﻿ 基于小波分解与GA-LSSVM的GPS可降水量短临预报
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 大地测量与地球动力学  2019, Vol. 39 Issue (5): 487-491  DOI: 10.14075/j.jgg.2019.05.009

### 引用本文

XIE Shaofeng, SU Yongning, LIU Chunli, et al. Short-Impending Prediction of GPS Precipitable Water Vapor Based on Wavelet Decomposition and GA-LSSVM[J]. Journal of Geodesy and Geodynamics, 2019, 39(5): 487-491.

### Foundation support

National Natural Science Foundation of China, No. 41864002, 41704027; Guangxi Natural Science Foundation, No.2018GXNSFAA281182, 2017GXNSFBA198139; Fund of Guangxi Key Laboratory of Spatial Information and Geomatics, No. 15-140-07-11.

### 第一作者简介

XIE Shaofeng, professor, majors in GNSS meteorology, E-mail:xieshaofeng111@126.com.

### 文章历史

1. 桂林理工大学测绘地理信息学院，桂林市雁山街319号，541006;
2. 广西空间信息与测绘重点实验室，桂林市雁山街319号，541006

1 算法原理 1.1 小波分解

 \left.\begin{aligned} c_{j+1} &=H c_{j} \\ d_{j+1} &=G d_{j} \end{aligned}\right\} (1)

 $C_{j}=H^{*} C_{j+1}+G^{*} D_{j+1}$ (2)

cJd1d2、…、dJ分别进行重构，重构后的信号为CJD1D2、…、DJ，则：

 $X=C_{J}+D_{1}+D_{2}+\cdots+D_{J}$ (3)

1.2 最小二乘支持向量机

 $f(\boldsymbol{x})=\boldsymbol{\omega}^{\mathrm{T}} \varphi(\boldsymbol{x})+b$ (4)

 $\min\limits_{\omega, b, e} J(\boldsymbol{\omega}, e)=\boldsymbol{\omega}^{\mathrm{T}} \boldsymbol{\omega}+\frac{1}{2} \gamma \sum\limits_{k=1}^{N} e_{k}^{2}$ (5)
 $y_{k}=\boldsymbol{\omega}^{\mathrm{T}} \varphi\left(x_{k}\right)+b+e_{k}, k=1, \cdots, N$ (6)

 $\begin{array}{c}{L(\boldsymbol{\omega}, b, e, \boldsymbol{a})=J(\boldsymbol{\omega}, e)-} \\ {\sum\limits_{k=1}^{N} a_{k}\left\{\boldsymbol{\omega}^{\mathrm{T}} \varphi\left(x_{k}\right)+b+e_{k}-y_{k}\right\}}\end{array}$ (7)

 $\left[ \begin{array}{cc}{0} & {\boldsymbol{E}_{v}^{\mathrm{T}}} \\ {\boldsymbol{E}_{v}^{\mathrm{T}}} & {\mathit{\boldsymbol{ \boldsymbol{\varOmega}}}+\frac{1}{\gamma} \boldsymbol{I}}\end{array}\right] \left[ \begin{array}{l}{b} \\ {\boldsymbol{a}}\end{array}\right]=\left[ \begin{array}{l}{0} \\ {\boldsymbol{y}}\end{array}\right]$ (8)

 $y(x)=\sum\limits_{k=1}^{N} a_{k} k\left(x, x_{k}\right)+b$ (9)

 $k\left(x_{k}, x_{l}\right)=\exp \left(\frac{-\left\|x_{k}-x_{l}\right\|}{2 \sigma^{2}}\right)$ (10)

σγ对LSSVM的泛化性能至关重要。实际应用中通常采用试凑法或凭经验确定，不但耗时耗力，而且效果还可能不好。所以，本文利用遗传算法的全局寻优能力来确定σγ

1.3 遗传算法

1) 输入训练数据。

2) 种群初始化。个体编码方法采用实数编码，对LSSVM的σ2γ进行编码。

3) 计算个体的适应度值。计算适应度目标函数，目的是寻找一组参数(γ, σ2)，通过LSSVM训练使得式(11)目标函数全局最小，也就是适应度值最大。然后判断是否满足精度要求，若满足则执行步骤7)，否则执行步骤4)。

