Two new approaches to smoothing over complex regions

Speaker: David Miller (University of Bath)

Abstract

Smoothing over complex 2-D regions is difficult. Several approaches have been proposed in past years including finite element analysis (Ramsay, 2002), within-area distance (Wang & Ranalli, 2007) and recently soap film smoothing (Wood, Bravington & Hedley, 2008.) Here I investigate two alternative methods.

The first (following from Eilers (2006)) is based on the Schwarz-Christoffel transform from complex analysis. This takes the region and ‘morphs’ it to a rectangle or disk in a prescribed way using a conformal mapping.

The second uses a “new” algorithm to create a set of within-area distances. These distances are then mapped to points in the new domain using multidimensional scaling.

In both cases the aim is to find a new configuration of points that takes into account the intrinsic structure of the domain. It is possible to then smooth over the transformed area using penalized regression splines and transform this smooth back to the original domain in order to perform analysis.