1 Introduction
2 Clustering ensemble methods
2.1 Clustering ensemble representation
2.2 A generic clustering ensemble framework
2.2.1 Ensemble member generation
2.2.2 Consensus function
2.2.3 Evaluation
3 Related work
4 The adaptive clustering ensemble (ACE)
4.1 Definitions of similarity measures
4.2 The ACE algorithm
4.2.1 Stage 1: transformation
4.2.2 Stage 2: generating new consensus clusters
4.2.3 Stage 3: enforce hard clustering
5 Experiments
Dataset | # Objects | # Features | # Clusters |
---|---|---|---|
Iris | 150 | 4 | 3 |
Wine | 178 | 13 | 3 |
Thyroid | 215 | 5 | 3 |
Mfeatures | 2000 | 2 | 10 |
Glass | 214 | 9 | 6 |
Bcw | 683 | 9 | 2 |
Soybean | 47 | 35 | 4 |
Ionosphere | 351 | 34 | 2 |
6 Results and analysis
6.1 Results of ensembles built with fixed k
6.1.1 Results of ARI index
6.1.2 Results of NMI index
CO-Average | ONCE-Average | DSCE | ACE | DICLENS | MCLA | Ave-mem | |
---|---|---|---|---|---|---|---|
Iris |
\(0.725 \pm 0.012\)
|
\(0.726 \pm \underline{0.009}\)
|
\(0.732 \pm 0.021\)
|
\(\mathbf{0.734 } \pm 0.023\)
|
\(0.680 \pm 0.077\)
|
\(0.723 \pm 0.012\)
|
\(0.702 \pm 0.038\)
|
Wine |
\(0.369 \pm 0.005\)
|
\(0.369 \pm 0.005\)
|
\(\mathbf{0.377 } \pm 0.025\)
|
\(0.371 \pm 0.008\)
|
\(0.369 \pm 0.005\)
|
\(0.372 \pm \underline{0.002}\)
|
\(0.366 \pm 0.004\)
|
Thyroid |
\(0.559 \pm 0.024\)
|
\(0.584 \pm 0.044\)
|
\(0.609 \pm 0.032\)
|
\(\mathbf{0.613 } \pm \underline{0.023}\)
|
\(0.582 \pm 0.044\)
|
\(0.563 \pm 0.025\)
|
\(0.473 \pm 0.036\)
|
Mfeatures |
\(0.315 \pm 0.006\)
|
\(\mathbf{0.316 } \pm 0.005\)
|
\(\mathbf{0.316 } \pm \underline{0.004}\)
|
\(0.314 \pm 0.008\)
|
\(0.290 \pm 0.069\)
|
\(0.308 \pm 0.021\)
|
\(0.293 \pm 0.029\)
|
Glass |
\(0.509 \pm 0.029\)
|
\(0.526 \pm 0.030\)
|
\(0.528 \pm 0.027\)
|
\(\mathbf{0.535 } \pm 0.029\)
|
\(0.392 \pm 0.123\)
| 0.534\(\pm \underline{0.020}\) |
\(0.501 \pm 0.009\)
|
Bcw |
\(\mathbf{0.849 } \pm 0.004\)
|
\(0.847 \pm \underline{0.003}\)
|
\(\mathbf{0.849 } \pm 0.004\)
|
\(\mathbf{0.849 } \pm 0.004\)
|
\(0.842 \pm 0.005\)
|
0.849
\(\pm 0.004\)
|
\(0.830 \pm 0.021\)
|
Soybean |
\(0.547 \pm \underline{0.006}\)
|
\(0.550 \pm 0.015\)
|
\(0.578 \pm 0.052\)
|
\(0.532 \pm 0.081\)
|
\(\mathbf{0.632 } \pm 0.046\)
|
\(0.548 \pm \underline{0.006}\)
|
\(0.566 \pm 0.025\)
|
Ionosphere |
\(0.163 \pm 0.014\)
|
\(0.166 \pm 0.008\)
|
\(\mathbf{0.169 } \pm \underline{0.005}\)
|
\(0.165 \pm 0.008\)
|
\(0.161 \pm 0.009\)
|
\(0.166 \pm 0.006\)
|
\(0.149 \pm 0.007\)
|
Ave-P | 0.505 | 0.511 |
\(\mathbf{0.520 }\)
| 0.514 | 0.493 | 0.508 | 0.443 |
Ave-C |
\(\underline{0.012}\)
