# Dynamic Soft Sensor Development for Time-Varying and Multirate Data Processes Based on Discount and Weighted ARMA Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

## 3. DSSMI-AMWPDD

#### 3.1. AMWPDD Sample Data Processing

_{k}to t

_{k+T}is achieved, and the problem that the weight of the dynamic weighted model is difficult to judge is solved.

- (1)
- λ
_{1}+ λ_{2}+ …… + λ_{T}= 1; - (2)
- λ
_{1}> λ_{2}> …… > λ_{T};

_{i}, i = 1, 2, ⋯, T based on different transition times T by Equations (10) and (11), realize the dynamic calculation of the DF λ value, and obtain more accurate data fusion weights.

_{k}− 1, and ${t}_{k}\left(k=1,2,\cdots ,M\right)$ indicates the sampling time at which the system outputs M sample points.

#### 3.2. LSSVM-Based SSMI

^{2}affects the regression performance of the LSSVM method. However, they are difficult to determine. To ensure the optimal regression performance of the LSSVM, this paper uses PSO to optimize the error term penalty parameter C and the kernel function parameter σ

^{2}.

## 4. Model Parameter Optimization Based on ωDPSO

_{id}, x

_{id}, and p

_{id}represent the velocity, position, and personal best value, respectively, of particle i in iteration t; rand() is a random number between [0, 1]; c

_{1}and c

_{2}are learning factors, which represent the weights of the statistical acceleration terms that push each particle to the pbest and gbest locations; ω is the inertia weight; t is the number of iterations.

^{2}are numerically optimized.

## 5. Simulation and Analysis

#### 5.1. CSTR Simulation Experiment and Result Analysis

_{A}of the raw material A in the reactor is considered to be the dominant variable of the SSM. The feed flow rate F

_{i}, the cooling water flow rate F

_{c}, and the reactor internal temperature T

_{r}are treated as auxiliary variables of the SSM.

_{0}is set for the data simulation based on the variation pattern in Figure 7, and the simulated data set is then normalized.

_{A}at time t

_{k}, ${u}_{j}\left({t}_{k-T}\right)\left(1\le j\le 3,0\le T\le 11\right)$ represents the sampled value of the j

^{th}auxiliary variable at time t

_{k-T}, ${v}_{j}\left({t}_{k}\right)$ is the discounted fusion value of ${u}_{j}\left({t}_{k-T}\right)\left(1\le j\le 3,0\le T\le 11\right)$ and $\widehat{y}\left({t}_{k}\right)$ represents the concentration in the reactor predicted by the SSM.

