The Online Soft Computing Models of key variables based on the Boundary Forest method

Abstract

The Online Soft Computing Models (OSCMs) based on ensemble methods are novel and quite effective data-driven tools for predicting key variables. The current challenge encountered by them is how to enhance the reliability caused by both the uncertainty from noise and the unsuitable specifications of models, on the premise of high predicting accuracy and low computational cost. To meet the current challenge, the OSCM based on the Boundary Forest (OSCM-BF) is proposed in this paper. The BF combines a set of the Tree-Structure Ensemble (TSE) models. In terms of the different values of θ (i.e., the minimum size of leaf nodes), the BF enhances the reliability of a single TSE not only by overlapping the gap segments of output range (i.e., connecting the discontinuous boundaries of leaf nodes), but also by possessing stronger robustness via producing enough diversity. Moreover, a theoretical range of the value of θ constructed by BF is provided. Since the simplicity, the nice interpretability and the flexibility on large-scale data, the moving-window strategy was adopted to realize the update of the BF models. The experiments on the noisy data from the industrial process of Ladle Furnace reveal that the OSCM-BF can enhance the reliability of the OSCM-TSE on the premise of high predicting accuracy and low computational cost.

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Abbreviations

BF:

Boundary Forest

CART:

Classification and Regression Tree

ELM:

Extreme Learning Machine

GRNN:

General Regression Neural Network

LF:

Ladle Furnace

LSSVR:

Least Squares Support Vector Regression

MAE:

Maximum Absolute Error

MSE:

Mean Square Errors

NN:

Neural Network

OSCM:

Online SCM

OSCM-BF:

OSCM based on the Boundary Forest

OSCM-TSE:

OSCM based on the TSE

pENsemble:

Parsimonious Ensemble

RF:

Random Forest

RMSE:

Root-Mean-Square Error

SCM:

Soft Computing Models

SVM:

Support Vector Machine

TSE:

Tree-Structure Ensemble

\( \varpi \) :

The width of a window

\( \vartheta \) :

The step for updating

Θ:

A learning set, and \( \varTheta = {\text{\{ (}}{\mathbf{X}},y )_{n} {\text{\} }}_{{n{ = }1}}^{N} \)

\( ({\mathbf{X}},y) \) :

A sample pair

y :

The output variable, or the real output, \( y \in {\mathbb{R}}^{1} \)

\( \hat{y} \) :

The prediction of a model

\( {\mathbf{X}} \) :

The input vector or a sample, and \( {\mathbf{X}} = (x_{1} , \ldots ,x_{M} ) \in {\mathbb{R}}^{M} \)

xi, i = 1, 2, …, M :

The ith input variable

N :

The number of the samples in Θ

M :

The dimension of the input variables

p(X):

The mapping of the piecewise function to X

\( \hbar_{i} , { }i = 1, \ldots ,M \) :

The threshold of the input variable \( x_{i} \)

Θleaf, Θright :

The sample subsets of the left and the right sub-branches

MSEleaf, MSEright :

The MSEs of the outputs in Θleaf and Θright

\( \bar{y}_{\text{left}} \), \( \bar{y}_{\text{left}} \) :

The mean values of the real outputs in Θleaf and Θright

Nleaf, Nright :

The numbers of samples in Θleaf and Θright

MSEmin :

The minimum sum of MSEleaf and MSEright

J :

The number of the possible thresholds of a input variable

θ :

The minimum size of leaf nodes in a TSE model

K :

The number of the TSE models in a BF model

T k :

The kth TSE models in a BF model, k = 1, …, K

θ k :

The minimum size of leaf nodes in the TSE model Tk

Φ k :

The set of leaf nodes in the TSE sub-model Tk, and \( \varPhi_{k} = \{ \varTheta_{1k}^{\text{leaf}} ,\varTheta_{2k}^{\text{leaf}} , \ldots ,\varTheta_{{\varGamma_{k} k}}^{\text{leaf}} \} \)

Гk :

The number of the leaf nodes in Φk

\( g_{1k}^{\text{leaf}} ({\mathbf{X}}),g_{2k}^{\text{leaf}} ({\mathbf{X}}), \ldots ,g_{{\varGamma_{k} k}}^{\text{leaf}} ({\mathbf{X}}) \) :

The mappings of the local TSE models learnt on Φk

fBF(X):

The mapping of a BF model

ω = [ω1, ω2, …, ωK] :

The weight vector of the TSE models{T1, T2, …, TK}

\( \omega_{k} \) :

The weight of the TSE sub-model Tk

f TSEk (X):

The mapping of the TSE sub-model Tk

Ω :

The covariance matrix with size K × K

Ω kj :

The element of Ω, j, k = 1, …, K

\( \hat{y}_{ki} \) :

