The Study Of Scatterplot Smoothing Based On Penalized Spline Method

Non-parametric models which relax the restriction of model parameters andimprove the traditional parametric models. It still attracts us to study and it also has a lotof development and application in many fields. There exist many other scatterplotsmoothers developed over the years, such as weighted least-squares, kernel smoother,local polynomial fitting and wavelets-based smoother and so on. All those approachescan be used effectively and have their devotes. For example, wavelets are more suited tohighly oscillatory functions. While, the splines continue to play a central role innon-parametric and semi-parametric regression modeling, such as regression spline,smoothing spline and penalized spline. In this literature, we mainly study the splinemethod. In regression splines, we use the polynomial spline as the basis functions tomodel target function. And it depends on the number and location of knots. Of course,the smoothing parameters also play an important role in the fitting procedure. While, weusually let the domain of definition be continuous. It could meet this condition easily insampling. However, there always exists missing value in practice. In smoothing splines,the number of bases functions roughly equals the sample size. However, there has beena great deal of research into more general spline/penalty strategies, most of which useconsiderably fewer basis functions. While, in penalized spline, there is no restrictions onthe number of bases functions, and more important is that the domain of target functionsneed not be continuous. We mainly consider the P-splines in the article and we need payattention to some parameters during the fitting procedure. That is, k (knot points), K (thenumber of knot), q (the degree of splines),(smoothing parameter) andm m (differencing order). While, it would likely be redundant to choose all of them in adata-driven way. And we devote to show the relationship between smoothingparameter and differencing order m. Meanwhile, we use an EM algorithm to optimizethe two of parameters. As for the rest of parameters, we could adopt an equal-spacedand a cubic B-spline basis. For most of the situations presented in the simulations,including the practical example, the new criteria out-perform the three existing criteria.