﻿ 优化背景值的非等间距线性时变参数GM(1, 1)幂模型在变形监测中的应用
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 大地测量与地球动力学  2022, Vol. 42 Issue (8): 823-828  DOI: 10.14075/j.jgg.2022.08.010

### 引用本文

WANG Bing, LI Peixian, ZHANG Jun, et al. Application of Non-Equidistant Linear Time-Varying Parameter GM(1, 1) Power Model with Optimized Background Value in Deformation Monitoring[J]. Journal of Geodesy and Geodynamics, 2022, 42(8): 823-828.

### Foundation support

The Joint Fund of Natural Science Foundation of Hebei Province Ecological Wisdom Mine, No. E2020402086;Fundamental Research Funds for the Central Universities, No. 2021YQDC01.

### Corresponding author

LI Peixian, PhD, associate professor, majors in mining subsidence and rock stratum movement, E-mail: lipx@cumtb.edu.cn.

### 第一作者简介

WANG Bing, postgraduate, majors in deformation data processing, E-mail: zqt2100204124@student.cumtb.edu.cn.

### 文章历史

1. 中国矿业大学(北京)地球科学与测绘工程学院，北京市学院路丁11号，100083;
2. 内蒙古农业大学理学院，呼和浩特市昭乌达路306号，010018;
3. 内蒙古农业大学计算机与信息工程学院，呼和浩特市鄂尔多斯东街29号，010011

1 线性时变参数非等间距GM(1, 1)幂模型

 $\frac{{{\rm{d}}{x^{(0)}}(t)}}{{{\rm{d}}t}} + a{x^{(0)}}(t) = (bt + c){\left[ {{x^{(0)}}(t)} \right]^\gamma }$ (1)

 ${\hat x^{(0)}}\left( {{t_k}} \right) = \left\{ {\begin{array}{*{20}{l}} {{x^{(0)}}\left( {{t_1}} \right), k = 1}\\ {{{\left\{ {\left[ {{{\left[ {{x^{(0)}}} \right]}^{1 - \gamma }} - \frac{b}{a}{t_1} + \frac{b}{{{a^2}(1 - \gamma )}} - \frac{c}{a}} \right]{{\rm{e}}^{a(1 - \gamma )\left( {{t_1} - {t_k}} \right)}} + \frac{b}{a}{t_k} - \frac{b}{{{a^2}(1 - \gamma )}} + \frac{c}{a}} \right\}}^{\frac{1}{{1 - \gamma }}}}, k = 2, \cdots , n} \end{array}} \right.$ (2)

 ${\left[ {{x^{(0)}}(t)} \right]^{ - \gamma }}\frac{{{\rm{d}}{x^{(0)}}(t)}}{{{\rm{d}}t}} + {\left[ {{x^{(0)}}(t)} \right]^{1 - \gamma }} = bt + c$ (3)

y(0)(t)=[x(0)(t)]1－γ，求导有$\frac{{{\rm{d}}{y^{(0)}}(t)}}{{{\rm{d}}t}} = (1 - \gamma ){\left[ {{x^{(0)}}(t)} \right]^{ - \gamma }}\frac{{{\rm{d}}{x^{(0)}}(t)}}{{{\rm{d}}t}}$代入式(3)并在方程两边同时乘以(1－γ)得：

 ${y^{(0)}}(t) = {C_1}{{\rm{e}}^{ - a(1 - \gamma )t}} + \frac{b}{a}t - \frac{b}{{{a^2}(1 - \gamma )}} + \frac{c}{a}$ (4)

 ${x^{(0)}}(t) = {\left\{ {{C_1}{{\rm{e}}^{ - a(1 - \gamma )t}} + \frac{b}{a}t - \frac{b}{{{a^2}(1 - \gamma )}} + \frac{c}{a}} \right\}^{\frac{1}{{1 - \gamma }}}}$ (5)

