﻿ 顾及邻近点的改进PSO-SVM模型在基坑沉降预测的应用研究
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 大地测量与地球动力学  2021, Vol. 41 Issue (3): 313-318  DOI: 10.14075/j.jgg.2021.03.017

### 引用本文

YUAN Zhiming, LI Peihong, LIU Xiaosheng. Study on the Application of Improved PSO-SVM Model Considering Neighbor-Point in the Settlement Prediction of Foundation Pit[J]. Journal of Geodesy and Geodynamics, 2021, 41(3): 313-318.

### Foundation support

National Natural Science Foundation of China, No.41561091; Key Project of Science and Technology Department of Jiangxi Province, No.20142BBE50024.

### Corresponding author

LI Peihong, professor, majors in precision engineering surveying and remote sensing, E-mail: 277649226@qq.com.

### 第一作者简介

YUAN Zhiming, postgraduate, majors in deformation monitoring and 3D laser scanning, E-mail: 277649226@qq.com.

### 文章历史

1. 江西理工大学土木与测绘工程学院，江西省赣州市红旗大道86号，341000

1 基本原理 1.1 SVM模型

SVM作为基坑沉降变形预测的常用模型，其原理是对给定的样本数据进行分类，通过确定决策边界来实现数据的归类处理。为了使分类精准度最高，需要通过某种衡量方式确定一条最好的决策边界(图 1)，称为最大边界[11]

 图 1 决策边界示意图 Fig. 1 Diagram of decision boundary

 ${y_i}\left( {{\mathit{\boldsymbol{\omega }}^{\rm{T}}}{x_i} + b} \right) = {\hat \gamma _i} > \hat \gamma ,i = 1, \cdots ,n$ (1)

 $\tilde \gamma = y\gamma = \frac{{\hat \gamma }}{{\left\| \mathit{\boldsymbol{\omega }} \right\|}}$ (2)

$\tilde{\gamma }$=1/‖ω‖，则目标函数可转化为：

 $\max \frac{1}{\|\boldsymbol{\omega}\|}, y_{i}\left(\boldsymbol{\omega}^{\mathrm{T}} x_{i}+b\right) \geqslant 1, i=1, \cdots, n$ (3)

 $\min \frac{1}{2}\|\boldsymbol{\omega}\|^{2}, y_{i}\left(\boldsymbol{\omega}^{\mathrm{T}} x_{i}+b\right) \geqslant 1, i=1, \cdots, n$ (4)

 $L(\boldsymbol{\omega}, b, \alpha)=\frac{1}{2}\|\boldsymbol{\omega}\|^{2}-\sum\limits_{i=1}^{n} \alpha_{i}\left(y_{i}\left(\boldsymbol{\omega}^{\mathrm{T}} x_{i}+b\right)-1\right)$ (5)

 ${\min \frac{1}{2}\sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {{\alpha _n}} } {\alpha _m}{y_i}{y_j}\left( {{x_i} \cdot {x_j}} \right) - \sum\limits_{i = 1}^N {{\alpha _i}} }$ (6)
 ${{\rm{ s}}{\rm{. t}},\sum\limits_{i = 1}^N {{\alpha _i}} {y_i} = 0,{\alpha _i} \ge 0,i = 1, \cdots ,N}$ (7)
1.2 PSO-SVM模型

D维空间中，包含n个粒子的种群X=(X1, X2, …, Xn)，第i个粒子表示D维向量空间Xi=[xi1, xi2, …, xiD]T，根据目标函数可计算得到每个粒子位置Xi对应的适应度值。第i个粒子的速度Vi=[Vi1, Vi2, …, ViD]T，个体极值Pi=[Pi1, Pi2, …, PiD]T，其种群的全局极值Pg=[Pg1, Pg2, …, PgD]T。粒子群的更新公式为：

 ${V_{id}^{k + 1} = \omega V_{id}^k + {c_1}{r_1}\left( {P_{id}^k - X_{id}^k} \right) + {c_2}{r_2}\left( {P_{gd}^k - X_{id}^k} \right)}$ (8)
 ${X_{id}^{k + 1} = X_{id}^k + V_{id}^{k + 1}}$ (9)

1.3 改进的PSO-SVM模型

 图 2 IPSO-SVM模型结构流程 Fig. 2 Flow chart of the IPSO-SVM model

1) 引入标准柯西分布密度函数对惯性权重w进行改进，可克服w为固定值的缺点。柯西分布密度函数[13]能够在算法前期取较大的w值，以提高算法的全局搜索能力；在后期取较小的w值，以提高算法的局部搜索能力：

 $w = \gamma p(x) = \frac{1}{{{\rm{ \mathsf{ π} }}\left( {1 + {x^2}} \right)}}$ (10)