#### 5.1.1. SSMI Based on Static Data

#### 5.1.2. SSMI Based on Dynamic Fusion Data

#### 5.1.3. Comparison and Analysis

#### 5.2. Simulation Experiment and Result Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Wu, Y.; Luo, X. A novel calibration approach of soft sensor based on multirate data fusion technology. J. Process Control
**2010**, 20, 1252–1260. [Google Scholar] [CrossRef] - Gopakumar, V.; Tiwari, S.; Rahman, I. A deep learning based data driven soft sensor for bioprocesses. Biol. Eng. J.
**2018**, 136, 28–39. [Google Scholar] [CrossRef] - Kadlec, P.; Gabrys, B.; Strandt, S. Data-driven Soft Sensors in the process industry. Comput. Chem. Eng.
**2009**, 33, 795–814. [Google Scholar] [CrossRef] - Yan, W.; Guo, P.; Tian, Y.; Gao, J. A framework and modeling method of data-driven soft sensors based on semi-supervised gaussian regression. Ind. Eng. Chem. Res.
**2016**, 55, 7394–7401. [Google Scholar] [CrossRef] - Zhao, Y.; Fatehi, A.; Huang, B. A data-driven hybrid arx and markov-chain modeling approach to process identification with time varying time delays. IEEE Trans. Ind. Electr.
**2017**, 64, 4226–4236. [Google Scholar] [CrossRef] - Wang, X.; Huang, L.; Yang, C. Prediction model of slurry ph based on mechanism and error compensation for mineral flotation process. Chin. J. Chem. Eng.
**2018**, 26, 174–180. [Google Scholar] [CrossRef] - Liu, S.M.; Wu, Y.P.; Che, H.; Yipeng, W.; Han, C. Monitoring data quality control for a water distribution system using data self-recognition. J. Tsinghua Univ. (Sci. Technol.)
**2017**, 57, 999–1003. [Google Scholar] [CrossRef] - Di, K.S.; Wang, Y.H.; Shang, C.; Huang, D.X. Dynamic soft sensor modeling based on nonlinear slow feature analysis. Comput. Appl. Chem.
**2016**, 33, 1160–1164. [Google Scholar] [CrossRef] - Liu, J.; Vitelli, V.; Zio, E.; Seraoui, R. A novel dynamic-weighted probabilistic support vector regression-based ensemble for prognostics of time series data. IEEE Trans. Reliab.
**2015**, 64, 1203–1213. [Google Scholar] [CrossRef] - Osorio, D.; Ricardo Pérez-Correa, J.; Agosin, E.; Cabrera, M. Soft-sensor for on-line estimation of ethanol concentrations in wine stills. J. Food Eng.
**2008**, 87, 571–577. [Google Scholar] [CrossRef] - Shang, C.; Gao, X.; Yang, F.; Huang, D.X. Novel bayesian framework for dynamic soft sensor based on support vector machine with finite impulse response. IEEE Trans. Control Syst. Technol.
**2014**, 22, 1550–1557. [Google Scholar] [CrossRef] - Gao, X.; Yang, F.; Huang, D.; Ding, Y. An iterative two-level optimization method for the modeling of wiener structure nonlinear dynamic soft sensors. Ind. Eng. Chem. Res.
**2014**, 53, 1172–1178. [Google Scholar] [CrossRef] - Wang, Z.; Luo, X. Modeling study of nonlinear dynamic soft sensors and robust parameter identification using swarm intelligent optimization CS-NLJ. J. Process Control
**2017**, 58, 33–45. [Google Scholar] [CrossRef] - Yuan, P.; Zhang, B.; Mao, Z. A self-tuning control method for wiener nonlinear systems and its application to process control problems. Chin. J. Chem. Eng.
**2017**, 25, 193–201. [Google Scholar] [CrossRef] - Zhang, L.; Peng, X. Time series estimation of gas sensor baseline drift using arma and kalman based models. Sens. Rev.
**2016**, 36, 34–39. [Google Scholar] [CrossRef] - Sun, W.; He, Y.J.; Chang, H. Forecasting Fossil Fuel Energy Consumption for Power Generation Using QHSA-Based LSSVM Model. Energies
**2015**, 8, 939–959. [Google Scholar] [CrossRef] - Hong, X.; Chen, S. The system identification and control of hammerstein system using non-uniform rational b-spline neural network and particle swarm optimization. Neurocomputing
**2012**, 82, 216–223. [Google Scholar] [CrossRef] - Kennedy, J.; Eberhart, R.C. Particle Swarm Optimization. In Proceedings of the 1995 IEEE International Conference on Neural Networks, Perth, Australia, 27 November–1 December 1995; IEEE: Piscataway, NJ, USA, 1995; pp. 1942–1948. [Google Scholar]
- Aburomman, A.A.; Reaz, M.B.I. A novel svm-knn-pso ensemble method for intrusion detection system. Appl. Soft Comput.
