e02ddc computes a bicubic spline approximation to a set of scattered data. The knots of the spline are located automatically, but a single argument must be specified to control the trade-off between closeness of fit and smoothness of fit.
The function may be called by the names: e02ddc, nag_fit_dim2_spline_sctr or nag_2d_spline_fit_scat.
3Description
e02ddc determines a smooth bicubic spline approximation $s(x,y)$ to the set of data points $({x}_{r},{y}_{r},{f}_{r})$ with weights ${w}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$.
The approximation domain is considered to be the rectangle $[{x}_{\mathrm{min}},{x}_{\mathrm{max}}]\times [{y}_{\mathrm{min}},{y}_{\mathrm{max}}]$, where ${x}_{\mathrm{min}}\left({y}_{\mathrm{min}}\right)$ and ${x}_{\mathrm{max}}\left({y}_{\mathrm{max}}\right)$ denote the lowest and highest data values of $x\left(y\right)$.
The spline is given in the B-spline representation
where ${M}_{i}\left(x\right)$ and ${N}_{j}\left(y\right)$ denote normalized cubic B-splines, the former defined on the knots ${\lambda}_{i}$ to ${\lambda}_{i+4}$ and the latter on the knots ${\mu}_{j}$ to ${\mu}_{j+4}$. 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 ${\lambda}_{1},\dots ,{\lambda}_{{n}_{x}}$ and ${\mu}_{1},\dots ,{\mu}_{{n}_{y}}$ are chosen automatically by the function. The knots ${\lambda}_{5},\dots ,{\lambda}_{{n}_{x}-4}$ and ${\mu}_{5},\dots ,{\mu}_{{n}_{y}-4}$ are the interior knots; they divide the approximation domain $[{x}_{\mathrm{min}},{x}_{\mathrm{max}}]\times [{y}_{\mathrm{min}},{y}_{\mathrm{max}}]$ into $({n}_{x}-7)\times ({n}_{y}-7)$ subpanels $[{\lambda}_{\mathit{i}},{\lambda}_{\mathit{i}+1}]\times [{\mu}_{\mathit{i}},{\mu}_{\mathit{i}+1}]$, for $\mathit{i}=4,5,\dots ,{n}_{x}-4$ and $\mathit{j}=4,5,\dots ,{n}_{y}-4$. Then, much as in the curve case (see e02bec);, the coefficients ${c}_{ij}$ are determined as the solution of the following constrained minimization problem:
where $\eta $ 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 (1981b) for details). ${\epsilon}_{r}$ denotes the weighted residual ${w}_{r}({f}_{r}-s({x}_{r},{y}_{r}))$, and $S$ is a non-negative number to be specified.
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 method would return an interpolating spline $(\theta =0)$ if $S$ were set to zero, and the least squares bicubic polynomial $(\eta =0)$ if $S$ is set very large. Experimenting with $S$ values between these two extremes should result in a good compromise. (See Section 9.3 for advice on choice of $S$.) Note however, that this function, unlike e02bec and e02dcc, does not allow $S$ to be set exactly to zero.
The method employed is outlined in Section 9.5 and fully described in Dierckx (1981a) and Dierckx (1981b). It involves an adaptive strategy for locating the knots of the bicubic spline (depending on the function underlying the data and on the value of $S$), and an iterative method for solving the constrained minimization problem once the knots have been determined.
Values and derivatives of the computed spline can subsequently be computed by calling e02dec,e02dfcande02dhc as described in Section 9.6.
4References
de Boor C (1972) On calculating with B-splines J. Approx. Theory6 50–62
Dierckx P (1981a) An improved algorithm for curve fitting with spline functions Report TW54 Department of Computer Science, Katholieke Univerciteit Leuven
Dierckx P (1981b) An algorithm for surface fitting with spline functions IMA J. Numer. Anal.1 267–283
Hayes J G and Halliday J (1974) The least squares fitting of cubic spline surfaces to general data sets J. Inst. Math. Appl.14 89–103
Peters G and Wilkinson J H (1970) The least squares problem and pseudo-inverses Comput. J.13 309–316
Reinsch C H (1967) Smoothing by spline functions Numer. Math.10 177–183
5Arguments
1: $\mathbf{start}$ – Nag_StartInput
On entry: start must be set to ${\mathbf{start}}=\mathrm{Nag\_Cold}$ or $\mathrm{Nag\_Warm}$.
