## 2 Quantile Regression  
Based on the conditional quantile regression of the explanatory variable \(X\), quantile regression obtains a regression model for all quantiles. Therefore, the use of quantile regression can more accurately reflect the impact of input variables on the explanatory variable. Through the analysis of these regression models, we can obtain different results of input variables on the explanatory variables in various ranges, providing more information about the effects of input variables on the position, distribution and shape of the explanatory variables.  
Quantile regression first defines the loss function; then, it estimates the optimal parameters to minimize the loss function and optimizes these parameters. The estimated parameters are constantly changing with different quantiles, allowing for a continuous reflection of the input variables on the response variables, providing a more comprehensive explanation and influence. The loss function is defined as:  
\[\rho_{\theta}(u)=u[\theta-I(u)] \tag{1}\]  
In Formula (1), \(I(u)=\begin{cases}0,&u,0\\ 1,&u<0\end{cases}\) is an indicator function. The nature of this function is a piecewise linear convex function.  
Quantile regression essentially assigns different weights to the absolute value of residuals through indicative functions, allowing for the acquisition of different parameter estimates by adjusting different quantiles. Through the analysis of these various parameter estimates, the influence of input variables on response variables under different quantiles can be obtained, providing more useful information for research. The quantile regression proposed by [PERSON] is still based on linear regression to analyze the relationship between input variables and response variables [20]. That is to say, if the response variable is \(Y\), the input variables are a series of factors \(X_{1}\), \(\ldots\), \(X_{n}\); a linear quantile function model is established firstly. This linear quantile function model is as follows:  
\[Q_{Y}(\theta|X)=\beta_{0}(\theta)+\beta_{1}(\theta)X_{1}+\beta_{2}(\theta)X_{2 }+\cdots+\beta_{n}(\theta)X_{n}\equiv X^{\prime}\beta(\theta) \tag{2}\]  
In Formula (2), \(X\) is a matrix consisting of explanatory variables; \(\theta\) is quantile, and \(\theta\in\) (0,1); and \(\beta(\theta)\) is a parameter matrix. Quantile regression uses a simplex method, interior point algorithm and other optimization methods to optimize Formula (3) and estimate the parameters.  
\[\min_{\beta}\sum_{i=1}^{n}\rho_{\theta}(Y_{i}-X_{i}^{T}\beta)=\min_{\beta} \sum_{i\in\Phi_{1}}\theta\Big{|}Y_{i}-X_{i}^{T}\beta\Big{|}+\min_{\beta}\sum_ {i\in\Phi_{2}}\left(1-\theta\right)\Big{|}Y_{i}-X_{i}^{T}\beta\Big{|} \tag{3}\]  
In Formula (3), \(X_{i}^{T}\) is the transposition of the \(i\)-th component of the input variable \(X\), \(Y_{i}\) is the \(i\)-th component of the response variable \(Y\), \(\Phi_{1}=\left\{i|i\in[1,n],Y_{i}-X_{i}^{T}\beta\geq 0\right\}\), \(\Phi_{2}=\left\{i|i\in[1,n],Y_{i}-X_{i}^{T}\beta<0\right\}\), where \(n\) represents the number of components of the input variable \(X\) or the response variable \(Y\).  
There are several advantages of quantile regression. Firstly, it does not assume any distribution of the random perturbation residuals in the model, which significantly differs from least squares regression, providing the entire regression model with good robustness. Secondly, as it encompasses regression for all quantiles, quantile regression has the resilience to handle non-stationary data, such as abnormal or singular points, in the dataset, thereby better reflecting the explanatory variables. Finally, the parameters estimated by quantile regression exhibit asymptotic superiority under large sample theory.