**2015**, 38, 360–372. [Google Scholar] [CrossRef] - Hu, W.; Yan, L.; Liu, K.; Wang, H. A short-term traffic flow forecasting method based on the hybrid PSO-SVR. Neur. Prof. Lett.
**2016**, 43, 155–172. [Google Scholar] [CrossRef] - Ma, D.; Tan, W.; Zhang, Z.; Hu, J. Gas emission source term estimation with 1-step nonlinear partial swarm optimization–Tikhonov regularization hybrid method. Chin. J. Chem. Eng.
**2018**, 26, 356–363. [Google Scholar] [CrossRef] - Wang, Y.; Hu, B.; Yang, X.; Wang, X.; Wang, J.; Huang, H. Prediction of flood season precipitation in southwest china based on improved pso-pls. J. Trop. Meteorol.
**2018**, 2, 163–175. [Google Scholar] [CrossRef] - Xiao, Y.; Kang, N.; Hong, Y.; Zhang, G. Misalignment Fault Diagnosis of DFWT Based on IEMD Energy Entropy and PSO-SVM. Entropy
**2017**, 19, 6. [Google Scholar] [CrossRef] - Guedria, N.B. Improved accelerated PSO algorithm for mechanical engineering optimization problems. Appl. Sofe Comput.
**2016**, 40, 455–467. [Google Scholar] [CrossRef] - Cao, P.; Luo, X. Modeling of soft sensor for chemical process. CIESC J.
**2013**, 64, 788–800. [Google Scholar] [CrossRef] - Zhang, D.; Cao, J.; Sun, L. Soft sensor modeling of moisture content in drying process based on LSSVM. Int. Conf. Electr. Meas. Instrum.
**2009**, 2, 989–993. [Google Scholar] [CrossRef] - Mejri, D.; Limam, M.; Weihs, C. A new dynamic weighted majority control chart for data streams. Soft Comput.
**2016**, 22, 511–522. [Google Scholar] [CrossRef] - Brekelmans, R.; Hertog, D.D.; Roos, K.; Eijgenraam, C. Safe dike heights at minimal costs: The nonhomogeneous case. Oper. Res.
**2012**, 60, 1342–1355. [Google Scholar] [CrossRef] - Triantafyllopoulos, K. Multivariate discount weighted regression and local level models. Comput. Stat. Data Anal.
**2006**, 50, 3702–3720. [Google Scholar] [CrossRef] - Suykens, J.; Johan, A.K. Least squares support vector machines. Int. J. Circ. Theor. Appl.
**2002**, 27, 605–615. [Google Scholar] [CrossRef] - Zhao, X.; Gao, Q.; Tang, C.; Liu, X.; Song, J.; Zhou, C. Prediction of reservoir parameters of delta lithologic reservoirs based on support vector regression and well-steering. Oil Geophys. Prospect.
**2016**, 51. [Google Scholar] [CrossRef] - Li, C.X.; Ding, X.D.; Zheng, X.F. Predicting non-Gaussian wind velocity using hybridizing intelligent optimization based LSSVM. J. Vib. Shock
**2017**, 36, 52–58. [Google Scholar] [CrossRef] - Suykens, J.A.K.; Brabanter, J.D.; Lukas, L.; Vandewalle, J. Weighted least squares support vector machines: Robustness and sparse approximation. Neurocomputing
**2002**, 48, 85–105. [Google Scholar] [CrossRef] - Roushangar, K.; Saghebian, S.M.; Mouaze, D. Predicting characteristics of dune bedforms using PSO-LSSVM. Int. J. Sediment Res.
**2017**, 32, 515–526. [Google Scholar] [CrossRef] - Jiang, C.; Zhao, S.; Shen, S.; Guo, L. Particle swarm optimization algorithm with sinusoidal changing inertia weight. Comput. Eng. Appl.
**2012**, 48, 40–42. [Google Scholar] [CrossRef] - Zhu, L.X.; Ling, J.; Wang, B.; Hao, J.; Ding, Y. Soft-sensing modeling of marine protease fermentation process based on improved PSO-RBFNN. CIESC J.
**2018**, 69, 1221–1227. [Google Scholar] [CrossRef] - Xu, B.; Wang, Y.; Wang, F.; He, M.; Wang, M.; Xie, Y. Prediction of package volume based on improved PSO-BP. Comput. Integr. Manuf. Syst.
**2018**, 24, 1871–1879. [Google Scholar] [CrossRef] - Sinha, A.; Mishra, R. Control of a nonlinear continuous stirred tank reactor via event triggered sliding modes. Chem. Eng. Sci.
**2018**, 187, 52–59. [Google Scholar] [CrossRef] [Green Version] - Shao, W. Adaptive Soft Sensor Modeling Based on Local Learning; China U Petrol: Qingdao, China, 2016. [Google Scholar]
- Khatibisepehr, S.; Huang, B.; Xu, F.; Espejo, A. A bayesian approach to design of adaptive multi-model inferential sensors with application in oil sand industry. J. Process Control
**2012**, 22, 1913–1929. [Google Scholar] [CrossRef]

