Research On Change Point Of Functional Linear Regression Model Based On Group Lasso

In recent years,there are numerous studies on functional data and change point problems,among which change point problems are widely used in many practical fields.Therefore,it is very valuable to study structural mutations of various functional models formed by the combination of the two.Among them,the functional linear regression model is more common.However,most of the existing studies on this model focus on the estimation of slope function,and some traditional change point detection methods cannot be well applied to the structural mutation detection of this model.Therefore,it is necessary to use efficient and stable methods to detect change points of functional data models.In this paper,we use the Group Lasso change-point detection method(GL)to study the change-point problem of functional linear regression models in which the response variables and covariates are functions.The change-point problem is transformed into an estimation problem,and the estimation of complex coefficient functions is transformed into the estimation of the coefficient values of the basis functions by basis expansion.The convergence rate of the parameter estimation of change point position and coefficient function is proved.In addition,considering that there may be over-estimation problem in one-step detection for single change points,a two-step estimation of GL is further proposed,and the efficiency and stability of the GL change point detection method for structural mutation detection of functional linear regression models are proved in simulation and empirical studies.The main contents of this paper are as follows:The first part,the change point estimation process of GL change point detection method applied to functional linear regression model is introduced in detail,including one-step estimation and two-step estimation.The one-step estimation is realized by transforming the change-point problem into an estimation problem,and the two-step estimation is based on the one-step estimation results to construct a new information criterion to remove the over-estimated change-points,so as to accurately detect the true change-points.In the second part,four theorems and assumptions of GL one-step change point detection are introduced.It is proved that when the estimated number of change points is greater than or equal to the real number of change points,the estimation of change point location parameters is consistent.The convergence of the estimator of the coefficient function is proved when the estimated number of change points is consistent with the real number of change points.We also find that the probability that the estimated number of change points is greater than or equal to the true number of change points tends to 1.The third part,The effect of GL change point detection method in structural mutation detection of functional linear regression model was verified by simulation.The simulation considers different functional data structures,single variable points and different detection methods.In the case of single variable point,the change point detection of univariate model and multivariate model is considered respectively.Different detection methods are simulated,and the comparison between GL method,Lasso method and Adaptive Group Lasso(hereinafter referred to as AGL)method is considered.The simulation results show that the detection performance of GL change-point detection method is better than the other two methods under different conditions.The fourth part,the effect of GL change point detection method in actual data is tested.Both univariate and multivariate models were studied.For the univariate model,structural mutation detection of Hangzhou temperature data was carried out.For the multivariate model,the rainfall data structure mutation detection is carried out.All of them produce interpretable results,which further prove the validity of GL method.