Adaptive Semiparametric Regression And Applications Of K-Nearest Neighbor Fused Lasso

In agriculture,biomedicine,economy and many other fields,we often collect a large number of diverse and complex datasets.The explanatory variables in these datasets have a certain order relationship,and some of them have a linear relationship with the response variables,others have an unknown functional relationship with the response variables.The semiparametric regression model effectively solves this kind of problem,and the estimation problem of the model is still a hot topic of current research.The estimation of semiparametric regression mainly involves the parameter estimation and nonparametric estimation.In this dissertation,we propose a new estimation method of semiparametric regression model which is K-nearest-neighbours fused lasso.K-nearest-neighbours fused lasso estimator is a local adaptive function estimator,which is composed of a parametric penalty term and a nonparametric first-order differential penalty term based on K-nearest-neighbours graph on the basis of minimizing the sum of squares of residuals.TheL₁ penalty term in the estimation gives the estimation of the parameter while the parameter is selected,the first-order differential penalty term based on K-nearest neighbours graph not only makes full use of the effective information in the dataset,but also reduces the waste of resources caused by the use of unnecessary information,It can also process the nonparametric part of high-dimensional data quickly.The estimation method proposed in this dissertation is applied to the K-nearest neighbours graph of sample points,which extends the estimation of design points based on regular network to irregular network.In the estimation method,the KKT condition is used to solve the parameter estimation part,while the coordinate descent method and the parameter maximum flow algorithm are used to solve the nonparameter estimation part.It can be verified that the estimation has good prediction consistency andL₁ consistency.In numerical simulation,this dissertation compares K-nearest-neighbours fused lasso estimation with traditional machine learning methods,and calculates the mean square error,mean absolute error and goodness of fit of various methods on small sample data sets.The experimental results show that K-nearest-neighbor fused lasso estimation has smaller mean square error and mean absolute error,and higher goodness of fit,which shows the effectiveness of K-nearest-neighbours fused fasso estimation.In empirical analysis,this dissertation compares the performance of K-nearest-neighbours fused fasso estimation method under nonparametric regression,K-nearest-neighbours fused lasso estimation method under semiparametric regression and generalized product kernel estimation method under semiparametric regression on California real estate dataset.The experimental results show that K-nearest-neighbours fused lasso is better than generalized product kernel method and nonparametric K-nearest-neighbours fused lasso,which shows the necessity of K-nearest-neighbours fused lasso extension to semiparametric regression.