# Robust Autoregression with Exogenous Input Model for System Identification and Predicting

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. SISO ARX

_{b}) are the model parameters for the output and input, respectively, ${n}_{a}$ and ${n}_{b}$ are the degree of the ARX model, and ${n}_{k}$ is the time delay.

_{k}+ 1).

#### 2.2. MISO ARX

#### 2.3. Lp (p ≤ 1) Norm-Based ARX Model

Algorithm 1 Lp (p ≤ 1) BFGS |

Require: Iteration number n, termination error $\epsilon \in \left[0,1\right]$, initialize W as a random nonzero 2q-dimensional vector, initial pseudo-Hessian matrix ${H}_{0}$.For k from 1 to n doCompute the gradient ${g}_{k}$ by $g=p{\displaystyle \sum _{i=1}^{N-q}{\left|{y}^{*}\left(i\right)-{\left({\psi}_{M}\left(i\right)\right)}^{T}{W}_{M}\right|}^{p-1}\mathrm{sgn}\left(i\right)\times \left(-{\psi}_{M}\left(i\right)\right)}$ if $\Vert {g}_{k}\Vert <\epsilon $ then${W}^{\ast}={W}_{k}$ breakend ifSolve the coupled linear equations ${H}_{k}{d}_{k}=-{g}_{k}$, and calculate ${d}_{k}$ Find the optimal learning velocity ${\alpha}_{k}$ by ${\alpha}_{k}=\underset{\alpha}{\mathrm{arg}\mathrm{min}}f({W}_{k}+\alpha {d}_{k})$ Update ARX parameter W by ${W}_{k+1}={W}_{k}+{\alpha}_{k}{d}_{k}$ Update ${H}_{k}$ by Equation (16) end for |

