# A Novel Isomap-SVR Soft Sensor Model and Its Application in Rotary Kiln Calcination Zone Temperature Prediction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Pellet Production Process and Soft Sensor Model

#### 2.1. Pellet Production Process

#### 2.2. Soft Sensor Model Structure

## 3. Data Dimension Reduction Processing Based on Isomap

^{M}. The geodesic distance on a manifold is approximated by the shortest distance between the points, Obtain the geodetic distance matrix D

^{M}= D

^{G}, the shortest distance ${D}_{ij}^{G}$ can be obtained by Floyd iterative algorithm [17,18].

## 4. SVR Soft Sensor Modeling Optimized by Improved Bat Algorithm

#### 4.1. SVR Algorithm

_{i}is the output of x

_{i}, $N$ is the number of samples). SVR solves the nonlinear relationship between ${x}_{i}$ and ${y}_{i}$ by mapping input vector ${x}_{i}$ into high-dimensional feature space through nonlinear mapping $\varphi (\cdot )$ and linear regression [20]. Its high-dimensional linear regression function is:

#### 4.2. Bat Algorithm and Its Improvement

#### 4.2.1. Bat Algorithm

^{0}, maximum pulse frequency R

^{0}, position moving range and initial velocity V

^{0}of each bat, search pulse frequency range [f

_{min}, f

_{max}], loudness attenuation factor $\alpha $, frequency enhancement factor $\gamma $, maximum iteration number T and search precision $\epsilon $ of the algorithm. The initial position vector ${\left\{{X}_{i}^{0}\right\}}_{i=1}^{M}$ of each bat is randomly determined. Set the current iterations t = 1.

_{i}is an acoustic frequency.

_{rand}between [0, 1]. For each bat, if R

_{rand}> R

_{i}, performs a local search, and ${X}_{i}^{t}$ performs a random perturbation to generate a new solution:

^{t}is the average loudness of the current bat population.

_{rand}between [0, 1]. If A

_{rand}< A

_{i}, and $f(new{X}_{i}^{t})<f({X}_{*}^{t})$, accept the new solution $new{X}_{i}^{t}$, then update ${R}_{i}^{t}$ and ${A}_{i}^{t}$ according to the following formula:

#### 4.2.2. Lévy Flight Strategy

#### 4.2.3. Cauchy Mutation Strategy

_{1}and f

_{2}are unimodal functions, f

_{3}to f

_{4}are multimodal functions, f

_{5}and f

_{6}are rotated and shifted functions. The corresponding dimensions D, range, global minimum values f

_{opt}of the functions are also listed in Table 1.

_{1}to f

_{5}. IBA has satisfactory performance on f

_{6}, and only loses in dBA algorithm. It indicates that the overall performance of IBA is much better than other algorithms.

#### 4.3. Soft Sensor Model Optimization Based on IBA Algorithm

^{2}and 5-fold cross verification accuracy Q

^{2}

_{cv5}as fitness function, namely:

## 5. Simulation Analysis

^{2}and Q

^{2}

_{cv5}are both greater than 0.8). When using BA to optimize parameters of Isomap and SVR, the range of $K$ is [1, 100], the range of $C$ is [1, 100], and that of $\delta $ is [1, 100]. The initial parameters of the BA are: Bat population size M = 30, algorithm maximum search times N = 100, loudness attenuation coefficient $\alpha =0.95$, search frequency enhancement coefficient $\gamma =0.95$, search frequency range [−2, 2], bat speed range [−5, 5], bat individual position range [1, 100]. When optimizing the model, we select variables whose standard deviation of output variables of Isomap algorithm is greater than 0.05 as SVR model inputs.

^{2}= 0.887 and model robustness index Q

^{2}

_{cv5}= 0.841. (it is generally believed that R

^{2}> 0.7 and Q

^{2}

_{cv5}> 0.7 can establish a robust and reliable soft measurement model). K = 28, C = 76.235, δ = 2.014.

