Parameter Estimation And Outlier Detection Based On Nonconvex Penalized Regressio

The presence of outliers in the data is a common problem in statistical analysis.Outliers are the data that are different from most of the entire data,usually referring to the data with two times standard deviation from the mean.Data samples with outliers can cause a large bias in the parameter estimation results.Therefore,estimating parameters from outlier data is a problem of practical significance.Robust estimation and outlier detection are critical for statistical learning.In order to obtain robust parameter estimation,this paper studies the problem of outlier detec-tion and parameter estimation based on non-convex penalized regression methods.For the outlier detection problem,a popular method is to introduce an outlier identification variable,which is known as the mean drift model.Moreover,the Huber loss function combines the advantages of least-absolute-deviation estimation and least-squares esti-mation and it is not only robust but also smooth.Therefore,in this paper,we build a regularization model combining Huber function and mean drift model for parameter estimation and outlier identification,study the optimization theory of this problem and propose the corresponding algorithm to solve it.The main contents are as follows:（1）First of all,because the mean drift model and Huber function have good anti-interference ability,this paper combines the Huber function and the mean drift model as the loss function,and then imposesl₀regularization to realize the function of variable selection to bulid the original problem.（2）Secondly,because the Capped-l₁regular function has good statistical proper-ties,we use the Capped-l₁regularization relaxation thel₀regularization to obtain the relaxation problem of the original problem,and define the directional stability point of the relaxation problem through the directional derivative to establish the first order optimality condition of the problem;describe the specific expression of the directional stability point of the relaxation problem,and analyze the lower bound theory;establish the equivalence of solutions between two problems,prove that the global solution of the two problems is equivalent under certain conditions.（3）Finally,for the relaxation problem,the alternating gradient descent algorith-m（AGD）and the accelerated neighbor alternating minimization algorithm（APAM）are proposed,and the effectiveness of the algorithm is verified through numerical experi-ments.