# A New Machine-Learning Prediction Model for Slope Deformation of an Open-Pit Mine: An Evaluation of Field Data

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## Abstract

**:**

## 1. Introduction

## 2. The Proposed Method

**w**is the weight vector. The generality of the proposed model is that one can select J suitable member algorithms for different prediction problems. Inspired by our literature review, in this article, five popular artificial intelligence algorithms (J = 5) were adopted as the algorithm members in the proposed ensemble learning model, including BPNN, SVM, recurrent neural network (RNN), adaptive network-based fuzzy inference system (ANFIS), and relevant vector machine (RVM).

**w**(= [w

_{1}, w

_{2}, …, w

_{J}]

^{T}) is the weight vector and f(•) is a predefined error criterion, such as root-mean-squared error (RSME). The optimal weight vector

**w**

^{o}(= [w

_{1}, w

_{2}, …, w

_{J}]

^{T}) was used for model testing. Then, an ensemble prediction was calculated using the weighted-sum formulation in Equation (2). It is expected that the resulting optimal weight vector will appropriately combine the five member models to enhance the deformation prediction accuracy and robustness.

**x**

_{t}is a test dataset; ${\widehat{D}}_{t}^{P}$ is the ensemble prediction for

**x**

_{t}; w

_{j}(j = 1, 2, …, J) is the weight assigned to the jth member model; and ${\widehat{D}}_{t}^{j}$(

**x**

_{t},

**X**) denotes the prediction of

**x**

_{t}by the jth member model trained by the training datasets

**X**.

#### 2.1. Relevance Vector Machine

**x**,

**x**

_{i}) is a kernel function (e.g., Gaussian kernel) centered at the training point x

_{i},

**ω**= (ω

_{0}, …, ω

_{N})

^{T}~$\mathcal{N}$(0, σ

^{−1}I) is the weight vector, and σ is the variance of the prior on

**ω**. The likelihood function of the complete dataset can be written as follows:

**Φ**is a kernel matrix with dimensions of N × (N + 1) and

**Φ**

_{ij}= φ (

**x**

_{i},

**x**

_{j}). The Bayesian inference is then applied to Equation (4) to approximate the weight vector

**ω**.

**γ**= (γ

_{0}, …, γ

_{N})

^{T}denotes a set of hyperparameters. To specify the Bayesian model,

**γ**and σ

^{2}are usually assumed to be subject to the Gamma prior distributions.

**ω**can be estimated by

**θ**(= σ

^{−2}

**ΣΦ**$\tilde{\mathbf{y}}$) is the mean of the posterior over the weight vector,

**Σ**= (σ

^{−2}

**Φ**

^{T}

**Φ+Λ**)

^{−1}, and

**Λ**= diag(

**γ**). Thus, the expectation maximization (EM) or other iterative optimization algorithms can be applied to the Bayesian model to obtain the optimal hyperparameters

**γ**

_{p}and σ

^{2}

_{p}. The distribution of $\tilde{y}$ can be calculated using the optimal values. In this study, the MATLAB program developed by Tipping [24] was employed to build the RVM predictive model. The EM algorithm was used to update the Bayesian model. The Gaussian kernel function with a basis width of 0.5 and a prior variance of σ = 0.2 were used in the RVM model.

#### 2.2. Adaptive Network-Based Fuzzy Inference System

**I**= [I

_{1}, I

_{2}, …, I

_{I}], the first layer fuzzifies the inputs into membership values.

^{1}denotes the output of layer 1, the subscript ij denotes the jth membership function of ith input variable, i (= 1, 2,…, I), and j (= 1, 2,…, F

_{i}) with F

_{i}being the membership function numbers of ith input variable. The membership function includes a triangular, bell, or Gaussian function. The parameters in the membership function are called premise parameters.

