拉斯噪声和均匀噪声下SVR的鲁棒性研究
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摘要
支持向量机SVM是实现统计学习理论的通用学习方法,其优异的泛化性能使得支持向量机在模式识别、回归分析和预测、密度估计等领域都得到了实际应用。当SVM用于回归分析和预测时,通常称其为支持向量回归机SVR。在回归分析中,样本数据通常含有噪声。如何选择合适的参数使得支持向量回归机SVR更具鲁棒性,从而对样本数据噪声产生尽可能强的抑制能力,是一个有着重要的理论价值和应用价值的课题。本文的主要目的就是研究当输入样本含有两种典型的噪声——拉斯噪声和均匀噪声的情况下如何选择SVR的参数使SVR具有更强的鲁棒性。
首先,本文研究了Huber-SVR的鲁棒性的问题,即着重研究了当输入样本噪声为拉斯模型和均匀模型时, Huber-SVR的参数选择问题,并在贝叶斯框架下推导出了以下结论:当Huber-SVR的鲁棒性最佳时,Huber-SVR中的参数μ与输入拉斯噪声和均匀噪声的标准差σ间均呈近似线性关系。
其次,本文研究了r-SVR的鲁棒性的问题,即着重研究了当输入样本噪声为拉斯模型和均匀模型时, r-SVR的参数选择问题,并在贝叶斯框架下推导出了以下结论:当r-SVR的鲁棒性最佳时, r-SVR中的参数r与输入拉斯噪声和均匀噪声的标准差σ间均呈近似线性反比关系。这两个结论亦得到了实验的证实。
最后,将Huber-SVR和r-SVR用于实际股市数据回归分析,并验证上述两个结论。
Support Vector Machine SVM is a general learning approach based on statistical learning theory, which has obtained its practical applications in many areas such as pattern recognition, regression and prediction, and density evaluation due to its excellent generalized capability. When applied to regression and prediction, we often call SVM as support vector regression machine SVR. In general, sample data in regression analysis often contain noise. Therefore, how to determine the optimal parameters such that SVR becomes as robust as possible is an important subject worth of studying. The main aim of this dissertation is to study the theoretical relationships between the SVR’s parameters and Laplacian and Uniform noisy inputs respectively.
Firstly, in this dissertation, the issue of Huber- SVR robustness is addressed. Focused on the parameter choice issues of Huber-SVR with Laplacian and Uniform noisy inputs respectively, and based on the Bayesian framework, we derived the first relationship: with the best robustness, the approximately linear relationship between the parameterμin Huber-SVR and the standard deviationσof Laplacian and Uniform noisy inputs is kept.
Secondly, the issue of r- SVR robustness is then studied. Focused on the parameter choice issues of r-SVR with Laplacian and Uniform noisy inputs respectively, and based on the Bayesian framework, we derived the second relationship: with the best robustness, the approximately inversely linear relationship between the parameter r in norm r-SVR and the standard deviationσof Laplacian and Uniform noisy inputs is kept. Meanwhile, our experimental results confirmed the above claims.
Finally, Huber- SVR and r-SVR were used to regress practical stock market data respectively, and the results confirmed the above two conclusions.
首先,本文研究了Huber-SVR的鲁棒性的问题,即着重研究了当输入样本噪声为拉斯模型和均匀模型时, Huber-SVR的参数选择问题,并在贝叶斯框架下推导出了以下结论:当Huber-SVR的鲁棒性最佳时,Huber-SVR中的参数μ与输入拉斯噪声和均匀噪声的标准差σ间均呈近似线性关系。
其次,本文研究了r-SVR的鲁棒性的问题,即着重研究了当输入样本噪声为拉斯模型和均匀模型时, r-SVR的参数选择问题,并在贝叶斯框架下推导出了以下结论:当r-SVR的鲁棒性最佳时, r-SVR中的参数r与输入拉斯噪声和均匀噪声的标准差σ间均呈近似线性反比关系。这两个结论亦得到了实验的证实。
最后,将Huber-SVR和r-SVR用于实际股市数据回归分析,并验证上述两个结论。
Support Vector Machine SVM is a general learning approach based on statistical learning theory, which has obtained its practical applications in many areas such as pattern recognition, regression and prediction, and density evaluation due to its excellent generalized capability. When applied to regression and prediction, we often call SVM as support vector regression machine SVR. In general, sample data in regression analysis often contain noise. Therefore, how to determine the optimal parameters such that SVR becomes as robust as possible is an important subject worth of studying. The main aim of this dissertation is to study the theoretical relationships between the SVR’s parameters and Laplacian and Uniform noisy inputs respectively.
