Environmental and Ecological Statistics

, Volume 21, Issue 4, pp 715–731 | Cite as

Finite area smoothing with generalized distance splines

  • David L. Miller
  • Simon N. Wood


Most conventional spatial smoothers smooth with respect to the Euclidean distance between observations, even though this distance may not be a meaningful measure of spatial proximity, especially when boundary features are present. When domains have complicated boundaries leakage (the inappropriate linking of parts of the domain which are separated by physical barriers) can occur. To overcome this problem, we develop a method of smoothing with respect to generalized distances, such as within domain distances. We obtain the generalized distances between our points and then use multidimensional scaling to find a configuration of our observations in a Euclidean space of 2 or more dimensions, such that the Euclidian distances between points in that space closely approximate the generalized distances between the points. Smoothing is performed over this new point configuration, using a conventional smoother. To mitigate the problems associated with smoothing in high dimensions we use a generalization of thin plate spline smoothers proposed by Duchon (Constructive theory of functions of several variables, pp 85–100, 1977). This general method for smoothing with respect to generalized distances improves on the performance of previous within-domain distance spatial smoothers, and often provides a more natural model than the soap film approach of Wood et al. (J R Stat Soc Ser B Stat Methodol 70(5):931–955, 2008). The smoothers are of the linear basis with quadratic penalty type easily incorporated into a range of statistical models.


Finite area smoothing Generalized additive model Multidimensional scaling Spatial modelling Splines 



We are especially grateful to Jean Duchon for generous help in understanding Duchon (1977). David wishes to thank EPSRC for financial support during his PhD at the University of Bath.

Supplementary material

10651_2014_277_MOESM1_ESM.pdf (141 kb)
Supplementary material 1 (pdf 140 KB)


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Centre for Research into Ecological and Environmental ModellingUniversity of St AndrewsThe ObservatoryScotland
  2. 2.Department of Mathematical SciencesUniversity of BathClaverton Down, BathUK

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