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may be particularly useful. In PSO the system is initialized with a population of random solutions, called particles. Optima are searched for by updating generations, with particles moving through the parameter space toward the current local and global optimum particles. At each time step the velocities of all particles are changed depending on the current optima. Although there are similarities with GAs, PSO systems tend to require fewer (possibly erroneous) design choices, such as the choice of evolutionary operators. For this application the particles will represent potential membership function de nitions de ned by sets of parameters. The initial population of particles could be generated by random parameter deviations from the original membership functions. Extra constraints will need to be enforced in order to restrict search to meaningful fuzzi cations. For example, only convex fuzzy sets should be considered. Particles are rated according to the fuzzy-rough dependency degree to provide a measure of tness. From this, the local and global optima can be determined and used to adjust particle velocities.
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SUMMARY
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This chapter has presented several potential directions for crisp and fuzzy-rough developments. One particular area of interest is that of reformulating the search for a reduct as a propositional satis ability problem. Solution techniques from this domain can then be applied to locate the smallest reducts. In addition to this and other areas of further research in feature selection, the possibility of inducing decision trees using the fuzzy-rough metric was discussed. The initial results show that the proposed method performs comparably to fuzzy ID3 for fuzzy datasets, and better than it does for crisp data. Further experimentation is to be carried out on a fuller range of datasets in the future. Other developments have been proposed that use fuzzy-rough set-based methodologies. Fuzzy-rough rule induction could be useful in extracting fuzzy rules from data in a manner similar to that of crisp rough set rule induction. By the extension of crisp concepts to the fuzzy-rough case, traditional rule induction algorithms can be extended for use with real-valued and noisy data. With these extensions new algorithms can be constructed for the purpose. Fuzzy-rough clustering can also be developed in a similar way, using fuzzy-rough methods. A related area is that of fuzzi cation optimization, where fuzzy-rough sets could prove bene cial in ne-tuning fuzzy set membership functions.
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APPENDIX A
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METRIC COMPARISON RESULTS: CLASSIFICATION DATASETS
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This section contains the full results for the tables in Section 8.7.2. The datasets were created by generating around 30 random feature values for 400 objects. Two or three features (referred to as x , y, or z ) are chosen to contribute to the nal Boolean classi cation by means of an inequality. The tables show the rating given to the features from the corresponding metric. For the data presented in the rst table, the rst feature, x , is used to determine the classi cation. The values of features y and z are derived from x : y = x, z = x 2 . Table cells are shaded to highlight the top ranked features determined by the feture metrics. Darker shading indicates a higher ranking.
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Computational Intelligence and Feature Selection: Rough and Fuzzy Approaches, by Richard Jensen and Qiang Shen Copyright 2008 Institute of Electrical and Electronics Engineers
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METRIC COMPARISON RESULTS: CLASSIFICATION DATASETS
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TABLE A.1 Feature x y z 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
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Feature evaluation for x > 0.5, y = FR 0.5257 0.5296 0.5809 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Re 0.31758 0.24586 0.32121 0.00276 0.00148 0.00268 0.00221 0.01002 0.00649 0.00889 0.00222 0.00182 0.00144 0.00475 0.01006 0.00613 0.00488 0.00563 0.01427 0.00467 0.01785 0.00327 0.00350 0.01339 0.00464 0.01334 0.01715 0.01742 0.00685 0.00206 0.00164 0.00171 0.00325 IG 0.997 0.997 0.997 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 GR 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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x, z = x2 1R 99.5 99.5 99.5 55.5 47.5 44.5 58.5 52.5 57.5 49.0 53.0 59.5 42.5 50.0 65.5 55.5 47.0 56.5 50.0 53.5 54.5 50.0 48.5 51.5 49.5 59.0 48.5 49.0 60.5 53.5 51.5 49.0 51.0 2 200 200 200 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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