| Partially linear models have both the interpretability of linear models and the flexibility of nonparametric models,which can model complex data.In fact,the performance of partially linear model depends largely on the choice of variable structure in the model,for example,which variables have a linear effect on the response and which variables have a nonlinear effect.Therefore,the problem of variable structure discovery for partially linear models has been widely studied.However,the existing models are usually limited to mean regression based on Gaussian noise assumption and sensitive to non-Gaussian noise.In order to improve the robustness of partially linear models,two kinds of robust partially linear models are constructed with modal regression,which is robust to non-Gaussian noise by learning conditional mode.The first model not only ensures the discovery of variable structure but also improves the flexibility of the model by using the gradient information of variables.The second model integrates trend filtering to discover the structure of variables.This model has low computational complexity and strong interpretability because it can adaptively select nodes in nonlinear approximation.In this paper,the statistical learning theory analysis and optimization algorithm of the proposed models are given.In theory,the generalization error bounds and consistency analysis of variable structure discovery of the first model,and the generalization error analysis of the second model and their properties are given,which establish the mathematical theoretical basis of the new models.In terms of calculation algorithm,based on half quadratic optimization,the quadratic programming technique and the generalized least absolute shrinkage and selection operator are respectively used for optimization this two models.Experiments on simulated data verify the robustness and competitiveness of the proposed two methods in variable structure discovery and estimation. |