## 3. Results

#### 3.1. Simulation Study

#### 3.1.1. Experimental Dataset

#### 3.1.2. Effect of Outlier Occurrence Rate

#### 3.1.3. Effect of Outlier Strength

#### 3.2. Real Data Studies

#### 3.2.1. Application to Actual EEG Recordings

#### 3.2.2. Application to Actual EEG Recordings

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Monden, Y.; Yamada, M.; Arimoto, S. Fast Algorithm for Identification of an Arx Model and Its Order Determination. IEEE Trans. Acoust. Speech Signal Proces.
**1982**, 30, 390–399. [Google Scholar] [CrossRef] - Isaksson, A.J. Identification of Arx-Models Subject to Missing Data. IEEE Trans. Autom. Control
**2002**, 38, 813–819. [Google Scholar] [CrossRef] - Jin, G.D.; Lu, L.B.; Zhu, X.F. A Method of Order Determination for Arx and Arma Models Based on Nonnegative Garrote. Appl. Mech. Mater.
**2014**, 721, 496–499. [Google Scholar] [CrossRef] - Nelles, O. Nonlinear System Identification: From Classical Approaches to Neuralnetworks and Fuzzy Models. Appl. Ther.
**2001**, 6, 717–721. [Google Scholar] - Xu, P.; Kasprowicz, M.; Bergsneider, M.; Hu, X. Improved Noninvasive Intracranial Pressure Assessment with Nonlinear Kernel Regression. IEEE Trans. Inform. Technol. Biomed.
**2010**, 14, 971–978. [Google Scholar] - Wang, Z.; Xu, P.; Liu, T.; Tian, Y.; Lei, X.; Yao, D. Robust Removal of Ocular Artifacts by Combining Independent Component Analysis and System Identification. Biomed. Signal Proces. Control
**2014**, 10, 250–259. [Google Scholar] [CrossRef] - Nguyen, V.T.; Breakspear, M.; Cunnington, R. Fusing Concurrent Eeg–Fmri with Dynamic Causal Modeling: Application to Effective Connectivity During Face Perception. NeuroImage
**2013**, 102, 60–70. [Google Scholar] [CrossRef] - Gourévitch, B.; Kay, L.M.; Martin, C. Directional Coupling from the Olfactory Bulb to the Hippocampus During a Go/No-Go Odor Discrimination Task. J. Neurophysiol.
**2010**, 103, 2633–2641. [Google Scholar] [CrossRef] - Zhao, Y.; Billings, S.A.; Wei, H.-L.; Sarrigiannis, P.G. A Parametric Method to Measure Time-Varying Linear and Nonlinear Causality with Applications to Eeg Data. IEEE Trans. Biomed. Eng.
**2013**, 60, 3141–3148. [Google Scholar] [CrossRef] - Siuly, S.; Li, Y. Discriminating the Brain Activities for Brain–Computer Interface Applications through the Optimal Allocation-Based Approach. Neural Comput. Appl.
**2015**, 26, 799–811. [Google Scholar] [CrossRef] - Burke, D.P.; Kelly, S.P.; De Chazal, P.; Reilly, R.B.; Finucane, C. A Parametric Feature Extraction and Classification Strategy for Brain-Computer Interfacing. IEEE Trans. Neural Syst. Rehabil. Eng.
**2005**, 13, 12–17. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Li, Y.; Wei, H.-L.; Billings, S.A.; Sarrigiannis, P. Time-Varying Model Identification for Time–Frequency Feature Extraction from Eeg Data. J. Neurosci. Meth.
**2011**, 196, 151–158. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Qidwai, U.; Shakir, M.; Malik, A.S.; Kamel, N. Parametric Modeling of Eeg Signals with Real Patient Data for Simulating Seizures and Pre-Seizures. In Proceedings of the 2013 International Conference on Human Computer Interactions, Chennai, India, 23–24 August 2013; pp. 1–5. [Google Scholar]
- Yu, H.; Guo, X.; Qin, Q.; Deng, Y.; Wang, J.; Liu, J.; Cao, Y. Synchrony Dynamics Underlying Effective Connectivity Reconstruction of Neuronal Circuits. Phys. Stat. Mech. Appl.
**2017**, 471, 674–687. [Google Scholar] [CrossRef] - Yu, H.; Wu, X.; Cai, L.; Deng, B.; Wang, J. Modulation of Spectral Power and Functional Connectivity in Human Brain by Acupuncture Stimulation. IEEE Trans. Neural Syst. Rehabil. Eng.
**2018**, 26, 977–986. [Google Scholar] [CrossRef] - Liao, W.; Mantini, D.; Zhang, Z.; Pan, Z.; Ding, J.; Gong, Q.; Yang, Y.; Chen, H. Evaluating the Effective Connectivity of Resting State Networks Using Conditional Granger Causality. Biol. Cybern.
**2010**, 102, 57–69. [Google Scholar] [CrossRef] [PubMed] - Kim, S.; Putrino, D.; Ghosh, S.; Brown, E.N. A Granger Causality Measure for Point Process Models of Ensemble Neural Spiking Activity. PLoS Comput. Biol.
**2011**, 7, e1001110. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kwak, N. Principal Component Analysis Based on L1-Norm Maximization. IEEE Trans. Pattern Anal. Mach. Intell.
**2008**, 30, 1672–1680. [Google Scholar] [CrossRef] - Xu, P.; Tian, Y.; Chen, H.; Yao, D. Lp Norm Iterative Sparse Solution for Eeg Source Localization. IEEE Trans. Biomed. Eng.
**2007**, 54, 400–409. [Google Scholar] [CrossRef] [PubMed] - Li, P.; Xu, P.; Zhang, R.; Guo, L.; Yao, D. L1 Norm Based Common Spatial Patterns Decomposition for Scalp Eeg Bci. Biomed. Eng. Online
**2013**, 12, 77. [Google Scholar] [CrossRef] [Green Version] - Mattsson, P.; Zachariah, D.; Stoica, P. Recursive Identification Method for Piecewise Arx Models: A Sparse Estimation Approach. IEEE Trans. Signal Proces.