^{2}represents the fitting ability of the model. The larger is R

^{2}, the stronger learning ability of the model to the training sample. Min_R

^{2}, Max_R

^{2}and Mean_R

^{2}, represent the maximum, minimum, and average values of R

^{2}of each algorithm in 30 experiments, respectively. Q

^{2}

_{CV5}represents the robustness of the model. The larger is Q

^{2}

_{CV5}, the better robustness of the model. Q

^{2}

_{ext}represents the prediction ability of the model for independent external test samples, and the value range is [0, 1]. The larger the is Q

^{2}

_{ext}, the stronger external prediction ability of the model. The formula of Q

^{2}

_{ext}is:

^{2}

_{ext}, Max_Q

^{2}

_{ext}and Mean_Q

^{2}

_{ext}, represent the maximum, minimum, and average values of Q

^{2}

_{ext}of each algorithm in 30 experiments, respectively.

- (1)
- For Isomap data segmentation method, the value of R
^{2}, Q^{2}_{CV5}and Q^{2}_{ext}of the IBA-SVR algorithm proposed in this paper are better than those of the comparison algorithm. Therefore, the optimization performance of IBA algorithm for SVR model is better than that of other comparison algorithms. - (2)
- By comparing the dimensionality reduction methods of Isomap and PCA data under IBA-SVR model, we can see that the model performance of dimensionality reduction using Isomap algorithm is better.
- (3)
- The joint to optimization of Isomap and SVR model using IBA algorithm effectively improves the prediction accuracy of the model. 98.6% of the prediction results meet the accuracy requirements of level 1.5 instruments, and 97.7% of the prediction results meet the accuracy requirements of level 2.5 instruments.

