Abstract
The traditional estimation of Gaussian mixture model is sensitive to heavy-tailed errors; thus we propose a robust mixture regression model by assuming that the error terms follow a Laplace distribution in this article. And for the variable selection problem in our new robust mixture regression model, we introduce the adaptive sparse group Lasso penalty to achieve sparsity at both the group-level and within-group-level. As numerical experiments show, compared with other alternative methods, our method has better performances in variable selection and parameter estimation. Finally, we apply our proposed method to analyze NBA salary data during the period from 2018 to 2019.
Highlights
The mixture regression model is a powerful tool to explain the relationships between the response variable and the covariates when the population is heterogeneous and consists of several homogeneous components, and the early research can trace back to [1]
In order to improve the robustness of the estimation procedure, we introduce a robust mixture regression model with a Laplace distribution
To evaluate the performances of variable selection and data fitting, we introduce the average number of selected non-zero variables without intercepts, average number of selected non-zero groups, frequency of correct identification of group sparsity structures, false negative rate (FNR) of missing important predictors, false positive rate (FPR) of selecting unimportant predictors and average value of root mean square errors (RMSE)
Summary
The mixture regression model is a powerful tool to explain the relationships between the response variable and the covariates when the population is heterogeneous and consists of several homogeneous components, and the early research can trace back to [1]. In order to achieve both the robustness of mixture regression model and correct identification of group structures, we assume random errors follow a Laplace distribution and consider a situation that covariates have natural grouping structures, where those in the same group are correlated. In this case, variable selection should be conducted at both the group-level and within-group-level; we use the adaptive sparse group Lasso [12] as a penalty function of our proposed mixture regression model and adopt EM algorithm to estimate the mixture regression parameters.
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