 $\begin{array}{*{20}{c}} {{{\hat x}^{(0)}}\left( {{t_k}} \right) = }\\ {\left\{ {\left[ {{{\left[ {{x^{(0)}}\left( {{t_1}} \right)} \right]}^{1 - \gamma }} - \frac{b}{a}{t_1} + \frac{b}{{{a^2}(1 - \gamma )}} - \frac{c}{a}} \right] \cdot } \right.}\\ {{{\left. {{{\rm{e}}^{ - a(1 - \gamma )\left( {{t_k} - {t_1}} \right)}} + \frac{b}{a}{t_k} - \frac{b}{{{a^2}(1 - \gamma )}} + \frac{c}{a}} \right\}}^{\frac{1}{{1 - \gamma }}}}} \end{array}$ (6)

 $\begin{gathered} C_{1}= \\ \left\{\left[x^{(0)}\left(t_{1}\right)\right]^{1-\gamma}-\frac{b}{a} t_{1}+\frac{b}{a^{2}(1-\gamma)}-\frac{c}{a}\right\} \mathrm{e}^{-a(\gamma-1) t_{1}} \end{gathered}$ (7)

2 线性时变参数非等间距GM(1, 1)幂模型背景值优化 2.1 模型最优背景值构建

 $z^{(1)}\left(t_{k}\right)=\lambda x^{(1)}\left(t_{k-1}\right)+(1-\lambda) x^{(1)}\left(t_{k}\right)$ (8)

2.2 优化背景值的时变参数非等间距GM(1, 1)幂模型参数估计

 \begin{aligned} &y^{(-1)}\left(t_{k}\right) \Delta t_{k}+a(1-\gamma) z^{(0)}\left(t_{i}\right) \Delta t_{k}= \\ &\frac{1}{2} b(1-\gamma)\left(t_{k}^{2}-t_{k-1}^{2}\right)+c(1-\gamma) \Delta t_{k} \end{aligned} (9)

 $\begin{gathered} (1-\gamma)\left[x^{(0)}(t)\right]^{-\gamma} \frac{\mathrm{d} x^{(0)}(t)}{\mathrm{d} t}+a(1-\gamma) \cdot \\ {\left[x^{(0)}(t)\right]^{1-\gamma}=(b t+c)(1-\gamma)} \end{gathered}$ (10)

y(0)(t)=[x(0)]1－γ，则式(10)可简化为：

 $\frac{\mathrm{d} y^{(0)}(t)}{\mathrm{d} t}+a(1-\gamma) y^{(0)}(t)=(b t+c)(1-\gamma)$ (11)

 \begin{aligned} &y^{(-1)}\left(t_{k}\right) \Delta t_{k}+a(1-\gamma) z^{(0)}\left(t_{i}\right) \Delta t_{k}= \\ &\frac{1}{2} b(1-\gamma)\left(t_{k}^{2}-t_{k-1}^{2}\right)+c(1-\gamma) \Delta t_{k} \end{aligned} (12)

 ${\left[ {\begin{array}{*{20}{l}} a&b&c \end{array}} \right]^{\rm{T}}} = {\left( {{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{B}}} \right)^{ - 1}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{Y}}$ (13)

 $\mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} {(\gamma - 1){z^{(0)}}\left( {{t_2}} \right)\Delta {t_2}}&{0.5(1 - \gamma )\left( {t_2^2 - t_1^2} \right)}&{(1 - \gamma )\Delta {t_2}}\\ {(\gamma - 1){z^{(0)}}\left( {{t_3}} \right)\Delta {t_3}}&{0.5(1 - \gamma )\left( {t_3^2 - t_2^2} \right)}&{(1 - \gamma )\Delta {t_3}}\\ \vdots & \vdots & \vdots \\ {(\gamma - 1){z^{(0)}}\left( {{t_n}} \right)\Delta {t_n}}&{0.5(1 - \gamma )\left( {t_n^2 - t_{n - 1}^2} \right)}&{(1 - \gamma )\Delta {t_n}} \end{array}} \right], \mathit{\boldsymbol{Y}} = \left[ {\begin{array}{*{20}{c}} {{y^{( - 1)}}\left( {{t_2}} \right)\Delta {t_2}}\\ {{y^{( - 1)}}\left( {{t_3}} \right)\Delta {t_3}}\\ \vdots \\ {{y^{( - 1)}}\left( {{t_n}} \right)\Delta {t_n}} \end{array}} \right]$ (14)
2.3 粒子群算法优化幂指数和背景值权重

