Research On Methods Of Smoothing Functional Data With Some Constraints Based B-Spline Basis With Roughness Penalty

In this paper, the criterion of B-spline with roughness penalty is used to smooth some constrained functional data through proper function transform in order to obtain better smoothing effect. Firstly,we introduce the method of getting B-spline basis and some excellent properties of B-spline. Then, the method of smoothing functonal data using B-spline basis by least square method is presented. But because we can't choose K continuously in real domain in this method, so we can't obtain satisfactory result when the data have strong local characters . For the sake of overcoming this problem, in the fourth chapter, we introduce the notions of roughness penalty of a function and smoothing parameter. Consequently, the criterion of smoothing functional data using B-spline basis with a roughness penalty by least square method is established. Through choosing proper smoothing parameter, this method can fix the number K of basis and controls the roughness degree to obtain the smoothing functions which can better accord with real condition or real requirement. In practical applications, because the data that we collected always have some special meanings, when the data have some specail constrains, we often can't get satisfactory result, so,in section 5 of this chapter, we smoothing some constrained functional data through the criterion of B-spline with roughness penalty after proper function transform, such as positive functional data,probability funcuional data and strictly monotone functional data. Finally, throuhg analysis and comparison, we disply the effect and merits of smoothing functional data which have some constrains using the methods introduced in this paper.