NAG Library Routine Document
G10ABF
1 Purpose
G10ABF fits a cubic smoothing spline for a given smoothing parameter.
2 Specification
| SUBROUTINE G10ABF ( | MODE, WEIGHT, N, X, Y, WT, RHO, YHAT, C, LDC, RSS, DF, RES, H, WK, IFAIL) |
| INTEGER | N, LDC, IFAIL |
| double precision | X(N), Y(N), WT(*), RHO, YHAT(N), C(LDC,3), RSS, DF, RES(N), H(N), WK(9*N+14) |
| CHARACTER*1 | MODE, WEIGHT |
3 Description
G10ABF fits a cubic smoothing spline to a set of n observations (xi, yi), for i=1,2,…,n. 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
f, with absolutely continuous first derivative and squared-integrable second derivative, which minimizes:
where
wi is the (optional) weight for the
ith observation and
ρ 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
ρ weights these two aspects; larger values of
ρ 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,
y^
=
y^1,y^2,…,y^nT
, and weighted residuals,
ri, can be written as
for a matrix
H. The residual degrees of freedom for the spline is
traceI-H and the diagonal elements of
H,
hii, are the leverages.
The parameter
ρ can be chosen in a number of ways. The fit can be inspected for a number of different values of
ρ. Alternatively the degrees of freedom for the spline, which determines the value of
ρ, can be specified, or the (generalized) cross-validation can be minimized to give
ρ; see
G10ACF for further details.
G10ABF requires the
xi to be strictly increasing. If two or more observations have the same
xi-value then they should be replaced by a single observation with
yi equal to the (weighted) mean of the
y values and weight,
wi, equal to the sum of the weights. This operation can be performed by
G10ZAF.
The computation is split into three phases.
| (i) |
Compute matrices needed to fit spline. |
| (ii) |
Fit spline for a given value of ρ. |
| (iii) |
Compute spline coefficients. |
When fitting the spline for several different values of
ρ, phase
(i) need only be carried out once and then phase
(ii) repeated for different values of
ρ. 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.
The algorithm is based on
Hutchinson (1986).
4 References
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. Software 12 150–153
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
5 Parameters
- 1: MODE – CHARACTER*1Input
On entry: indicates in which mode the routine is to be used.
- MODE='P'
- 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.
- MODE='Q'
- Fitting only is performed. Initialization must have been performed previously by a call to G10ABF with MODE='P'. This quick fit may be called repeatedly with different values of RHO without re-initialization.
- MODE='F'
- Initialization and full fitting is performed and the function coefficients are calculated.
Constraint:
MODE='P', 'Q' or 'F'.
- 2: WEIGHT – CHARACTER*1Input
On entry: indicates whether user-defined weights are to be used.
- WEIGHT='W'
- User-defined weights should be supplied in WT.
- WEIGHT='U'
- The data is treated as unweighted.
Constraint:
WEIGHT='W' or 'U'.
- 3: N – INTEGERInput
On entry:
n, the number of distinct observations.
Constraint:
N≥3.
- 4: X(N) – double precision arrayInput
On entry: the distinct and ordered values xi for i=1,2,…,n.
Constraint:
Xi<Xi+1, for i=1,2,…,n-1.
- 5: Y(N) – double precision arrayInput
-
On entry: the values yi, for i=1,2,…,n.
- 6: WT(*) – double precision arrayInput
Note: the dimension of the array
WT
must be at least
1 if
WEIGHT='U' and at least
N if
WEIGHT='W'.
On entry: if
WEIGHT='W',
WT must contain the
n weights.
If
WEIGHT='U',
WT is not referenced and unit weights are assumed.
Constraint:
if WEIGHT='W', WTi>0.0, for i=1,2,…,n.
- 7: RHO – double precisionInput
On entry: ρ, the smoothing parameter.
Constraint:
RHO≥0.0.
- 8: YHAT(N) – double precision arrayOutput
-
On exit: the fitted values, y^i, for i=1,2,…,n.
- 9: C(LDC,3) – double precision arrayInput/Output
On entry: if
MODE='Q',
C must be unaltered from the previous call to G10ABF with
MODE='P'. Otherwise
C need not be set.
On exit: if
MODE='F',
C contains the spline coefficients. More precisely, the value of the spline at
t is given by
Ci3×d+Ci2×d+Ci1×d+y^i, where
xi≤t<xi+1 and
d=t-xi.
If
MODE='P' or
'Q',
C contains information that will be used in a subsequent call to G10ABF with
MODE='Q'.
- 10: LDC – INTEGERInput
-
On entry: the first dimension of the array
C as declared in the (sub)program from which G10ABF is called.
Constraint:
LDC≥N-1.
- 11: RSS – double precisionOutput
-
On exit: the (weighted) residual sum of squares.
- 12: DF – double precisionOutput
On exit: the residual degrees of freedom.
- 13: RES(N) – double precision arrayOutput
-
On exit: the (weighted) residuals, ri, for i=1,2,…,n.
- 14: H(N) – double precision arrayOutput
-
On exit: the leverages, hii, for i=1,2,…,n.
- 15: WK(9×N+14) – double precision arrayInput/Output
On entry: if
MODE='Q',
WK must be unaltered from the previous call to G10ABF with
MODE='P'. Otherwise
WK is used as workspace and need not be set.
On exit: if
MODE='P' or
'Q',
WK contains information that will be used in a subsequent call to G10ABF with
MODE='Q'.
- 16: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
6 Error Indicators and Warnings
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
- IFAIL=1
-
| On entry, | N<3, |
| or | LDC<N-1, |
| or | RHO<0.0, |
| or | MODE≠'Q', 'P' or 'F', |
| or | WEIGHT≠'W' or 'U'. |
- IFAIL=2
-
| On entry, | WEIGHT='W' and at least one element of WT≤0.0. |
- IFAIL=3
-
| On entry, | Xi≥Xi+1, for some i, i=1,2,…,n-1. |
7 Accuracy
Accuracy depends on the value of ρ and the position of the x values. The values of xi-xi-1 and wi are scaled and ρ is transformed to avoid underflow and overflow problems.
8 Further Comments
The time taken by G10ABF is of order n.
Regression splines with a small
<n number of knots can be fitted by
E02BAF and
E02BEF.
9 Example
The data, given by
Hastie and Tibshirani (1990), is the age,
xi, and C-peptide concentration (pmol/ml),
yi, 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 spline is fitted by G10ABF with
ρ=10.0. The fitted values and residuals are printed.
9.1 Program Text
Program Text (g10abfe.f)
9.2 Program Data
Program Data (g10abfe.d)
9.3 Program Results
Program Results (g10abfe.r)