An Interpolation-Based Evolutionary Algorithm for Bi-Objective Feature Selection in Classification
Abstract
:1. Introduction
2. Related Works
3. Proposed Algorithm
3.1. General Framework
Algorithm 1 |
|
Algorithm 2 |
|
3.2. Initialization Process
Algorithm 3 |
|
Algorithm 4 |
|
Algorithm 5 |
|
3.3. Reproduction Process
Algorithm 6 |
|
4. Experimental Setup
4.1. Comparison Algorithms
4.2. Classification Datasets
4.3. Performance Indicators
4.4. Parameter Settings
5. Experimental Studies
5.1. Empirical Result Analyses
5.2. Classification Performance Analyses
5.3. Nondominated Solution Distributions
5.4. Computational Time Analyses
6. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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No. | Dataset | Feature | Sample | Class |
---|---|---|---|---|
1 | Sonar | 60 | 208 | 2 |
2 | HillValley | 100 | 606 | 2 |
3 | Synthetic | 100 | 200 | 5 |
4 | Arrhythmia | 278 | 452 | 13 |
5 | Yale | 1024 | 165 | 15 |
6 | Colon | 2000 | 62 | 2 |
7 | SRBCT | 2308 | 83 | 4 |
8 | AR10P | 2400 | 130 | 10 |
9 | PIE10P | 2420 | 210 | 10 |
10 | Leukemia1 | 5327 | 72 | 3 |
11 | Tumor9 | 5726 | 60 | 9 |
12 | Brain1 | 5920 | 90 | 5 |
13 | Leukemia2 | 7070 | 72 | 2 |
14 | CNS | 7129 | 60 | 2 |
15 | ALLAML | 7129 | 72 | 2 |
16 | Nci9 | 9712 | 60 | 9 |
17 | Pixraw10P | 10,000 | 100 | 10 |
18 | Orlraws10P | 10,304 | 100 | 10 |
19 | Brain2 | 10,367 | 50 | 4 |
20 | Prostate | 10,509 | 102 | 2 |
Dataset | IPEA | NSGA-II | MOEA/D | MOEA/HD | PMEA |
---|---|---|---|---|---|
Sonar | 8.072e−01 | ★ 7.955e−01 | ★ 7.940e−01 | ★ 8.027e−01 | 7.740e−01 |
±2.36e−02 | ±2.59e−02 | ±2.87e−02 | ±2.53e−02 | ±3.09e−02 | |
HillValley | 6.310e−01 | 5.849e−01 | ★ 6.259e−01 | ★ 6.259e−01 | 5.819e−01 |
±1.08e−02 | ±2.51e−02 | ±1.04e−02 | ±8.54e−03 | ±2.18e−02 | |
Synthetic | 4.343e−01 | 3.846e−01 | 3.852e−01 | 3.956e−01 | 3.860e−01 |
±3.76e−02 | ±3.88e−02 | ±4.16e−02 | ±5.93e−02 | ±3.74e−02 | |
Arrhythmia | 6.932e−01 | 6.308e−01 | 6.655e−01 | 4.826e−01 | 4.715e−01 |
±1.64e−02 | ±2.66e−02 | ±1.58e−02 | ±1.69e−02 | ±4.16e−02 | |
Yale | 7.043e−01 | 4.878e−01 | 5.127e−01 | 4.920e−01 | 4.868e−01 |