 $\begin{array}{*{20}{c}} {\min f(\gamma , \lambda ) = \frac{1}{{n - 1}}\sum\limits_{k = 2}^n {\left| {\frac{{{{\hat x}^{(0)}}\left( {{t_k}} \right) - {x^{(0)}}\left( {{t_k}} \right)}}{{{x^{(0)}}\left( {{t_k}} \right)}}} \right|} , }\\ {\gamma \in [ - 3, 3], \lambda \in [0, 1]} \end{array}$ (15)

1) 设置运行参数的初始化值，计算初始种群在搜索域内每个粒子的位置Xid1和速度Vid1

2) 根据目标函数计算初始化粒子群各粒子的适应度fitness，寻找初始搜索域内参数γλ的个体最佳位置Pbest1Pbest2和全局最佳位置Gbest1Gbest2

3) 在种群规模和进化次数的搜索条件下，通过迭代不断更新粒子速度Vid1和位置Xid1，即

 $\begin{array}{l} V_{id}^{k + 1} = \omega V_{id}^k + {c_1}{\alpha _1}\left( {{P_{{\rm{best }}}} - X_{id}^k} \right) + \\ {c_2}{\alpha _2}\left( {{G_{{\rm{best }}}} - X_{id}^k} \right)\\ X_{id}^{k + 1} = X_{id}^k + V_{id}^{k + 1} \end{array}$ (16)

4) 若某粒子个体位置适应度优于Pbest1Pbest2，则设置新位置为Pbest1Pbest2；若粒子全局位置适应度优于Gbest1Gbest2，则设置新位置为Gbest1Gbest2

5) 输出运行结果Gbest1为幂指数γ的最优解；参数Gbest2为背景值权重λ的最优解。此时优化背景值的线性时变非等间距GM(1, 1)幂模型的MAPE最小。

2.4 模型精度验证

 $\varepsilon\left(k_{k}\right)=x^{(0)}\left(k_{k}\right)-\hat{x}^{(0)}\left(k_{k}\right), k=2, 3, \cdots, n$ (17)

 $P\left(k_{k}\right)=\frac{\varepsilon\left(k_{k}\right)}{x^{(0)}\left(k_{k}\right)} \times 100 \%, k=2, 3, \cdots, n$ (18)

MAPE为：

 ${\rm{MAPE}} = \frac{1}{{n - 1}}\sum\limits_{k = 2}^n {\left| {P\left( {{k_k}} \right)} \right|}$ (19)
3 工程应用及结果验证

3.1 宁东矿区GNSS沉降监测工程

 $\begin{array}{*{20}{c}} {{{\hat x}^{(0)}}\left( {{t_k}} \right) = \left\{ { - 13.642\;18{{\rm{e}}^{0.168691{t_k}}} + } \right.}\\ {{{\left. {0.180\;92{t_k} + 26.148\;27} \right\}}^{\frac{1}{{1.067794}}}}, k = 2, 3, \cdots , n} \end{array}$ (20)

 图 1 MAPE与非线性参数的关系 Fig. 1 Relationship between nonlinear parameters and MAPE
3.2 梁家矿铁路岩移沉降监测工程

 $\begin{array}{*{20}{c}} {{{\hat x}^{(0)}}\left( {{t_k}} \right) = \left\{ { - 2.438\;10{{\rm{e}}^{0.175198{t_k}}} + } \right.}\\ {{{\left. {0.021\;02{t_k} + 5.007\;57} \right\}}^{\frac{1}{{0.534187}}}}, k = 2, 3, \cdots , n} \end{array}$ (21)

 图 2 MAPE与非线性参数关系 Fig. 2 Relationship between nonlinear parameters and MAPE

4 结语

1) 针对沉降监测小样本非等间距原始数据序列建模问题，构建优化背景值的非等间距线性时变参数GM(1, 1)幂模型。基于粒子群算法，以最小MAPE为目标函数对非线性参数进行优化选取，可有效避免局部极小值陷阱，增强非线性参数选取的可靠性。

