Robust identification of linear ARX models with recursive EM algorithm based on Student’s t-distribution
Introduction
For industrial processes, incomplete data problems, such as unknown time-delay, randomly missing observations, multiple-mode of the dynamics, and etc., are commonly encountered. To deal with the problem of incomplete data, Dempster et al. [1] proposed the expectation-maximization (EM) algorithm, which has exhibited its superiority in various fields. As for the identification problem, the EM algorithm has been extensively discussed owing to its ability to deal with incomplete data. For example, the EM algorithm has been used in the identification problem of the state-space models [2], the piece-wise affine models [3], the Wiener-Hammerstein models [4], and the linear parameter varying (LPV) models [5].
However, most of the works mentioned above are of the batch form, in which the historical observations over a period of time are taken into account during data processing. This feature leads to a defect that the EM algorithm of a batch manner can hardly adapt to unknown varying dynamics. When a new observation arrives, the algorithm should be completely updated, increasing the computational burden. Consequently, intense efforts have been invested in an online/recursive variant of the EM algorithm. The earliest version of the recursive EM algorithm (REM) was raised by Titterington et al. [6] in a fashion of stochastic approximation. In Titterington’s algorithm, parameters were recursively updated based on the Fisher information matrix (FIM) of complete data and the gradient of the likelihood of observed data. Later, Wang et al. [7] proved the almost sure convergence of Titterington’s algorithm. Krishnamurthy et al. [8] developed a recursive estimation approach for hidden Markov models (HMMs) according to Kullback-Leibler information measure, in which parameters were estimated in a stochastic approximation form. Moreover, the recursive identification problem of the autoregressive (AR) models with Markov regime was solved by Krishnamurthy et al [9].
To avoid calculating the inverse of FIM, a new variant of the online EM algorithm was designed by Cappé et al. [10] based on the recursion of sufficient statistics. Although an earlier thought of the sufficient statistics’ recursion appeared in the work of Mongillo et al. [11], Cappé provided a systematic expression of the online EM algorithm based on the exponential family of distributions. Later, Cappé’s algorithm [12] was also applied to the parameter estimation of the hidden Markov models. For the state-space models, several works involving the online EM algorithm were reported. For example, an online EM algorithm based on the split-data likelihood function was developed for state-space models by Andrieu et al. [13]. Özkan et al. [14] proposed a recursive EM approach for the identification of jump Markov nonlinear state-space systems using the recursion of sufficient statistics. Zhao et al. [15] solved the identification problem of the hybrid autoregressive exogenous (ARX) models with Markov-Chain-typed time-varying time-delay, using the recursive EM algorithm.
In practical industrial engineering, data quality is a significant issue that lain ahead identification and modeling. The reliability of data is always damaged by outliers. Therefore, a robust identification algorithm aiming at outliers is considered in this paper. The outliers often appear in practice due to sensor failure, transmission interference, and other unknown disturbances. If not be appropriately dealt with in modeling, they may lead to performance degradation of the model. The simple approaches to deal with outliers are trimming and smoothing. Although these approaches are intuitive, they generally suffer from information loss and biased estimation [16]. To overcome these drawbacks, a piecewise loss function approach was proposed by Huber et al. [17], which is known as Huber’s robust regression. Similar to the piecewise loss function, the contaminated Gaussian distribution was also employed in some works to describe the measurement noise corrupted with outliers. For example, two weighted Gaussian distributions were adopted in [18] for robust identification of the piecewise-affine systems. A Bayesian approach was developed by Khatibisepehr et al. [19] with the contaminated Gaussian distribution for robust identification of linear systems. Besides, due to the Laplace distribution’s property of heavy tails, it was also used to describe the process noise that contaminated with outliers [20]. As an alternative approach, the Student’s t-distribution, which is a powerful extension of the contaminated Gaussian distribution, is also employed for robust identification. In comparison to Gaussian distribution, the probability density function (pdf) of the t-distribution has heavier tails, which assign more capacity to outliers [21]. Due to its reliability and analytic property, the t-distribution has been applied in various ranges of research, such as clustering [22], statistical process monitoring[23], image processing [24], state estimation [25], and so on. In the field of identification, several results entailing Student’s t-distribution have been reported. For example, a robust identification approach for the LPV systems was developed in [26] using Student’s t-distribution. A generalized EM algorithm was proposed by Yang et al. [27] for global identification of the LPV systems, in which the noise was depicted with the t-distribution. A robust identification approach for the switched Markov autoregressive exogenous systems was proposed by Fan et al. [28] based on Student’s t-distribution. Moreover, Student’s t-distribution has been considered in the framework of the variational Bayesian inference in the identification field [29], [30].