±2.72e−02 | ±1.34e−02 | ±2.62e−02 | ±3.39e−02 | ±1.92e−02 | |
Colon | 8.990e−01 | 5.500e−01 | 6.078e−01 | 5.239e−01 | 5.387e−01 |
±3.06e−02 | ±2.65e−02 | ±4.78e−02 | ±3.30e−02 | ±2.68e−02 | |
SRBCT | 9.230e−01 | 2.841e−01 | 3.178e−01 | 2.554e−01 | 2.759e−01 |
±5.64e−02 | ±2.05e−03 | ±2.03e−03 | ±1.81e−03 | ±2.03e−03 | |
AR10P | 7.778e−01 | 3.631e−01 | 3.709e−01 | 3.449e−01 | 3.536e−01 |
±3.52e−02 | ±2.01e−02 | ±2.26e−02 | ±1.75e−02 | ±1.48e−02 | |
PIE10P | 9.347e−01 | 6.023e−01 | 6.458e−01 | 5.883e−01 | 5.902e−01 |
±2.07e−02 | ±1.06e−02 | ±1.13e−02 | ±1.22e−02 | ±9.87e−03 | |
Leukemia1 | 9.647e−01 | 5.290e−01 | 5.431e−01 | 5.132e−01 | 5.160e−01 |
±2.43e−02 | ±1.81e−02 | ±3.10e−02 | ±2.40e−02 | ±1.93e−02 | |
Tumor9 | 4.897e−01 | 2.790e−01 | 2.832e−01 | 2.673e−01 | 2.789e−01 |
±5.65e−02 | ±2.80e−02 | ±2.54e−02 | ±2.35e−02 | ±2.24e−02 | |
Brain1 | 7.844e−01 | 4.718e−01 | 4.906e−01 | 4.535e−01 | 4.698e−01 |
±4.93e−02 | ±3.11e−03 | ±1.09e−02 | ±1.00e−02 | ±2.77e−03 | |
Leukemia2 | 9.337e−01 | 5.360e−01 | 5.450e−01 | 5.126e−01 | 5.270e−01 |
±6.48e−02 | ±8.95e−03 | ±1.94e−02 | ±1.79e−02 | ±1.55e−02 | |
CNS | 6.842e−01 | 3.781e−01 | 3.743e−01 | 3.707e−01 | 3.663e−01 |
±6.93e−02 | ±3.28e−02 | ±3.32e−02 | ±3.25e−02 | ±3.52e−02 | |
ALLAML | 9.813e−01 | 5.205e−01 | 5.358e−01 | 5.060e−01 | 5.148e−01 |
±3.66e−02 | ±1.52e−02 | ±1.34e−02 | ±1.44e−02 | ±1.52e−02 | |
Nci9 | 4.712e−01 | 2.406e−01 | 2.616e−01 | 2.370e−01 | 2.447e−01 |
±8.13e−02 | ±2.54e−02 | ±2.94e−02 | ±2.21e−02 | ±2.46e−02 | |
Pixraw10P | 9.802e−01 | 5.795e−01 | 5.911e−01 | 5.640e−01 | 5.778e−01 |
±2.83e−02 | ±2.22e−03 | ±9.88e−03 | ±7.64e−03 | ±3.00e−03 | |
Orlraws10P | 9.605e−01 | 5.390e−01 | 5.447e−01 | 5.297e−01 | 5.392e−01 |
±3.28e−02 | ±7.53e−03 | ±9.58e−03 | ±8.00e−03 | ±6.12e−03 | |
Brain2 | 7.181e−01 | 3.903e−01 | 3.824e−01 | 3.782e−01 | 3.790e−01 |
±5.65e−02 | ±2.15e−02 | ±2.43e−02 | ±2.12e−02 | ±1.80e−02 | |
Prostate | 9.486e−01 | 4.629e−01 | 4.599e−01 | 4.559e−01 | 4.598e−01 |
±2.31e−02 | ±1.29e−02 | ±1.51e−02 | ±1.16e−02 | ±2.09e−02 |
Dataset | IPEA | NSGA-II | MOEA/D | MOEA/HD | PMEA |
---|---|---|---|---|---|