 $\left\{\begin{array}{l}{\min f\left(\gamma, \sigma^{2}\right)=\frac{1}{2 n} \sum\limits_{i=1}^{n}\left(y_{i}-\hat{y}_{i}\right)^{2}} \\ {\text { s.t }~~ \gamma \in\left[\gamma_{\min }, \gamma_{\max }\right], \quad \sigma^{2} \in\left[\sigma_{\min }^{2}, \sigma_{\max }^{2}\right]}\end{array}\right.$ (11)

4) 选择。筛选适应度优良的个体遗传给下一代，以提高全局收敛性和计算速率。

5) 交叉。随机选择第k个染色体αk和第l个染色体αlj位的基因进行交换，产生新的优秀个体：

 \left.\begin{aligned} \alpha_{k j} &=\alpha_{k j}(1-\beta)+\alpha_{l j} \beta \\ \alpha_{l j} &=\alpha_{l j}(1-\beta)+\alpha_{k j} \beta \end{aligned}\right\} (12)

6) 变异。从种群中任意选取第i个个体的第j个染色体变异为更加优秀的个体：

 $\alpha_{i j}=\left\{\begin{array}{l}{\alpha_{i j}+\left(\alpha_{i j}-\alpha_{\max }\right) f(g), r \geqslant 0.5} \\ {\alpha_{i j}+\left(\alpha_{\min }-\alpha_{i j}\right) f(g), r<0.5}\end{array}\right.$ (13)

7) 终止循环，得到最佳染色体。将遗传算法优化得到的最优个体分解为σ2γ，用于预报。

2 基于WD-GA-LSSVM的可降水量短临预报

1) 把GPS可降水量的原始时间序列x(t)通过小波分解得到1个近似分量cji个细节分量d1d2、…、di

2) 对分解后的各分量分别进行LSSVM建模，并利用遗传算法优化其参数，得到各分量的预报值。本文设置遗传算法的参数为：迭代次数100，种群规模20，交叉概率0.9，变异概率0.2。

3) 对各分量的预报值进行叠加重构，得到可降水量的最终预报结果。

3 实验分析

 图 1 年积日186 d时间序列及其小波分解结果 Fig. 1 The time series of doy 186 and its decomposition results

 图 2 年积日186 d各模型预报值和实际值对比 Fig. 2 Comparison between the predicted values of each model and the actual value for doy 186

 图 3 年积日186 d各模型残差 Fig. 3 Residual of each model for doy 186

 图 4 年积日325 d时间序列及其小波分解结果 Fig. 4 The time series of doy 325 and its decomposition results

 图 5 年积日325 d各模型预报值和实际值对比 Fig. 5 Comparison between the predicted values of each model and the actual value for doy 325

 图 6 年积日325 d各模型残差 Fig. 6 Residual of each model for doy 325

4 结语

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Short-Impending Prediction of GPS Precipitable Water Vapor Based on Wavelet Decomposition and GA-LSSVM
XIE Shaofeng1,2     SU Yongning1     LIU Chunli1     LIU Lilong1,2
1. College of Geomatics and Geoinformation, Guilin University of Technology, 319 Yanshan Street, Guilin 541006, China;
2. Guangxi Key Laboratory of Spatial Information and Geomatics, 319 Yanshan Street, Guilin 541006, China
Abstract: Aiming at the random and nonlinear characteristic of the time series of GPS precipitable water vapor(PWV), this paper proposes a new short-impending prediction method of GPS PWV based on wavelet decomposition(WD), genetic algorithm(GA) and least squares support vector machine(LSSVM). First, WD is used to decompose the GPS PWV time series into low frequency and high frequency components, which are easy to predict. Second, GA is used to optimize the parameters of LSSVM, and the prediction model of each component is established. Finally, the results of each component prediction are superimposed and reconstructed to get the final prediction results. In this paper, two groups of data are selected for experiments, and the prediction results are compared with those of LSSVM and genetic wavelet neural network(GA-WNN). The results show that the combined model has good generalization ability, can effectively solve the problem that neural network tends to trap in local minimum, and improves global prediction accuracy.
Key words: GPS precipitable water vapor; wavelet decomposition; genetic algorithm; least squares support vector machine; short-impending prediction