| 0.015 | 0.017 | 0.023 | 0.048 |
\(\underline{0.012}\)
| 0.031 |
CO-Average | ONCE-Average | DSCE | ACE | DICLENS | MCLA | Ave-mem | |
---|---|---|---|---|---|---|---|
Iris |
\(0.751 \pm 0.015\)
|
\(0.752 \pm 0.012\)
|
\(0.763 \pm 0.024\)
|
\(\mathbf{0.766 } \pm 0.028\)
|
\(0.757 \pm \underline{0.008}\)
|
\(0.749 \pm 0.015\)
|
\(0.737 \pm 0.025\)
|
Wine |
\(0.428 \pm 0.003\)
|
\(0.428 \pm 0.003\)
|
\(\mathbf{0.432 } \pm 0.014\)
|
\(0.429 \pm 0.006\)
|
\(0.427 \pm 0.004\)
|
\(0.429 \pm \underline{0.001}\)
|
\(0.428 \pm 0.003\)
|
Thyroid |
\(0.434 \pm 0.047\)
|
\(0.473 \pm 0.062\)
|
\(0.480 \pm 0.056\)
|
\(\mathbf{0.531 } \pm 0.042\)
|
\(0.501 \pm 0.053\)
|
\(0.418 \pm \underline{0.033}\)
|
\(0.403 \pm 0.026\)
|
Mfeatures |
\(\mathbf{0.479 } \pm \underline{0.002}\)
|
\(\mathbf{0.479 } \pm 0.003\)
|
\(\mathbf{0.479 } \pm \underline{0.002}\)
|
\(0.478 \pm 0.007\)
|
\(0.468 \pm 0.026\)
|
\(0.475 \pm 0.009\)
|
\(0.460 \pm 0.027\)
|
Glass |
\(0.712 \pm 0.027\)
|
\(0.725 \pm 0.029\)
|
\(0.725 \pm 0.021\)
|
\(0.726 \pm 0.022\)
|
\(0.617 \pm 0.107\)
|
\(\mathbf{0.728 } \pm \underline{0.017}\)
|
\(0.704 \pm 0.007\)
|
Bcw |
\(0.750 \pm 0.005\)
|
\(0.749 \pm \underline{0.004}\)
|
\(0.750 \pm 0.005\)
|
\(\mathbf{0.751 } \pm 0.005\)
|
\(0.742 \pm 0.006\)
|
\(\mathbf{0.751 } \pm 0.005\)
|
\(0.731 \pm 0.023\)
|
Soybean |
\(0.717 \pm \underline{0.002}\)
|
\(0.723 \pm 0.024\)
|
\(0.756 \pm 0.064\)
|
\(0.712 \pm 0.076\)
|
\(\mathbf{0.822 } \pm 0.056\)
|
\(0.717 \pm \underline{0.002}\)
|
\(0.736 \pm 0.019\)
|
Ionosphere |
\(0.122 \pm 0.014\)
|
\(0.124 \pm 0.009\)
|
\(\mathbf{0.128 } \pm \underline{0.005}\)
|
\(0.123 \pm 0.008\)
|
\(0.119 \pm 0.009\)
|
\(0.124 \pm 0.006\)
|
\(0.108 \pm 0.006\)
|
Ave-P | 0.549 | 0.557 |
\(\mathbf{0.564 }\)
|
\(\mathbf{0.564 }\)
| 0.557 | 0.549 | |
Ave-C | 0.015 | 0.018 | 0.024 | 0.024 | 0.034 |
\(\underline{0.011}\)
|
6.2 Results of ensembles built with random variable k
6.2.1 Results of ARI index
6.2.2 Results of NMI index
6.2.3 Identifying the true number of clusters in DICLENS
CO-Average | ONCE-Average | ACE | DICLENS | MCLA | Ave-mem | |
---|---|---|---|---|---|---|
Iris |
\(0.669 \pm 0.065\)
|
\(0.674 \pm 0.057\)
|
\(0.696 \pm 0.038\)
|
\(0.565 \pm \underline{0.009}\)
|
\(\mathbf{0.722 } \pm 0.043\)
|
\(0.605 \pm 0.029\)
|
Wine |
\(0.324 \pm 0.045\)
|
\(0.344 \pm 0.060\)
|
\(\mathbf{0.403 } \pm 0.014\)
|
\(0.367 \pm 0.024\)
|
\(0.393 \pm \underline{0.008}\)
|
\(0.326 \pm 0.011\)
|
Thyroid |
\(0.252 \pm 0.175\)
|
\(0.189 \pm 0.121\)
|
\(0.303 \pm \underline{0.032}\)
|
\(0.308 \pm 0.118\)
|
\(\mathbf{0.448 } \pm 0.119\)