Sonar | 5.352e−02 | ★ 6.239e−02 | ★ 6.637e−02 | ★ 5.961e−02 | 7.886e−02 |
±1.54e−02 | ±2.31e−02 | ±2.18e−02 | ±1.70e−02 | ±2.94e−02 | |
HillValley | 2.504e−02 | 6.517e−02 | ★ 2.614e−02 | ★ 2.688e−02 | 7.049e−02 |
±6.73e−03 | ±2.45e−02 | ±9.07e−03 | ±1.06e−02 | ±2.43e−02 | |
Synthetic | 9.448e−02 | 1.408e−01 | 1.341e−01 | 1.328e−01 | 1.402e−01 |
±2.35e−02 | ±3.14e−02 | ±3.52e−02 | ±4.19e−02 | ±3.06e−02 | |
Arrhythmia | 1.767e−02 | 6.646e−02 | 3.616e−02 | 2.092e−01 | 2.317e−01 |
±7.17e−03 | ±1.92e−02 | ±8.31e−03 | ±1.40e−02 | ±4.63e−02 | |
Yale | 2.232e−02 | 3.175e−01 | 2.643e−01 | 3.197e−01 | 3.166e−01 |
±1.19e−02 | ±1.06e−02 | ±1.39e−02 | ±1.47e−02 | ±8.91e−03 | |
Colon | 6.077e−02 | 3.889e−01 | 3.241e−01 | 4.155e−01 | 4.011e−01 |
±2.96e−02 | ±1.69e−02 | ±3.84e−02 | ±2.42e−02 | ±2.01e−02 | |
SRBCT | 3.704e−02 | 6.225e−01 | 5.751e−01 | 6.693e−01 | 6.352e−01 |
±3.28e−02 | ±3.15e−03 | ±2.55e−03 | ±3.12e−03 | ±3.24e−03 | |
AR10P | 5.458e−02 | 4.639e−01 | 4.353e−01 | 4.810e−01 | 4.725e−01 |
±2.13e−02 | ±1.59e−02 | ±1.92e−02 | ±1.51e−02 | ±1.27e−02 | |
PIE10P | 1.366e−02 | 3.730e−01 | 3.164e−01 | 3.867e−01 | 3.806e−01 |
±8.19e−03 | ±8.95e−03 | ±1.00e−02 | ±9.94e−03 | ±6.47e−03 | |
Leukemia1 | 1.942e−02 | 4.343e−01 | 4.077e−01 | 4.502e−01 | 4.415e−01 |
±1.34e−02 | ±7.64e−03 | ±2.30e−02 | ±1.30e−02 | ±9.49e−03 | |
Tumor9 | 9.052e−02 | 4.515e−01 | 4.323e−01 | 4.650e−01 | 4.516e−01 |
±5.05e−02 | ±1.79e−02 | ±1.43e−02 | ±1.20e−02 | ±1.36e−02 | |
Brain1 | 6.266e−02 | 4.300e−01 | 4.036e−01 | 4.558e−01 | 4.328e−01 |
±3.95e−02 | ±4.38e−03 | ±1.53e−02 | ±1.41e−02 | ±3.90e−03 | |
Leukemia2 | 7.277e−02 | 4.499e−01 | 4.349e−01 | 4.738e−01 | 4.571e−01 |
±7.13e−02 | ±6.09e−03 | ±1.71e−02 | ±1.31e−02 | ±1.05e−02 | |
CNS | 1.806e−01 | 5.032e−01 | 4.974e−01 | 5.127e−01 | 5.165e−01 |
±7.62e−02 | ±2.82e−02 | ±3.43e−02 | ±2.91e−02 | ±3.02e−02 | |
ALLAML | 2.056e−02 | 4.608e−01 | 4.411e−01 | 4.758e−01 | 4.636e−01 |
±4.03e−02 | ±1.01e−02 | ±9.58e−03 | ±1.06e−02 | ±9.55e−03 | |
Nci9 | 9.879e−02 | 4.775e−01 | 4.513e−01 | 5.029e−01 | 4.756e−01 |
±6.21e−02 | ±1.43e−02 | ±1.49e−02 | ±1.07e−02 | ±1.19e−02 | |
Pixraw10P | 1.172e−02 | 4.431e−01 | 4.300e−01 | 4.607e−01 | 4.450e−01 |
