# Research on PM2.5 Concentration Prediction Based on the CE-AGA-LSTM Model
## 3 Methods  
### Correlation Analysis of PM2.5 and Influencing Factors  
CE is a mathematical concept strictly defined by [PERSON] et al. [23] in 2008 to measure the relationship of statistical independence. CE is an ideal measure of statistical independence as a correlation analysis tool for any nonlinear, non-Gaussian correlations and is a good method for causal discovery, since no assumptions need to be made. As pollutant factors affecting PM2.5 interact with each other and reflect nonlinear characteristics, the traditional analysis method suitable for linear correlations will affect the accuracy of the prediction [24]. The CE values of each factor and PM2.5 were calculated for 8 meteorological factors (air temperature, body temperature, air pressure, humidity, rainfall, wind direction, wind force, wind speed) and 5 kinds of atmospheric pollutants (PM10, SO2, NO2, CO, O3), and the relevance of the influencing factors was ranked according to the size of the correlation coefficient. The top 4 factors with greater relevance to PM2.5 were taken as the main influencing factors; CE was used to analyze the correlation between the 8 main influencing factors and PM2.5 again, and the correlation coefficients were normalized as the weights of each main influencing factor and introduced into the prediction model to realize the prediction of PM2.5 pollutant concentration. The mutual information (MI) indicator that describes the interrelation between different variables originates from the information theory and reflects the size of common information regarding different variables, i.e., the larger MI is, the stronger the correlation between the variables, and vice versa, the weaker the correlation. Assuming that there is a certain connection between the random variables \(X\) and \(Y\), the MI between them can be calculated by Equation (1)  
\[\text{MI}=\frac{1}{N}\sum_{i=1}^{N}p(x_{i},y_{i})\ln\frac{p(x_{i},y_{i})}{p(x_ {i})p\left(y_{j}\right)} \tag{1}\]  
where \(N\) is the sample size, \(x_{i}\) and \(y_{i}\) are the samples of the random variables \(X\) and \(Y\), respectively. From Equation (1), it is easy to find that when \(X\) and \(Y\) are uncorrelated, the value of MI is close to 0, while when \(X\) and \(Y\) present a functional relationship, the value of MI will be close to positive infinity. Compared with other similarity indexes, the MI index has the advantage that in addition to reflect the nonlinear correlation between variables, it does not change in size under the influence of any reversible transformation of random variables. MI has strong information mining ability, but the joint distributions of different random variables in practical studies are often skewed and non-homogeneous, so it is difficult to find the appropriate distribution type to fit it. To solve this problem, this paper introduces the Copula theory. The Copula theory is a representation of multivariate dependencies by Copula functions [25; 26]. **Definition 1 (Ce). **_Let \(X\) be the random variable of the edge distribution \(u\) and the association density\(c(u)\); the CE of \(X\) is defined as:_  
\[H_{c}(X)=-\int_{u}c(u)\log c(u)du \tag{2}\]  
In information theory, MI and entropy are two different concepts [27]. However, [PERSON] and [PERSON] proved that they are essentially the same [23], and MI is also entropy with the following relationship. **Theorem 1**. : _The MI of a random variable is equivalent to negative CE, i.e.,_  
\[I(X)=-H_{c}(X) \tag{3}\]  
**Proof of Theorem 1. **  
\[I(x)=\int\limits_{x}p(x)\log\frac{p(x)}{\prod\limits_{i}p_{i}(x_{i})}dx=\int \limits_{x}c(u_{x})\prod\limits_{i}p_{i}(x_{i})\log c(u_{x})dx \tag{4}\]  
\[=\int\limits_{x}c(u_{x})\log c(u_{x})du_{x}=-H_{c}(x)\]  
Based on Theorem 1, a simple nonparametric method is proposed which requires only two steps to estimate CE or MI from data. Step 1. Estimate the empirical Copula density function (ECD). Step 2. Estimate the CE.