Bayesian Locally Penalized Spline Regression Model

Because of concise form and easy calculation, parametric regression model has been used in various fields. When the relationship between explanatory variables and explained variables is complex and difficult to be expressed by functions, it is improper to use parametric regression model. Nonparametric regression model has gained much attention due to the fact that it has the abil-ity of fitting complex data, and depicting the nonlinear characters underlying the data. This paper focuses on the Bayesian locally penalized spline regres-sion model. Based on the idea of Bayesian globally penalized spline regression model, the local penalty via the range of local region of data is constructed and added into the model. Simulations show that Bayesian locally penalized spline regression model makes better adaptivity and holds better fitting ability than Bayesian globally penalized spline regression model. This paper is organized as follows:(1) Firstly, the parametric regression model and nonparametric regression model are introduced. It is pointed out that the nonparametric regression model holds better fitting ability than the parametric regression model when the data is complex.(2) Secondly, for the penalized spline regression, the construction of penalties and the process of solving parameters are discussed in detail. The smoothing parameter is selected by GCV to keep the balance between goodness of fit and smoothness of regression function.(3) Lastly, the Bayesian locally penalized spline regression model is given. The basic Bayesian statistic idea and the Bayesian globally penalized spline regres- sion model are introduced. Then the local penalty based on the range of local region of data is added into the model to gain the Bayesian locally penalized spline regression model. Simulations of three examples and one application of real data are presented.