Research on Robust Probabilistic System Identification Method With Asymmetric and Skewed Noise
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摘要: 在现有的系统辨识算法中, 常用的高斯、学生氏t (Student＇s t, St)、拉普拉斯等噪声分布均呈现出对称的统计特性, 难以描述非对称性、有偏的输出噪声, 使得在非对称偏斜噪声条件下算法的性能下降. 基于此, 研究一类广义双曲倾斜学生氏t (Generalized hyperbolic skew student＇s t, GHSkewt)分布, 并在非对称偏斜噪声条件下, 提出一种线性系统鲁棒辨识算法. 首先, 对GHSkewt分布的重尾特性和偏斜特性进行详细阐述, 数学上证明了标准学生氏t分布可看作是GHSkewt分布的一个特例; 其次, 引入隐含变量将GHSkewt分布进行数学分解, 以方便算法的推导和实现; 最后, 在期望最大化(Expectation-maximization, EM)算法下, 重构具有隐含变量系统的代价函数, 通过迭代优化的方式, 不断从被污染数据集中学习过程的动态特性和噪声分布, 实现噪声参数和模型参数的联合估计.
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关键词:
- 鲁棒系统辨识 /
- 非对称偏斜噪声 /
- 广义双曲倾斜学生氏t分布 /
- 期望最大化算法
Abstract: In the existing system identification algorithms, the commonly used Gaussian, student＇s t (St) and Laplace distributions all show symmetric statistical characteristics which makes them difficult to describe the asymmetric and skewed noise, therefore the performance of the corresponding algorithms may largely degrade with the skewed noise. To this end, this paper introduces the generalized hyperbolic skew student＇s t (GHSkewt) distribution and proposes a robust identification algorithm for linear systems with the asymmetric and skewed noise. Firstly, the thick-tailed and skewed characteristics of the GHSkewt distribution are introduced detailedly and it is also proved that the standard student＇s t-distribution can be regarded as a special case of the GHSkewt distribution; Secondly, the latent variables are introduced to mathematically decompose the GHSkewt distribution in order to facilitate the derivation and implementation of the algorithm; Finally, the system cost function with the latent variables is reconstructed under the expectation-maximization (EM) algorithm. The dynamic characteristics and noise distribution of the system are continuously learned from the contaminated data with iterative optimization, then the estimation of noise parameters and model parameters are realized. -
图 1 对称分布与参数值$\beta $和$\upsilon $不同的GHSkewt分布对比
Fig. 1 Comparison of the symmetric distribution and the GHSkewt distribution with different parameter values of $\beta $ and $\upsilon $
图 2 真实输出和含15%异常值比例输出的对比
Fig. 2 Comparison of the real output and output with 15% of outlier points
图 3 偏斜噪声${{\boldsymbol{G}}_0}$的概率密度曲线
Fig. 3 The probability density curve of skewed noise${{\boldsymbol{G}}_0}$
图 4 偏斜噪声${{\boldsymbol{G}}_0}$下参数估计轨迹
Fig. 4 Trajectories of the parameter estimates with skewed noise${{\boldsymbol{G}}_0}$
图 5 偏斜噪声${{\boldsymbol{G}}_0}$下3种算法实验结果的对比
Fig. 5 Comparison of experimental results of three algorithms under skewed noise ${{\boldsymbol{G}}_0}$
图 7 不同偏斜噪声下蒙特卡洛仿真实验的RPEE曲面
Fig. 7 The RPEE surface for the Monte Carlo with different levels of skewed noise
图 8 参数初始值不同时, 蒙特卡洛仿真实验的参数估计轨迹
Fig. 8 Trajectories of the parameter estimates for the Monte Carlo with different parameter initial values
图 11 质量弹簧阻尼系统的真实输出和含偏斜噪声输出
