﻿ 基于模糊分类的UBGM(1,1)-Markov模型在露天矿边坡沉降预测中的应用
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 大地测量与地球动力学  2020, Vol. 40 Issue (12): 1259-1262  DOI: 10.14075/j.jgg.2020.12.010

### 引用本文

MA Zhanwu, HE Bing, LU Mingxing, et al. The Prediction of Slope Settlement of Open-Pit Mine Using UBGM(1, 1)-Markov Model Based on Fuzzy Classification[J]. Journal of Geodesy and Geodynamics, 2020, 40(12): 1259-1262.

### Foundation support

National Key Research and Development Program of China, No.2016YFC0801603; Natural Science Foundation of Liaoning Province, No.20170540456;Experimental Education Reform and Laboratory Construction Project of University of Science and Technology Liaoning, No. SYJG202045; Innovation and Entrepreneurship Project of University of Science and Technology Liaoning, No. 201910146368.

### 第一作者简介

MA Zhanwu, assistant experimenter, majors in geodesy, deformation monitoring and data processing, and spatio-temporal data mining, E-mail:923511109@qq.com.

### 文章历史

1. 辽宁科技大学土木工程学院，辽宁省鞍山市千山中路189号，114051;
2. 中国科学院深圳先进技术研究院，深圳市学苑大道1068号，518055;
3. 辽宁科技大学矿业工程学院，辽宁省鞍山市千山中路189号，114051

UBGM(1, 1)-Markov模型广泛应用于沉降预测[1]。该模型利用UBGM(1, 1)模型消除GM(1, 1)-Markov模型中的灰色固有偏差，提高预测精度。然而，UBGM(1, 1)-Markov预测模型存在2个邻近值可能被归属到不同状态、导致预测值产生偏差的问题。针对该问题，本文引入模糊分类理论，构建了FC-UBGM(1, 1)-Markov预测模型。

1 方法 1.1 模糊分类

 图 1 三角法模糊空间 Fig. 1 Trigonometric fuzzy space
1.2 构建FC-UBGM(1, 1)-Markov预测模型

 ${\hat x^{\left( 0 \right)}}\left( k \right) = \left\{ \begin{array}{l} {x^{\left( 0 \right)}}\left( 1 \right), k = 1\\ A{{\rm{e}}^{E\left( {k - 1} \right)}}, k = 2, 3, \cdots , n \end{array} \right.$ (1)

 $\hat \varepsilon \left( {k + 1} \right) = \left( { - {a_\varepsilon }} \right)\left( {{\varepsilon ^{\left( 0 \right)}}\left( {{k_0}} \right) - \frac{{{b_\varepsilon }}}{{{a_\varepsilon }}}} \right){{\rm{e}}^{ - {a_ \in }\left( {k - {k_0}} \right)}}$ (2)

 ${\hat x^{\left( 0 \right)}}\left( {k + 1} \right) = \left\{ \begin{array}{l} {{\hat x}^{\left( 0 \right)}}, k = 0\\ A{{\rm{e}}^{Ek}}, 0 ＜ k ＜ {k_0}\\ A{{\rm{e}}^{Ek}} \pm \hat \varepsilon \left( {k + 1} \right), k \ge {k_0} \end{array} \right.$ (3)

 ${\mathit{\boldsymbol{V}}^{\left( l \right)}} = {\mathit{\boldsymbol{V}}^{\left( 0 \right)}}{\left( {{\mathit{\boldsymbol{p}}^1}} \right)^l}$ (4)

 ${Y_{i1}} = \frac{{{{\hat x}^{\left( 0 \right)}}\left( k \right)}}{{1 - {\delta _{i - 1}}}}, {Y_{i2}} = \frac{{{{\hat x}^{\left( 0 \right)}}\left( k \right)}}{{1 - {\delta _i}}}$ (5)

 $\hat x\left( l \right) = \frac{{{Y_{i1}} + {Y_{i2}}}}{2}$ (6)

2 案例分析 2.1 研究区概况与数据源

2.2 算法验证

 $d\left( i \right) = \frac{{\mathop \sum \limits_{i = k}^n x\left( i \right)}}{{n - k - 1}}, k = 1, 2, \cdots , n$

 图 2 弱化缓冲算子处理前后累积沉降量对比 Fig. 2 The comparison of cumulative settlements before and after the weakening of the buffer operator

