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Bayesian curve fitting with roughness penalty prior distributions Rahman, Md Mushfiqur

Abstract

In statistical research with populations having a multilevel structure, hierarchical models can play significant roles. The use of the Bayesian approach to hierarchical models has numerous advantages over the classical approach. For example, a spline with roughness penalties can easily be expressed as a hierarchical model and the model parameters can be estimated by the Bayesian techniques. Splines are sometimes useful to express the rapid fluctuating relationship between response and the covariate. In smoothing spline problems, usually one smoothing parameter (variance component in Bayesian context) is considered for the whole data set. But to deal with rapidly fluctuating or wiggly data sets, it is more logical to consider different smoothing parameters at different knot points in order to find more efficient estimates of the the regression functions under consideration. In this study, we have proposed the roughness penalty prior distribution considering local variance components at different knot points and call it Prior 2. Prior 2 is compared with Prior 1, where a single global variance component is considered for the whole data set, and with Prior 3, where no roughness penalty terms are considered ( i.e., the parameters at different knot points are assumed independent). Performance of the proposed prior distributions are checked for three different data sets of different curvature. Similar performance of Prior 1 and Prior 2 is observed for all three data sets under the assumption of piecewise linear spline. The application has been extended to the case of natural cubic spline, where the modification of Prior 1 and Prior 3 are straightforward. However, for Prior 2, the modification becomes very tedious. We have proposed an approximate roughness penalty matrix for Prior 2. Parameters corresponding to the smoothing splines are estimated using MCMC techniques. We carefully compare the inferential procedures in simulation studies and illustrate them for two data sets. Similarity among the curves produced by Prior 1 and Prior 2 are observed, and they are much smoother than the curve estimated by Prior 3 for both piecewise linear and natural cubic splines. Therefore, in the context of Bayesian curve fitting, both local and global roughness penalty priors produce equally smooth curves in dealing with wiggly data.

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