^{2}denotes the output of layer 2, subscript k (= 1, 2,…, K) corresponds to the kth fuzzy rule, K is the total number of rules, and ∏μ(I

_{k}) only involves the membership values of those inputs that contribute the kth fuzzy rule. Layer 3 takes the normalization operator to the firing strength. The output of layer 3 (O

^{3}) for the kth rule is described by Equation (10).

^{4}

_{k}denotes the output of kth fuzzy rule,

**A**= [

**A**

_{1},

**A**

_{1},…,

**A**

_{K}]

^{T}and

**B**= [B

_{1}, B

_{1},…, B

_{K}]

^{T}are the consequent parameters for the Takagi–Sugeno fuzzy model. The sum of O

^{4}will give the output of the ANFIS. In this study, the Gaussian function was used as the membership function for the input variables, and each variable had 12 memberships. The fuzzy rule k was 10 in the ANFIS model.

#### 2.3. Recurrent Neural Network

**I**

^{(t)}= [I

^{(t)}

_{1}, I

^{(t)}

_{2}, …, I

^{(t)}

_{I}] (I is the neural node number in the input layer), and the previous hidden state

**H**

^{(t−1)}= [H

^{(t−1)}

_{1}, H

^{(t−1)}

_{2}, …, H

^{(t1)}

_{H}] (H is the number of neural nodes in the hidden layer) are connected to the hidden layer by the weights

**W**

^{HI}and

**W**

^{HM}, respectively. The current hidden state

**H**

^{(t)}= [H

^{(t)}

_{1}, H

^{(t)}

_{2}, …, H

^{(t)}

_{H}] is calculated by

^{−x})

^{−1}. The hidden layer

**H**

^{(t)}is connected to the output layer

**O**

^{(t)}= [O

^{(t)}

_{1}, O

^{(t)}

_{2}, …, O

^{(t)}

_{O}] (O is the neural node number in the output layer) by the weights

**W**

^{OH}.

**H**

^{(t−1)}by feeding back the current activities of neural nodes through the recurrent weights

**W**

^{HM}, and hence, the output

**H**

^{(t)}is the contribution of current and previous hidden states. The connections between the inputs at different timesteps can be included in the prediction behavior of the network. In this study, the MATLAB program developed by Cernansky [26] was used to establish the RNN predictive model, in which the number of nodes in the hidden layer was 10 and the output node number was 1.

## 3. Field Data

#### 3.1. Geological Survey of Anjialing Mine

^{2}and its mining area is about 7842 m from east to west and 6556 m from north to south. Anjialing mine belongs to a temperate, semi-arid, continental, monsoon climate zone. It is drought-prone, cold, and windy in spring and winter and rainy, cool, and less windy in summer and fall [4]. The average annual rainfall in this area is about 435 mm. The highest annual rainfall in historical records was 757.4 mm, and the lowest annual rainfall was 195.6 mm. In the Anjialing mining area, 75% of the annual rainfall is in summer. The annual average temperature is about 6 °C. The temperature difference between day and night is about 20 °C, the maximum temperature is about 37.9 °C, and the minimum temperature is −32.4 °C.

^{−5}m/s, and there is no water outlet point on the slope. There are several water outlet points in the bedrock joints and fissures in the northern end, among which the no. 4 coal seam is the most serious, indicating that there is static water in the bedrock. However, from the perspective of the static water level in the borehole, the water level is in the floor of the no. 4 coal seam, and the water level is relatively low, indicating that the water content in the rock mass is very low.

#### 3.2. Data Acquisition

#### 3.3. Field Geology Tests

## 4. Result and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The flowchart of the proposed method for ensemble prediction. Where J is the number of member algorithms and w is the weight vector.

**Figure 2.**A typical architecture of type-3 adaptive network-based fuzzy inference system (ANFIS), where II denotes the algebraic product operator and D denotes the normalization operator, MF

_{Fi}is the membership function of ith input variable.