**2016**, 64, 5082–5093. [Google Scholar] [CrossRef] - Guo, S.; Wang, Z.; Ruan, Q. Enhancing Sparsity Via ℓp (0<P<1) Minimization for Robust Face Recognition. Neurocomputing
**2013**, 99, 592–602. [Google Scholar] - Chartrand, R.; Staneva, V. Restricted Isometry Properties and Nonconvex Compressive Sensing. Inverse Probl.
**2008**, 24, 035020. [Google Scholar] [CrossRef] [Green Version] - Chartrand, R. Exact Reconstruction of Sparse Signals Via Nonconvex Minimization. IEEE Signal Process. Lett.
**2007**, 14, 707–710. [Google Scholar] [CrossRef] - Foucart, S.; Lai, M.-J. Sparsest Solutions of Underdetermined Linear Systems Via ℓq-Minimization for 0 <Q ⩽ 1. Comput. Harmon. Anal.
**2009**, 26, 395–407. [Google Scholar] - Nie, F.; Huang, Y.; Wang, X.; Huang, H. New Primal Svm Solver with Linear Computational Cost for Big Data Classifications. In Proceedings of the 31st International Conference on International Conference on Machine Learning, Beijing, China, 22–24 June 2014; Volume 32, pp. II-505–II-513. [Google Scholar]
- Ye, Q.; Fu, L.; Zhang, Z.; Zhao, H.; Naiem, M. Lp-and Ls-Norm Distance Based Robust Linear Discriminant Analysis. Neural Netw.
**2018**, 105, 393–404. [Google Scholar] [CrossRef] - Wang, H.; Nie, F.; Cai, W.; Huang, H. Semi-Supervised Robust Dictionary Learning Via Efficient L-Norms Minimization. In Proceedings of the IEEE International Conference on Computer Vision, Sydney, Australia, 1–8 December 2013; pp. 1145–1152. [Google Scholar]
- Lustig, M.; Donoho, D.; Pauly, J.M. Sparse Mri: The Application of Compressed Sensing for Rapid Mr Imaging. Magn. Reson. Med.
**2007**, 58, 1182–1195. [Google Scholar] [CrossRef] - Chartrand, R. Fast Algorithms for Nonconvex Compressive Sensing: Mri Reconstruction from Very Few Data. In Proceedings of the 2009 IEEE International Symposium on Biomedical Imaging: From Nano to Macro, Boston, MA, USA, 28 June–1 July 2009; pp. 262–265. [Google Scholar]
- Li, P.; Wang, X.; Li, F.; Zhang, R.; Ma, T.; Peng, Y.; Lei, X.; Tian, Y.; Guo, D.; Liu, T. Autoregressive Model in the Lp Norm Space for Eeg Analysis. J. Neurosci. Methods
**2015**, 240, 170–178. [Google Scholar] [CrossRef] [PubMed] - Li, P.; Huang, X.; Li, F.; Wang, X.; Zhou, W.; Liu, H.; Ma, T.; Zhang, T.; Guo, D.; Yao, D. Robust Granger Analysis in Lp Norm Space for Directed Eeg Network Analysis. IEEE Trans. Neural Syst. Rehabil. Eng.
**2017**, 25, 1959–1969. [Google Scholar] [CrossRef] [PubMed] - Rahim, M.A.; Ramasamy, M.; Tufa, L.D.; Faisal, A. Iterative Closed-Loop Identification of Mimo Systems Using Arx-Based Leaky Least Mean Square Algorithm. In Proceedings of the 2014 IEEE International Conference on Control System, Computing and Engineering, Penang, Malaysia, 28–30 November 2014; pp. 611–616. [Google Scholar]
- Broyden, C.G. Quasi-Newton Methods and Their Application to Function Minimisation. Math. Comput.
**1993**, 21, 368–381. [Google Scholar] [CrossRef] - Pavon, M. A Variational Derivation of a Class of Bfgs-Like Methods. Optimization
**2018**, 67, 2081–2089. [Google Scholar] [CrossRef] [Green Version] - Goldfarb, D. A Family of Variable-Metric Methods Derived by Variational Means. Math. Comput.
**1970**, 24, 23–26. [Google Scholar] [CrossRef] - Fletcher, R. A New Variational Result for Quasi-Newton Formulae. SIAM J. Optim.
**1991**, 1, 18–21. [Google Scholar] [CrossRef] - Shanno, D.F. Conditioning of Quasi-Newton Methods for Function Minimization. Math. Comput.
**1970**, 24, 647–656. [Google Scholar] [CrossRef] - Broyden, C.G. A Class of Methods for Solving Nonlinear Simultaneous Equations. Math. Comput.
**1965**, 19, 577–593. [Google Scholar] [CrossRef] - Robitaille, B.; Marcos, B.; Veillette, M.; Payre, G. Quasi-Newton Methods for Training Neural Networks. WIT Trans. Inform. Commun. Technol.
**1993**, 2. [Google Scholar] [CrossRef] - Broyden, C.G. The Convergence of a Class of Double-Rank Minimization Algorithms 1. General Considerations. IMA J. Appl. Math.
**1970**, 6, 76–90. [Google Scholar] [CrossRef] - Nagasaka, Y.; Shimoda, K.; Fujii, N. Multidimensional Recording (Mdr) and Data Sharing: An Ecological Open Research and Educational Platform for Neuroscience. PLoS ONE
**2011**, 6, e22561. [Google Scholar] [CrossRef] [Green Version] - Hardin, J.W. Generalized Estimating Equations (Gee); John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2005. [Google Scholar]
- Wu, H.; Sun, D.; Zhou, Z. Model Identification of a Micro Air Vehicle in Loitering Flight Based on Attitude Performance Evaluation. IEEE Trans. Robot.
**2004**, 20, 702–712. [Google Scholar] [CrossRef]

