Smoothing under Diffeomorphic Constraints with Homeomorphic Splines

In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms. This diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time-dependent vector field. This solution has a closed form expression which can be computed using classical unconstrained spline smoothing techniques. This method does not require the use of quadratic or linear programming under inequality constraints and has therefore a low computational cost. In a one dimensional setting incorporating diffeomorphic constraints is equivalent to impose monotonicity. Thus, as an illustration, it is shown that such a monotone spline can be used to monotonize any unconstrained estimator of a regression function, and that this monotone smoother inherits the convergence properties of the unconstrained estimator. Some numerical experiments are proposed to illustrate its finite sample performances, and to compare them with another monotone estimator. We also provide a two-dimensional application on the computation of diffeomorphisms for landmark and image matching

Smoothing with Curvature Constraints based on Boosting Techniques

In many applications it is known that the underlying smooth function is constrained to have a specific form. In the present paper, we propose an estimation method based on the regression spline approach, which allows to include concavity or convexity constraints in an appealing way. Instead of using linear or quadratic programming routines, we handle the required inequality constraints on basis coefficients by boosting techniques. Therefore, recently developed componentwise boosting methods for regression purposes are applied, which allow to control the restrictions in each iteration. The proposed approach is compared to several competitors in a simulation study. We also consider a real world data set

Spline-based self-controlled case series method

The self-controlled case series (SCCS) method is an alternative to study designs such as cohort and case control methods and is used to investigate potential associations between the timing of vaccine or other drug exposures and adverse events. It requires information only on cases, individuals who have experienced the adverse event at least once, and automatically controls all fixed confounding variables that could modify the true association between exposure and adverse event. Time-varying confounders such as age, on the other hand, are not automatically controlled and must be allowed for explicitly. The original SCCS method used step functions to represent risk periods (windows of exposed time) and age effects. Hence, exposure risk periods and/or age groups have to be prespecified a priori, but a poor choice of group boundaries may lead to biased estimates. In this paper, we propose a nonparametric SCCS method in which both age and exposure effects are represented by spline functions at the same time. To avoid a numerical integration of the product of these two spline functions in the likelihood function of the SCCS method, we defined the first, second, and third integrals of I-splines based on the definition of integrals of M-splines. Simulation studies showed that the new method performs well. This new method is applied to data on pediatric vaccines

Modelling the Load Curve of Aggregate Electricity Consumption Using Principal Components

Since oil is a non-renewable resource with a high environmental impact, and its most common use is to produce combustibles for electricity, reliable methods for modelling electricity consumption can contribute to a more rational employment of this hydrocarbon fuel. In this paper we apply the Principal Components (PC) method to modelling the load curves of Italy, France and Greece on hourly data of aggregate electricity consumption. The empirical results obtained with the PC approach are compared with those produced by the Fourier and constrained smoothing spline estimators. The PC method represents a much simpler and attractive alternative to modelling electricity consumption since it is extremely easy to compute, it significantly reduces the number of variables to be considered, and generally increases the accuracy of electricity consumption forecasts. As an additional advantage, the PC method is able to accommodate relevant exogenous variables such as daily temperature and environmental factors, and it is extremely versatile in computing out-of-sample forecasts.Electricity, Load curves, Principal components, Fourier estimator, Constrained smoothing estimator, Temperature, Non-renewable resources, Hydrocarbon fuels, Environment