|
\(0.285 \pm 0.053\)
|
Mfeatures |
\(0.325 \pm 0.002\)
|
\(\mathbf{0.326 } \pm \underline{0.001}\)
|
\(0.325 \pm 0.005\)
|
\(0.324 \pm 0.006\)
|
\(0.277 \pm 0.013\)
|
\(0.321 \pm 0.005\)
|
Glass |
\(0.265 \pm 0.006\)
|
\(0.259 \pm 0.008\)
|
\(\mathbf{0.269 } \pm \underline{0.004}\)
|
\(0.200 \pm 0.048\)
|
\(0.152 \pm 0.022\)
|
\(0.258 \pm 0.005\)
|
Bcw |
\(0.866 \pm 0.018\)
|
\(0.860 \pm 0.016\)
|
\(\mathbf{0.869 } \pm \underline{0.014}\)
|
\(0.853 \pm 0.031\)
|
\(0.864 \pm \underline{0.014}\)
|
\(0.773 \pm 0.037\)
|
Soybean |
\(0.534 \pm \underline{0.000}\)
|
\(0.534 \pm \underline{0.000}\)
|
\(\mathbf{0.578 } \pm 0.160\)
|
\(0.575 \pm 0.070\)
|
\(0.547 \pm 0.039\)
|
\(0.547 \pm 0.036\)
|
Ionosphere |
\(0.076 \pm 0.047\)
|
\(0.037 \pm 0.035\)
|
\(\mathbf{0.084 } \pm 0.034\)
|
\(0.076 \pm 0.039\)
|
\(0.061 \pm \underline{0.019}\)
|
\(0.117 \pm 0.014\)
|
Ave-P | 0.414 | 0.403 |
\(\mathbf{0.441 }\)
| 0.409 | 0.433 | 0.404 |
Ave-C | 0.045 | 0.037 | 0.038 | 0.043 |
\(\mathbf{0.035 }\)
| 0.024 |
CO-Average | ONCE-Average | ACE | DICLENS | MCLA | Ave-mem | |
---|---|---|---|---|---|---|
Iris |
\(0.753 \pm \underline{0.017}\)
|
\(0.749 \pm 0.027\)
|
\(\mathbf{0.766 } \pm 0.032\)
|
\(0.753 \pm 0.026\)
|
\(0.755 \pm 0.037\)
|
\(0.706 \pm 0.012\)
|
Wine |
\(0.406 \pm 0.010\)
|
\(0.415 \pm 0.022\)
|
\(0.421 \pm 0.014\)
|
\(\mathbf{0.435 } \pm 0.018\)
|
\(0.415 \pm \underline{0.005}\)
|
\(0.410 \pm 0.010\)
|
Thyroid |
\(0.293 \pm 0.077\)
|
\(0.250 \pm 0.066\)
|
\(0.308 \pm 0.050\)
|
\(0.331 \pm \underline{0.040}\)
|
\(\mathbf{0.356 } \pm 0.048\)
|
\(0.302 \pm 0.035\)
|
Mfeatures |
\(0.486 \pm \underline{0.002}\)
|
\(0.487 \pm \underline{0.002}\)
|
\(0.490 \pm 0.008\)
|
\(\mathbf{0.493 } \pm 0.005\)
|
\(0.464 \pm 0.007\)
|
\(0.484 \pm 0.005\)
|
Glass |
\(0.441 \pm 0.018\)
|
\(\mathbf{0.449 } \pm \underline{0.016}\)
|
\(0.430 \pm \underline{0.016}\)
|
\(0.389 \pm 0.032\)
|
\(0.307 \pm 0.032\)
|
\(0.423 \pm 0.011\)
|
Bcw |
\(0.773 \pm 0.024\)
|
\(0.765 \pm 0.021\)
|
\(\mathbf{0.776 } \pm \underline{0.019}\)
|
\(0.759 \pm 0.032\)
|
\(0.770 \pm \underline{0.019}\)
|
\(0.687 \pm 0.028\)
|
Soybeans |
\(0.710 \pm \underline{0.000}\)
|
\(0.710 \pm \underline{0.000}\)
|
\(0.722 \pm 0.127\)
|
\(\mathbf{0.767 } \pm 0.070\)
|
\(0.716 \pm 0.018\)
|
\(0.734 \pm 0.020\)
|
Ionosphere |
\(0.043 \pm 0.035\)
|
\(0.023 \pm \underline{0.012}\)
|
\(\mathbf{0.048 } \pm 0.026\)
|
\(0.043 \pm 0.029\)
|
\(0.030 \pm 0.013\)
|
\(0.099 \pm 0.016\)
|
Ave-P | 0.488 | 0.481 | 0.495 |
0.496
| 0.477 | 0.480 |
Ave-C | 0.023 |
0.021
| 0.036 | 0.032 | 0.022 | 0.017 |