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Focusing on the problem that the echo state network is easily influenced by outliers, in this paper we propose a robust model based on the Laplace prior distribution. This is achieved by replacing the Gaussian distribution with the Laplace distribution as the prior of the model output, the Laplace prior is less sensitive to the outliers and can enhance the capbility of the model to restrict outliers. Furthermoer, to solve the problem arising from the introduction of the Laplace distribution, which makes the solving process of the method difficlut, the bound optimization algorithm is employed and a suitable surrogate function is established. Based on the bound optimization algorithm, the Laplace prior can be equivalently transformed into the form of Gaussian prior, which is easily computed, and it can also be use to estimate the model parameters adaptively. Simulation results illustrate that the proposed method can be robust when outliers exist, while remaining acceptable prediction accuracy.
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Keywords:
- echo state network /
- robust model /
- surrogate function /
- Laplace distribution
[1] Chen S M, Hwang J R 2000 IEEE Trans. Systems, Man and Cybernetics-Part B 30 263
[2] Dhanya C T, Kumar D N 2010 Advances in Warer Resources 33 327
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[8] Cai J W, Hu S S, Tao H F 2007 Acta Phys. Sin. 56 6820 (in Chinese) [蔡俊伟、 胡寿松、 陶洪峰 2007 56 6820]
[9] [10] [11] Ma Q L, Zheng Q L, Peng H, Tan J W 2009 Acta Phys. Sin. 58 1410 (in Chinese) [马千里、 郑启伦、 彭 宏、 覃姜维 2009 58 1410]
[12] Farsa M A, Zolfaghari S 2010 Neurocomputing 73 2540
[13] [14] Song Q S, Feng Z R, Li R H 2009 Acta Phys. Sin. 58 5057 (in Chinese) [宋青松、 冯祖仁、 李人厚 2009 58 5057]
[15] [16] [17] Jaeger H, Haas H 2004 Science 304 78
[18] [19] Shi Z W, Han M 2007 IEEE Trans. Neural Networks 18 359
[20] [21] Shi Z W, Han M 2007 Control and Decision 22 258 (in Chinese) [史志伟、 韩 敏 2007 控制与决策 22 258]
[22] Han M, Mu D Y 2010 Control and Decision 25 531 (in Chinese) [韩 敏、 穆大芸 2010 控制与决策 25 531]
[23] [24] [25] Ting J A, Dsouza A, Schaal S 2007 ICRA 2007 2489
[26] [27] Zhong M J 2006 Neurocomputing 69 2351
[28] Tipping M E 2001 Journal of Machine Learning Research 1 211
[29] [30] Hong X, Chen S 2005 IEEE Trans. Systems, Man and Cybernetics-Part B 35 155
[31] [32] Ding T, Zhou H C, Huang J H 2004 2010 Journal of Hydraulic Engineering 12 15 (in Chinese) [丁 涛、 周惠成、 黄健辉 2004 水力学报 12 15]
[33] [34] [35] Islam M N, Sivakumar B 2002 Advances in Water Resources 25 179
[36] [37] Fraser A M, Swinney H L 1986 Phys. Rev. A 33 1134
[38] Kennel M B, Brown R, Abarbanel H D I 1992 Phys. Rev. A 45 3403
[39] -
[1] Chen S M, Hwang J R 2000 IEEE Trans. Systems, Man and Cybernetics-Part B 30 263
[2] Dhanya C T, Kumar D N 2010 Advances in Warer Resources 33 327
[3] [4] Du J, Cao Y J Liu Z J, Xu L Z, Jiang Q Y, Guo C X, Lu J G 2009 Acta Phys. Sin. 58 5997 (in Chinese) [杜 杰、 曹一家、 刘志坚、 徐立中、 江全元、 郭创新、 陆金桂 2009 58 5997]
[5] [6] [7] Leung H, Lo T, Wang S C 2001 IEEE Trans. Neural Network 12 1163
[8] Cai J W, Hu S S, Tao H F 2007 Acta Phys. Sin. 56 6820 (in Chinese) [蔡俊伟、 胡寿松、 陶洪峰 2007 56 6820]
[9] [10] [11] Ma Q L, Zheng Q L, Peng H, Tan J W 2009 Acta Phys. Sin. 58 1410 (in Chinese) [马千里、 郑启伦、 彭 宏、 覃姜维 2009 58 1410]
[12] Farsa M A, Zolfaghari S 2010 Neurocomputing 73 2540
[13] [14] Song Q S, Feng Z R, Li R H 2009 Acta Phys. Sin. 58 5057 (in Chinese) [宋青松、 冯祖仁、 李人厚 2009 58 5057]
[15] [16] [17] Jaeger H, Haas H 2004 Science 304 78
[18] [19] Shi Z W, Han M 2007 IEEE Trans. Neural Networks 18 359
[20] [21] Shi Z W, Han M 2007 Control and Decision 22 258 (in Chinese) [史志伟、 韩 敏 2007 控制与决策 22 258]
[22] Han M, Mu D Y 2010 Control and Decision 25 531 (in Chinese) [韩 敏、 穆大芸 2010 控制与决策 25 531]
[23] [24] [25] Ting J A, Dsouza A, Schaal S 2007 ICRA 2007 2489
[26] [27] Zhong M J 2006 Neurocomputing 69 2351
[28] Tipping M E 2001 Journal of Machine Learning Research 1 211
[29] [30] Hong X, Chen S 2005 IEEE Trans. Systems, Man and Cybernetics-Part B 35 155
[31] [32] Ding T, Zhou H C, Huang J H 2004 2010 Journal of Hydraulic Engineering 12 15 (in Chinese) [丁 涛、 周惠成、 黄健辉 2004 水力学报 12 15]
[33] [34] [35] Islam M N, Sivakumar B 2002 Advances in Water Resources 25 179
[36] [37] Fraser A M, Swinney H L 1986 Phys. Rev. A 33 1134
[38] Kennel M B, Brown R, Abarbanel H D I 1992 Phys. Rev. A 45 3403
[39]
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