Multiple Change Points Detection Of Reduced-rank Regression Model

Dynamic models are often used to detect whether the observation sequence has a heterogeneous structure,which have a wide range of applications in the fields of EEG analysis,DNA segmentation and financial stock market analysis.Heterogeneity indicates that the observed data comes from different sampling distributions,multiple different models need to be considered for doing statistical inference.With the popularization of high-dimensional data,detecting the heterogeneity of such data is an important and challenging problem.In recent years,a large amount of literature has studied how to accurately and quickly detect change points,and most of the research results have focused on the mean shift model.There is little research on the change point detection of high-dimensional reduced-rank regression,this thesis will discuss this kind of issues.The following is a brief introduction to the main content of each part of this thesis.The first chapter of this thesis is an introduction,including some background knowledge.Firstly,it shows the motivation and content of this thesis.Then we briefly reviews the common methods of detecting change points and the current research status in recent years.Finally,the reduced-rank regression model and the estimation method of low-rank matrices are introduced.The second chapter of this thesis studies the detection of multiple change points in reduced-rank regression model.This chapter considers a dynamic high-dimensional regression model,the coefficient matrices are subject to abrupt changes of unknown magnitudes at unknown locations.We adopt a procedure that minimizes a penalized least-squares loss function via a dynamic programming algorithm for estimating the locations of change points.In change points regression model,while estimating the change points,the coefficient matrices are also estimated,which leads to excessive calculation tasks.To this end,we propose a prescreening procedure,which deletes a large number of irrelevant points before implementing estimation procedure.On this basis,a candidate set containing true change points is obtained,which greatly improves the detection efficiency.Under mild assumptions,we establish the consistency of the proposed estimators,and further provide error bounds for estimated parameters which achieve almost-optimal rate.The effectiveness of the proposed method is verified by numerical simulation and actual data application.The third chapter of this thesis discusses the robust change points detection method in reduced-rank regression.In the second chapter,we consider the problem of change points detection in which the observation data comes from a light-tailed distribution.However,in different subject areas,heavy-tailed or a part of arbitrarily contaminated data are often encountered,which leads to unreliable inferences based on squared loss.To address this problem,this chapter considers a robust method to detect multiple change points.Through a kind of robust loss function whose derivative is bounded to replace the squared loss,this can reduce the influence of contaminated data.This method is suitable for situations where the response variable and predictor variable are contaminated.Under reasonable conditions,we further study that the proposed method produces consistent estimators with ideal statistical rates.Both numerical simulation and actual data show the superiority of our method.The conclusions and results of this thesis are summarized in the last chapter,and the future research work is prospected.The innovations of this thesis are as follows:(1)Based on marginal least squares estimation,we propose a prescreening algorithm for detecting change points.A candidate set containing true change points can beobtained via this algorithm,which greatly reduces the computational cost.In par-ticular,the advantages of this algorithm are particularly obvious in robust change point detection.(2)When the observation data comes from the light-tailed distribution,a method fordetecting multiple change points is given in reduced-rank regression model.The asymptotic consistency of the proposed estimators is studied,and they enjoy the same convergence rate as the estimators of stationary model.(3)For data with heavy-tailed or outliers,a robust method of detecting multiple change points is proposed.This method has good performance in terms of robustness and estimation accuracy.