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EM algorithm-based identification of a class of nonlinear Wiener systems with missing output data

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Abstract

This paper is concerned with the problem of parameter estimation for nonlinear Wiener systems in the stochastic framework. Based on the expectation–maximization (EM) algorithm in dealing with the incomplete data, it is applied to estimate the parameters of nonlinear Wiener models considering the randomly missing outputs. By means of the EM approach, the parameters and the missing outputs can be estimated simultaneously. To obtain the noise-free output in the linear subsystem of the Wiener model, the auxiliary model identification idea is adopted here. The simulation results indicate the effectiveness of the proposed approach for identification of a class of nonlinear Wiener models.

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Author information

Authors and Affiliations

  1. Key Laboratory of Advanced Process Control for Light Industry, Ministry of Education, Jiangnan University, Wuxi, 214122, People’s Republic of China

    Weili Xiong, Liang Ke & Baoguo Xu

  2. Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, 150080, Heilongjiang, People’s Republic of China

    Xianqiang Yang

Authors
  1. Weili Xiong
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  2. Xianqiang Yang
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  3. Liang Ke
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  4. Baoguo Xu
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Corresponding authors

Correspondence to Weili Xiong or Xianqiang Yang.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 21206053, 21276111, 61273131), the PAPD of Jiangsu Higher Education Institutions and the 111 Project (B12018).

Appendix: Detailed derivation of Eqs.(18) and (19)

Appendix: Detailed derivation of Eqs.(18) and (19)

The Q-function in Eq. (17) can be further written as

$$\begin{aligned}&Q(\varTheta |\varTheta ^s)=\sum _{t=m_1}^{m_\alpha }\left\{ -\frac{1}{2}\log (2\pi \sigma ^{2})-\frac{1}{2\sigma ^{2}}((\sigma ^s)^{2}\right. \nonumber \\&\quad \left. +(\varphi ^{T}(t)\vartheta -\varphi ^{T}(t)\vartheta ^s)^T(\varphi ^{T}(t)\vartheta -\varphi ^{T}(t)\vartheta ^s)\right\} \nonumber \\&\quad +\sum _{t=o_1}^{o_\beta }\left\{ -\frac{1}{2}\log (2\pi \sigma ^{2})-\frac{1}{2\sigma ^{2}}(y_t-\varphi ^T(t)\vartheta )^T\right. \nonumber \\&\quad \left. (y_t-\varphi ^T(t)\vartheta )\right\} +\log C. \end{aligned}$$
(26)

Taking the gradient of \(Q(\varTheta |\varTheta ^s)\) with respect to \(\vartheta \) and setting it to zero,

$$\begin{aligned} \frac{\partial Q(\varTheta |\varTheta ^s)}{\partial \vartheta }&=\sum _{t=m_1}^{m_\alpha }\left\{ -\frac{1}{2\sigma ^{2}}(2 \varphi (t)\varphi ^{T}(t)\vartheta \right. \nonumber \\&\quad \left. -2\varphi (t)\varphi ^{T}(t)\vartheta ^s)\right\} \nonumber \\&\quad +\sum _{t=o_1}^{o_\beta }\left\{ -\frac{1}{2\sigma ^{2}}(2\varphi (t)\varphi ^{T}(t)\vartheta \right. \nonumber \\&\quad \left. -2\varphi (t)y_t)\right\} \nonumber \\&=0 \end{aligned}$$
(27)

Through keeping the terms that not related with \(\vartheta \) at the right side, Eq. (27) can be written as

$$\begin{aligned}&\sum _{t=m_1}^{m_\alpha } \varphi (t)\varphi ^{T}(t)\vartheta +\sum _{t=o_1}^{o_\beta } \varphi (t)\varphi ^{T}(t)\vartheta \nonumber \\&\quad = \sum _{t=m_1}^{m_\alpha } \varphi (t)\varphi ^{T}(t)\vartheta ^s+\sum _{t=o_1}^{o_\beta }\varphi _t y_t \end{aligned}$$
(28)

Then, the new estimate of parameter \(\vartheta \) can be obtained as:

$$\begin{aligned} \vartheta ^{s+1}=\frac{\sum _{t=m_1}^{m_\alpha }\varphi (t)\varphi ^{T}(t)\vartheta ^s+\sum _{t=o_1}^{o_\beta }\varphi (t)y(t)}{\sum _{t=1}^{N}\varphi (t)\varphi ^{T}(t)} \end{aligned}$$
(29)

Taking the gradient of \(Q(\varTheta |\varTheta ^s)\) with respect to \(\sigma ^2\) and setting it to zero,

$$\begin{aligned} \frac{\partial Q(\varTheta |\varTheta ^s)}{\partial \sigma ^2}&=\sum _{t=m_1}^{m_\alpha }\left\{ -\frac{1}{2\sigma ^{2}} + \frac{1}{2\sigma ^4} \left[ (\sigma ^s)^{2}\right. \right. \nonumber \\&\quad \left. \left. +(\varphi ^{T}(t)\vartheta -\varphi ^{T}(t)\vartheta ^s)^{2}\right] \right\} \nonumber \\&\quad +\,\sum _{t=o_1}^{o_\beta } \left\{ -\frac{1}{2\sigma ^{2}} + \frac{1}{2\sigma ^4}(y_t-\varphi ^T(t)\vartheta )^2 \right\} \nonumber \\&=0. \end{aligned}$$
(30)

Through keeping the two terms including \(\sigma ^2\) at the left side, the Eq. (30) can be written as:

$$\begin{aligned}&\sum _{t=m_1}^{m_\alpha } \sigma ^2 + \sum _{t=o_1}^{o_\beta } \sigma ^2=N \cdot \sigma ^2\nonumber \\&\quad =\sum _{t=m_1}^{m_\alpha } \left\{ (\sigma ^s)^{2}+(\varphi ^{T}(t)\vartheta -\varphi ^{T}(t)\vartheta ^s)^{2}\right\} \nonumber \\&\qquad +\sum _{t=o_1}^{o_\beta } (y_t-\varphi ^T(t)\vartheta )^2 \end{aligned}$$
(31)

Then, the estimation of parameter \(\sigma ^2\) can be obtained as:

$$\begin{aligned} (\sigma ^{2})^{s+1}=\frac{\sum _{t=m_1}^{m_\alpha } ( (\sigma ^s)^2+(\varphi ^T(t)\vartheta ^{s+1}-\varphi ^T(t)\vartheta ^s )^2 )+ \sum _{t=o_1}^{o_\beta } ( y_t-\varphi ^T(t)\vartheta ^{s+1})^2 }{N}\end{aligned}$$
(32)

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Xiong, W., Yang, X., Ke, L. et al. EM algorithm-based identification of a class of nonlinear Wiener systems with missing output data. Nonlinear Dyn 80, 329–339 (2015). https://doi.org/10.1007/s11071-014-1871-6

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