**Figure 6.**Meteorological sensor module: (

**1**) rain collector, (

**2**) solar panel, (

**3**) radiation shield, (

**4**) sensors mounting shelf, and (

**5**) module base.

Input: | Slope Deformation Datasets |
---|---|

1. | Model training: Calculate the optimal weight vector for member algorithms |

1.1 | Select appropriate sensory measurements from the ground-based interferometric radar (GB-SAR) |

1.2 | Train each of the member algorithms to build J member prediction models |

1.3 | Perform model validation to obtain the predicted deformation of each member model |

1.4 | Compute the optimal weight vector using nonlinear optimization |

2. | Model testing: Perform predictions using ensemble learning |

2.1 | Predict the slop deformation of the online testing unit using each base learner |

2.2 | Carry out ensemble prognostics using the optimal weight vector |

Output: | Predicted deformation amount |

Prediction Model | Inputs | Outputs |
---|---|---|

BPNN | 12 parameters from the geographical, climatic, and hydrographic aspects | Deformation |

SVM | ||

RNN | ||

ANFIS | ||

RVM | ||

Ensemble |

Learner | Parameter |
---|---|

BPNN | Number of hidden neurons is 20 |

SVM | Gaussian kennel with 0.5 kennel width |

RNN | Number of hidden notes is 8 |

ANFIS | Number of Fuzzy rules is 10 |

RVM | Prior variance σ = 0.1 |

Learner | Optimal Weights | Validation Results (RMSE) | Testing Results (RMSE) |
---|---|---|---|

BPNN | 0 | 4.42 mm | 4.58 mm |

SVM | 0.01 | 3.93 mm | 3.87 mm |

RNN | 0.35 | 2.79 mm | 3.00 mm |

ANFIS | 0.12 | 3.26 mm | 3.16 mm |

RVM | 0.52 | 2.65 mm | 2.64 mm |

Ensemble | 2.01 mm | 2.23 mm |

Learner | Predicted Deformation (mm) | Absolute Error (mm) | Average Absolute Error |
---|---|---|---|

BPNN | [0.69 16.35 10.06 9.87 15.94] | [4.73 10.62 2.53 0.23 2.50] | 4.122 mm |

SVM | [3.81 2.35 3.99 5.32 8.69] | [1.61 3.38 3.54 4.78 4.75] | 3.612 mm |

RNN | [5.33 6.51 8.25 10.76 14.08] | [0.09 0.78 0.72 0.66 0.64] | 0.578 mm |

ANFIS | [6.35 3.42 6.76 8.29 15.92] | [0.93 2.31 0.77 1.81 2.48] | 1.660 mm |

RVM | [5.22 5.65 7.10 9.82 12.22] | [0.20 0.08 0.43 0.28 1.22] | 0.442 mm |

Ensemble | [5.38 5.65 7.4300 9.92 13.28] | [0.04 0.08 0.10 0.18 0.16] | 0.112 mm |

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**MDPI and ACS Style**

Du, S.; Feng, G.; Wang, J.; Feng, S.; Malekian, R.; Li, Z.
A New Machine-Learning Prediction Model for Slope Deformation of an Open-Pit Mine: An Evaluation of Field Data. *Energies* **2019**, *12*, 1288.
https://doi.org/10.3390/en12071288

**AMA Style**

Du S, Feng G, Wang J, Feng S, Malekian R, Li Z.
A New Machine-Learning Prediction Model for Slope Deformation of an Open-Pit Mine: An Evaluation of Field Data. *Energies*. 2019; 12(7):1288.
https://doi.org/10.3390/en12071288

**Chicago/Turabian Style**

Du, Sunwen, Guorui Feng, Jianmin Wang, Shizhe Feng, Reza Malekian, and Zhixiong Li.
2019. "A New Machine-Learning Prediction Model for Slope Deformation of an Open-Pit Mine: An Evaluation of Field Data" *Energies* 12, no. 7: 1288.
https://doi.org/10.3390/en12071288