**Figure 1.**The output and input waveforms mingled with outliers. (

**a**) The system input. (

**b**) The system output.

**Figure 2.**The waveforms of the desired output and the predicted results from six ARXs when outliers were mixed in the original data. (

**a**) The predictive performance of the LS-ARX model. (

**b**–

**f**) The predictive performance of the Lp-ARX model (p = 1, 0.8, 0.6, 0.4, 0.2).

**Figure 3.**The input and output waveforms. (

**a**) The system input. (

**b**) The system output. The time series on the left side of the green dotted line is used for parameter estimation, and the data on the right side indicate the fitting results of both ARX models.

**Figure 4.**The predicted results of the model. The red curve is the desired system output, the green curve is the predicted output by L1-ARX, and the blue curve represents the output predicted by LS-ARX. For the convenience of display, we only draw the prediction results for a certain period of time.

**Figure 5.**The system output and system input for parameter estimation. (

**a**) The original output waveform with outlier influence. (

**b**) The original input waveform with outlier influence.

**Figure 6.**The system output and system input for performance evaluation. (

**a**) The waveform of the second system output segment. (

**b**) The waveform of the second system input segment.

**Figure 7.**The system output waveforms. The waveform depicted by the red curve is the desired system output, the blue curve depicts the predicted results by LS-ARX, and the green curve depicts the predicted results by L1-ARX. For the convenience of display, we only draw the prediction results for a certain period of time.

**Table 1.**Parameter estimation bias of the single-input–single-output (SISO) system with different numbers of outliers.

Method | Number of Outliers | ||||
---|---|---|---|---|---|

8 | 12 | 16 | 20 | 24 | |

LS | 0.9961 ± 0.012 | 0.9977 ± 0.008 | 0.9986 ± 0.008 | 0.9985 ± 0.008 | 0.9991 ± 0.008 |

Huber | 0.9626 ± 0.055 | 0.9614 ± 0.059 | 0.9424 ± 0.070 | 0.9981 ± 0.061 | 0.9999 ± 0.064 |

