Change Point Estimation Of Smooth Spline Regression Model

In recent years,the change point problem has become one of the important research topics in statistics.In this paper,we use smooth spline change point estimation and weighted iterative smooth spline change point estimation to realize the mean function estimation and the variance change point detection of the residual sequence.Combined with the dichotomy method,the two models are extended from a single change point to multiple change points.The specific work is as follows:In the study of smoothing spline change point estimation,taking into account the goodness of fitting of the curve and the degree of rough punishment,using Generalized Cross-Validation to select smoothing parameters,and then using the Reinsch algorithm to implement the mean function estimation;then,based on the improved Schwarz Information Criterion(SIC),a likelihood test statistic is constructed,and the relationship between the statistic and the variance under the null hypothesis and alternative hypothesis is derived;finally,the variance change point detection is implemented in the residual sequence with the mean trend removed.Numerical simulations show that smooth spline regression change point estimation performs well in the three aspects of estimating the mean function,the position of the change point,and the variance before and after the change point,and the estimation effect is better as the sample size increases.In the case of a large sample size,the constructed confidence interval has a high change point coverage.Taking traffic flow data as an example,the results show that the method can accurately capture the change points in the data under different time and space situations,which can provide a more accurate decision basis for the traffic control department.On the basis of smooth spline change point estimation,a weight matrix is introduced,and combined with the dichotomy method to obtain a weighted iterative smooth spline multiple change point estimation model to achieve a more accurate estimation of the mean function and variance change point.The model steps are as follows: first ignore the heterogeneity of variance and use a smooth spline to estimate the initial mean function;secondly,the change point detection is performed on the residual sequence with the mean trend removed,and the position of the change point is obtained,and the variance before and after the change point is estimated using the maximum likelihood method,and then the distribution of the test statistics and the convergence of all estimated parameters are derived speed;next,the estimated value of variance before and after the change point and the position of the change point are constructed into a diagonal matrix and is substituted into a weighted iterative smooth spline,then combine the Generalized Cross Validation again to select the smoothing parameters and achieve the update of the mean function;finally,the dichotomy method is introduced to extend the method to the case of multiple change points.The simulation results show that compared with some existing change point detection methods,weighted iterative smooth spline change point estimation and smooth spline change point estimation have obvious advantages in mean function estimation and change point detection.Taking the monthly trading volume of Apple's stock as an example,its internal laws can be tapped through accurate change point detection,which has strong practical value.