To the best knowledge of the authors, however, there is no report about the robust recursive EM algorithm based on the Student’s t-distribution up to now. Although a robust recursive least square algorithm was developed for the spectral estimation, under the assumption that the time-series follow the t-distribution [31], the scale factor for the residual of the estimator was obtained with a moving window median method. In this work, a robust identification approach is derived based on the recursive EM algorithm, through which the parameter vector, the variance, and the degree of freedom are recursively estimated. With the implementation of the Student’s t-distribution, the robustness of the algorithm against outliers is enhanced.
The robust recursive EM algorithm can be applied in the identification of the real-life linear industrial processes, in which the measurements are contaminated with irregular outliers. With the implementation of the proposed algorithm, pretreatment method for outlier is not necessary. Thereby, the information loss caused by the pretreatment is eradicated. Furthermore, the robust recursive EM algorithm makes it possible for the online updating of the model of a system in the presence of outliers. Compared with the robust iterative algorithms, the robust recursive EM algorithm is more flexible in dealing with the unknown dynamic variation by incorporating the new samples. On the other hand, the influence of the outliers can be minimized in the robust recursive EM algorithm in contrast to the conventional recursive algorithms.
The remainders of this paper are organized as follows: The identification model and the mathematical description of the Student’s t-distribution are detailed in the next section. In Section 3, the derivation of the robust recursive EM algorithm is provided. The effectiveness and adaptiveness of the robust recursive EM algorithm are verified with a numerical example and a simulated continuous stirred tank reactor (CSTR) process in Section 4. In Section 5, the conclusion is finally drawn.
Section snippets
Problem formulation
For the identification problem, the most natural description of a linear system is the autoregressive exogenous (ARX) model. Considering the approximation capability and analytical property of the ARX model, it is adopted in this work to describe general linear processes. The mathematical formulation of the ARX model is expressed aswhere is the time index of the process, denotes the measurements at time k, denotes the regression vector, which is specified
Derivation of recursive EM algorithm
The expectation-maximization framework is proposed in [1] for the estimation of the systems with unobserved variables based on the maximum likelihood principle. Generally, the batch EM (BEM) algorithm is an iterative algorithm. In the EM algorithm, an expectation step (E-step) and a maximization step (M-step) are iteratively executed. In E-step, the expectation of complete data log-likelihood function with respect to the missing data, which is also named as Q-function, is calculated as follows:
A numerical example
In this section, the effectiveness of the robust recursive EM algorithm is verified using a simulated numerical ARX system with second-order. The corresponding governing equation is given as follows:where represents the regression vector, the input signal uk is subject to a uniform distribution as denotes the parameter vector. A series of noise that follows a Gaussian distribution as is generated and
Conclusion
In this paper, a robust recursive EM algorithm is derived based on the recursion of Q-function, through which the robust identification problem of the linear ARX systems is solved. Considering the outliers, which are frequently encountered in practice, the Student’s t-distribution is employed in the framework of recursive EM algorithm to depict the statistical characteristics of process data. The effectiveness of the robust recursive identification approach is verified with a numerical example
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (grant number 61773183, 61833007, 61801323), supported by national first-class discipline program of Light Industry Technology and Engineering (grant number LITE2018-25), and supported by the Science and Technology Projects Fund of Suzhou (grant number SS2019029).
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