L1 | 0.9754 ± 0.012 * | 0.9739 ± 0.011 * | 0.9709 ± 0.011 * | 0.9684 ± 0.010 * | 0.9675 ± 0.011 * |

L0.8 | 0.9912 ± 0.007 * | 0.9884 ± 0.009 * | 0.9855 ± 0.008 * | 0.9815 ± 0.008 * | 0.9773 ± 0.010 * |

L0.6 | 0.9929 ± 0.007 * | 0.9918 ± 0.008 * | 0.9906 ± 0.007 * | 0.9881 ± 0.008 * | 0.9863 ± 0.009 * |

L0.4 | 0.9935 ± 0.007 * | 0.9927 ± 0.008 * | 0.9922 ± 0.007 * | 0.9906 ± 0.007 * | 0.9900 ± 0.009 * |

L0.2 | 0.9940 ± 0.007 * | 0.9936 ± 0.008 * | 0.9936 ± 0.008 * | 0.9929 ± 0.008 * | 0.9923 ± 0.009 * |

Method | Number of Outliers | ||||
---|---|---|---|---|---|

8 | 12 | 16 | 20 | 24 | |

LS | 2.8701 ± 0.128 | 3.1407 ± 0.156 | 3.3440 ± 0.152 | 3.4941 ± 0.149 | 3.5629 ± 0.173 |

Huber | 3.7933 ± 1.624 | 4.7454 ± 5.712 | 3.0067 ± 0.826 | 3.3647 ± 2.065 | 2.8149 ± 0.861 |

L1 | 1.4601 ± 0.339 * | 1.6705 ± 0.268 * | 1.8201 ± 0.218 * | 1.9369 ± 0.187 * | 2.0424 ± 0.213 * |

L0.8 | 2.1322 ± 0.284 * | 2.0543 ± 0.288 * | 2.0326 ± 0.243 * | 2.0279 ± 0.217 * | 2.0043 ± 0.233 * |

L0.6 | 2.3604 ± 0.151 * | 2.3212 ± 0.227 * | 2.3171 ± 0.207 * | 2.2776 ± 0.243 * | 2.2330 ± 0.293 * |

L0.4 | 2.4589 ± 0.177 * | 2.4635 ± 0.227 * | 2.4936 ± 0.258 * | 2.4601 ± 0.276 * | 2.4058 ± 0.336 * |

L0.2 | 2.5288 ± 0.220 * | 2.5825 ± 0.280 * | 2.6723 ± 0.319 * | 2.7355 ± 0.393 * | 2.6981 ± 0.491 * |

**Table 3.**Parameter estimation bias of the two-input–single-output (TISO) system with different numbers of outliers.

Method | Number of Outliers | ||||
---|---|---|---|---|---|

8 | 12 | 16 | 20 | 24 | |

LS | 1.1100 ± 0.024 | 1.1488 ± 0.043 | 1.1619 ± 0.046 | 1.1779 ± 0.041 | 1.1893 ± 0.042 |

Huber | 0.4743 ± 0.671 | 0.5245 ± 0.742 | 0.5490 ± 0.776 | 0.5860 ± 0.829 | 0.6524 ± 0.923 |

L1 | 0.8564 ± 0.062 * | 0.9028 ± 0.060 * | 0.9172 ± 0.059 * | 0.9282 ± 0.062 * | 0.9214 ± 0.065 * |

L0.8 | 0.9539 ± 0.110 * | 0.9779 ± 0.115 * | 1.0017 ± 0.096 * | 1.0333 ± 0.076 * | 1.0432 ± 0.063 * |

L0.6 | 1.0828 ± 0.042 * | 1.0926 ± 0.060 * | 1.1043 ± 0.056 * | 1.1107 ± 0.047 * | 1.1146 ± 0.059 * |

L0.4 | 1.1001 ± 0.028 * | 1.1376 ± 0.054 * | 1.1464 ± 0.051 * | 1.1713 ± 0.046 * | 1.1824 ± 0.047 * |

L0.2 | 1.1076 ± 0.025 * | 1.1444 ± 0.047 * | 1.1553 ± 0.050 * | 1.1768 ± 0.046 * | 1.1887 ± 0.046 * |

Method | Number of Outliers | ||||
---|---|---|---|---|---|

8 | 12 | 16 | 20 | 24 | |

LS | 10.3674 ± 0.171 | 11.6089 ± 0.253 | 12.6547 ± 0.466 | 13.3863 ± 0.461 | 14.1104 ± 0.549 |

Huber | 10.0984 ± 0.470 | 10.9375 ± 0.470 | 11.5306 ± 0.495 | 11.9261 ± 0.389 | 11.4484 ± 0.625 |