2) 针对基坑沉降监测的样本数据具有随机性和非平稳性的特点[14]，本文采用小波分解函数对样本数据进行粗差检验和去噪处理。小波分解函数的阈值取[C, L]=wavedec(y, 3, ‘sym4’)，其中C为分解结构变量，L为样本数据长度变量，y为样本数据，分解层数为3次，小波类型为sym4。

3) 随机初始化粒子位置和粒子速度，并根据适应度函数计算粒子适应度值。设置PSO算法的运行参数，包括迭代次数和种群规模以及个体和速度的最大最小值。

4) 迭代寻优。首先进行粒子位置和速度更新，然后根据新粒子的适应度值进行个体极值和群体极值更新。

5) 最优解计算。yi(ωTxi+b)≥1, i=1, …, n为约束条件，目标函数为：

 $\min \frac{1}{2}{\left\| \mathit{\boldsymbol{\omega }} \right\|^2},{y_i}\left( {{\mathit{\boldsymbol{\omega }}^{\rm{T}}}{x_i} + b} \right) \ge 1,i = 1, \cdots ,n$ (11)

 $\begin{array}{*{20}{l}} {\min \frac{1}{2}\sum\limits_{n = 1}^N {\sum\limits_{m = 1}^N {{\alpha _n}} } {\alpha _m}{y_i}{y_j}\left( {{x_i} \cdot {x_j}} \right) - \sum\limits_{i = 1}^N {{\alpha _i}} }\\ {{\rm{ s}}{\rm{.t}}{\rm{. }}\sum\limits_{i = 1}^N {{\alpha _i}} {y_i} = 0,{\alpha _i} \ge 0,i = 1, \cdots ,N} \end{array}$ (12)
2 实例应用研究

2.1 最优训练样本数量研究

2.2 顾及邻近点的PSO-SVM模型

 图 3 3种样本数量下的最佳适应度曲线 Fig. 3 Best fitness curves under three kinds of sample size

2.3 顾及邻近点的改进PSO-SVM模型

 图 4 小波分解效果 Fig. 4 Wavelet decomposition effect

3 结语

1) 针对传统模型未对最优训练样本数据量进行研究的缺点，选取PSO-SVM模型在4种不同样本数量情况下的结果进行分析。实验结果表明，在20期数据条件下，PSO-SVM模型的拟合性能最优。

2) 针对传统PSO-SVM模型基于单点数据建模的缺点，引入时间、历史数据和邻近点沉降变形值改进PSO-SVM模型。结果表明，在最佳训练样本数据下，顾及邻近点的PSO-SVM模型优于单因素的PSO-SVM模型，且在短期样本数据下拟合效果最佳，但不适用于中长期样本数据下的拟合预测。

3) 针对顾及邻近点的PSO-SVM模型在中长期样本数据下预测精度不佳的缺点，本文提出组合多尺度一维小波分解函数和柯西分布函数改进顾及邻近点的PSO-SVM模型。研究结果表明，在不同样本期数下，顾及邻近点的IPSO-SVM模型预测结果的均方根误差平均减小85.1%，平均相对误差减小84.7%，表明其适应性强、拟合精度高。

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Study on the Application of Improved PSO-SVM Model Considering Neighbor-Point in the Settlement Prediction of Foundation Pit
YUAN Zhiming1     LI Peihong1     LIU Xiaosheng1
1. School of Civil and Surveying and Mapping Engineering, Jiangxi University of Science and Technology, 86 Hongqi Road, Ganzhou 341000, China
Abstract: In view of the disadvantages of the SVM model, such as difficulty in parameter selection and single-point data modeling in the field of settlement prediction of foundation pit, we establish a neighbor-point PSO-SVM model. Selecting the PSO-SVM model for the optimal training sample quantity research, the results show that short-term samples have the best prediction effect. We introduce the settlement deformation value of neighbor points as a factor affecting the settlement of foundation pit into the improved PSO-SVM model. The example shows that the fitting accuracy of the PSO-SVM model considering the neighbor points under short-term sample data is better than that of the PSO-SVM model. The prediction accuracy is poor under medium and long-term sample conditions. Aiming at this shortcoming, we propose a combination of multi-scale one-dimensional wavelet decomposition function and Cauchy distribution function to improve the PSO-SVM model that takes into account the neighbor points. The experimental results show that the improved PSO-SVM model effectively solves the difficulty in parameter selection and single-point data modeling. The model is suitable for the prediction of settlement deformation under different sample sizes, and has high prediction accuracy.
Key words: support vector machine; wavelet analysis; particle swarm optimization algorithm; IPSO-SVM; foundation pit deformation monitoring