nag_smooth_spline_fit (g10abc) 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 nag_smooth_spline_estim (g10acc) for further details.
nag_smooth_spline_fit (g10abc) 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 nag_order_data (g10zac).
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
mode – Nag_SmoothFitTypeInput
On entry: indicates in which mode the function 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 nag_smooth_spline_fit (g10abc) or when the coefficients are not required.
Fitting only is performed. Initialization must have been performed previously by a call to nag_smooth_spline_fit (g10abc) 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.
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 nag_order_data (g10zac) and a series of splines fit using a range of values for the smoothing parameter, .