The prediction of the sample Xi from the TSE sub-model Tk, j, k = 1, …, K

\( y_{i} \) :

The real output of the sample Xi

\( {\hat{\mathbf{\varLambda }}} \) :

The prediction matrix of the training samples from the K TSE models

X q :

The query sample

\( \hat{y}_{{1{\text{q}}}} ,\hat{y}_{{2{\text{q}}}} , \ldots ,\hat{y}_{{K{\text{q}}}} \) :

The predictions of Xq from the K TSE models in a BF model

χ jk :

The size of the jth leaf node in Tk, j = 1, …, Гk, k = 1, …, K

References

  1. Breiman L (1996) Bagging predictors. Mach Learn 24(2):123–140

    MATH  Google Scholar 

  2. Breiman L (2001) Random forests. Mach Learn 45(1):5–32

    Article  Google Scholar 

  3. Breiman L, Friedman JH, Olshen RA, Stone CJ (1984) Classification and regression trees. Wadsworth Int Group 40(3):17–23

    MATH  Google Scholar 

  4. Demsǎr J (2006) Statistical comparisons of classifiers over multiple data sets. J Mach Learn Res 7:1–30

    MathSciNet  MATH  Google Scholar 

  5. García S, Fernandez A, Luengo J, Herrera F (2009) A study statistical of techniques and performance measures for genetics-based machine learning: accuracy and interpretability. Soft Comput 13(10):959–977

    Article  Google Scholar 

  6. Huang GB, Zhou H, Ding X, Zhang R (2012) Extreme learning machine for regression and multiclass classification. IEEE Trans Syst Man Cybern B Cybern 42(2):513–529

    Article  Google Scholar 

  7. Jaramillo F, Orchard M, Muñoz C, Antileo C, Sáez D, Espinoza P (2018) On-line estimation of the aerobic phase length for partial nitrification processes in SBR based on features extraction and SVM classification. Chem Eng J 331:114–123

    Article  Google Scholar 

  8. Kadlec P, Gabrys B (2011) Local learning-based adaptive soft sensor for catalyst activation prediction. AIChE J 57(5):1288–1301

    Article  Google Scholar 

  9. Kadlec P, Gabrys B, Strandt S (2009) Data-driven soft sensor in the process industry. Comput Chem Eng 33(4):795–814

    Article  Google Scholar 

  10. Kadlec P, Grbić R, Gabrys B (2011) Review of adaptation mechanisms for data-driven soft sensors. Comput Chem Eng 35(1):1–24

    Article  Google Scholar 

  11. Kazienko P, Lughofer E, Trawinski B (2015) Editorial on the special issue “Hybrid and ensemble techniques in soft computing: recent advances and emerging trends”. Soft Comput 19:3353–3355

    Article  Google Scholar 

  12. Liu Y, Gao Z, Chen J (2013) Development of soft-sensors for online quality prediction of sequential-reactor-multi-grade industrial processes. Chem Eng Sci 102(11):602–612

    Article  Google Scholar 

  13. Liukkonen M, Hälikkä E, Hiltunen T, Hiltunen Y (2013) Adaptive soft sensor for fluidized bed quality: applications to combustion of biomass. Fuel Process Technol 105(1):46–51

    Article  Google Scholar 

  14. Lughofer E, Macian V, Guardiola C, Klement EP (2011) Identifying static and dynamic prediction models for NOx emissions with evolving fuzzy systems. Appl Soft Comput 11(2):2487–2500

    Article  Google Scholar 

  15. Marković D, Petković D, Nikolić V, Milovančević M, Petković B (2017) Soft computing prediction of economic growth based in science and technology factors. Phys A 465:217–220

    Article  Google Scholar 

  16. Moayedi H, Hayati S (2018) Modelling and optimization of ultimate bearing capacity of strip footing near a slope by soft computing methods. Appl Soft Comput 29:1393–1409

    Google Scholar 

  17. Parsaie A, Haghiabi AH, Saneie M, Torabi H (2018) Applications of soft computing techniques for prediction of energy dissipation on stepped spillways. Neural Comput Appl 29:1393–1409

    Article  Google Scholar 

  18. Peng X, Tang Y, Du W, Qian F (2017) Online performance monitoring and modeling paradigm based on just-in-time learning and ELM for a non-Gaussian chemical process. Ind Eng Chem Res 56(23):6671–6684

    Article  Google Scholar 

  19. Perrone MP, Cooper LN (1993) When networks disagree: ensemble methods for hybrid neural networks. In: Mammone RJ (ed) Artificial neural networks for speech and vision. Chapman & Hall, London, pp 126–142

