## 3 Results and Discussion
### Coupling Analysis at Water Mixing and Variable Excess Air Coefficient Conditions
#### 3.3.1 Correlation Analysis  
In order to further explain the relative relationship between the small molecules of the free radicals and the NO\({}_{x}\), the correlation coefficient algorithm is introduced to illustrate this problem. Number of pairs, such as \((x_{1},y_{1})\), \((x_{2},y_{2})\),..., \((x_{n},y_{n})\) are given. It is necessary to check whether the two variables, X and Y, are related. The traditional Pearson product-moment correlation coefficient (PPMCC) is a measure method of the linear relationship between X and Y. However, there are 2 limitations when using Pearson product-moment correlation coefficient. On one side, the data must be obtained from normal distribution in pairs. On the other side, the data are at least equidistant in logical range. Whereas the non-parameter Spearman rank correlation coefficient (SRCC) is used to measure the more generalized correlation relation (not necessarily linear). Let \((x,y)\) be a sample from two different entire populations: \((X,Y)\) and its observation value is \((x_{1},y_{1})\), \((x_{2},y_{2})\),..., \((x_{n},y_{n})\). If \(x_{i}\) and \(y_{i}\) are sorted by value individually, the new rank of \(x_{i}\) and \(y_{i}\) in the two sequential samples is recorded as \(R_{x,i}\) and \(R_{y,i}\)\((0\leq i\leq n)\). So, n pairs of rank numbers, \((R_{x,1},R_{y,1})\), \((R_{x,2},R_{y,2})\),..., \((R_{x,n},R_{y,n})\) is obtained from 1 to n. Generally, the Spearman rank correlation coefficient formula can be recorded as follows [37]:  
\[r_{s}=\frac{\sum\limits_{i=1}^{n}\left(R_{x,i}-\overline{R_{x}}\right)\left(R _{y,i}-\overline{R_{y}}\right)}{\sqrt{\sum\limits_{i=1}^{n}\left(R_{x,i}- \overline{R_{x}}\right)^{2}\sum\limits_{i=1}^{n}\left(R_{y,i}-\overline{R_{y} }\right)^{2}}} \tag{8}\]  
\[\overline{R_{x}}=\frac{1}{n}{\sum\limits_{i=1}^{n}R_{x,i}} \tag{9}\]  
\[\overline{R_{y}}=\frac{1}{n}{\sum\limits_{i=1}^{n}R_{y,i}} \tag{10}\]  
If there is no same data in the X, Y sequence, then the mean value can be recorded as:  
\[\overline{R_{x}}=\overline{R_{y}}=\frac{n+1}{2} \tag{11}\]  
So the formula of \(r_{s}\) can be further simplified to:  
\[r_{s}=1-\frac{6\sum\limits_{i}^{n}(R_{x,i}-R_{y,i})^{2}}{n(n^{2}-1)} \tag{12}\]  
\(r_{s}>0\) represents positive correlation and \(r_{s}<0\) represents negative correlation. \(|r_{s}|\) is more close to 1, the correlation between the samples is higher. Whereas \(|r_{s}|\) is closer to 0, the correlation between the samples is lower. Sometimes the sample with very high correlation coefficient may also come from the uncorrelated population. In order to exclude such situation, a significance test of the correlation coefficient is needed. The Student's \(t\) test provides the excellent ability to test the significance of \(r_{s}\). The test formula of student's t is as follows [38]:  
\[t=r_{s}\sqrt{\frac{n-2}{1-r_{s}^{2}}}\sim t(n-2). \tag{13}\]  
The general step of the significance test for the correlation coefficient is to establish the null hypothesis that H0: X and Y are related, and the alternative hypothesis that H1: X and Y is not related. Then give the confidence level \(\alpha\). Based on \(\alpha\), find out the corresponding critical value. At the end make a comparison: if \(|t|\geq t^{\alpha}\), it is assumed that H0 is tenable and if \(|t|<t^{\alpha}\), it is assumed that H1 is tenable. Table 3 shows the SRCC matrix of free radicals mole fraction and maximum reaction rate of NO\({}_{x}\). Figure 8a shows that there is a close relation between the maximum production rate of NO and the mole fraction of O, OH, H\({}_{2}\) (coefficient of SRCC, OH > O > H\({}_{2}\)). Figure 8b shows that there is a close relation between the maximum production rate of N\({}_{2}\)O and the mole fraction of O, OH, HO\({}_{2}\), H\({}_{2}\) (coefficient of SRCC, O > H\({}_{2}\) > OH > HO\({}_{2}\)).