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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_smooth_fit_spline (g10ab)

## Purpose

nag_smooth_fit_spline (g10ab) fits a cubic smoothing spline for a given smoothing parameter.

## Syntax

[yhat, c, rss, df, res, h, comm, ifail] = g10ab(mode, x, y, rho, c, comm, 'n', n, 'wt', wt)
[yhat, c, rss, df, res, h, comm, ifail] = nag_smooth_fit_spline(mode, x, y, rho, c, comm, 'n', n, 'wt', wt)
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 24: weight was removed from the interface; wt was made optional

## Description

nag_smooth_fit_spline (g10ab) fits a cubic smoothing spline to a set of $n$ observations (${x}_{i}$, ${y}_{i}$), for $i=1,2,\dots ,n$. 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:
 $∑i=1nwiyi-fxi2+ρ ∫-∞∞f′′x2dx,$
where ${w}_{i}$ is the (optional) weight for the $i$th observation and $\rho$ 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 $\rho$ weights these two aspects; larger values of $\rho$ 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, $\stackrel{^}{y}={\left({\stackrel{^}{y}}_{1},{\stackrel{^}{y}}_{2},\dots ,{\stackrel{^}{y}}_{n}\right)}^{\mathrm{T}}$, and weighted residuals, ${r}_{i}$, can be written as
 $y^=Hy and ri=wiyi-y^i$
for a matrix $H$. The residual degrees of freedom for the spline is $\mathrm{trace}\left(I-H\right)$ and the diagonal elements of $H$, ${h}_{ii}$, are the leverages.
The parameter $\rho$ can be chosen in a number of ways. The fit can be inspected for a number of different values of $\rho$. Alternatively the degrees of freedom for the spline, which determines the value of $\rho$, can be specified, or the (generalized) cross-validation can be minimized to give $\rho$; see nag_smooth_fit_spline_parest (g10ac) for further details.
nag_smooth_fit_spline (g10ab) 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 nag_smooth_data_order (g10za).
The computation is split into three phases.
 (i) Compute matrices needed to fit spline. (ii) Fit spline for a given value of $\rho$. (iii) Compute spline coefficients.
When fitting the spline for several different values of $\rho$, phase (i) need only be carried out once and then phase (ii) repeated for different values of $\rho$. 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).

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{mode}$ – string (length ≥ 1)
Indicates in which mode the function is to be used.
${\mathbf{mode}}=\text{'P'}$
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 nag_smooth_fit_spline (g10ab) or when the coefficients are not required.
${\mathbf{mode}}=\text{'Q'}$
Fitting only is performed. Initialization must have been performed previously by a call to nag_smooth_fit_spline (g10ab) with ${\mathbf{mode}}=\text{'P'}$. This quick fit may be called repeatedly with different values of rho without re-initialization.
${\mathbf{mode}}=\text{'F'}$
Initialization and full fitting is performed and the function coefficients are calculated.
Constraint: ${\mathbf{mode}}=\text{'P'}$, $\text{'Q'}$ or $\text{'F'}$.
2:     $\mathrm{x}\left({\mathbf{n}}\right)$ – double array
The distinct and ordered values ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
Constraint: ${\mathbf{x}}\left(\mathit{i}\right)<{\mathbf{x}}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,n-1$.
3:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
The values ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
4:     $\mathrm{rho}$ – double scalar
$\rho$, the smoothing parameter.
Constraint: ${\mathbf{rho}}\ge 0.0$.
5:     $\mathrm{c}\left(\mathit{ldc},3\right)$ – double array
ldc, the first dimension of the array, must satisfy the constraint $\mathit{ldc}\ge {\mathbf{n}}-1$.
If ${\mathbf{mode}}=\text{'Q'}$, c must be unaltered from the previous call to nag_smooth_fit_spline (g10ab) with ${\mathbf{mode}}=\text{'P'}$. Otherwise c need not be set.
6:     $\mathrm{comm}\left(9×{\mathbf{n}}+14\right)$ – double array
If ${\mathbf{mode}}=\text{'Q'}$, comm must be unaltered from the previous call to nag_smooth_fit_spline (g10ab) with ${\mathbf{mode}}=\text{'P'}$. Otherwise comm need not be set.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the arrays x, y, comm. (An error is raised if these dimensions are not equal.)
$n$, the number of distinct observations.
Constraint: ${\mathbf{n}}\ge 3$.
2:     $\mathrm{wt}\left(:\right)$ – double array
The dimension of the array wt must be at least ${\mathbf{n}}$ if $\mathit{weight}=\text{'W'}$
If $\mathit{weight}=\text{'W'}$, wt must contain the $n$ weights. Otherwise wt is not referenced and unit weights are assumed.
Constraint: if $\mathit{weight}=\text{'W'}$, ${\mathbf{wt}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,n$.

