NAG Library Routine Document

G10ABF

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,

\hat{y} = {({\hat{y}}_{1}, {\hat{y}}_{2}, \dots, {\hat{y}}_{n})}^{T}

, and weighted residuals,

r_{i}

, can be written as

\hat{y} = H y and r_{i} = \sqrt{w_{i}} (y_{i} - {\hat{y}}_{i})

for a matrix

H

. The residual degrees of freedom for the spline is

trace (I - H)

and the diagonal elements of

H

h_{i i}

, 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

x_{i}

to be strictly increasing. If two or more observations have the same

x_{i}

-value then they should be replaced by a single observation with

y_{i}

equal to the (weighted) mean of the

y

values and weight,

w_{i}

, 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(1)Input

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'

'F'

2: WEIGHT – CHARACTER(1)Input

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'

'U'

3: N – INTEGERInput

On entry:

n

, the number of distinct observations.

Constraint:

N \geq 3

4: X(N) – REAL (KIND=nag_wp) arrayInput

On entry: the distinct and ordered values

x_{i}

, for

i = 1, 2, \dots, n

Constraint:

X (i) < X (i + 1)

, for

i = 1, 2, \dots, n - 1

5: Y(N) – REAL (KIND=nag_wp) arrayInput

On entry: the values

y_{i}

, for

i = 1, 2, \dots, n

6: WT( $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array WT must be at least

N

WEIGHT ='W'

On entry: if

WEIGHT ='W'

, WT must contain the

n

weights. Otherwise WT is not referenced and unit weights are assumed.

Constraint: if

WEIGHT ='W'

WT (i) > 0.0

, for

i = 1, 2, \dots, n

7: RHO – REAL (KIND=nag_wp)Input

On entry:

ρ

, the smoothing parameter.

Constraint:

RHO \geq 0.0

8: YHAT(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the fitted values,

{\hat{y}}_{i}

, for

i = 1, 2, \dots, n

9: C(LDC, $3$ ) – REAL (KIND=nag_wp) 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

((C (i, 3) \times d + C (i, 2)) \times d + C (i, 1)) \times d + {\hat{y}}_{i}

, where

x_{i} \leq t < x_{i + 1}

and

d = t - x_{i}

MODE ='P'

'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 \geq N - 1

11: RSS – REAL (KIND=nag_wp)Output

On exit: the (weighted) residual sum of squares.

12: DF – REAL (KIND=nag_wp)Output

On exit: the residual degrees of freedom.

13: RES(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the (weighted) residuals,

r_{i}

, for

i = 1, 2, \dots, n

14: H(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the leverages,

h_{i i}

, for

i = 1, 2, \dots, n

15: COMM( $9 \times N + 14$ ) – REAL (KIND=nag_wp) arrayCommunication Array

On entry: if

MODE ='Q'

, COMM must be unaltered from the previous call to G10ABF with

MODE ='P'

. Otherwise COMM need not be set.

On exit: if

MODE ='P'

'Q'

, COMM 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.

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.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 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	$MODE \neq'Q'$ , $'P'$ or $'F'$ ,
or	$WEIGHT \neq'W'$ or $'U'$ .

$IFAIL = 2$

On entry,

WEIGHT ='W'

and at least one element of

WT \leq 0.0

$IFAIL = 3$

On entry,

X (i) \geq X (i + 1)

, for some

i

i = 1, 2, \dots, n - 1

7 Accuracy

Accuracy depends on the value of

ρ

and the position of the

x

values. The values of

x_{i} - x_{i - 1}

and

w_{i}

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,

x_{i}

, and C-peptide concentration (pmol/ml),

y_{i}

, 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,

ρ

NAG Library Routine DocumentG10ABF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G10ABF