# Refine Decision Boundaries of a Statistical Ensemble by Active Learning

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2 prediction to any instances x, x X. There are various kinds of classifiers that do not output the pure -of-m representation but offer a confidence for each class. Such classifiers can be converted into a probabilistic form by the following transformation: ŷ i (x) = M e j= 2 ( yi + ) e ( y j + ) 2 probability that the input pattern (being tested) belongs to a specific class. For decision-making, the maximum a posterior (MAP) rule is applied such that * C = arg max yˆ (2) j M ( i =,, M ) () where y i is the ith output component of a classifier. From the probabilistic point of view, each can be interpreted by the By the MAP rule, traditional statistical ensemble methods, e.g. boost, divides training patterns into two categories: easy and hard portions based on whether a correct class label is assigned to a pattern. Apparently, the MAP decision-making ŷ i j rule is suitable for testing an unknown pattern and tends to be necessary. When such a rule is used in a training stage, however, it would incur losses of useful information. Fig. depicts an example to demonstrate such a problem. For two patterns belonging to the same class (class 5), both of them have been correctly classified by a classifier. However, the two patterns convey unequal information. Apparently, the classifier can more likely produce the correct label for the pattern corresponding in Fig. (a) than that shown in Fig. (b) in terms of the probabilistic justification. In other words, the pattern shown in Fig. (b) is more informative given that it tends to be closer to decision boundaries [3]. Unfortunately, the previous resampling criteria in statistical ensemble methods merely focus on the misclassified patterns and fail to consider the distinction among patterns that have been correctly classified. Motivated by our previous work in active learning where such information has turned to be useful for refining decision boundaries [3], we propose a resampling criterion to find all possible informative patterns for construction of statistical ensemble classifiers, which is expected to provide a more active date selection procedure for refining the decision boundaries. B. Active Difficulty Measure (a) According to Bayesian decision theory [4], the zero-one loss function provides a criterion to obtain the minimum error rate or to make the maximum correct prediction. According to the zero-one loss criterion, an ideal classifier always outputs the correct -of-m representation, where the ith component corresponds to class C i, such that for x Ci, if j = i d j (x) = (3) 0 if j i In other words, the ith element of such an ideal output vector is one only while other elements are zero. Fig. 2 shows an example where a pattern belonging to class 5 is perfectly classified (c.f. Fig. ). (b) Fig.. The outputs of two patterns belonging to the same class. Fig. 2. The ideal output of a probabilistic classifier for a pattern belonging to class 5.

5 Recongnition Rate (%) Recongnition Rate (%) Recongnition Rate (%) (a) Results on digit 0 (b) Results on digit (c) Results on digit 2 Recongnition Rate (%) Recongnition Rate (%) Recongnition Rate (%) (d) Results on digit 3 (e) Results on digit 4 (f) Results on digit 5 Recongnition Rate (%) Recongnition Rate (%) Recongnition Rate (%) (g) Results on digit 6 (h) Results on digit 7 (i) Results on digit 8 Recongnition Rate (%) Baseline AdaBoost Active AdaBoost (j) Results on digit 9 Fig. 4. Comparative results for the handwritten digit recognition problem. (a)-(j) Results corresponding to ten digits from 0 to 9.

6 information conveyed in the patterns being classified correctly, which leads to the effect of refining decision boundaries as shown in Fig. 4. The results also indicate that the idea underlying our method is highly consistent with the use of active data selection for refining the decision boundaries of a strong classifier [3]. IV. CONCLUSION In this paper, we have presented an alternative resampling criterion for active selection of informative patterns to construct a statistical ensemble classifier. In comparison with the existing error-based resampling criteria in statistical ensemble learning, our criterion makes better use of information conveyed in training patterns. Comparative results on two real world problems based on Adaboost, along with others with different statistical ensemble methods (e.g. [0],[]) not reported here, demonstrate that for pattern classification our method yields the better generalization performance by refining decision boundaries with additional information conveyed in training patterns. REFERENCES The Annals of Statistics, vol. 26, pp , 8. [2] Y. Freund and R. E. Schapire, Experiments with a new boosting algorithm, Proceedings of International Conference of Machine Learning, pp , 9. [3] L. Wang, K. Chen, and H. Chi, Capture interspeaker information with a neural network for speaker identification, IEEE Transactions on Neural Networks, vol. 3, pp , [4] R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification (2nd Edition), Wiley-Interscience, 200. [5] Y. Freund and R. E. Schapire, A decision-theoretic generalization of on-line learning and an application to boosting, Journal of Computer and System Sciences, vol. 55, pp.9-39, 9. [6] D. S. Luo and K. Chen, A comparative study of statistical ensemble methods on mismatch conditions, Proceedings of International Joint Conference on Neural Networks, pp.59-64, [7] URL: [8] Y. Linde, A. Buzo, and R. M. Gray, An algorithm for vector quantizer design, IEEE Transactions on Communications, vol. 28, pp. 84-, 0. [9] F. K. Soong, A. E. Rosenberg, L. R. Rabiner,, and B. H. Zhuang, A vector quantization approach to speaker identification, Proceedings of International Conference on Acoustics, Speech and Signal Processing, pp , 5. [0] R. E. Schapire, The strength of weak learnability, Machine Learning, vol. 5, pp. -227, 9. [] C. Y. Ji and S. Ma, Combinations of weak classifiers, IEEE Transactions on Neural Networks, vol. 8, pp 32-42, 9. [] R. E. Schapire, Y. Freund, P. Bartlett, and W. S. Lee, Boosting the margin: A new explanation for the effectiveness of voting methods,

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