196 6±0.041 1 0.196 6±0.041 1
Scene 0.100 8±0.012 4 0.084 0±0.010 6 0.190 0±0.012 4 0.180 9±0.014 9 0.172 0±0.014 5 0.172 7±0.017 4
Yeast 0.185 1±0.016 0 0.135 3±0.010 9 0.193 2±0.013 0 0.200 3±0.014 3 0.189 5±0.016 9 0.187 1±0.015 2
One Error(↓)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 0.779 9±0.043 0 0.831 0±0.050 9 0.713 3±0.056 5 0.717 9±0.057 2 0.728 8±0.050 6 0.728 8±0.052 5
Cal500 0.119 3±0.065 0 0.651 4±0.040 8 0.119 3±0.061 7 0.119 3±0.065 5 0.119 2±0.069 9 0.117 3±0.069 8
Emotions 0.296 8±0.049 8 0.430 1±0.050 6 0.349 0±0.036 0 0.393 0±0.067 3 0.382 9±0.033 2 0.357 7±0.057 2
Flags 0.272 6±0.070 7 0.608 4±0.098 6 0.258 4±0.054 0 0.201 1±0.048 9 0.196 3±0.099 3 0.196 3±0.099 3
Scene 0.277 1±0.022 2 0.630 7±0.030 9 0.472 4±0.032 6 0.454 9±0.043 4 0.448 7±0.039 4 0.441 6±0.042 0
Yeast 0.248 7±0.027 0 0.254 0±0.029 8 0.254 9±0.030 1 0.250 3±0.030 5 0.244 9±0.024 4 0.247 0±0.031 0
Coverage(↓)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 3.154 9±0.521 9 0.941 3±0.246 1 2.388 0±0.395 9 2.414 7±0.403 7 2.421 8±0.415 3 2.417 4±0.389 3
Cal500 129.451 6±2.226 7 74.915 2±2.068 7 129.751 7±1.949 0 129.841 6±1.706 7 129.482 6±1.941 1 129.806 3±1.883 3
Emotions 1.796 6±0.152 1 0.327 6±0.092 4 2.035 3±0.154 2 2.062 7±0.147 4 2.081 0±0.126 9 2.084 5±0.123 3
Flags 3.793 7±0.502 8 0.993 7±0.284 4 3.662 9±0.457 4 3.757 4±0.324 5 3.667 9±0.359 9 3.667 9±0.359 9
Scene 0.592 0±0.055 2 0.275 1±0.034 8 1.035 3±0.051 7 0.988 3±0.064 4 0.945 6±0.062 8 0.945 1±0.080 4
Yeast 6.523 3±0.256 9 2.441 4±0.155 4 6.644 0±0.224 0 6.727 3±0.195 2 6.632 5±0.279 3 6.582 5±0.251 8
Average Precision(↑)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 0.512 7±0.063 7 0.594 9±0.081 7 0.605 3±0.065 3 0.597 9±0.067 6 0.591 7±0.056 7 0.590 8±0.067 6
Cal500 0.487 5±0.012 5 0.328 3±0.010 4 0.490 4±0.012 1 0.488 6±0.013 0 0.487 9±0.010 6 0.489 0±0.013 9
Emotions 0.790 1±0.022 6 0.897 8±0.027 4 0.748 6±0.028 5 0.728 9±0.035 0 0.735 7±0.023 0 0.741 5±0.026 3
Flags 0.805 1±0.032 3 0.713 3±0.067 3 0.818 4±0.025 3 0.820 8±0.023 8 0.826 3±0.045 8 0.826 3±0.045 8
Scene 0.831 6±0.014 8 0.814 1±0.025 6 0.706 8±0.018 7 0.718 6±0.025 7 0.724 7±0.022 9 0.728 4±0.025 9
Yeast 0.739 3±0.021 7 0.772 7±0.017 8 0.727 0±0.020 9 0.719 8±0.021 8 0.733 9±0.021 3 0.735 6±0.022 7

Table 12

Performance comparison under BRkNN classifier"

