e02dc:: Curve and Surface Fitting (NAG Toolbox)

The spline is given in the B-spline representation

s (x, y) = \sum_{i = 1}^{n_{x} - 4} \sum_{j = 1}^{n_{y} - 4} c_{i j} M_{i} (x) N_{j} (y),

(1)

where

M_{i} (x)

and

N_{j} (y)

denote normalized cubic B-splines, the former defined on the knots

λ_{i}

λ_{i + 4}

and the latter on the knots

μ_{j}

μ_{j + 4}

. For further details, see Hayes and Halliday (1974) for bicubic splines and de Boor (1972) for normalized B-splines.

The total numbers

n_{x}

and

n_{y}

of these knots and their values

λ_{1}, \dots, λ_{n_{x}}

and

μ_{1}, \dots, μ_{n_{y}}

are chosen automatically by the function. The knots

λ_{5}, \dots, λ_{n_{x} - 4}

and

μ_{5}, \dots, μ_{n_{y} - 4}

are the interior knots; they divide the approximation domain

[x_{1}, x_{m_{x}}] \times [y_{1}, y_{m_{y}}]

into

(n_{x} - 7) \times (n_{y} - 7)

subpanels

[λ_{i}, λ_{i + 1}] \times [μ_{j}, μ_{j + 1}]

, for

i = 4, 5, \dots, n_{x} - 4

and

j = 4, 5, \dots, n_{y} - 4

. Then, much as in the curve case (see nag_fit_1dspline_auto (e02be)), the coefficients

c_{i j}

are determined as the solution of the following constrained minimization problem:

minimize η,

(2)

subject to the constraint

θ = \sum_{q = 1}^{m_{x}} \sum_{r = 1}^{m_{y}} ε_{q, r}^{2} \leq S,

(3)

where	$η$	is a measure of the (lack of) smoothness of $s (x, y)$ . Its value depends on the discontinuity jumps in $s (x, y)$ across the boundaries of the subpanels. It is zero only when there are no discontinuities and is positive otherwise, increasing with the size of the jumps (see Dierckx (1982) for details).
	$ε_{q, r}$	denotes the residual $f_{q, r} - s (x_{q}, y_{r})$ ,
and	$S$	is a non-negative number specified by you.

By means of the argument

S

, ‘the smoothing factor’, you will then control the balance between smoothness and closeness of fit, as measured by the sum of squares of residuals in (3). If

S

is too large, the spline will be too smooth and signal will be lost (underfit); if

S

is too small, the spline will pick up too much noise (overfit). In the extreme cases the function will return an interpolating spline

(θ = 0)

S

is set to zero, and the least squares bicubic polynomial

(η = 0)

S

is set very large. Experimenting with

S

-values between these two extremes should result in a good compromise. (See Choice of s for advice on choice of

S

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Timing

Weighting of Data Points

Choice of s

If the standard deviation of

f_{q, r}

is the same for all

q

and

r

(the case for which nag_fit_2dspline_grid (e02dc) is designed – see Weighting of Data Points.) and known to be equal, at least approximately, to

σ

, say, then following Reinsch (1967) and choosing the argument s in the range

σ^{2} (m \pm \sqrt{2 m})

, where

m = m_{x} m_{y}

, is likely to give a good start in the search for a satisfactory value. If the standard deviations vary, the sum of their squares over all the data points could be used. Otherwise experimenting with different values of s will be required from the start, taking account of the remarks in Description.

In that case, in view of computation time and memory requirements, it is recommended to start with a very large value for s and so determine the least squares bicubic polynomial; the value returned for fp, call it

{fp}_{0}

, gives an upper bound for s. Then progressively decrease the value of s to obtain closer fits – say by a factor of

10

in the beginning, i.e.,

s = {fp}_{0} / 10

s = {fp}_{0} / 100

, and so on, and more carefully as the approximation shows more details.

The number of knots of the spline returned, and their location, generally depend on the value of

s

and on the behaviour of the function underlying the data. However, if nag_fit_2dspline_grid (e02dc) is called with

start ='W'

, the knots returned may also depend on the smoothing factors of the previous calls. Therefore if, after a number of trials with different values of

s

and

start ='W'

, a fit can finally be accepted as satisfactory, it may be worthwhile to call nag_fit_2dspline_grid (e02dc) once more with the selected value for

s

but now using

start ='C'

. Often, nag_fit_2dspline_grid (e02dc) then returns an approximation with the same quality of fit but with fewer knots, which is therefore better if data reduction is also important.

