Finite area smoothing with generalized distance splines
- 444 Downloads
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.
KeywordsFinite 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.
- Bernstein M, De Silva V, Langford J, Tenenbaum J (2000) Graph approximations to geodesics on embedded manifolds. Technical report, Department of Psychology, Stanford University. ftp://ftp-sop.inria.fr/prisme/boissonnat/ImageManifolds/isomap.pdf
- Chatfield C, Collins AJ (1980) Introduction to multivariate analysis. Science paperbacks, Chapman and HallGoogle Scholar
- Curriero F (2005) On the use of non-euclidean isotropy in geostatistics. Technical report 94, Johns Hopkins University, Department of Biostatistics. http://www.bepress.com/cgi/viewcontent.cgi?article=1094&context=jhubiostat
- Duchon J (1977) Splines minimizing rotation-invariant semi-norms in Sobolev spaces. Constructive theory of functions of several variables, pp 85–100Google Scholar
- Hastie TJ, Tibshirani RJ (1990) Generalized additive models. Monographs on statistics and applied probability. Taylor & Francis, New YorkGoogle Scholar
- Miller DL (2012) On smooth models for complex domains and distances. PhD thesis, University of BathGoogle Scholar
- Miller DL, Burt ML, Rexstad EA (2013) Spatial models for distance sampling data: recent developments and future directions. Methods in Ecology and EvolutionGoogle Scholar
- Scott-Hayward LAS, MacKenzie ML, Donovan CR, Walker CG, Ashe E (2013) Complex region spatial smoother (CReSS). J Comput Graph. Stat. doi: 10.1080/10618600.2012.762920
- Vretblad A (2003) Fourier analysis and its applications. Graduate texts in mathematics. Springer, BerlinGoogle Scholar
- Williams R, Hedley SL, Branch TA, Bravington MV, Zerbini AN, Findlay KP (2011) Chilean blue whales as a case study to illustrate methods to estimate abundance and evaluate conservation status of rare species. Conserv Biol 25(3):526–535. doi: 10.1111/j.1523-1739.2011.01656.x PubMedCrossRefGoogle Scholar
- Wood SN (2006) Generalized additive models: an introduction with R. Chapman & Hall/CRC, LondonGoogle Scholar