Hamming Loss(↓)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 0.053 2±0.006 1 0.035 9±0.008 3 0.049 4±0.005 8 0.050 1±0.005 7 0.049 8±0.005 8 0.049 4±0.006 6
Cal500 0.140 1±0.002 7 0.100 4±0.002 5 0.140 2±0.002 4 0.142 2±0.003 4 0.141 5±0.001 8 0.142 0±0.003 3
Emotions 0.198 2±0.016 8 0.130 0±0.021 0 0.223 1±0.022 6 0.227 7±0.021 5 0.238 0±0.014 6 0.231 6±0.016 6
Flags 0.273 1±0.027 2 0.208 5±0.048 3 0.260 6±0.030 4 0.267 0±0.024 2 0.254 8±0.027 0 0.254 8±0.027 0
Scene 0.109 5±0.005 1 0.123 5±0.007 6 0.155 7±0.007 5 0.149 1±0.008 4 0.151 3±0.006 8 0.150 2±0.009 2
Yeast 0.210 7±0.011 7 0.151 2±0.009 7 0.218 4±0.010 1 0.219 9±0.009 7 0.217 9±0.011 2 0.215 9±0.009 6
Ranking Loss(↓)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 0.151 7±0.023 5 0.070 8±0.016 1 0.099 7±0.022 8 0.099 3±0.018 8 0.100 8±0.020 2 0.099 9±0.022 0
Cal500 0.201 1±0.007 9 0.237 3±0.013 1 0.212 5±0.007 7 0.215 7±0.008 9 0.220 2±0.008 6 0.214 9±0.008 6
Emotions 0.172 1±0.033 1 0.057 7±0.017 6 0.202 5±0.024 8 0.219 6±0.023 7 0.225 2±0.024 2 0.225 0±0.022 5
Flags 0.210 4±0.032 9 0.228 0±0.073 3 0.190 2±0.037 7 0.191 0±0.032 0 0.184 3±0.024 8 0.184 3±0.024 8
Scene 0.113 7±0.012 1 0.092 6±0.017 9 0.191 3±0.013 1 0.186 7±0.012 6 0.183 9±0.011 4 0.181 4±0.017 1
Yeast 0.197 8±0.016 4 0.145 6±0.012 1 0.201 1±0.014 2 0.214 1±0.013 1 0.199 0±0.015 2 0.194 1±0.011 3
One Error(↓)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 0.815 5±0.034 9 0.823 2±0.054 0 0.707 0±0.048 9 0.696 3±0.052 2 0.717 9±0.045 6 0.702 3±0.047 1
Cal500 0.139 3±0.066 2 0.684 9±0.061 1 0.157 1±0.079 3 0.159 3±0.067 2 0.173 2±0.073 3 0.157 1±0.067 8
Emotions 0.293 6±0.055 1 0.441 9±0.051 4 0.335 6±0.047 9 0.362 4±0.048 6 0.377 7±0.041 7 0.364 4±0.056 6
Flags 0.237 6±0.054 3 0.628 9±0.103 3 0.200 8±0.061 7 0.180 5±0.047 9 0.180 3±0.057 0 0.180 3±0.057 0
Scene 0.300 8±0.026 2 0.644 8±0.026 8 0.477 3±0.044 1 0.457 8±0.036 1 0.454 5±0.036 7 0.450 3±0.040 2
Yeast 0.256 1±0.026 9 0.263 5±0.027 2 0.268 1±0.025 8 0.258 6±0.024 9 0.246 2±0.029 5 0.254 9±0.022 6
Coverage(↓)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 3.907 8±0.458 0 1.074 6±0.231 7 2.793 8±0.475 7 2.820 8±0.405 1 2.851 1±0.374 3 2.810 7±0.409 3
Cal500 138.659 1±1.945 6 86.374 3±3.427 6 144.799 6±2.470 5 144.950 9±3.038 8 147.130 3±2.659 7 144.871 8±2.685 4
Emotions 1.845 6±0.186 7 0.332 7±0.085 1 1.968 1±0.158 2 2.052 4±0.117 0 2.048 7±0.127 1 2.089 5±0.137 0
Flags 3.776 8±0.425 0 0.977 4±0.292 8 3.639 5±0.424 7 3.679 7±0.290 6 3.616 6±0.318 7 3.616 6±0.318 7
Scene 0.656 0±0.054 4 0.302 4±0.059 6 1.039 0±0.048 7 1.017 4±0.050 1 1.006 6±0.050 6 0.991 7±0.082 9
Yeast 6.837 8±0.254 5 2.553 9±0.162 1 6.917 5±0.223 7 7.126 1±0.182 0 6.839 5±0.212 6 6.782 4±0.202 4
Average Precision(↑)
Dataset Original CMSVP IG-BR IG-LP RF-BR RF-LP
Birds 0.455 7±0.050 5 0.595 9±0.080 8 0.607 6±0.058 4 0.615 3±0.058 6 0.598 1±0.056 9 0.605 9±0.061 9
Cal500 0.476 7±0.011 8 0.306 8±0.014 0 0.475 5±0.014 1 0.467 0±0.014 2 0.466 6±0.011 6 0.466 3±0.014 3
Emotions 0.788 4±0.033 0 0.891 1±0.030 2 0.761 4±0.028 8 0.743 7±0.025 8 0.739 6±0.025 3 0.741 6±0.024 1
Flags 0.815 2±0.024 8 0.706 2±0.049 1 0.837 7±0.032 1 0.835 1±0.030 3 0.839 6±0.021 8 0.839 6±0.021 8
Scene 0.814 7±0.015 6 0.800 5±0.029 0 0.704 3±0.024 4 0.714 6±0.020 6 0.716 8±0.020 7 0.719 1±0.025 3
Yeast 0.732 2±0.019 8 0.766 9±0.017 3 0.723 7±0.019 8 0.713 8±0.0195 0.730 7±0.019 6 0.733 9±0.016 6
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