±1.88e−02 | ±2.51e−03 | ±1.12e−02 | ±8.65e−03 | ±3.40e−03 | |
Orlraws10P | 1.684e−02 | 4.491e−01 | 4.386e−01 | 4.603e−01 | 4.498e−01 |
±1.71e−02 | ±3.95e−03 | ±4.68e−03 | ±5.75e−03 | ±3.23e−03 | |
Brain2 | 1.001e−01 | 4.813e−01 | 4.828e−01 | 4.934e−01 | 4.888e−01 |
±4.86e−02 | ±1.33e−02 | ±1.94e−02 | ±1.34e−02 | ±1.11e−02 | |
Prostate | 1.465e−02 | 4.890e−01 | 4.865e−01 | 4.968e−01 | 4.929e−01 |
±1.95e−02 | ±1.09e−02 | ±1.20e−02 | ±9.49e−03 | ±1.55e−02 |
Dataset | IPEA | NSGA-II | MOEA/D | MOEA/HD | PMEA |
---|---|---|---|---|---|
Sonar | 1.957e−01 | ★ 2.000e−01 | ★ 2.065e−01 | ★ 2.032e−01 | ★ 2.097e−01 |
NSF: 6 | NSF: 7 | NSF: 6 | NSF: 5 | NSF: 7 | |
HillValley | 3.984e−01 | 4.211e−01 | ★ 4.048e−01 | ★ 4.038e−01 | 4.218e−01 |
NSF: 4 | NSF: 10 | NSF: 3 | NSF: 4 | NSF: 11 | |
Synthetic | 6.161e−01 | 6.439e−01 | 6.600e−01 | 6.433e−01 | 6.406e−01 |
NSF: 8 | NSF: 15 | NSF: 9 | NSF: 12 | NSF: 15 | |
Arrhythmia | 3.317e−01 | 3.775e−01 | 3.530e−01 | 4.650e−01 | 4.566e−01 |
NSF: 8 | NSF: 17 | NSF: 12 | NSF: 51 | NSF: 58 | |
Yale | 3.185e−01 | 3.519e−01 | 3.711e−01 | 3.504e−01 | 3.459e−01 |
NSF: 10 | NSF: 329 | NSF: 265 | NSF: 332 | NSF: 323 | |
Colon | 1.088e−01 | 2.070e−01 | 1.947e−01 | 2.246e−01 | 2.123e−01 |
NSF: 3 | NSF: 703 | NSF: 552 | NSF: 735 | NSF: 720 | |
SRBCT | 7.600e−02 | 6.400e−01 | 6.400e−01 | 6.400e−01 | 6.400e−01 |
NSF: 4 | NSF: 810 | NSF: 610 | NSF: 990 | NSF: 866 | |
AR10P | 2.408e−01 | 4.958e−01 | 5.150e−01 | 5.150e−01 | 5.075e−01 |
NSF: 7 | NSF: 909 | NSF: 780 | NSF: 930 | NSF: 931 | |
PIE10P | 7.389e−02 | 9.722e−02 | 1.044e−01 | 1.028e−01 | 1.050e−01 |
NSF: 11 | NSF: 917 | NSF: 767 | NSF: 933 | NSF: 927 | |
Leukemia1 | 4.697e−02 | 1.636e−01 | 1.833e−01 | 1.712e−01 | 1.773e−01 |
NSF: 3 | NSF: 2236 | NSF: 2052 | NSF: 2308 | NSF: 2251 | |
Tumor9 | 5.537e−01 | 6.019e−01 | 6.148e−01 | 6.037e−01 | 5.944e−01 |
NSF: 6 | NSF: 2455 | NSF: 2329 | NSF: 2517 | NSF: 2466 | |
Brain1 | 2.370e−01 | 2.593e−01 | 2.593e−01 | 2.593e−01 | 2.593e−01 |
NSF: 4 | NSF: 2499 | NSF: 2321 | NSF: 2640 | NSF: 2512 | |
Leukemia2 | 6.818e−02 | 1.318e−01 | 1.470e−01 | 1.439e−01 | 1.500e−01 |