    Google Scholar 

  20. Polikar R, Upda L, Upda SS, Honavar V (2001) Learn ++: an incremental learning algorithm for supervised neural networks. IEEE Trans Syst Man Cybern C Appl Rev 31(4):497–508

    Article  Google Scholar 

  21. Pratama M, Pedrycz W, Lughofer E (2018) Evolving ensemble fuzzy classifier. IEEE Trans Fuzzy Syst 26(5):2552–2567

    Article  Google Scholar 

  22. Shen K-Y, Tzeng G-H (2015) A decision rule-based soft computing model for supporting financial performance improvement of the banking industry. Soft Comput 19:859–874

    Article  Google Scholar 

  23. Specht DF (1991) A general regression neural network. IEEE Trans Neural Netw 2(6):568–576

    Article  Google Scholar 

  24. Suykens JAK, Gestel TV, De Brabanter J, De Moor B, Vandewalle J (2002) Least squares support vector machines. World Scientific, Singapore

    Google Scholar 

  25. Tatinati S, Veluvolu KC, Wei TA (2015) Multistep prediction of physiological tremor based on machine learning for robotics assisted microsurgery. IEEE Trans Cybern 45(2):328–339

    Article  Google Scholar 

  26. Tian H-X, Mao Z-Z (2010) An ensemble ELM based on modified AdaBoost.RT algorithm for predicting the temperature of molten steel in ladle furnace. IEEE Trans Autom Sci Eng 7(1):73–85

    Article  Google Scholar 

  27. Vandechali MR, Abbaspour-Fard MH, Rohani A (2018) Development of a prediction model for estimating tractor engine torque based on soft computing and low cost sensors. Measurement 121:83–95

    Article  Google Scholar 

  28. Vapnik VN (1999) The nature of statistical learning theory, 2nd edn. Springer, New York

    Google Scholar 

  29. Wang X (2017) Ladle furnace temperature prediction model based on large-scale data with random forest. IEEE/CAA J Autom Sin 4(4):770–774

    Article  Google Scholar 

  30. Wang L, Jin H, Chen X, Dai J, Yang K, Zhang D (2016a) Soft sensor development based on the hierarchical ensemble of Gaussian process regression models for nonlinear and non-Gaussian chemical processes. Ind Eng Chem Res 55(28):7704–7719

    Article  Google Scholar 

  31. Wang X, You M, Mao Z, Yuan P (2016b) Tree-structure ensemble general regression neural networks applied to predict the molten steel temperature in ladle furnace. Adv Eng Inform 30(3):368–375

    Article  Google Scholar 

  32. Wang X, Yuan P, Mao Z, You M (2016c) Molten steel temperature prediction model based on bootstrap feature subsets ensemble regression trees. Knowl Based Syst 101:48–59

    Article  Google Scholar 

  33. Wang XJ, Wang XY, Zhang Q, Mao ZZ (2018) The soft sensor of the molten steel temperature using the modified maximum entropy based pruned bootstrap feature subsets ensemble method. Chem Eng Sci 189:401–412

    Article  Google Scholar 

  34. Weigl E, Heidl W, Lughofer E, Radauer T, Eitzinger C (2016) On improving performance of surface inspection systems by on-line active learning and flexible classifier updates. Mach Vis Appl 27(1):103–127

    Article  Google Scholar 

  35. Yan Y, Wang L, Wang T, Wang X, Hu Y, Duan Q (2018) Application of soft computing techniques to multiphase flow measurement: a review. Measurement 60:30–43

    Google Scholar 

  36. Yuan X, Ge Z, Huang B, Song Z, Wang Y (2017) Semisupervised JITL framework for nonlinear industrial soft sensing based on locally semisupervised weighted PCR. IEEE Trans Industr Inf 13(2):99

    Article  Google Scholar 

  37. Zou QY, Wang XJ, Zhou CJ, Zhang Q (2018) The memory degradation based online sequential extreme learning machine. Neurocomputing 275:2864–2879

    Article  Google Scholar 

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Acknowledgements

The authors would like to acknowledge Professor Zhi-Zhong Mao for providing the data and suggestions. He is a PhD Supervisor at Northeastern University, and his research interests include control and optimization in complex industrial system.

Funding

This study was funded by the National Natural Science Foundation of China (No. 61702070) and the Research Projects of Liaoning Marine Fisheries Office (No. 201512).

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Correspondence to Xiao-Jun Wang.

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Deng, C., Wang, X., Gu, J. et al. The Online Soft Computing Models of key variables based on the Boundary Forest method. Soft Comput 24, 10815–10828 (2020). https://doi.org/10.1007/s00500-019-04584-1

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Keywords

  • Industrial process
  • Key variables
  • Soft computing
  • Machine learning
  • Online prediction
  • Big data