INTRODUCTION
The aim of the ANUSPLIN package is to provide a facility for transparent analysis and
interpolation of noisy multi-variate data using thin plate smoothing splines. The package
supports this process by providing comprehensive statistical analyses, data diagnostics
and spatially distributed standard errors. It also supports flexible data input and surface
interrogation procedures.
The original thin plate (formerly Laplacian) smoothing spline surface fitting technique
was described by Wahba (1979), with modifications for larger data sets due to Bates and
Wahba (1982), Elden (1984), Hutchinson (1984) and Hutchinson and de Hoog (1985).
The package also supports the extension to partial thin plate splines based on Bates
et
al
. (1987). This allows for the incorporation of parametric linear sub-models (or
covariates), in addition to the independent spline variables. This is a robust way of
allowing for additional dependencies, provided a parametric form for these
dependencies can be determined. In the limiting case of no independent spline variables
(not currently permitted), the procedure would become simple multi-variate linear
regression.
Thin plate smoothing splines can in fact be viewed as a generalisation of standard multi-
variate linear regression, in which the parametric model is replaced by a suitably smooth
non-parametric function. The degree of smoothness, or inversely the degree of
complexity, of the fitted function is usually determined automatically from the data by
minimising a measure of predictive error of the fitted surface given by the generalised
cross validation (GCV). Theoretical justification of the GCV and demonstration of its
performance on simulated data have been given by Craven and Wahba (1979).
An alternative criterion is to minimise the generalised maximum likelihood (GML)
developed by Wahba (1985,1990). It is based on a Bayesian formulation for the thin
plate smoothing spline model and has been found to be superior to GCV in some cases
(Kohn
et al
. 1991). Both criteria are offered in this version of ANUSPLIN.
A comprehensive introduction to the technique of thin plate smoothing splines, with
various extensions, is given in Wahba (1990). A brief overview of the basic theory and
applications to spatial interpolation of monthly mean climate is given in Hutchinson
(1991a). More comprehensive discussion of the algorithms and associated statistical
analyses, and comparisons with kriging, are given in Hutchinson (1993) and Hutchinson
Gessler (1994). Recent applications to annual, monthly and daily precipitation data have
been described by Hutchinson (1995, 1998ab) and Price
et al.
(2000). The book by
Schimek (2000) provides a good overview of the subject of smoothing and non-
parametric regression with extensive references.
It is often convenient, particularly when processing climate data, to process several
surfaces simultaneously. If the independent variables and the relative weightings of the
data are the same for each surface then many surfaces can be calculated for little more
computation than one surface. ANUSPLIN allows for arbitrarily many such surfaces
with significant savings in computation. ANUSPLIN also introduces the concept of
"surface independent variables", to accommodate independent variables that change
systematically from surface to surface. ANUSPLIN permits systematic interrogation of
these surfaces, and their standard errors, in both point and grid form.
ANUSPLIN also permits transformations of both independent and dependent variables
and permits processing of data sets with missing data values. When a transformation is