Choice of nxest and nyest

The number of knots may also depend on the upper bounds nxest and nyest. Indeed, if at a certain stage in nag_fit_2dspline_grid (e02dc) the number of knots in one direction (say

n_{x}

) has reached the value of its upper bound (nxest), then from that moment on all subsequent knots are added in the other

(y)

direction. Therefore you have the option of limiting the number of knots the function locates in any direction. For example, by setting

nxest = 8

(the lowest allowable value for nxest), you can indicate that you want an approximation which is a simple cubic polynomial in the variable

x

Outline of Method Used

S > 0

, suitable knot sets are built up in stages (starting with no interior knots in the case of a cold start but with the knot set found in a previous call if a warm start is chosen). At each stage, a bicubic spline is fitted to the data by least squares, and

θ

, the sum of squares of residuals, is computed. If

θ > S

, new knots are added to one knot set or the other so as to reduce

θ

at the next stage. The new knots are located in intervals where the fit is particularly poor, their number depending on the value of

S

and on the progress made so far in reducing

θ

. Sooner or later, we find that

θ \leq S

and at that point the knot sets are accepted. The function then goes on to compute the (unique) spline which has these knot sets and which satisfies the full fitting criterion specified by (2) and (3). The theoretical solution has

θ = S

. The function computes the spline by an iterative scheme which is ended when

θ = S

within a relative tolerance of

0.001

. The main part of each iteration consists of a linear least squares computation of special form, done in a similarly stable and efficient manner as in nag_fit_1dspline_knots (e02ba) for least squares curve-fitting.

Evaluation of Computed Spline

The values of the computed spline at the points

(x_{r}, y_{r})

, for

r = 1, 2, \dots, m

, may be obtained in the double array ff (see nag_fit_2dspline_evalv (e02de)), of length at least

m

, by the following call:

[ff, ifail] = e02de(x, y, lamda, mu, c);

where

M = m

and the coordinates

x_{r}

y_{r}

are stored in

X (k)

Y (k)

. PX and PY have the same values as nx and ny as output from nag_fit_2dspline_grid (e02dc), and LAMDA, MU and C have the same values as lamda, mu and c output from nag_fit_2dspline_grid (e02dc). WRK is a double workspace array of length at least

PY - 4

, and IWRK is an integer workspace array of length at least

PY - 4

To evaluate the computed spline on a

m_{x}

m_{y}

rectangular grid of points in the

x

y

plane, which is defined by the

x

coordinates stored in

X (q)

, for

q = 1, 2, \dots, m_{x}

, and the

y

coordinates stored in

Y (r)

, for

r = 1, 2, \dots, m_{y}

, returning the results in the double array ff (see nag_fit_2dspline_evalm (e02df)) which is of length at least

mx \times my

, the following call may be used:

[ff, ifail] = e02df(x, y, lamda, mu, c);

where

MX = m_{x}

MY = m_{y}

. LAMDA, MU and C have the same values as lamda, mu and c output from nag_fit_2dspline_grid (e02dc). WRK is a double workspace array of length at least

LWRK = \min (nwrk1, nwrk2)

, where

nwrk1 = MX \times 4 + PX

and

nwrk2 = MY \times 4 + PY

. IWRK is an integer workspace array of length at least

LIWRK = MY + PY - 4

nwrk1 \geq nwrk2

, or

MX + PX - 4

otherwise.

Example

This example reads in values of mx, my,

x_{q}

, for

q = 1, 2, \dots, mx

, and

y_{r}

, for

r = 1, 2, \dots, my

, followed by values of the ordinates

f_{q, r}

defined at the grid points

(x_{q}, y_{r})

. It then calls nag_fit_2dspline_grid (e02dc) to compute a bicubic spline approximation for one specified value of s, and prints the values of the computed knots and B-spline coefficients. Finally it evaluates the spline at a small sample of points on a rectangular grid.

function e02dc_example


fprintf('e02dc example results\n\n');

start = 'C';
x = [0:0.5:5];
y = [0:0.5:4];
f = [ 1         1.5      2.06     2.57     3        3.5;
      0.88758   1.3564   1.7552   2.124    2.6427   3.1715;
      0.5403    0.82045  1.0806   1.3508   1.6309   1.8611;
      0.070737  0.10611  0.15147  0.17684  0.21221  0.24458;
     -0.41515  -0.62422 -0.83229 -1.0404  -1.2484  -1.4565;
     -0.80114  -1.2317  -1.6023  -2.0029  -2.2034  -2.864;
     -0.97999  -1.485   -1.97    -2.475   -2.97    -3.265;
     -0.93446  -1.3047  -1.8729  -2.3511  -2.8094  -3.2776;
     -0.65664  -0.98547 -1.4073  -1.6741  -1.9809  -2.2878];