- (1)
- Considering the fitting ability, different data segmentation methods have little influence on the fitting ability of the model.
- (2)
- Considering the robustness of the model, the model established by the method of randomly segmented data fluctuates more than that established by the method of SOM neural network.
- (3)
- Considering the external prediction ability of the model, the external prediction ability of the model established by the method of SOM neural network segmentation data is more stable.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Boateng, A.A.; Barr, P.V. A Thermal-Model for the Rotary Kiln Including Heat-Transfer within the Bed. Int. J. Heat Mass Transf.
**1996**, 39, 2131–2147. [Google Scholar] [CrossRef] - Tian, Z.; Li, S.; Wang, Y.; Wang, X. A multi-model fusion soft sensor modelling method and its application in rotary kiln calcination zone temperature prediction. Trans. Inst. Meas. Control.
**2015**, 38, 110–124. [Google Scholar] - Zhang, L.; Gao, X.W.; Wang, J.S.; Zhao, J. Soft-Sensing for Calcining Zone Temperature in Rotary Kiln Based on Model Migration. J. Northeast. Univ.
**2011**, 32, 175–178. [Google Scholar] - Li, M.W.; Hong, W.C.; Kang, H.G. Urban traffic flow forecasting using Gauss–SVR with cat mapping, cloud model and PSO hybrid algorithm. Neurocomputing
**2013**, 99, 230–240. [Google Scholar] [CrossRef] - Shi, X.; Chi, Q.; Fei, Z.; Liang, J. Soft-Sensing Research on Ethylene Polymerization Based on PCA-SVR Algorithm. Asian J. Chem.
**2013**, 25, 4957–4961. [Google Scholar] [CrossRef] - Kaneko, H.; Funatsu, K. Application of online support vector regression for soft sensors. AIChE J.
**2014**, 60, 600–612. [Google Scholar] [CrossRef] - Edwin, R.D.J.; Kumanan, S. Evolutionary fuzzy SVR modeling of weld residual stress. Appl. Soft Comput.
**2016**, 42, 423–430. [Google Scholar] [CrossRef] - Hou, Y.; Wei, S. Method for Mass Production of Phosphoric Acid with Rotary Kiln. U.S. Patent 10,005,669, 26 June 2018. [Google Scholar]
- Phummiphan, I.; Horpibulsuk, S.; Rachan, R.; Arulrajah, A.; Shen, S.-L.; Chindaprasirt, P. High calcium fly ash geopolymer stabilized lateritic soil and granulated blast furnace slag blends as a pavement base material. J. Hazard. Mater.
**2018**, 341, 257–267. [Google Scholar] [CrossRef] - Konrád, K.; Viharos, Z.J.; Németh, G. Evaluation, ranking and positioning of measurement methods for pellet production. Measurement
**2018**, 124, 568–574. [Google Scholar] [CrossRef] [Green Version] - Tian, Z.; Li, S.; Wang, Y.; Wang, X. SVM predictive control for calcination zone temperature in lime rotary kiln with improved PSO algorithm. Trans. Inst. Meas. Control.
**2017**, 40, 3134–3146. [Google Scholar] - Yin, Q.; Du, W.-J.; Ji, X.-L.; Cheng, L. Optimization design and economic analyses of heat recovery exchangers on rotary kilns. Appl. Energy
**2016**, 180, 743–756. [Google Scholar] [CrossRef] - Kim, K.I.; Jung, K.; Kim, H.J. Face recognition using kernel principal component analysis. IEEE Signal Process. Lett.
**2002**, 9, 40–42. [Google Scholar] - Bengio, Y.; Paiement, J.; Vincent, P.; Delalleau, O.; Roux, N.; Ouimet, M. Out-of-sample extensions for lle, isomap, mds, eigenmaps, and spectral clustering. In Advances in Neural Information Processing Systems; 2004; pp. 177–184. Available online: https://dl.acm.org/doi/10.5555/2981345.2981368 (accessed on 12 December 2019).
- Hannachi, A.; Turner, A.G. Isomap nonlinear dimensionality reduction and bimodality of Asian monsoon convection. Geophys. Res. Lett.
**2013**, 40, 1653–1658. [Google Scholar] [CrossRef] [Green Version] - Hannachi, A.; Turner, A. Monsoon convection dynamics and nonlinear dimensionality reduction vis Isomap. In Proceedings of the EGU General Assembly Conference, Vienna, Austria, 22–27 April 2012; EGU General Assembly Conference Abstracts. p. 8534. [Google Scholar]
- Jing, L.; Shao, C. Selection of the Suitable Parameter Value for Isomap. J. Softw.
**2011**, 6, 1034–1041. [Google Scholar] [CrossRef] - Orsenigo, C.; Vercellis, C. Linear versus nonlinear dimensionality reduction for banks’ credit rating prediction. Knowl. Based Syst.
**2013**, 47, 14–22. [Google Scholar] [CrossRef] - Smola, A.J.; Schölkopf, B. A tutorial on support vector regression. Stat. Comput.
**2004**, 14, 199–222. [Google Scholar] [CrossRef] [Green Version] - Sammon, J.W. A nonlinear mapping for data structure analysis. IEEE Trans. Comput.
**1969**, 100, 401–409. [Google Scholar] [CrossRef] - Bertsekas, D.P. Constrained Optimization and Lagrange Multiplier Methods; Academic Press: Cambridge, MA, USA, 2014. [Google Scholar]
- Golub, G.H.; Welsch, J.H. Calculation of Gauss Quadrature Rules. Math. Comput.
**1969**, 23, 221–230. [Google Scholar] [CrossRef] - Yang, X.S. A New Metaheuristic Bat-Inspired Algorithm. Comput. Knowl. Technol.
**2010**, 284, 65–74. [Google Scholar] - Subsequently. A Novel Hybrid Bat Algorithm with Harmony Search for Global Numerical Optimization. J. Appl. Math.
**2013**, 2013, 233–256. [Google Scholar] - Yang, X.S.; Deb, S. Cuckoo Search via Lévy Flights. In Proceedings of the World Congress on Nature & Biologically Inspired Computing (NaBIC), Coimbatore, India, 9–11 December 2009; IEEE: Piscataway, NJ, USA, 2010; pp. 210–214. [Google Scholar]
- Wang, G.-G.; Gandomi, A.H.; Zhao, X.; Chu, H.C.E. Hybridizing harmony search algorithm with cuckoo search for global numerical optimization. Soft Comput.
**2014**, 20, 273–285. [Google Scholar] [CrossRef] - Wang, Y.K.; Chen, X.B. Improved multi-area search and asymptotic convergence PSO algorithm with independent local search mechanism. Kongzhi Yu Juece/Control Decis.
**2018**, 33, 1382–1390. [Google Scholar] - Ghanem, W.A.H.M.; Jantan, A. An enhanced Bat algorithm with mutation operator for numerical optimization problems. Neural Comput. Appl.
**2019**, 31, 617–651. [Google Scholar] [CrossRef] - Chakri, A.; Khelif, R.; Benouaret, M.; Yang, X.-S. New directional bat algorithm for continuous optimization problems. Expert Syst. Appl.
**2017**, 69, 159–175. [Google Scholar] [CrossRef] [Green Version] - Shen, X.; Fu, X.; Zhou, C. A combined algorithm for cleaning abnormal data of wind turbine power curve based on change point grouping algorithm and quartile algorithm. IEEE Trans. Sustain. Energy
**2018**, 10, 46–54. [Google Scholar] [CrossRef] - Gupta, S.; Perlman, R.M.; Lynch, T.W.; McMinn, B.D. Normalizing Pipelined Floating Point Processing Unit. U.S. Patent 5,058,048, 15 October 1991. [Google Scholar]
- Pedrycz, W. Conditional fuzzy clustering in the design of radial basis function neural networks. IEEE Trans. Neural Netw.
**1998**, 9, 601–612. [Google Scholar] [CrossRef]

