Group LASSO For Change-point Problem Of Mean Function In Functional Data Analysis

Principal component analysis is a really classical and useful tool in functional data analysis.But we need to assume that sample come from one population with the common mean function when we use principal component analysis in functional data analysis.However,in the real data analysis,this assumption is not satisfied.Thus,this paper researches on this problem.We consider the model Xi(t)=?i(t)+?i(t),i = 1,2,...,n.In this model,Xi(t)means sample function,?i(t)is a mean function,?i(t)is a error function.For convenience,we consider t ?[0,1].For all values of i,we need to test whether ?i(t)is the same or not.That means,we need to find if mean function changes at some point(s).Besides,it is important to note that ?i(t)is unknown.In other words,we do not know its concrete form and we also do not know where change point(s)is(are).In this paper,in order to test whether structural changes of mean function occur,firstly,we transform test problem to estimation problem.And in this process,we re-represent function by basis function.Then,we use Group Lasso to estimate.At the same time,consider Lasso usually will be overestimated.For that,we create a new Information criterion for two-step estimation.And we also do some sample simulations and real data analysis to prove that our method is really efficient and effective.