NSF: 2 | NSF: 3056 | NSF: 2908 | NSF: 3181 | NSF: 3067 | |
CNS | 3.537e−01 | 4.056e−01 | 4.370e−01 | 4.167e−01 | 4.352e−01 |
NSF: 2 | NSF: 3104 | NSF: 2956 | NSF: 3172 | NSF: 3122 | |
ALLAML | 1.818e−02 | 1.576e−01 | 1.652e−01 | 1.606e−01 | 1.621e−01 |
NSF: 2 | NSF: 3081 | NSF: 2957 | NSF: 3181 | NSF: 3077 | |
Nci9 | 5.860e−01 | 6.491e−01 | 6.421e−01 | 6.404e−01 | 6.526e−01 |
NSF: 6 | NSF: 4278 | NSF: 4092 | NSF: 4606 | NSF: 4292 | |
Pixraw10P | 1.778e−02 | 3.333e−02 | 3.333e−02 | 3.333e−02 | 3.333e−02 |
NSF: 2 | NSF: 4428 | NSF: 4277 | NSF: 4628 | NSF: 4455 | |
Orlraws10P | 3.667e−02 | 1.033e−01 | 1.100e−01 | 1.044e−01 | 1.022e−01 |
NSF: 7 | NSF: 4582 | NSF: 4463 | NSF: 4702 | NSF: 4590 | |
Brain2 | 3.089e−01 | 3.778e−01 | 3.978e−01 | 3.933e−01 | 3.844e−01 |
NSF: 4 | NSF: 4637 | NSF: 4572 | NSF: 4733 | NSF: 4656 | |
Prostate | 5.484e−02 | 2.462e−01 | 2.591e−01 | 2.452e−01 | 2.462e−01 |
NSF: 3 | NSF: 4718 | NSF: 4585 | NSF: 4797 | NSF: 4721 |
Dataset | IPEA | NSGA-II | MOEA/D | MOEA/HD | PMEA |
---|---|---|---|---|---|
Sonar | 3.881e+01 | 3.923e+01 | 4.050e+01 | 3.944e+01 | 3.937e+01 |
±3.25e−01 | ±4.31e−01 | ±3.27e−01 | ±4.60e−01 | ±3.51e−01 | |
HillValley | 1.455e+02 | 1.723e+02 | 1.506e+02 | 1.634e+02 | 1.758e+02 |
±2.17e+00 | ±7.25e+00 | ±1.46e+00 | ±4.00e+00 | ±3.90e+00 | |
Synthetic | 3.990e+01 | 4.278e+01 | 4.311e+01 | 4.353e+01 | 4.257e+01 |
±5.41e−01 | ±1.25e+00 | ±6.22e−01 | ±8.03e−01 | ±6.76e−01 | |
Arrhythmia | 1.333e+02 | 1.571e+02 | 1.516e+02 | 1.957e+02 | 1.887e+02 |
±1.21e+00 | ±1.18e+01 | ±6.88e+00 | ±5.01e+00 | ±7.70e+00 | |
Yale | 2.251e+02 | 3.871e+02 | 3.698e+02 | 4.042e+02 | 3.789e+02 |
±4.75e+00 | ±9.78e+00 | ±9.55e+00 | ±1.49e+01 | ±1.05e+01 | |
Colon | 8.224e+01 | 1.926e+02 | 1.666e+02 | 1.946e+02 | 1.823e+02 |
±3.08e+00 | ±6.56e+00 | ±7.77e+00 | ±8.58e+00 | ±7.88e+00 | |
SRBCT | 2.754e+02 | 4.754e+02 | 4.368e+02 | 5.117e+02 | 4.727e+02 |
±3.46e+00 | ±6.34e+00 | ±5.39e+00 | ±7.46e+00 | ±5.93e+00 | |
AR10P | 6.210e+02 | 1.087e+03 | 1.053e+03 | 1.119e+03 | 1.075e+03 |
±1.78e+01 | ±2.46e+01 | ±3.21e+01 | ±3.56e+01 | ±3.68e+01 | |
PIE10P | 1.167e+03 | 2.145e+03 | 2.087e+03 | 2.309e+03 | 2.113e+03 |