L1 | 10.2015 ± 0.406 * | 10.4782 ± 0.421 * | 10.7154 ± 0.501 * | 11.1291 ± 0.616 * | 11.3014 ± 0.640 * |

L0.8 | 10.5792 ± 1.750 * | 11.0256 ± 1.770 * | 11.4188 ± 1.823 * | 12.6198 ± 2.701 * | 12.9624 ± 2.641 * |

L0.6 | 10.7649 ± 2.207 * | 11.3840 ± 2.504 * | 11.2274 ± 2.540 * | 11.4422 ± 2.738 * | 12.0195 ± 3.306 * |

L0.4 | 10.3466 ± 2.070 * | 10.1870 ± 1.922 * | 10.4557 ± 1.920 * | 10.6500 ± 1.153 * | 10.7433 ± 0.843 * |

L0.2 | 9.8747 ± 1.312 * | 9.9064 ± 0.753 * | 10.3452 ± 1.039 * | 10.6899 ± 1.406 * | 10.7704 ± 0.852 * |

Method | Outliers Strengths | ||||
---|---|---|---|---|---|

1.5 | 2.0 | 2.5 | 3.0 | 3.5 | |

LS | 0.9949 ± 0.008 | 0.9972 ± 0.007 | 0.9989 ± 0.007 | 1.0002 ± 0.007 | 1.0011 ± 0.007 |

Huber | 0.9674 ± 0.056 | 0.9734 ± 0.063 | 0.9838 ± 0.070 | 0.9908 ± 0.074 | 0.9880 ± 0.082 |

L1 | 0.9720 ± 0.014 * | 0.9720 ± 0.013 * | 0.9723 ± 0.010 * | 0.9718 ± 0.008 * | 0.9714 ± 0.008 * |

L0.8 | 0.9899 ± 0.008 * | 0.9880 ± 0.008 * | 0.9863 ± 0.008 * | 0.9846 ± 0.008 * | 0.9818 ± 0.009 * |

L0.6 | 0.9927 ± 0.007 * | 0.9914 ± 0.007 * | 0.9904 ± 0.007 * | 0.9903 ± 0.007 * | 0.9897 ± 0.008 * |

L0.4 | 0.9933 ± 0.008 * | 0.9922 ± 0.008 * | 0.9914 ± 0.008 * | 0.9911 ± 0.008 * | 0.9909 ± 0.008 * |

L0.2 | 0.9936 ± 0.008 * | 0.9932 ± 0.008 * | 0.9926 ± 0.008 * | 0.9919 ± 0.008 * | 0.9922 ± 0.009 * |

Method | Outliers Strengths | ||||
---|---|---|---|---|---|

1.5 | 2.0 | 2.5 | 3.0 | 3.5 | |

LS | 2.7784 ± 0.149 | 3.1592 ± 0.155 | 3.4438 ± 0.162 | 3.6502 ± 0.168 | 3.8041 ± 0.173 |

Huber | 3.9342 ± 1.399 | 3.6513 ± 0.925 | 3.1422 ± 0.570 | 3.0822 ± 0.459 | 3.0151 ± 0.444 |