### Output Parameters

1:     $\mathrm{yhat}\left({\mathbf{n}}\right)$ – double array
The fitted values, ${\stackrel{^}{y}}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
2:     $\mathrm{c}\left(\mathit{ldc},3\right)$ – double array
If ${\mathbf{mode}}=\text{'F'}$, c contains the spline coefficients. More precisely, the value of the spline at $t$ is given by $\left(\left({\mathbf{c}}\left(i,3\right)×d+{\mathbf{c}}\left(i,2\right)\right)×d+{\mathbf{c}}\left(i,1\right)\right)×d+{\stackrel{^}{y}}_{i}$, where ${x}_{i}\le t<{x}_{i+1}$ and $d=t-{x}_{i}$.
If ${\mathbf{mode}}=\text{'P'}$ or $\text{'Q'}$, c contains information that will be used in a subsequent call to nag_smooth_fit_spline (g10ab) with ${\mathbf{mode}}=\text{'Q'}$.
3:     $\mathrm{rss}$ – double scalar
The (weighted) residual sum of squares.
4:     $\mathrm{df}$ – double scalar
The residual degrees of freedom.
5:     $\mathrm{res}\left({\mathbf{n}}\right)$ – double array
The (weighted) residuals, ${r}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
6:     $\mathrm{h}\left({\mathbf{n}}\right)$ – double array
The leverages, ${h}_{\mathit{i}\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
7:     $\mathrm{comm}\left(9×{\mathbf{n}}+14\right)$ – double array
If ${\mathbf{mode}}=\text{'P'}$ or $\text{'Q'}$, comm contains information that will be used in a subsequent call to nag_smooth_fit_spline (g10ab) with ${\mathbf{mode}}=\text{'Q'}$.
8:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<3$, or $\mathit{ldc}<{\mathbf{n}}-1$, or ${\mathbf{rho}}<0.0$, or ${\mathbf{mode}}\ne \text{'Q'}$, $\text{'P'}$ or $\text{'F'}$, or $\mathit{weight}\ne \text{'W'}$ or $\text{'U'}$.
${\mathbf{ifail}}=2$
 On entry, $\mathit{weight}=\text{'W'}$ and at least one element of ${\mathbf{wt}}\le 0.0$.
${\mathbf{ifail}}=3$
 On entry, ${\mathbf{x}}\left(i\right)\ge {\mathbf{x}}\left(i+1\right)$, for some $i$, $i=1,2,\dots ,n-1$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Accuracy depends on the value of $\rho$ and the position of the $x$ values. The values of ${x}_{i}-{x}_{i-1}$ and ${w}_{i}$ are scaled and $\rho$ is transformed to avoid underflow and overflow problems.

The time taken by nag_smooth_fit_spline (g10ab) is of order $n$.
Regression splines with a small $\left( number of knots can be fitted by nag_fit_1dspline_knots (e02ba) and nag_fit_1dspline_auto (e02be).

## 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 nag_smooth_data_order (g10za) and a series of splines fit using a range of values for the smoothing parameter, $\rho$.
function g10ab_example

fprintf('g10ab example results\n\n');

x =  [ 5.2  8.8 10.5 10.6 10.4  1.8 12.7 15.6  5.8  1.9 ...
2.2  4.8  7.9  5.2  0.9 11.8  7.9 11.5 10.6  8.5 ...
11.1 12.8 11.3  1.0 14.5 11.9  8.1 13.8 15.5  9.8 ...
11.0 12.4 11.1  5.1  4.8  4.2  6.9 13.2  9.9 12.5 ...
13.2  8.9 10.8];
y =  [ 4.8  4.1  5.2  5.5  5.0  3.4  3.4  4.9  5.6  3.7 ...
3.9  4.5  4.8  4.9  3.0  4.6  4.8  5.5  4.5  5.3 ...
4.7  6.6  5.1  3.9  5.7  5.1  5.2  3.7  4.9  4.8 ...
4.4  5.2  5.1  4.6  3.9  5.1  5.1  6.0  4.9  4.1 ...
4.6  4.9  5.1];