±2.69e+01 | ±1.28e+02 | ±1.08e+02 | ±1.18e+02 | ±8.12e+01 | |
Leukemia1 | 8.660e+02 | 1.586e+03 | 1.499e+03 | 1.614e+03 | 1.499e+03 |
±3.40e+01 | ±1.12e+02 | ±1.00e+02 | ±1.13e+02 | ±5.18e+01 | |
Tumor9 | 7.800e+02 | 1.441e+03 | 1.379e+03 | 1.458e+03 | 1.338e+03 |
±4.19e+01 | ±1.05e+02 | ±1.09e+02 | ±1.07e+02 | ±5.66e+01 | |
Brain1 | 1.269e+03 | 2.641e+03 | 2.542e+03 | 2.668e+03 | 2.408e+03 |
±5.98e+01 | ±2.46e+02 | ±2.36e+02 | ±2.35e+02 | ±4.55e+01 | |
Leukemia2 | 1.210e+03 | 2.529e+03 | 2.402e+03 | 2.541e+03 | 2.286e+03 |
±5.99e+01 | ±2.35e+02 | ±2.17e+02 | ±2.37e+02 | ±4.97e+01 | |
CNS | 1.022e+03 | 1.874e+03 | 1.792e+03 | 1.905e+03 | 1.769e+03 |
±5.68e+01 | ±1.34e+02 | ±1.42e+02 | ±1.50e+02 | ±5.26e+01 | |
ALLAML | 1.241e+03 | 2.579e+03 | 2.461e+03 | 2.598e+03 | 2.306e+03 |
±6.69e+01 | ±2.69e+02 | ±2.53e+02 | ±2.61e+02 | ±1.78e+01 | |
Nci9 | 1.490e+03 | 2.924e+03 | 2.832e+03 | 3.042e+03 | 2.678e+03 |
±8.17e+01 | ±2.76e+02 | ±3.03e+02 | ±3.01e+02 | ±2.47e+01 | |
Pixraw10P | 2.721e+03 | 5.213e+03 | 5.096e+03 | 5.320e+03 | 4.774e+03 |
±1.87e+02 | ±4.87e+02 | ±4.98e+02 | ±5.02e+02 | ±4.74e+01 | |
Orlraws10P | 2.841e+03 | 5.396e+03 | 5.272e+03 | 4.905e+03 | 4.833e+03 |
±1.78e+02 | ±5.12e+02 | ±5.59e+02 | ±1.51e+02 | ±1.25e+02 | |
Brain2 | 1.438e+03 | 2.469e+03 | 2.313e+03 | 2.493e+03 | 2.438e+03 |
±1.05e+02 | ±6.16e+01 | ±5.46e+01 | ±7.12e+01 | ±5.03e+01 | |
Prostate | 3.098e+03 | 5.129e+03 | 4.943e+03 | 5.198e+03 | 5.107e+03 |
±2.09e+02 | ±1.59e+02 | ±1.24e+02 | ±1.78e+02 | ±1.31e+02 |
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Xu, H. An Interpolation-Based Evolutionary Algorithm for Bi-Objective Feature Selection in Classification. Mathematics 2024, 12, 2572. https://doi.org/10.3390/math12162572
Xu H. An Interpolation-Based Evolutionary Algorithm for Bi-Objective Feature Selection in Classification. Mathematics. 2024; 12(16):2572. https://doi.org/10.3390/math12162572
Chicago/Turabian StyleXu, Hang. 2024. "An Interpolation-Based Evolutionary Algorithm for Bi-Objective Feature Selection in Classification" Mathematics 12, no. 16: 2572. https://doi.org/10.3390/math12162572