Locally Adaptive Bayesian P-Splines with a Normal-Exponential-Gamma Prior

The necessity to replace smoothing approaches with a global amount of smoothing arises in a variety of situations such as effects with highly varying curvature or effects with discontinuities. We present an implementation of locally adaptive spline smoothing using a class of heavy-tailed shrinkage priors. These priors utilize scale mixtures of normals with locally varying exponential-gamma distributed variances for the differences of the P-spline coefficients. A fully Bayesian hierarchical structure is derived with inference about the posterior being based on Markov Chain Monte Carlo techniques. Three increasingly flexible and automatic approaches are introduced to estimate the spatially varying structure of the variances. In an extensive simulation study, the performance of our approach on a number of benchmark functions is shown to be at least equivalent, but mostly better than previous approaches and fits both functions of smoothly varying complexity and discontinuous functions well. Results from two applications also reflecting these two situations support the simulation results

Partial and Interaction Spline Models for the Semiparametric Estimation of Functions of Several Variables

A partial spline model is a model for a response as a function of several variables, which is the sum of a smooth function of several variables and a parametric function of the same plus possibly some other variables. Partial spline models in one and several variables, with direct and indirect data, with Gaussian errors and as an extension of GLIM to partially penalized GLIM models are described. Application to the modeling of change of regime in several variables is described. Interaction splines are introduced and described and their potential use for modeling non-linear interactions between variables by semiparametric methods is noted. Reference is made to recent work in efficient computational methods

Function estimation with locally adaptive dynamic models

We present a nonparametric Bayesian method for fitting unsmooth and highly oscillating functions, which is based on a locally adaptive hierarchical extension of standard dynamic or state space models. The main idea is to introduce locally varying variances in the state equations and to add a further smoothness prior for this variance function. Estimation is fully Bayesian and carried out by recent MCMC techniques. The whole approach can be understood as an alternative to other nonparametric function estimators, such as local or penalized regression with variable bandwidth or smoothing parameter selection. Performance is illustrated with simulated data, including unsmooth examples constructed for wavelet shrinkage, and by an application to sales data. Although the approach is developed for classical Gaussian nonparametric regression, it can be extended to more complex regression problems

A new Edge Detector Based on Parametric Surface Model: Regression Surface Descriptor

In this paper we present a new methodology for edge detection in digital images. The first originality of the proposed method is to consider image content as a parametric surface. Then, an original parametric local model of this surface representing image content is proposed. The few parameters involved in the proposed model are shown to be very sensitive to discontinuities in surface which correspond to edges in image content. This naturally leads to the design of an efficient edge detector. Moreover, a thorough analysis of the proposed model also allows us to explain how these parameters can be used to obtain edge descriptors such as orientations and curvatures. In practice, the proposed methodology offers two main advantages. First, it has high customization possibilities in order to be adjusted to a wide range of different problems, from coarse to fine scale edge detection. Second, it is very robust to blurring process and additive noise. Numerical results are presented to emphasis these properties and to confirm efficiency of the proposed method through a comparative study with other edge detectors.Comment: 21 pages, 13 figures and 2 table