2) 以2组煤矿开采变形监测数据为例，对4种模型分别进行拟合与预测。计算结果表明，优化背景值的非等间距线性时变参数GM(1, 1) 幂模型在宁东矿区开采沉陷GNSS监测工程和梁家矿铁路岩移沉降监测工程的MAPE分别为2.33%、4.70%，预测误差分别为2.10%、6.38%，均优于其他3种常用的非等间距GM(1, 1)优化模型。

3) 相比于传统方法，优化背景值的非等间距线性时变参数GM(1, 1)幂模型可提升变形监测数据的拟合与预测精度，适用于短期沉降监测数据的预测分析，可拓宽GM(1, 1)幂模型在沉降监测数据处理中的应用范围。

 [1] 郑渊茂, 王翠平, 王豪伟, 等. 厦门市地面沉降影响分析与风险评价[J]. 生态学报, 2021, 41(1): 388-400 (Zheng Yuanmao, Wang Cuiping, Wang Haowei, et al. Impact Analysis and Risk Assessment of Urban Land Subsidence in Xiamen City[J]. Acta Ecologica Sinica, 2021, 41(1): 388-400) (0) [2] 邓聚龙. 灰预测与灰决策[M]. 武汉: 华中科技大学出版社, 2002 (Deng Julong. Grey Prediction and Grey Decision[M]. Wuhan: Huazhong University of Science and Technology Press, 2002) (0) [3] 魏家猛, 刁建鹏, 姜华根. 非等间距GM(1, 1)幂指数优化模型在变形监测中的应用[J]. 辽东学院学报: 自然科学版, 2020, 27(4): 253-257 (Wei Jiameng, Diao Jianpeng, Jiang Huagen. Application of Non-Equidistant GM(1, 1) Power Exponent Optimization Model in Deformation Monitoring[J]. Journal of Eastern Liaoning University: Natural Science Edition, 2020, 27(4): 253-257) (0) [4] 郭金海, 杨锦伟. GM(1, 1)模型初始条件和初始点的优化[J]. 系统工程理论与实践, 2015, 35(9): 2 333-2 338 (Guo Jinhai, Yang Jinwei. Optimizing the Initial Condition and the Initial Point of GM(1, 1)[J]. Systems Engineering-Theory and Practice, 2015, 35(9): 2 333-2 338) (0) [5] 胡攀. 优化背景值的GM(1, 1)幂模型及其应用[J]. 数学的实践与认识, 2017, 47(19): 99-104 (Hu Pan. To Optimize the Background Value of GM(1, 1) Power Model and Its Application[J]. Mathematics in Practice and Theory, 2017, 47(19): 99-104) (0) [6] 马光红, 魏勇. GM(1, 1)幂模型的幂指数计算新方法[J]. 统计与决策, 2021, 37(11): 16-20 (Ma Guanghong, Wei Yong. A New Method for Power Exponent Calculation of GM(1, 1) Power Model[J]. Statisticsand Decision, 2021, 37(11): 16-20) (0) [7] 王正新, 党耀国, 练郑伟. 无偏GM(1, 1)幂模型其及应用[J]. 中国管理科学, 2011, 19(4): 144-151 (Wang Zhengxin, Dang Yaoguo, Lian Zhengwei. Unbiased GM(1, 1) Power Model and Its Application[J]. Chinese Journal of Management Science, 2011, 19(4): 144-151) (0) [8] 王正新. 振荡型GM(1, 1)幂模型及其应用[J]. 控制与决策, 2013, 28(10): 1 459-1 464 (Wang Zhengxin. Oscillating GM(1, 1) Power Model and Its Application[J]. Control and Decision, 2013, 28(10): 1 459-1 464) (0) [9] 陈鹏宇. GM(1, 1)幂模型的改进及其在沉降预测中的应用[J]. 大地测量与地球动力学, 2020, 40(5): 464-469 (Chen Pengyu. Improvement of GM(1, 1) Power Model and Its Application on Settlement Prediction[J]. Journal of Geodesy and Geodynamics, 2020, 40(5): 464-469) (0) [10] 李希灿. 灰色系统GM(1, 1)模型适用范围拓广[J]. 系统工程理论与实践, 1999, 19(1): 97-101 (Li Xican. Widening of Suitable Limits of Grey System GM(1, 1) Model[J]. Systems Engineering-Theory and Practice, 1999, 19(1): 97-101) (0) [11] 徐华锋, 刘思峰, 方志耕. GM(1, 1)模型灰色作用量的优化[J]. 数学的实践与认识, 2010, 40(2): 26-32 (Xu Huafeng, Liu Sifeng, Fang Zhigeng. Optimum Grey Action Quantity for GM(1, 1)[J]. Mathematics in Practice and Theory, 2010, 40(2): 26-32) (0) [12] 王正新. 时变参数GM(1, 1)幂模型及其应用[J]. 