Variable Selection In Finite Mixture Of Time-Varying Regression Models

In recent years,there has been a growing interest in finite mixture models which provide a flexible tool for modeling data that arise from a multi-modal(heterogeneous)population.However,for time series data with heterogeneity,as for the data contains the time factor,there may be the dependence of variables on time and the heteroscedasticity of dependent variables changing with time.The finite mixture generalized autoregressive con-ditional heteroscedasticity(GARCH)model can be used to describe the heteroscedasticity of time series data which has heterogeneous.On the other hand,the time dependence of time series data is often expressed by the laged distribution regression model.Therefore,this paper studies the regression model of the time series data which has the heterogene-ity and the time dependence.We propose two kinds of finite mixture regression models of time-varying(MIX-TVR).One is the finite mixture laged regression model(ML-AR)which is consisted of the combination of the finite mixture regression model and the laged distri-bution regression model.Another is the finite mixture laged GARCH model(ML-GARCH)which is obtained by combining the GARCH model and the finite mixture regression model The variable selection methods of these two models are further discussed in this paper,and a new method of penalized maximum logarithmic quasi-likelihood function is proposed.On the base of the common penalty functions such as LASSO,MCP,and SCAD,the mixture penalty function consists of l1 penalty and l2 penalty that proportion to the mixture weight of each component.It is also proved that this regularization method is consistent and sparse for the MIX-TVR model.Due to the complexity of the parameter solution,we adapt the Block-wise MM algorithm with global convergence.Respectively find the minimization function of each parameter block and obtain the estimated value of the parameter by max-imizing the minimization function.The derivation process of Block-wise MM algorithm based on Gaussian ML-GARCH model is also given.The main research results of this paper are summarized as follows:1.In the Gaussian ML-AR(1)simulation,it is found that the three penalty functions in the paper can screen out all the non-zero coefficient explanatory variables when the explanatory variable dimension is 10 or 20,among which the variable selection effect of MIXLASSO-ML2 is the worst.When the explanatory variable dimension is 100,the number of correctly identified non-zero coefficients of MIXSCAD-ML2 is more than that of the other two penalty functions.As the dimension of explanatory variables increases,more explanatory variables with insignificant effects are misestimated in the model.2.According to the simulation of variable selection of ML-GARCH(1,1,1)models,the performance of MIXLASSO-ML2 penalty function is outperform than the other two penalty functions which indicate that the variable selection effect of this penalty func-tion is the best.However,its variable selection effect is worse than that of ML-AR model because of the addition of GARCH effect in ML-GARCH model3.In the empirical analysis of the urban vehicle traffic behavior in Sao Paulo during December 18,2009(solstice),we established the Gaussian ML-AR(1),the Gaussian ML-GARCH(1,1,1),and general Gaussian FMR models.And the MIXSCAD-ML2 penalty was used for variable selection of these models.The results show that the ML-GARCH(1,1,1)model has the lowest BIC value among the three models,and the BIC value of the general Gaussian FMR model is largest.Moreover,the mean square error(MSE)of the response variable(the traffic slowness rate)predicted by ML-GARCH(1,1,1)model is the smallest.These results show that the proposed model is superior to the FMR model and has a better explanatory effect on this actual data setIt can be seen from the research results that the MIX-TVR model and variable selection framework proposed in this paper has better performance in practical application.