﻿ 大坝变形的奇异谱分析预测
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 大地测量与地球动力学  2019, Vol. 39 Issue (10): 1081-1085  DOI: 10.14075/j.jgg.2019.10.018

### 引用本文

ZHANG Donghua, LI Zhijuan, LIU Quanming, et al. Singular Spectrum Analysis for Analyzing and Forecasting the Dam Deformation[J]. Journal of Geodesy and Geodynamics, 2019, 39(10): 1081-1085.

### Foundation support

National Natural Science Foundation of China, No.51569018, 51509131.

### Corresponding author

LIU Quanming, PhD, associate professor, majors in geodesy data processing and remote sensing inversion, E-mail:61654346@qq.com.

### 第一作者简介

ZHANG Donghua, lecturer, majors in geodesy data processing and GIS algorithm and program development, E-mail:zdh@imau.edu.cn.

### 文章历史

1. 内蒙古农业大学水利与土木建筑工程学院，呼和浩特市昭乌达路306号，010018;
2. 内蒙古自治区航空遥感测绘院，呼和浩特市兴安南路42号，010010

1 SSA预测方法 1.1 SSA原理

SSA是根据所观测到的时间序列构造出轨迹矩阵，对矩阵进行分解、重构，提取代表原时间序列不同成分的信号，通过分析时间序列的结构进行预测[3]。首先，对于中心化后的时间序列xi(1≤iN)，根据窗口L按照式(1)构造L×(NL+1)的轨迹矩阵X

 $\mathit{\boldsymbol{X}} = \left[ {\begin{array}{*{20}{c}} {{x_1}}&{{x_2}}& \cdots &{{x_{i + 1}}}& \cdots &{{x_{N - L + 1}}}\\ {{x_2}}&{{x_3}}& \cdots &{{x_{i + 2}}}& \cdots &{{x_{N - L + 2}}}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ {{x_L}}&{{x_{L + 1}}}& \cdots &{{x_{i + 1}}}& \cdots &{{x_N}} \end{array}} \right]$ (1)

 $\mathit{\boldsymbol{C}} = \mathit{\boldsymbol{V \boldsymbol{\varLambda} }}{\mathit{\boldsymbol{V}}^{\rm{T}}}$ (2)

 $\mathit{\boldsymbol{A}}{\rm{ = }}\mathit{\boldsymbol{VX}}$ (3)

A的第k行(ak)即为第k个主成分，其中每个数值ak, i为：

 ${a_{k, i}} = \sum\limits_{j = 1}^L {{x_{i + j - 1}}} {v_{k, j}}, 1 \le i \le N - L + 1$ (4)

 $x_{i}^{k}=\\\left\{ {\begin{array}{*{20}{l}} {\frac{1}{i}\sum\limits_{j = 1}^i {{a_{k, i - j + 1}}} {v_{k, j}}, 1 \le i \le L - 1}\\ {\frac{1}{L}\sum\limits_{j = 1}^L {{a_{k \cdot - j + 1}}} {v_{k, j}}, L \le i \le N - L + 1}\\ {\frac{1}{{N - i + 1}}\sum\limits_{j - i - N + L}^L {{a_{k, i - j + 1}}} {v_{k, j}}, \quad N - L + 2 \le i \le N} \end{array}} \right.$ (5)

SSA的特征值是按降序排列的，因此时间序列可由前几个主成分分量重构，其余则视为噪声，即信号${\hat x}$可以由前q个主成分得到：

 $\hat x = \sum\limits_{k = 1}^q {x_i^k} , i = 1, 2, \cdots , N$ (6)
1.2 SSA预测原理

SSA迭代预测法建立在重构序列基础上，具体步骤如下。

 $\boldsymbol{\mathit{R}}=\frac{1}{1-{{v}^{2}}}\sum\limits_{i\in I}{{{\pi }_{i}}}\underline{{{\boldsymbol{\mathit{P}}}_{i}}}$ (7)

 ${{\boldsymbol{\mathit{Z}}}_{i}}=\left\{ \begin{array}{*{35}{l}} {{{\hat{X}}}_{i}}, i=1, \cdots , K \\ {{p}_{vex}}{{\boldsymbol{\mathit{Z}}}_{i-1}}, i=K+1, \cdots , K+M \\ \end{array} \right.$ (8)

 ${{y}_{i}}=\left\{ \begin{array}{*{35}{l}} {{{\tilde{x}}}_{i}}, i=1, \cdots , N \\ \sum\limits_{j=1}^{L-1}{{{a}_{j}}}{{y}_{i-j}}, i=N+1, \cdots , N+M \\ \end{array} \right.$ (9)

2 大坝变形成因分析

 图 1 大坝变形数据序列 Fig. 1 The data series of dam deformation

 图 2 W-correlations中前30个分量图 Fig. 2 The first 30 PCs of W-correlations

 图 3 大坝变形奇异谱分析周期趋势项 Fig. 3 The trend and periodic terms of dam deformation

 图 4 变形量、水位和温度变化周年分量对比 Fig. 4 Comparison of annual components of deformation, water level and temperature

 图 5 变形量、水位和温度变化半年分量对比 Fig. 5 Comparison of semi-year components of deformation, water level and temperature
3 大坝变形预测

 图 6 预测与残差绝对值对比 Fig. 6 The comparison of predictions and absolute errors

 $\text{RMSE}=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}{{{\left( {{s}_{i}}-{{{\hat{s}}}_{i}} \right)}^{2}}}}$ (10)
 $\text{MAE}=\frac{1}{N}\sum\limits_{i=1}^{N}{\left| {{s}_{i}}-{{{\hat{s}}}_{i}} \right|}$ (11)

4 结语

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Singular Spectrum Analysis for Analyzing and Forecasting the Dam Deformation
ZHANG Donghua1     LI Zhijuan2     LIU Quanming1     LUO Yanyun1
1. Water Conservancy and Civil Engineering College, Inner Mongolia Agricultural University, 306 Zhaowuda Road, Hohhot 010018, China;
2. Institute of Aerial Remote Sensing Surveying and Mapping of Inner Mongolia Autonomous Region, 42 South-Xing'an Road, Hohhot 010010, China
Abstract: We use singular spectrum analysis to extract the corresponding components and compute the correlation coefficients, under the influence of factors. The results show that the trend term of dam deformation mainly relates to water level and aging factor. For seasonal terms, the temperature factor contributes more than water level. Experimental results of dam deformation show that SSA can extract the trends and periodic signal effectively and is useful to forecast dam deformation. Then, recurrent forecasting method of SSA is used for the prediction of dam deformation. Compared with Gaussian process and multiple-regression analysis, the results show that SSA is an effective method with a higher predictive accuracy for analyzing and forecasting dam deformation.
Key words: singular spectrum analysis(SSA); dam deformation; deformation prediction