控制与决策, 2014, 29(10): 1 828-1 832 (Wang Zhengxin. GM(1, 1) Power Model with Time-Varying Parameters and Its Application[J]. Control and Decision, 2014, 29(10): 1 828-1 832) (0) [13] 袁德宝, 张振超, 张军. PSO优化的分数阶PFDGM(1, 1)模型在变形监测预报中的应用[J]. 测绘科学技术学报, 2019, 36(4): 340-345 (Yuan Debao, Zhang Zhenchao, Zhang Jun. Application of Fractional PFDGM(1, 1) Model Based on PSO in Deformation Monitoring and Forecasting[J]. Journal of Geomatics Science and Technology, 2019, 36(4): 340-345) (0) [14] 熊萍萍, 党耀国, 姚天祥. 基于初始条件优化的一种非等间距GM(1, 1)建模方法[J]. 控制与决策, 2015, 30(11): 2 097-2 102 (Xiong Pingping, Dang Yaoguo, Yao Tianxiang. Modeling Method of Non-Equidistant GM(1, 1) Model Based on Optimization Initial Condition[J]. Control and Decision, 2015, 30(11): 2 097-2 102) (0) [15] 罗友洪, 陈友军. 线性时变参数非等间距GM(1, 1)幂模型及其应用[J]. 系统工程, 2021, 39(5): 152-158 (Luo Youhong, Chen Youjun. Linear Time-Varying Parameter Non-Equidistant GM(1, 1) Power Model and Its Application[J]. Systems Engineering, 2021, 39(5): 152-158) (0)
Application of Non-Equidistant Linear Time-Varying Parameter GM(1, 1) Power Model with Optimized Background Value in Deformation Monitoring
WANG Bing1     LI Peixian1     ZHANG Jun2     HAO Dengcheng1     SUN Zhiming3     ZHOU Shoubao1
1. School of Geoscience and Surveying Engineering, China University of Mining and Technology-Beijing, D11 Xueyuan Road, Beijing 100083, China;
2. College of Science, Inner Mongolia Agricultural University, 306 Zhaowuda Road, Hohhot 010018, China;
3. College of Computer and Information Engineering, Inner Mongolia Agricultural University, 29 East-Erdos Street, Hohhot 010011, China
Abstract: In order to fill in the gaps of the traditional GM(1, 1) power model with equal-weight construction for background values, a non-equidistance linear time-varying parametric GM(1, 1) power model with weighted optimization of background values is constructed for the non-equidistance spaced oscillation characteristics of the original deformation sequences. In addition, we use the particle swarm optimization(PSO) algorithm with fast convergence and high precision to solve the power exponent and background value weight. Taking the cumulative settlement observation data of monitoring points in two mining areas as examples, we use the constructed model for settlement analysis and prediction. The results show that average absolute percentage fitting errors of the model in this paper are 2.33% and 4.70% respectively, and the prediction errors are 2.10% and 6.38% respectively, which are better than other three GM(1, 1) power models. The engineering application shows that the proposed optimization model has applicability and superiority to deal the small-sample non-equidistant oscillation sequences, and that it is suitable for short-term prediction and time-varying analysis in coal mining deformation monitoring engineering.
Key words: deformation monitoring; GM(1, 1) power model; non-equidistance; background value optimization; PSO