**Figure 3.**The convergence comparisons of all algorithms. (

**a**) Convergence Curve of sphere Function (f

_{1}) (

**b**) Convergence Curve of Schwefel 2.22 Function (f

_{2}) (

**c**) Convergence Curve of Ackley Function (f

_{3}) (

**d**) Convergence Curve of Griewank Function (f

_{4}) (

**e**) Convergence Curve of Shifted Sum Square Function (f

_{5}) (

**f**) Convergence Curve of Rotated Griewank Function (f

_{6}).

Name | Function | D | Range | f_{opt} |
---|---|---|---|---|

Shpere | ${f}_{1}(x)={\displaystyle {\sum}_{i=1}^{D}{x}_{i}^{2}}$ | 30 | [−10,10]^{D} | 0 |

Schwefel 2.22 | ${f}_{2}(x)={\displaystyle {\sum}_{i=1}^{D}\left|{x}_{i}\right|}+{\displaystyle {\prod}_{i=1}^{D}\left|{x}_{i}\right|}$ | 30 | [−10,10]^{D} | 0 |

Ackley | ${f}_{3}(x)=-20\mathrm{exp}(-0.2\sqrt{1/D{\displaystyle {\sum}_{i=1}^{D}{x}_{i}^{2}}})-\mathrm{exp}(\frac{1}{D}{\displaystyle {\sum}_{i=1}^{D}\mathrm{cos}(2\pi {x}_{i})})+20+e$ | 30 | [−32,32]^{D} | 0 |

Griewank | ${f}_{4}(x)=1/4000{\displaystyle {\sum}_{i=1}^{D}{x}_{i}^{2}}-{\displaystyle {\prod}_{i=1}^{D}\mathrm{cos}({x}_{i}/\sqrt{i})}+1$ | 30 | [−600,600]^{D} | 0 |

Shifted Sum Square | ${f}_{5}(x)={\displaystyle {\sum}_{i=1}^{D}i{({x}_{i}-\sqrt{i})}^{2}}$ | 30 | [−100,100]^{D} | 0 |

Rotated Griewank | ${f}_{6}(x)=1/4000{\displaystyle {\sum}_{i=1}^{D}{y}_{i}^{2}}-{\displaystyle {\prod}_{i=1}^{D}\mathrm{cos}({y}_{i}/\sqrt{i})}+1,$$where\begin{array}{c}\end{array}y=Mx\text{}M\text{}\mathrm{is}\text{}\mathrm{an}\text{}\mathrm{orthogonal}\text{}\mathrm{matrix}$ | 30 | [−600,600]^{D} | 0 |