% Reorder x, remove ties and weight accordingly
[n, x, y, wt, rss, ifail] = g10za( ...
x, y);
x = x(1:n);
y = y(1:n);

rho  = [1 10 100];
nrho = numel(rho);

c    = zeros(n, 3);
comm = zeros(9*n+14, 1);
yhat = zeros(n,nrho);
df   = zeros(nrho,1);

% Initialize and fit for rho(1)
mode = 'P';
[yhat(:,1), c, rss(1), df(1), res, h, comm, ifail] = ...
g10ab(mode, x, y, rho(1), c, comm, 'wt', wt);

% Fit for subsequent rhos
mode = 'Q';
for j = 2:nrho
[yhat(:,j), c, rss(j), df(j), res, h, comm, ifail] = ...
g10ab( ...
mode, x, y, rho(j), c, comm, 'wt', wt);
end

%  Display results
fprintf('Smoothing coefficient (rho) = ');
fprintf('  %8.2f', rho);
fprintf('\nResidual sum of squares     = ');
fprintf('\nDegrees of freedom          = ');
fprintf('%10.3f', df);
fprintf('\n\n    x       y                            Fitted Values\n');
fprintf('%8.4f%8.4f%24.4f%10.4f%10.4f\n', [x y yhat]');

fig1 = figure;
plot(x,y,'+',x,yhat(:,1),x,yhat(:,2),x,yhat(:,3));
legend('Raw data', '\rho = 1', '\rho = 10', '\rho = 100', ...
'Location','NorthWest');
xlabel('Age (years)');
ylabel('C-peptide concentration (pmol/ml)');
title({'Cubic smoothing spline', ...
'Factors affecting insulin-dependent diabetis mellitus', ...
'in children; Hastie and Tibshirani (1990)'});


g10ab example results

Smoothing coefficient (rho) =       1.00     10.00    100.00
Residual sum of squares     =      9.118    11.288    11.881
Degrees of freedom          =     22.505    27.785    31.191

x       y                            Fitted Values
0.9000  3.0000                  3.3784    3.3674    3.3699
1.0000  3.9000                  3.4173    3.4008    3.4063
1.8000  3.4000                  3.6144    3.6642    3.6973
1.9000  3.7000                  3.6639    3.7016    3.7341
2.2000  3.9000                  3.8607    3.8214    3.8449
4.2000  5.1000                  4.7441    4.5265    4.5194
4.8000  4.2000                  4.4914    4.6471    4.6746
5.1000  4.6000                  4.6708    4.7561    4.7470
5.2000  4.8500                  4.7704    4.7993    4.7702
5.8000  5.6000                  5.3426    5.0458    4.8879
6.9000  5.1000                  5.1728    5.1204    4.9753
7.9000  4.8000                  4.9467    4.9590    4.9537
8.1000  5.2000                  4.9556    4.9262    4.9452
8.5000  5.3000                  4.8742    4.8595    4.9276
8.8000  4.1000                  4.7305    4.8172    4.9168
8.9000  4.9000                  4.7024    4.8095    4.9143
9.8000  4.8000                  4.8394    4.8676    4.9170
9.9000  4.9000                  4.8746    4.8818    4.9191
10.4000  5.0000                  4.9971    4.9445    4.9303
10.5000  5.2000                  4.9997    4.9521    4.9321
10.6000  5.0000                  4.9921    4.9572    4.9335
10.8000  5.1000                  4.9603    4.9613    4.9354
11.0000  4.4000                  4.9396    4.9614    4.9363
11.1000  4.9000                  4.9494    4.9618    4.9366
11.3000  5.1000                  4.9926    4.9623    4.9366
11.5000  5.5000                  5.0116    4.9568    4.9355
11.8000  4.6000                  4.9372    4.9338    4.9315
11.9000  5.1000                  4.9042    4.9251    4.9300
12.4000  5.2000                  4.7929    4.8943    4.9240
12.5000  4.1000                  4.8042    4.8944    4.9237
12.7000  3.4000                  4.9020    4.9051    4.9244
12.8000  6.6000                  4.9752    4.9138    4.9252
13.2000  5.3000                  5.0173    4.9239    4.9276
13.8000  3.7000                  4.6164    4.8930    4.9304
14.5000  5.7000                  5.1883    4.9938    4.9518
15.5000  4.9000                  4.9854    4.9773    4.9687
15.6000  4.9000                  4.9167    4.9682    4.9697 