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Documentation for SPLINA and SPLINB VERSION 3.1

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SPLINA is a FORTRAN program which fits a partial thin plate smoothing

spline to multi-variate noisy data.

SPLINB is an approximate version of SPLINA designed for larger data

sets. It fits a partial thin plate smoothing spline, with user defined

knots, to multi-variate noisy data.

Author: M.F.Hutchinson, CRES, Australian National University,

surfaces, including residuals and standard error estimates.

The command file should only contain the program directives. The program

is then executed by typing, for example,

splina < cmdfile > cmdfile.lis

The original thin plate (formerly Laplacian) surface fitting technique

was described by Wahba (1979), with modifications for larger data sets

due to Bates and Wahba (1982), Elden (1984) and Hutchinson (1984). The

extension to partial splines is based on Bates et al. (1987). This

allows the incorporation of parametric linear sub-models (or covariates)

into the fitted spline, in addition to the usual independent spline

(1990). A brief overview of the basic theory and applications is given

in Hutchinson (1991a). More comprehensive discussion of the algorithms

and the associated statistical analyses are given in Hutchinson and

Gessler (1994) and Hutchinson (1993,1995).

Program Input

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This includes:- the number of independent spline variables, the number

of independent covariates, the lower and upper limits for each

independent variable, the order of derivative to be minimized, the

number of surfaces, the data error standard deviation estimate for each

variables in the data file. An error in these specifications would be

indicated if fewer than the expected number data points are selected.

SPLINA may currently fit up to 12 surfaces to data at up to several

hundred sites. SPLINB, the approximate version of the thin plate smoothing

procedure, may fit up to 12 surfaces to data at up to a few thousand sites

using up to several hundred knots.

For SPLINA, the main storage requirements of the program are proportional

to the square of the number of data points and the processing time is

proportional to cube of the number of data points. For SPLINB, storage

requirements are proportional to the square of the number of knots and

processing time is approximately proportional to the cube of the number of

knots, since a singular value decomposition of a matrix of this order is

required. The required linear algebra routines are contained within the

double precision LINPACK library (Dongarra et al., 1979). The required

LINPACK routines can be supplied with ANUSPLIN.

coincident data points. Coincident knots are not permitted. Simplified

fitted values with estimated standard errors may be written to output

parameters used to calculate the optimum smoothing parameter may be

written. The surface coefficients are used to calculate values of the

LAPPNT. The ranked list of residuals is particularly useful in

data set.

Knot Selection for SPLINB

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The following, somewhat heuristic, procedure works well in practice:

identical. Choose an initial set of knots about 1/3 the size

of the data set. The number of knots required depends on the spatial

complexity of the data being fitted - more knots for more spatially

variable data. If the signal of the final result is within 10% of the

re-done starting with a larger initial knot set.

2. Run SPLINB, with the output list of data and fitted values, and

examine the largest residuals for data errors. Re-fit the surface if

necessary after data errors have been corrected, usually without the

output list of data and fitted values. Add to the knot index file the

knot are identified by a minus sign, both in the output ranked residual

list, and in the list of data and fitted values.

3. Repeat the procedure of adding to the knot list the indices of the

largest 15-25 residuals and re-fitting the surface until the solution

stabilises or the variance estimates output by the program are in

approximate agreement with a priori estimates. This should normally

only be done two or three times since there is a risk of overfitting the

data if it is done too many times.

Interpretation of Output Statistics

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The output statistics are best interpreted in relation to the partial

spline model for n observed data values zi given by

zi = f(xi) + b'yi + ei (i=1,...,n)

where each xi is a d-dimensional vector of independent spline variables,

vector of parameters or covariates, b is a p-dimensional vector of

error term with variance = wi*sig2 where wi is termed the relative

the programs.

The signal degrees of freedom and the error degrees of freedom for each

surface should add up to n (the number of data points). The surface

genuine optimum value of the smoothing parameter rho if either rho is

number which depends on the number of independent variables and the

order of the derivative). Both of these conditions are flagged by an

asterisk in the output.

The generalized cross validation (GCV), mean square residual (MSR) and

the error variance estimate (VAR) are given together with their square

of the error variance (so that sig2=1), then the variance estimate

provided by the procedure should be approximately 1. The goodness of

fit of the fitted model may be checked by comparing the residual sum of

squares (n*MSR where n is the number of data points) with the critical

df = the error degrees of freedom and sig2^ is an a priori estimate of

sig2.

When an estimate of sig2 is available an alternative strategy is to

provide the corresponding standard deviation estimate (sqrt(sig2)) to

the program (instead of -1). The program then minimises an unbiased

preferable when fitting surfaces to very small data sets (less than say

20-30 data points). Note that SPLINA should be used for small data

sets.

The procedure also outputs the coefficients of any covariates (standard

errors of these coefficients are not yet available) as well as an

estimate of the true mean square error of the smoothed data values.