Algorithm | Reference | Parameters |
---|---|---|

BA | Ref. [23] | $\alpha =0.95$, $\gamma =0.95$, ${f}_{\mathrm{min}}=-2$, ${f}_{\mathrm{max}}=2$ |

PSO | Ref. [27] | ${c}_{1}=2$, ${c}_{2}=2$, ${\omega}_{\mathrm{max}}=1.0$, ${\omega}_{\mathrm{min}}=0.3$ |

EBA | Ref. [28] | ${\xi}_{init}=0.6$, $n=3$ |

dBA | Ref. [29] | ${r}_{0}=0.1$, ${r}_{\infty}=0.7$, ${A}_{0}=0.9$, ${A}_{\infty}=0.6$ |

IBA | Present | $\alpha =0.95$, $\gamma =0.95$, ${f}_{\mathrm{min}}=-2$, ${f}_{\mathrm{max}}=2$ |

Dimensionality Reduction Method | Isomap | PCA | |||
---|---|---|---|---|---|

Modeling Method | GA-SVR | PSO-SVR | BA-SVR | IBA-SVR | IBA-SVR |

$R{a}_{AD}<22.5{\text{}}^{\xb0}\mathrm{C}$ | 95.3% | 95.7% | 96.3% | 98.6% | 96.9% |

$R{a}_{AD}<37.5{\text{}}^{\xb0}\mathrm{C}$ | 98.3% | 98.8% | 98.8% | 99.7% | 99.1% |

Min_R^{2} | 0.794 | 0.822 | 0.816 | 0.841 | 0.842 |

Max_R^{2} | 0.911 | 0.921 | 0.891 | 0.922 | 0.921 |

Mean_R^{2} | 0.855 | 0.853 | 0.852 | 0.862 | 0.859 |

Min_Q^{2}_{CV5} | 0.741 | 0.757 | 0.755 | 0.813 | 0.786 |

Max_Q^{2}_{CV5} | 0.866 | 0.839 | 0.821 | 0.871 | 0.857 |

Mean_Q^{2}_{CV5} | 0.821 | 0.823 | 0.829 | 0.851 | 0.836 |

Min_Q^{2}_{ext} | 0.731 | 0.817 | 0.822 | 0.831 | 0.811 |

Max_Q^{2}_{ext} | 0.832 | 0.855 | 0.869 | 0.874 | 0.862 |

Mean_Q^{2}_{ext} | 0.811 | 0.831 | 0.847 | 0.851 | 0.836 |

Method of Data Segmentation | Random | SOM | |
---|---|---|---|

Training Data | Min_R^{2} | 0.835 | 0.841 |

Max_R^{2} | 0.912 | 0.922 | |

Mean_R^{2} | 0.859 | 0.862 | |

Min_Q^{2}_{CV5} | 0.753 | 0.813 | |

Max_Q^{2}_{CV5} | 0.891 | 0.871 | |

Mean_Q^{2}_{CV5} | 0.821 | 0.851 | |

Test Data | Min_Q^{2}_{ext} | 0.741 | 0.831 |

Max_Q^{2}_{ext} | 0.896 | 0.874 | |

Mean_Q^{2}_{ext} | 0.816 | 0.851 |

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**MDPI and ACS Style**

Liu, J.; Wang, Y.; Zhang, Y.
A Novel Isomap-SVR Soft Sensor Model and Its Application in Rotary Kiln Calcination Zone Temperature Prediction. *Symmetry* **2020**, *12*, 167.
https://doi.org/10.3390/sym12010167

**AMA Style**

Liu J, Wang Y, Zhang Y.
A Novel Isomap-SVR Soft Sensor Model and Its Application in Rotary Kiln Calcination Zone Temperature Prediction. *Symmetry*. 2020; 12(1):167.
https://doi.org/10.3390/sym12010167

**Chicago/Turabian Style**

Liu, Jialun, Yukun Wang, and Yong Zhang.
2020. "A Novel Isomap-SVR Soft Sensor Model and Its Application in Rotary Kiln Calcination Zone Temperature Prediction" *Symmetry* 12, no. 1: 167.
https://doi.org/10.3390/sym12010167