**Figure 8.**Modeling data training for the different SSMI methods. (

**a**) The training curve based on LSSVM; (

**b**) The training curve based on PSO-LSSVM; (

**c**) The training curve based on ωDPSO-LSSVM.

**Figure 9.**Prediction of test data by the different SSMI methods. (

**a**) The prediction curve based on LSSVM; (

**b**) The prediction curve based on PSO-LSSVM; (

**c**) The prediction curve based on ωDPSO-LSSVM.

**Figure 10.**Modeling data training for the different SSMI methods. (

**a**) The training curve based on LSSVM; (

**b**) The training curve based on PSO-LSSVM; (

**c**) The training curve based on ωDPSO-LSSVM.

**Figure 11.**Prediction of test data by the different SSMI methods. (

**a**) The prediction curve based on LSSVM; (

**b**) The prediction curve based on PSO-LSSVM; (

**c**) The prediction curve based on ωDPSO-LSSVM.

Parameter | Description | Steady-State Value |
---|---|---|

F_{i} | Feed flow rate | 100 L/min |

C_{Ai} | Reactant concentration in the feed | 1 mol/L |

T_{i} | Feed temperature | 350 K |

V | Reactor volume | 100 L |

k_{0} | Reaction speed | 7.2 × 10^{10} min^{−1} |

C_{p} | Reactant specific heat capacity | 1 cal/g/k |

hA | Thermal conductivity | 7 × 10^{5} cal/min/K |

T_{ci} | Cooling water inlet temperature | 350 K |

C_{pc} | Cooling water specific heat capacity | 1 cal/g/k |

Soft Sensor | RMSE | Running Time(s) |
---|---|---|

LSSVM | 0.0446 | 0.851 |

PSO-LSSVM | 0.0440 | 26.520 |

ωDPSO -LSSVM | 0.0439 | 22.744 |

Soft Sensor | MAE | RMSE |
---|---|---|

LSSVM | 0.0820 | 0.0853 |

PSO-LSSVM | 0.0807 | 0.0839 |

ωDPSO -LSSVM | 0.0794 | 0.0823 |

Soft Sensor | RMSE | Running Time(s) |
---|---|---|

LSSVM | 0.0341 | 0.793 |

PSO-LSSVM | 0.0328 | 28.94 |

ωDPSO -LSSVM | 0.0327 | 22.143 |

Soft Sensor | MAE | RMSE |
---|---|---|

LSSVM | 0.0529 | 0.0683 |

PSO-LSSVM | 0.0521 | 0.0650 |

ωDPSO -LSSVM | 0.0511 | 0.0632 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, L.; Dai, Y.
Dynamic Soft Sensor Development for Time-Varying and Multirate Data Processes Based on Discount and Weighted ARMA Models. *Symmetry* **2019**, *11*, 1414.
https://doi.org/10.3390/sym11111414

**AMA Style**

Li L, Dai Y.
Dynamic Soft Sensor Development for Time-Varying and Multirate Data Processes Based on Discount and Weighted ARMA Models. *Symmetry*. 2019; 11(11):1414.
https://doi.org/10.3390/sym11111414

**Chicago/Turabian Style**

Li, Longhao, and Yongshou Dai.
2019. "Dynamic Soft Sensor Development for Time-Varying and Multirate Data Processes Based on Discount and Weighted ARMA Models" *Symmetry* 11, no. 11: 1414.
https://doi.org/10.3390/sym11111414