Firstly, in this dissertation, the issue of Huber- SVR robustness is addressed. Focused on the parameter choice issues of Huber-SVR with Laplacian and Uniform noisy inputs respectively, and based on the Bayesian framework, we derived the first relationship: with the best robustness, the approximately linear relationship between the parameterμin Huber-SVR and the standard deviationσof Laplacian and Uniform noisy inputs is kept.
Secondly, the issue of r- SVR robustness is then studied. Focused on the parameter choice issues of r-SVR with Laplacian and Uniform noisy inputs respectively, and based on the Bayesian framework, we derived the second relationship: with the best robustness, the approximately inversely linear relationship between the parameter r in norm r-SVR and the standard deviationσof Laplacian and Uniform noisy inputs is kept. Meanwhile, our experimental results confirmed the above claims.
Finally, Huber- SVR and r-SVR were used to regress practical stock market data respectively, and the results confirmed the above two conclusions.
引文
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2. Vladimir N. Vapnik, The Nature of Statistical Learning Theory[J]. Springer, 1995
3. R. Jain, R. Kasturi, B. G. Schunck. Machine Vision[M]. Beijing:China Machine Press,2003
4.徐晨,赵瑞珍,甘小冰.小波分析.应用算法[M].北京:科学出版社,2004
5.阎平凡,张长水.人工神经网络与模拟进化计算[M].北京:清华大学出版社,2000
6.钱松迪等.运筹学[M].北京:清华大学出版社,1990
7.沈永欢,梁在中等.实用数学手册[M].北京:科学出版社,2002
8.程云鹏.矩阵论[M].西安:西北工业大学出版社,1989
9.朱嘉钢,王士同,杨静宇.鲁棒r-支持向量回归机中参数r的选择研究[J].控制与决策,1001-0920(2004)12-1383-04.
10.朱嘉钢,王士同,杨静宇. r-SVR中参数r与输入噪声间线性反比关系的仿真研究[J].计算机科学,2005,.32(9):205-207.
11.朱嘉钢,王士同. Huber-SVR中参数μ与输入噪声间关系的研究[J].复旦学报,2004,43(5):793-796.(ZHU Jia-gang, WANG Shi-tong. Research on the Dependency Betweenμand the Input Noise in Huber-Support Vector Regression[J]. Journal of Fudan University,2004,43(5):793-796.)
12. Wang Shitong, Zhu Jiagang, F.L.Chung,Lin Qing,Hu Dewen.Theoretically optimal parameter choices for support vector regression machines Huber-SVR and Norm_r-SVR with noisy input[J]. Soft Computing.2005,9(10): 732 - 741.
13. James T. Kwok, Ivor W. Tsang. Linear dependency betweenεand the input noise inε-support vector regression[J]. IEEE Transaction on Neural Networks. 2003,14(3):544-553
14. A. J. Smola, N. Murata, B.Sch?lkopf, K. R.Müller. Asymptotically optimal choice ofε-loss for support vector machines[C]. In Proceedings of the International Conference on Artificial Neural Networks.1998
15. M.H.Law, J.T. Kwok. Bayesian support vector regression[C]. In Proceedings of the English International Workshop on Artificial Intelligence and Statistics.2001: 239-244
16. J.B.Gao, S.R.Gunn, C.J. Ham’s. A probabilistic framework for SVM regression and Error Bar Estimation[J]. Machine Learning.2002,46(1-3):71-89
17. D.J.C. MacKay. Bayesian interpolation[J]. Neural Computation.1992(5):415-447
18. O.L.Mangasarian, D.R.Musicant. Robust linear and support vector regression[J]. IEEE Transactions on Pattern Analysis and Machine Learning.2000,22(9):1-6
19. Jiagang Zhu, Shitong Wang, Xisheng Wu, F. L. Chung. A Novel Adaptive SVR Based Filter ASBF for Image Restoration[J]. Soft Computing. 2006,10(8):665-672.
