G10ABF fits a cubic smoothing spline to a set of observations (, ), for . The spline provides a flexible smooth function for situations in which a simple polynomial or nonlinear regression model is unsuitable.
Cubic smoothing splines arise as the unique real-valued solution function , with absolutely continuous first derivative and squared-integrable second derivative, which minimizes:
where is the (optional) weight for the th observation and is the smoothing parameter. This criterion consists of two parts: the first measures the fit of the curve, and the second the smoothness of the curve. The value of the smoothing parameter weights these two aspects; larger values of give a smoother fitted curve but, in general, a poorer fit. For details of how the cubic spline can be estimated see Hutchinson and de Hoog (1985) and Reinsch (1967).
The fitted values, , and weighted residuals, , can be written as
for a matrix . The residual degrees of freedom for the spline is and the diagonal elements of , , are the leverages.
The parameter can be chosen in a number of ways. The fit can be inspected for a number of different values of . Alternatively the degrees of freedom for the spline, which determines the value of , can be specified, or the (generalized) cross-validation can be minimized to give ; see G10ACF for further details.
G10ABF requires the to be strictly increasing. If two or more observations have the same -value then they should be replaced by a single observation with equal to the (weighted) mean of the values and weight, , equal to the sum of the weights. This operation can be performed by G10ZAF.
The computation is split into three phases.
Compute matrices needed to fit spline.
Fit spline for a given value of .
Compute spline coefficients.
When fitting the spline for several different values of , phase (i) need only be carried out once and then phase (ii) repeated for different values of . If the spline is being fitted as part of an iterative weighted least squares procedure phases (i) and (ii) have to be repeated for each set of weights. In either case, phase (iii) will often only have to be performed after the final fit has been computed.
Hastie T J and Tibshirani R J (1990) Generalized Additive Models Chapman and Hall
Hutchinson M F (1986) Algorithm 642: A fast procedure for calculating minimum cross-validation cubic smoothing splines ACM Trans. Math. Software12 150–153
Hutchinson M F and de Hoog F R (1985) Smoothing noisy data with spline functions Numer. Math.47 99–106
Reinsch C H (1967) Smoothing by spline functions Numer. Math.10 177–183
1: MODE – CHARACTER(1)Input
On entry: indicates in which mode the routine is to be used.
Initialization and fitting is performed. This partial fit can be used in an iterative weighted least squares context where the weights are changing at each call to G10ABF or when the coefficients are not required.
Fitting only is performed. Initialization must have been performed previously by a call to G10ABF with . This quick fit may be called repeatedly with different values of RHO without re-initialization.
Initialization and full fitting is performed and the function coefficients are calculated.
, or .
2: WEIGHT – CHARACTER(1)Input
On entry: indicates whether user-defined weights are to be used.
15: COMM() – REAL (KIND=nag_wp) arrayCommunication Array
On entry: if , COMM must be unaltered from the previous call to G10ABF with . Otherwise COMM need not be set.
On exit: if or , COMM contains information that will be used in a subsequent call to G10ABF with .
16: IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to , . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is . When the value is used it is essential to test the value of IFAIL on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6 Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
, or ,
and at least one element of .
, for some , .
Accuracy depends on the value of and the position of the values. The values of and are scaled and is transformed to avoid underflow and overflow problems.
8 Further Comments
The time taken by G10ABF is of order .
Regression splines with a small number of knots can be fitted by E02BAF and E02BEF.
The data, given by Hastie and Tibshirani (1990), is the age, , and C-peptide concentration (pmol/ml), , from a study of the factors affecting insulin-dependent diabetes mellitus in children. The data is input, reduced to a strictly ordered set by G10ZAF and a series of splines fit using a range of values for the smoothing parameter, .