# 基于矩阵优化填充和结构性先验统计信息的气象数据恢复Meteorological Data Restoration Based on Matrix Completion and Prior Features

DOI: 10.12677/SA.2018.72024, PDF, HTML, XML, 下载: 924  浏览: 1,224  国家自然科学基金支持

Abstract: Because of the limitation of observation means and background, combined with the complex environment, only some observation data are available. For the sake of better weather forecast, the research of meteorological data restoration based on part of observation data and matrix completion would have important scientific significance. This paper aims to, through part of real-time observation data, according to the low rank of a matrix, with applying SVT (Singular Value Thresholding) algorithm of matrix completion, obtain the deficient data so that one can make weather forecast better. The experimental result shows that the accuracy of forecast with matrix completion method is obviously higher than that with classical statistical method. When available data proportion is higher than the critical sampling proportion, errors of data filling can be controlled within 10%, which meet the requirements of meteorological data.

1. 引言

2. 矩阵填充

$\mathrm{min}\text{ }{‖X‖}_{*},\text{s}.\text{t}.\text{}{X}_{i,j}={M}_{i,j},\text{}\left(i,j\right)\in \Omega ,$ (1)

$\mathrm{min}\text{ }rank\left(X\right),\text{s}.\text{t}.\text{}{X}_{i,j}={M}_{i,j},\text{}\left(i,j\right)\in \Omega .$ (2)

$\mathrm{min}\text{ }rank\left(X\right),\text{s}.\text{t}.\text{}{P}_{\Omega }\left({X}_{i,j}\right)={P}_{\Omega }\left({M}_{i,j}\right),\text{}\left(i,j\right)\in \Omega .$ (3)

$\left\{\begin{array}{l}{X}^{k}=shrink\left({Y}^{k-1},\tau \right)\\ {Y}^{k}={Y}^{k-1}+{\delta }_{k}{P}_{\Omega }\left(M-{X}^{k}\right)\end{array}$(4)

3. 气象数据的先验特征

Table 1. Historical Weather Data of Dalian (Part)

3.1. 数据的规律统计特性

Figure 1. The historical statistical weather class data of Dalian

Figure 2. The historical statistical wind direction data of Dalian

Figure 3. The historical statistical wind power data of Dalian

Figure 4. The historical statistical highest temperature data of Dalian

Figure 5. The historical statistical lowest temperature data of Dalian

Figure 6. The historical statistical temperature difference data of Dalian

3.2. 数据的分类

Table 2. Historical weather data of Dalan (Part)

1) 天气类型分类

2) 风向分类

3) 风力分类

4)气温温差分类

3.3. 数据的结构性等级划分

Table 3. Historical weather data after classification (Part)

1) 按月处理(表11)

1.1) 竖排结构

1.2) 横排结构

2) 按季处理(表12)

2.1) 竖排结构

2.2) 横排结构

Table 4. Digitization of highest temperature

Table 5. Digitization of Lowest Temperature

Table 6. Digitization of Temperature Difference

Table 7. Digitization of Weather Class

Table 8. Digitization of Wind Direction

Table 9. Digitization of Wind Power

4. 基于部分数据和先验特征的气象数据填充

Table 10. Digitization of historical weather data of Dalian (Part)

Table 11. Processing the data according to months

Table 12. Processing the data according to seasons

5. 按月横排矩阵格式最优化分析

Table 13. The error rate under different sampling rate

Table 14. The error rate according to months

Table 15. The error rate according to seasons

Table 16. The threshold sampling rate

Figure 7. The error rate under different sampling rate

$\xi \propto \frac{M}{m}\cdot \frac{r}{n}$(5)

Figure 8. The error rate under different sampling rate (refined)

${X}_{Tp×q}=\left[\begin{array}{l}{x}_{1,1}\text{\hspace{0.17em}}{x}_{1,2}\text{\hspace{0.17em}}\cdots \text{\hspace{0.17em}}\text{ }{x}_{1,q}\\ {x}_{2,1}\text{\hspace{0.17em}}{x}_{2,2}\text{\hspace{0.17em}}\cdots \text{\hspace{0.17em}}\text{ }{x}_{2,q}\\ ⋮\\ {x}_{p,1}\text{\hspace{0.17em}}{x}_{p,2}\text{\hspace{0.17em}}\cdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{p,q}\\ ⋮\\ {x}_{Tp,1}\text{\hspace{0.17em}}{x}_{Tp,2}\text{\hspace{0.17em}}\cdots \text{\hspace{0.17em}}{x}_{Tp,q}\end{array}\right]$ , (6)

${X}_{mon,}{}_{Tp×q}={X}_{Tp×q}$ , (7)

${X}_{mon,}{}_{p×Tq}=\left[\begin{array}{l}{x}_{1,1}\text{\hspace{0.17em}}{x}_{1,2}\text{\hspace{0.17em}}\cdots \text{\hspace{0.17em}}\text{ }{x}_{1,q}\text{\hspace{0.17em}}{x}_{p+1,1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdots \text{\hspace{0.17em}}\text{ }{x}_{\left(T-1\right)p+1,q}\\ {x}_{2,1}\text{\hspace{0.17em}}{x}_{2,2}\text{\hspace{0.17em}}\cdots \text{\hspace{0.17em}}\text{ }{x}_{2,q}\text{\hspace{0.17em}}{x}_{p+2,1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdots \text{\hspace{0.17em}}\text{ }{x}_{\left(T-1\right)p+2,q}\\ ⋮\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{\hspace{0.17em}}\text{ }⋮\\ {x}_{p,1}\text{\hspace{0.17em}}{x}_{p,2}\text{\hspace{0.17em}}\cdots \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{p,q}\text{\hspace{0.17em}}{x}_{2p,1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdots \text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{Tp,q}\end{array}\right]$ , (8)

${X}_{se,}{}_{Tp×q}={X}_{Tp×q}$ , (9)

${X}_{se,}{}_{p×Tq}={X}_{mon,}{}_{p×Tq}$ . (10)

${X}_{mon,}{}_{Tp×q}={U}_{mon,Tp×Tp}\left[\begin{array}{l}{\sigma }_{1}\\ \text{ }\ddots \\ \text{ }\text{ }\text{ }{\sigma }_{r1}\end{array}\right]{{V}^{\prime }}_{mon,q×q}^{}$ . (11)

6. 仿真试验

Table 17. The comparison of restored measures (Average)

Figure 9. The error rate under different sampling rate

Figure 10. The error rate under different sampling rate (refined)

7. 结论

NOTES

*通讯作者。

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