 图 3 弱化缓冲算子处理前后沉降速率对比 Fig. 3 The comparison of settlement rates before and after the weakening of the buffer operator

 $\begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\hat x}^{\left( 0 \right)}}\left( {k + 1} \right) = \\ \left\{ \begin{array}{l} 39.81, k = 0\\ 44.020{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 9{{\rm{e}}^{0.016{\kern 1pt} {\kern 1pt} {\kern 1pt} 7\left( k \right)}}, 0 ＜ k ＜ 19\\ 44.020{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 9{{\rm{e}}^{0.016{\kern 1pt} {\kern 1pt} {\kern 1pt} 7\left( k \right)}} - 0.990{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 2{{\rm{e}}^{0.063{\kern 1pt} {\kern 1pt} {\kern 1pt} 7\left( {k - 19} \right)}}, k \ge 19 \end{array} \right. \end{array}$

 图 4 预测值的残差与传统模型对比 Fig. 4 The residual errors of the prediction values compared with traditional models

 图 5 Q3监测点预测值的相对残差与传统模型对比 Fig. 5 The relative residuals of prediction values compared with traditional models
3 结语

 [1] 杨帆, 赵增鹏, 王小兵. 改进灰色马尔科夫模型在基坑预测中的研究[J]. 测绘与空间地理信息, 2017, 40(7): 15-18 (Yang Fan, Zhao Zengpeng, Wang Xiaobing. Prediction of Foundation Settlement Prediction Based on Improved Grey Markov Model[J]. Geomatics and Spatial Information Technology, 2017, 40(7): 15-18) (0) [2] 谢建文, 张元标, 王志伟. 基于无偏灰色模糊马尔可夫链法的铁路货运量预测研究[J]. 铁道学报, 2009, 31(1): 1-7 (Xie Jianwen, Zhang Yuanbiao, Wang Zhiwei. Railway Freight Volume Forecasting Based on Unbiased Grey-Fuzzy-Markov Chain Method[J]. Journal of China Railway Society, 2009, 31(1): 1-7) (0) [3] 冯子帆, 成枢, 董娟. 残差修正的GM(1, 1)模型在滑坡位移预测中的应用[J]. 地理空间信息, 2019, 17(9): 113-115 (Feng Zifan, Cheng Shu, Dong Juan. Application of GM(1, 1) Residual Modified Model in Landslide Displacement Prediction[J]. Geospatial Information, 2019, 17(9): 113-115) (0) [4] 王池, 刘超锋, 刘琪. 弱化缓冲算子修正的离散灰色预测[J]. 测绘与空间地理信息, 2019, 42(12): 223-225 (Wang Chi, Liu Chaofeng, Liu Qi. Discrete Grey Prediction Model Revised by Weakening the Buffer Operators[J]. Geomatics and Spatial Information Technology, 2019, 42(12): 223-225) (0)
The Prediction of Slope Settlement of Open-Pit Mine Using UBGM(1, 1)-Markov Model Based on Fuzzy Classification
MA Zhanwu1     HE Bing2     LU Mingxing1     XU Zhenyang3
1. School of Civil Engineering, University of Science and Technology Liaoning, 189 Mid-Qianshan Road, Anshan 114051, China;
2. Shenzhen Institutes of Advanced Technology, CAS, 1068 Xueyuan Road, Shenzhen 518055, China;
3. School of Mining Engineering, University of Science and Technology Liaoning, 189 Mid-Qianshan Road, Anshan 114051, China
Abstract: Predicting slope settlement in open-pit mines is an important means to grasp the trend of slope movement and guarantee the safe operation of the mine. Aiming at the problem that for the UBGM(1, 1)-Markov model, two neighboring values may be assigned to different states, leading to deviations in predicted values; combined with fuzzy classification theory, we propose an unbiased grey-Markov model based on fuzzy classification model(FC-UBGM(1, 1)-Markov). First, the residual correction is performed on the UBGM(1, 1) model; the relative residual sequence of the fitted values after the correction is used as the Markov chain to divide the interval, and the membership function of the fuzzy classification is used to calculate the fuzzy vector of the relative residual. The accuracy of this prediction model is analyzed through actual cases. The experiment results show that, compared with the traditional UBGM(1, 1)-Markov model, the predictive power of the model in this paper is better.
Key words: fuzzy classification; UGM(1, 1)-Markov model; Markov state transition probability matrix; residual error correction; open-pit mine