L1 | 1.4513 ± 0.353 * | 1.6540 ± 0.282 * | 1.8031 ± 0.226 * | 1.9235 ± 0.194 * | 2.0272 ± 0.193 * |

L0.8 | 2.1394 ± 0.316 * | 2.0702 ± 0.305 * | 2.0400 ± 0.256 * | 2.0166 ± 0.247 * | 2.0053 ± 0.210 * |

L0.6 | 2.3761 ± 0.200 * | 2.3551 ± 0.221 * | 2.3181 ± 0.228 * | 2.2893 ± 0.224 * | 2.2462 ± 0.274 * |

L0.4 | 2.4730 ± 0.203 * | 2.4843 ± 0.234 * | 2.4732 ± 0.269 * | 2.4393 ± 0.262 * | 2.4093 ± 0.276 * |

L0.2 | 2.5262 ± 0.201 * | 2.6099 ± 0.274 * | 2.6290 ± 0.329 * | 2.5707 ± 0.335 * | 2.6121 ± 0.392 * |

Method | Outliers Strengths | ||||
---|---|---|---|---|---|

1.5 | 2.0 | 2.5 | 3.0 | 3.5 | |

LS | 1.1100 ± 0.024 | 1.1488 ± 0.043 | 1.1619 ± 0.046 | 1.1779 ± 0.041 | 1.1893 ± 0.042 |

Huber | 0.5642 ± 0.798 | 0.5820 ± 0.823 | 0.5899 ± 0.834 | 0.5859 ± 0.829 | 0.5774 ± 0.817 |

L1 | 0.8564 ± 0.062 * | 0.9028 ± 0.060 * | 0.9172 ± 0.059 * | 0.9282 ± 0.062 * | 0.9214 ± 0.065 * |

L0.8 | 0.9539 ± 0.110 * | 0.9779 ± 0.115 * | 1.0017 ± 0.096 * | 1.0333 ± 0.076 * | 1.0432 ± 0.063 * |

L0.6 | 1.0828 ± 0.042 * | 1.0926 ± 0.060 * | 1.1043 ± 0.056 * | 1.1107 ± 0.047 * | 1.1146 ± 0.059 * |

L0.4 | 1.1001 ± 0.028 * | 1.1376 ± 0.054 * | 1.1464 ± 0.051 * | 1.1713 ± 0.046 * | 1.1824 ± 0.047 * |

L0.2 | 1.1076 ± 0.025 * | 1.1444 ± 0.047 * | 1.1553 ± 0.050 * | 1.1768 ± 0.046 * | 1.1887 ± 0.046 * |

Method | Outliers Strengths | ||||
---|---|---|---|---|---|

1.5 | 2.0 | 2.5 | 3.0 | 3.5 | |

LS | 11.2180 ± 0.294 | 12.6363 ± 0.419 | 13.9806 ± 0.577 | 15.3493 ± 0.762 | 16.7532 ± 0.924 |

Huber | 10.7821 ± 0.425 | 11.5455 ± 0.484 | 11.5870 ± 0.561 | 11.2759 ± 0.759 | 11.4137 ± 0.712 |