Fig. 11 The real output and output with skewed noise of the mass-spring-damper system
表 1 异常值比例为15%时, 蒙特卡洛仿真实验参数估计结果的均值和标准差
Table 1 The mean and standard deviation of the Monte Carlo parameter estimation results with 15% of outlier points
算法 参数$ {a_1} $ 参数$ {a_2} $ 参数$ {b_1} $ 参数$ {b_2} $ 均值 标准差 均值 标准差 均值 标准差 均值 标准差 Laplace-Iden − 0.5751 $ {3.6 \times 10^{ - 4}} $ 0.7739 $ {1.9 \times 10^{-4}} $ 0.5083 $ {2.8 \times 10^{-4}} $ 0.4085 $ {2.7 \times 10^{-4}} $ St-Iden − 0.5878 $ {3.2 \times 10^{-8}} $ 0.7899 $ {1.1 \times 10^{-8}} $ 0.5043 $ {6.5 \times 10^{-8}} $ 0.4028 $ {2.0 \times 10^{-8}} $ GHSkewt-Iden − 0.5876 $ {6.5 \times 10^{-7}} $ 0.7899 $ {2.5 \times 10^{-7}} $ 0.5042 $ {1.1 \times 10^{-7}} $ 0.4030 $ {4.0 \times 10^{-7}} $ 真实值 − 0.6000 — 0.8000 — 0.5000 — 0.4000 — 
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表 2 不同异常值比例下蒙特卡洛仿真实验的平均RPEE、RMSE和$ {\rm{R}}^2 $
Table 2 The averaged RPEE、RMSE and$ {\rm{R}}^2 $for the Monte Carlo with different ratios of outlier points
异常值比例 Laplace-Iden St-Iden GHSkewt-Iden RPEE (%) RMSE $ {\rm{R}}^2 $ RPEE (%) RMSE $ {\rm{R}}^2 $ RPEE (%) RMSE $ {\rm{R}}^2 $ 5% 0.71 0.0144 0.9995 0.61 0.0104 0.9998 0.39 0.0072 0.9999 10% 2.44 0.0475 0.9949 1.20 0.0186 0.9992 1.13 0.0192 0.9992 15% 3.20 0.0564 0.9928 1.40 0.0267 0.9984 1.42 0.0271 0.9983 20% 5.33 0.0880 0.9826 2.70 0.0499 0.9944 2.74 0.0503 0.9943
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表 3 不同偏斜噪声下蒙特卡洛仿真的平均RPEE、RMSE和$ {\rm{R}}^2 $
Table 3 The averaged RPEE、RMSE and$ {\rm{R}}^2 $for the Monte Carlo with different levels of skewed noise
偏斜噪声 Laplace-Iden St-Iden GHSkewt-Iden RPEE (%) RMSE $ {\rm{R}}^2 $ RPEE (%) RMSE $ {\rm{R}}^2 $ RPEE (%) RMSE $ {\rm{R}}^2 $ $ {\boldsymbol{G}}_1 $ 10.93 0.2615 0.8416 7.19 0.1889 0.9175 1.79 0.0315 0.9978 $ {\boldsymbol{G}}_2 $ 16.02 0.3381 0.7356 12.39 0.2825 0.8155 3.99 0.0699 0.9884 $ {\boldsymbol{G}}_3 $ 11.78 0.2607 0.8426 9.40 0.2184 0.8886 4.41 0.0779 0.9855
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表 4 参数初始值不同时蒙特卡洛仿真的均值和标准差
Table 4 The mean and standard deviation of the Monte Carlo with different parameter initial values
模型参数 真实值 均值 标准差 $ {{a_1}} $ − 0.6000 − 0.5860 0.0045 $ {{a_2}} $ 0.8000 0.7872 0.0033 $ {{b_1}} $ 0.5000 0.5028 0.0068 $ {{b_2}} $ 0.4000 0.4037 0.0080
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表 5 自我验证的RMSE和$ {\rm{R}}^2 $
Table 5 The RMSE and$ {\rm{R}}^2 $for self-verification
质量块位置 Laplace-Iden St-Iden GHSkewt-Iden RMSE $ {\rm{R}}^2 $ RMSE $ {\rm{R}}^2 $ RMSE $ {\rm{R}}^2 $ $ {{S}} \ ({\rm{m}})$ 0.0795 0.9373 0.0607 0.9635 0.0137 0.9981
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表 6 交叉验证的RMSE和$ {\rm{R}}^2 $
Table 6 The RMSE and$ {\rm{R}}^2 $for cross-verification
质量块位置 Laplace-Iden St-Iden GHSkewt-Iden RMSE $ {\rm{R}}^2 $ RMSE $ {\rm{R}}^2 $ RMSE $ {\rm{R}}^2 $ $ {{S}} \ ({\rm{m}})$ 0.0894 0.9345 0.0674 0.9627 0.0148 0.9982
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