f(:,7:11) = [   4.04     4.5      5.04     5.505    6;
                3.5103   3.9391   4.3879   4.8367   5.2755;  
                2.0612   2.4314   2.7515   2.9717   3.2418;  
                0.28595  0.31632  0.35369  0.38505  0.42442; 
               -1.6946  -1.8627  -2.0707  -2.2888  -2.4769;  
               -3.2046  -3.6351  -4.0057  -4.4033  -4.8169;  
               -3.96    -4.455   -4.97    -5.445   -5.93;    
               -3.7958  -4.2141  -4.6823  -5.1405  -5.6387;  
               -2.6146  -2.9314  -3.2382  -3.595   -3.9319];

% Input argument initializations
s     = 0.1;
nx    = int64(0); ny = nx;
mx    = size(x,2);
my    = size(y,2);
lamda = zeros(mx+4, 1);
mu    = zeros(my+4, 1);
wrk   = zeros(1000, 1);
iwrk = zeros(100, 1, 'int64');

% Compute bi-cubic spline for grid data
[nx, lamda, ny, mu, c, fp, wrk, iwrk, ifail] = ...
e02dc( ...
       start, x, y, f, s, nx, lamda, ny, mu, wrk, iwrk);

% Print details of spline
fprintf('\nCalling with smoothing factor S = %5.2f\n\n', s);
fprintf('Knots:   lamda      mu\n');
for j = 4:max(nx,ny)-3
  if j<=min(nx,ny)-3
    fprintf('%4d%10.4f%10.4f\n', j, lamda(j), mu(j));
  elseif j<=nx-3
    fprintf('%4d%10.4f\n', j, lamda(j));
  else
    fprintf('%4d%20.4f\n', j, mu(j));
  end
end

cp = c(1:(ny-4)*(nx-4));
cp = reshape(cp,[ny-4,nx-4]);
fprintf('\nB-spline coefficients:\n');
disp(cp);

fprintf('Weighted sum of squared residuals = %7.4f\n', fp);
if fp==0
  fprintf('(The spline is an interpolating spline)\n');
elseif nx==8 && ny==8
  fprintf('(The spline is the weighted least-squares cubic polynomial)\n');
end
fprintf('\n');

% Evaluate spline on mesh
mx = [0:0.2:5];
my = [0:0.2:4];
[ff, ifail] = e02df( ...
                     mx, my, lamda(1:nx), mu(1:ny), c);

fig1 = figure;
ff = reshape(ff,[21,26]);
meshc(mx,my,ff);
xlabel('x');
ylabel('y');
title('Least-squares bi-cubic spline fit');

e02dc example results


Calling with smoothing factor S =  0.10

Knots:   lamda      mu
   4    0.0000    0.0000
   5    1.5000    1.0000
   6    2.5000    2.0000
   7    5.0000    2.5000
   8              3.0000
   9              3.5000
  10              4.0000

B-spline coefficients:
    0.9918    1.5381    2.3913    3.9845    5.2138    5.9965
    1.0546    1.5270    2.2441    4.2217    5.0860    6.0821
    0.6098    0.9557    1.5587    2.3458    3.3860    3.7716
   -0.2915   -0.4199   -0.7399   -1.1763   -1.5527   -1.7775
   -0.8476   -1.3296   -1.8521   -3.3468   -4.3628   -5.0085
   -1.0168   -1.5952   -2.4022   -3.9390   -5.4680   -6.1656
   -0.9529   -1.3381   -2.2844   -3.9559   -5.0032   -5.8709
   -0.7711   -1.0914   -1.8488   -3.2549   -3.9444   -4.7297
   -0.6476   -1.0373   -1.5936   -2.5887   -3.3485   -3.9330

Weighted sum of squared residuals =  0.1000

On entry,	$start \neq'C'$ or $'W'$ ,
or	$mx < 4$ ,
or	$my < 4$ ,
or	$s < 0.0$ ,
or	$s = 0.0$ and $nxest < mx + 4$ ,
or	$s = 0.0$ and $nyest < my + 4$ ,
or	$nxest < 8$ ,
or	$nyest < 8$ ,
or	$lwrk < 4 \times (mx + my) + 11 \times (nxest + nyest) + nxest \times my + \max (my, nxest) + 54$ ,
or	$liwrk < 3 + mx + my + nxest + nyest$ .

NAG Toolbox: nag_fit_2dspline_grid (e02dc)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Timing

Weighting of Data Points

Choice of s

Choice of nxest and nyest

Outline of Method Used

Evaluation of Computed Spline

Example