L1 | 10.5468 ± 0.595 * | 10.7873 ± 0.498 * | 11.0528 ± 0.538 * | 11.3524 ± 0.634 * | 11.6935 ± 0.668 * |

L0.8 | 11.6983 ± 0.803 * | 12.0348 ± 0.472 * | 12.5147 ± 0.415 * | 12.9623 ± 0.586 * | 13.1487 ± 0.549 * |

L0.6 | 9.6700 ± 0.185 * | 10.2408 ± 0.240 * | 10.7195 ± 0.306 * | 11.0724 ± 0.376 * | 11.2908 ± 0.374 * |

L0.4 | 9.6880 ± 0.178 * | 10.2577 ± 0.244 * | 10.7121 ± 0.253 * | 11.0420 ± 0.304 * | 11.2194 ± 0.277 * |

L0.2 | 9.7605 ± 0.283 * | 10.3073 ± 0.278 * | 10.7745 ± 0.369 * | 11.1013 ± 0.423 * | 11.3517 ± 0.561 * |

Subject | Method | |
---|---|---|

LS-ARX | L1-ARX | |

1 | 9.2295 | 7.7953 |

2 | 7.3647 | 6.4325 |

3 | 7.3430 | 6.4876 |

4 | 5.0670 | 4.0245 |

5 | 4.3612 | 3.3372 |

6 | 5.5014 | 4.5819 |

7 | 6.8686 | 5.4962 |

8 | 7.5475 | 6.5210 |

9 | 5.3146 | 4.1594 |

10 | 5.6730 | 4.5378 |

11 | 9.7528 | 8.3152 |

Average | 6.7294 ± 1.73 | 5.6081 ± 1.62 * |

Subject | Method | ||
---|---|---|---|

LS (mmHg) | L1 (mmHg) | Real (mmHg) | |

1 | 13.7831 | 13.8759 | 14.2000 |

2 | 18.5624 | 18.6203 | 19.0778 |

3 | 8.6183 | 8.7711 | 9.3489 |

4 | 23.7407 | 23.8564 | 24.4172 |

5 | 17.1373 | 17.2725 | 17.8372 |

6 | 24.0823 | 24.2048 | 25.0012 |

7 | 19.7513 | 19.7813 | 20.8477 |

8 | 19.8864 | 19.9744 | 20.7015 |

9 | 21.5216 | 21.5523 | 21.9293 |

10 | 17.0425 | 17.0672 | 17.1580 |

11 | 14.9890 | 15.1406 | 16.3673 |

12 | 7.8933 | 7.9316 | 8.4423 |

13 | 3.9604 | 4.0621 | 4.5802 |

14 | 7.7528 | 7.9307 | 8.5877 |

15 | 9.3349 | 9.5021 | 10.0645 |

**Table 11.**The error (mmHg) and correlation coefficient (CC) of the predicted signal with the ARX models.

Subject | Method | |||
---|---|---|---|---|

LS ARX | L1 ARX | |||

Error | CC | Error | CC | |

1 | 0.4482 | 0.9152 | 0.3754 | 0.9766 |

2 | 0.5306 | 0.9502 | 0.4794 | 0.9839 |

3 | 0.6162 | 0.8519 | 0.4943 | 0.9790 |

4 | 0.6807 | 0.9894 | 0.5715 | 0.9989 |

5 | 0.3293 | 0.9151 | 0.2718 | 0.9936 |

6 | 0.9217 | 0.9484 | 0.8011 | 0.9951 |

7 | 1.0964 | 0.9024 | 1.0664 | 0.9347 |

8 | 0.3779 | 0.8940 | 0.3384 | 0.9954 |

9 | 0.4364 | 0.9610 | 0.4118 | 0.9936 |

10 | 0.2154 | 0.9127 | 0.2069 | 0.9525 |

11 | 1.3784 | 0.9018 | 1.2268 | 0.9776 |

12 | 0.3072 | 0.8706 | 0.2888 | 0.9895 |

13 | 0.3116 | 0.8814 | 0.2654 | 0.9251 |

14 | 0.5458 | 0.8570 | 0.4354 | 0.9846 |

15 | 0.4512 | 0.9392 | 0.3528 | 0.9871 |

Mean Result | 0.5765 ± 0.32 | 0.9127 ± 0.04 | 0.5057 ± 0.30 * | 0.9778 ± 0.02 * |

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## Share and Cite

**MDPI and ACS Style**

Xie, J.; Li, C.; Li, N.; Li, P.; Wang, X.; Gao, D.; Yao, D.; Xu, P.; Yin, G.; Li, F.
Robust Autoregression with Exogenous Input Model for System Identification and Predicting. *Electronics* **2021**, *10*, 755.
https://doi.org/10.3390/electronics10060755

**AMA Style**

Xie J, Li C, Li N, Li P, Wang X, Gao D, Yao D, Xu P, Yin G, Li F.
Robust Autoregression with Exogenous Input Model for System Identification and Predicting. *Electronics*. 2021; 10(6):755.
https://doi.org/10.3390/electronics10060755

**Chicago/Turabian Style**

Xie, Jiaxin, Cunbo Li, Ning Li, Peiyang Li, Xurui Wang, Dongrui Gao, Dezhong Yao, Peng Xu, Gang Yin, and Fali Li.
2021. "Robust Autoregression with Exogenous Input Model for System Identification and Predicting" *Electronics* 10, no. 6: 755.
https://doi.org/10.3390/electronics10060755