﻿ 基于EMD的北京与天津PM2.5相关结构探究

# 基于EMD的北京与天津PM2.5相关结构探究Study on Correlation Structures between Beijing and Tianjin PM2.5 Based on EMD

Abstract: In order to explore the correlation between air quality in Beijing and Tianjin, the most representative air quality indexes in Beijing and Tianjin were found to be PM2.5 by calculating the correlation coefficient matrix. Empirical mode decomposition (EMD) was used to decompose the PM2.5 time series in Beijing and Tianjin, and modal reconstruction was carried out based on the period. The results show that PM2.5 in Beijing and PM2.5 in Tianjin are highly correlated in trend terms and strongly correlated in high-frequency parts. In the low-frequency part of PM2.5, Beijing lags behind Tianjin, and the two sequences reach the maximum positive correlation when lagging 1 day.

1. 引言

2. 数据与方法

2.1. 数据来源

2.2. 研究方法

2.2.1. EMD分解

1) 对于给定的时间序列 $x\left(t\right)$ 找到其局部极大值 ${e}_{\mathrm{max}}\left(t\right)$ 和局部极小值 ${e}_{\mathrm{min}}\left(t\right)$ ，构造上包络和下包络，分别用三次样条局部极大值和局部极小值算法；

2) 估计两个包络的平均值 ${M}_{1}\left(t\right)=\frac{{e}_{\mathrm{max}}\left(t\right)+{e}_{\mathrm{min}}\left(t\right)}{2}$

3) 令 ${h}_{1}\left(t\right)=x\left(t\right)-{M}_{1}\left(t\right)$ ，若 ${h}_{1}\left(t\right)$ 满足上述两个条件，则为第一个IMF (固有模态函数)，如果 ${h}_{1}\left(t\right)$ 不满足上述条件，将其作为新的时间序列重复以上步骤直到第k次，即 ${h}_{1k}\left(t\right)$ 为固有模态函数，则：

4) 一直重复前面步骤，直到 $r\left(t\right)$ 为单调函数或者常数，EMD分解结束.这样，原始信号 $X\left(t\right)$ 被分解成 $n-1$ 个固有模态函数和趋势项： $X\left(t\right)={\sum }_{m=1}^{n-1}{C}_{m}\left(t\right)+r\left(t\right)$

2.2.2. 互相关函数

3. 分析与讨论

(a) (b)

Figure 1. EMD diagram of Beijing PM2.5 and Tianjin PM2.5 time series. (a) EMD results of PM2.5 time series in Beijing; (b) EMD results of PM2.5 time series in Tianjin

Table 1. IMF cycles of PM2.5 in Beijing

Table 2. IMF cycles of PM2.5 in Tianjin

Table 3. IMF correlation and correlation test between Beijing and Tianjin PM2.5

Figure 2. Comparison of IMF cycles of PM2.5 in Beijing and Tianjin

Figure 3. Comparison of PM2.5 high-frequency parts between Beijing and Tianjin

Figure 4. Comparison of PM2.5 low-frequency parts between Beijing and Tianjin

Table 4. Cross correlation results of reconstructed PM2.5 parts in Beijing and Tianjin

4. 结论与讨论

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