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


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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Fig. 1
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Boundary Forest


Classification and Regression Tree


Extreme Learning Machine


General Regression Neural Network


Ladle Furnace


Least Squares Support Vector Regression


Maximum Absolute Error


Mean Square Errors


Neural Network


Online SCM


OSCM based on the Boundary Forest


OSCM based on the TSE


Parsimonious Ensemble


Random Forest


Root-Mean-Square Error


Soft Computing Models


Support Vector Machine


Tree-Structure Ensemble

\( \varpi \) :

The width of a window

\( \vartheta \) :

The step for updating


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


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


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


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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.


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).

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