| Longitudinal data is an important data type in the fields of finance,medicine,and biology,and it has always been a frontier and hot issue in modern statistical research.Nowadays,the quantile regression model has been widely applied to the modeling and analysis of longitudinal data.However,research on the quantile regression model of longitudinal data is still rare under high-dimensional data.In this paper,we first propose a dual regularized quantile regression model for longitudinal data based on a simple linear stochastic intercept model with random effects.That is,in the loss function of the model,both the fixed effect and the random effect are enforced with a L1 regularization penalty.Based on the principle of Bayesian inference,it is proved that when the prior distribution of the parameters to be estimated is the conditional Laplace distribution,the solution of the parameter to be estimated in the dual regularized quantile regression model can be regarded as a post-conventional number solution.In the model solving method,taking into account the limitations of the traditional quantile regression model algorithm,a combination of MM algorithm and alternate iterative algorithm is proposed.First,given the initial value of the random effect parameters,then based on the MM algorithm to find the optimal function of the loss function,and then use the Gauss-Newton iteration method to solve the optimization function parameter value of the fixed effect.Then based on the obtained fixed effect parameter estimate,the estimate of the random effect parameter is removed and iteratively proceeds until a threshold stop iteration is reached.The algorithm greatly improves the efficiency and stability of the model.In the selection of regularization parameters,based on the SIC criterion,we use the ten-fold cross validation to select the regularization parameters,which further improves the accuracy of the model prediction and proves the asymptotic property of the method.Considering that under actual circumstances,sometimes the individual effect not only affects the intercept of the model,it may also affect the slope of the model.Second,although the Lasso penalty method can select important predictor variables in the model,its estimated variable parameters are biased.Therefore,based on the theory of the double regularized quantile regression model,this paper then improves the model and extends it to the case of multiple random effects.The parameter solution of the double regularized quantile regression model is used as the new model loss.The weights of function penalty terms are presented,and a double-adaptive Lasso penalty quantile regression model for longitudinal data is proposed,and its Bayesian interpretation is given.The choice of model solving algorithm and regularization parameters is consistent with the dual regularized quantile regression model.Then,the results of numerical simulations show that the parameter estimation results obtained by the dual regularized quantile regression model and the dual adaptive Lasso penalty quantile regression model based on the MM algorithm are better than those obtained by the interior point algorithm.The MM algorithm is not affected by the initial iterative value.The larger the sample size,the better the model fitting effect.Finally,we select a set of unbalanced measurement data released by ACTG of the United States AIDS medical testing institution to establish the ordinary linear quantile regression model,the dual regularized quantile regression model and the dual adaptive Lasso penalty quantile regression model,respectively.The results of the three models were compared and analyzed.It was found that the dual-adaptive Lasso penalized quantile regression model takes into account the limitations of the double-regularized quantile regression model.The stability of the model and the prediction accuracy are all above it..By observing the results of the double penalty quantile regression model,we found that for the treatment of AIDS,age,therapy,and measurement time are all significant variables affecting the efficacy,and there are interactions between variables,so it is recommended that patients of different ages should use different Therapies,and different therapies have different measurement times. |