20. T.B.Trafalis, A. M. Malyscheff. An analytic Center Machine[J]. Machine Learning.2002,46(1-3):203-223
21. Alexander M., Malyscheff, Theodore B. Trafalis. An analytic center machine for regression. Central European Journal of Operational Research. In press
22. Herbrich R, Graepel T.,Campdell C.. Robust bayes point machines[C]. In Proceedings of ESANN 2000.2000:49-54
23. Ruján P..Playing billard in version space[J]. Neural Computation.1997,9:99-122
24. J.T.Kwok.The evidence framework applied to support vector machines[J]. IEEE Transactions on Neural Networks. 2000,11(5):1162-1173
25. C.J.C.Burges. A tutorial on support vector machines for pattern recognition[J]. Data Mining Knowledge Discovery.1998,2(2):955-974
26. J.T.Kwok. Moderating the outputs of support vector machine classifiers[J]. IEEE Transactions on Neural Networks.1999(10):1018-1031
27. Christopher J.C. Burges, David J. Crisp.Uniqueness theorems for kernel methods. Neurocomputing. 2003, 55(1-2): 187-220.
28. T.c.Lin, P.T.Yu. Adaptive Two-Pass Median Filter Based on Support Vector Machines for Image Restoration[J]. Neural Computation.2004,16:333-354
29. Arakawa K.. Median filters based on fuzzy rules and its application to image restoration[J]. Fuzzy Sets and Systems.1996,77:3-13
30. S.S. Keerthi, S.K. Shevade, C.Bhattacharyya, K.R.K. Murthy. A fast iterative nearest point algorithm for support vector machine classi1er design[J]. IEEE Transaction on Neural Networks. 2000,11 (1):124-136
31. V. David, Sánchez A.. Advanced support vector machines and kernel methods[J]. Neurocomputing.2003,55(1-2):5-20
32. S. Amari, S. Wu. Improving support vector machine classifiers by modifying kernel functions[J]. Neural Networks.1999,12 (6) :783-789
33. R. Burbidge, M. Trotter, B. Buxton, S. Holden. Drug design by machine learning: support vector machines for pharmaceutical data analysis[J]. Computer Chemistry. . 2001,26 (1): 5-14
34. L. Cao. Support vector machines experts for time series forecasting[J].Neurocomputing.2003,51:321-339
35. G.C. Cawley, N.L.C. Talbot. Improved sparse least-squares support vector machines[J]. Neurocomputing. 2002,48 (4):1025-1031
36. C.-C. Chang, C.-J. Lin. Trainingυ-support vector classifiers: theory and algorithms[J]. Neural Computation. 2001,13 (9) : 2119-2147
37. O. Chapelle, V.N. Vapnik, O. Bousquet, S. Mukherjee. Choosing multiple parameters for support vector machines[J]. Machine Learning. 2002,46 (1-3): 131-159
38. R. Collobert, S. Bangio. SVMTorch: a support vector machine for large-scale regression and classification problems[J]. Machine Learning. 2001,1(2):143-160
39. C. Cortes, V. Vapnik. Support vector networks[J]. Machine Learning. 1995,20(3): 273-297
40. H. Drucker, D. Wu, V. Vapnik.Support vector machines for span categorization[J]. Neural Networks.1999,10 (5):1048-1054
41. K. Duan, S.S. Keerthi, A.N. Poo.Evaluation of simple performance measures for tuning SVM hyperparameters[J].Neurocomputing. 2003,51: 41-59
42. H. Eghbalnia, A. Assadi. An application of support vector machines and symmetry to computational modeling of perception through visual attention[J]. Neurocomputing. 2001, 38-40: 1193-1201
43. G.W. Flake, S. Lawrence. Efficient SVM regression training with SMO.Mach. Learn[J]. 2002,46(1-3) : 271-290
44. J.B. Gao, S.R. Gunn, C.J. Harris. Mean field methods for the support vector machine regression[J]. Neurocomputing. 2003,50 (4): 391-405
45. C.-W. Hsu, C.-J. Lin. A comparison of methods for multiclass support vector machines[J]. IEEE Transaction on Neural Networks. 2002 ,13 (2) : 415-425
46. S.S. Keerthi, S.K. Shevade. SMO algorithm for least-squares SVM formulations[J]. Neurocomputing . 2003,15 (2): 487-507
47. S. Keerthi, C.B.S.K. Shevade, K.R.K. Murphy. A fast iterative nearest point algorithm for support vector machine classifier design[J]. IEEE Transaction on Neural Networks. 2000,11 (1):124-136
48. P. Laskov. An improved decomposition algorithm for regression support vector machines[J]. Machine Learning. 2002,46(1-3):315-350
49. S.-P. Liao, H.-T. Liu, C.-J. Lin. A note on the decomposition methods for support vector regression[J]. Neural Computation .2002,14: 1267-1281
50. Jonathan Robinson, Vojislav Kecman. Combining support vector machine learning with the discrete cosine transform in image compression[J]. IEEE Trans on Neural Networks. 2003,14(4):950-958
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