# Weight Assignment Method and Application of Key Parameters in Shale Gas Resource Evaluation
## 4 Weight Conversion Model
The specific method for assigning weights to key parameters is as follows:  
Firstly, calculate the complex correlation coefficient and partial correlation coefficient. Perform a regression analysis on \(X_{1}\), \(X_{2}\), \(\ldots\), \(X_{k}\) using \(\hat{y}\):  
\[\hat{y}=\widehat{\beta_{0}}+\widehat{\beta_{1}}X_{1}+\widehat{\beta_{2}}X_{2} +\cdots+\widehat{\beta_{k}}X_{k}+\varepsilon_{i} \tag{1}\]  
In the formula, \(\widehat{\beta_{0}}\) is a constant term, and \(\widehat{\beta_{1}}\), \(\widehat{\beta_{2}}\), \(\ldots\), \(\widehat{\beta_{m}}\) is called the partial regression coefficient. \(\varepsilon_{i}\) is a random error, also known as the residual, which represents the part of the variation in Y that cannot be explained by the independent variable \(X_{1}\) (i = 1, 2, \(\ldots\), m). Next, use the least squares method to estimate the vector parameter \(\widehat{B}\),  
\[X= \begin{bmatrix}1&x_{11}&x_{12}&\ldots&x_{1p}\\ 1&x_{21}&x_{22}&\ldots&x_{2p}\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ 1&x_{n1}&x_{n2}&\ldots&x_{np}\end{bmatrix} \]  
\[y= \begin{bmatrix}y_{1}\\ y_{2}\\ \ldots\\ y_{n}\end{bmatrix} \]  
\[\widehat{B}= \begin{bmatrix}\beta_{0}\\ \beta_{1}\\ \ldots\\ \beta_{p}\end{bmatrix} \]  
Organize and obtain the estimated vector parameter \(\widehat{B}\), where \(\widehat{B}\) is the partial correlation coefficient matrix:  
\[\widehat{B}=\left(X^{\prime}X\right)^{-1}X^{\prime}Y \tag{2}\]  
Then, calculate the simple correlation coefficient between Y and \(\hat{y}\), which is the complex correlation coefficient between Y and \(X_{1}\), \(X_{2}\), \(\ldots\), \(X_{k}\), where R is the complex correlation coefficient. \[R=\frac{\sum(y-\overline{y})(\hat{y}-\overline{y})}{\sqrt{\sum(y-\overline{y })^{2}\cdot\sum(y-\overline{y})^{2}}} \tag{3}\]  
Based on the \(\beta\) matrix in the calculation, the complex correlation coefficients of the key parameters can be obtained. After obtaining the complex correlation coefficients of the variables, significance testing should be conducted to check whether they conform to the correlation. The purpose of significance testing here is to eliminate interference and determine their true correlation rather than low probability events with accidental correlation and to eliminate human errors. The F-test is generally used, and the specific methods are as follows:  
Make an alternative hypothesis: the complex correlation coefficient between the dependent variable y and the independent variables X\({}_{1}\), X\({}_{2}\), \(\ldots\), X\({}_{k}\) is 0, indicating complete independence. where M is the weight conversion coefficient. The weight of a certain parameter in all key parameters is determined based on its partial correlation coefficient, which is the proportion of the degree of correlation to the partial correlation coefficients of all parameters, and the M value between 0 and 1 is calculated. The closer the value of a certain parameter, M, is to 1, the greater its weight in the resource estimation method. After assigning weights to each parameter, a resource estimation evaluation can be conducted again to obtain a resource with high reliability.