Abstract
Penalized spline has been a popular method for estimating an unknown function in the non-parametric regression due to their use of low-rank spline bases, which make computations tractable. However its performance is poor when estimating functions that are rapidly varying in some regions and are smooth in other regions. This is contributed by the use of a global smoothing parameter that provides a constant amount of smoothing across the function. In order to make this spline spatially adaptive we have introduced hierarchical penalized splines which are obtained by modelling the global smoothing parameter as another spline.
Highlights
Non parametric smoothing involves letting the data determine the amount of smoothing
In order to make this spline spatially adaptive we have introduced hierarchical penalized splines which are obtained by modelling the global smoothing parameter as another spline
When homogeneity of the smoothness cannot be reasonably assumed across the whole domain of the function, a natural extension is to allow the smoothing parameter to vary over the domain as a penalty function of independent variable, adapting to the change of roughness [1] [2]
Summary
Non parametric smoothing involves letting the data determine the amount of smoothing. When homogeneity of the smoothness cannot be reasonably assumed across the whole domain of the function, a natural extension is to allow the smoothing parameter to vary over the domain as a penalty function of independent variable, adapting to the change of roughness [1] [2]. Modeling the smoothing parameter as a penalty function of independent variable can be used to achieve adaptiveness This involves formulating the adaptive smoothing as a minimization problem with a new penalty function in which the estimate has the same form as the smoothing spline and method developed for classical smoothing splines can be used. A complexity penalty was added to the generalized likelihood in selecting the best step function from a collection of candidate This approach was very computational expensive due to the large number of candidate models and proposed search algorithm and has a serious limitation. In this research we aim at developing a Hierarchical penalty model using p-splines which will result in more adaptive smoothing
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