The function will build up the knot set starting with no interior knots. No values need be assigned to $\mathbf{spline}\mathbf{\to}\mathbf{nx}$ and $\mathbf{spline}\mathbf{\to}\mathbf{ny}$ and memory will be internally allocated to $\mathbf{spline}\mathbf{\to}\mathbf{lamda}$, $\mathbf{spline}\mathbf{\to}\mathbf{mu}$ and $\mathbf{spline}\mathbf{\to}\mathbf{c}$.
The function will restart the knot-placing strategy using the knots found in a previous call of the function. In this case, all arguments except s must be unchanged from that previous call. This warm start can save much time in searching for a satisfactory value of $S$.
Constraint:
${\mathbf{start}}=\mathrm{Nag\_Cold}$ or $\mathrm{Nag\_Warm}$.
2: $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of data points.
The number of data points with nonzero weight (see weights) must be at least 16.
On entry: ${\mathbf{x}}\left[\mathit{r}-1\right]$, ${\mathbf{y}}\left[\mathit{r}-1\right]$, ${\mathbf{f}}\left[\mathit{r}-1\right]$ must be set to the coordinates of $({x}_{\mathit{r}},{y}_{\mathit{r}},{f}_{\mathit{r}})$, the $\mathit{r}$th data point, for $\mathit{r}=1,2,\dots ,m$. The order of the data points is immaterial.
On entry: ${\mathbf{weights}}\left[\mathit{r}-1\right]$ must be set to ${w}_{\mathit{r}}$, the $\mathit{r}$th value in the set of weights, for $\mathit{r}=1,2,\dots ,m$. Zero weights are permitted and the corresponding points are ignored, except when determining ${x}_{\mathrm{min}}$, ${x}_{\mathrm{max}}$, ${y}_{\mathrm{min}}$ and ${y}_{\mathrm{max}}$ (see Section 9.4). For advice on the choice of weights, see the E02 Chapter Introduction.
Constraint:
the number of data points with nonzero weight must be at least 16.
7: $\mathbf{s}$ – doubleInput
On entry: the smoothing factor, $S$. For advice on the choice of $S$, see Section 3 and Section 9.2.
Constraint:
${\mathbf{s}}>0.0$.
8: $\mathbf{nxest}$ – IntegerInput
9: $\mathbf{nyest}$ – IntegerInput
On entry: an upper bound for the number of knots ${n}_{x}$ and ${n}_{y}$ required in the $x$ and $y$ directions respectively. In most practical situations, ${\mathbf{nxest}}={\mathbf{nyest}}=5+\sqrt{m}$ is sufficient. See also Section 9.3.
Constraint:
${\mathbf{nxest}}\ge 8$ and ${\mathbf{nyest}}\ge 8$.
10: $\mathbf{fp}$ – double *Output
On exit: the weighted sum of squared residuals, $\theta $, of the computed spline approximation. fp should equal $S$ within a relative tolerance of $0.001$ unless $\mathbf{spline}\mathbf{\to}\mathbf{nx}=\mathbf{spline}\mathbf{\to}\mathbf{ny}=8$, when the spline has no interior knots and so is simply a bicubic polynomial. For knots to be inserted, $S$ must be set to a value below the value of fp produced in this case.
11: $\mathbf{rank}$ – Integer *Output
On exit: rank gives the rank of the system of equations used to compute the final spline (as determined by a suitable machine-dependent threshold). When ${\mathbf{rank}}=(\mathbf{spline}\mathbf{\to}\mathbf{nx}-4)\times (\mathbf{spline}\mathbf{\to}\mathbf{ny}-4)$, the solution is unique; otherwise the system is rank-deficient and the minimum-norm solution is computed. The latter case may be caused by too small a value of $S$.
12: $\mathbf{warmstartinf}$ – double *Output
On exit: if the warm start option is used, its value must be left unchanged from the previous call.
13: $\mathbf{spline}$ – Nag_2dSpline *
Pointer to structure of type Nag_2dSpline with the following members:
nx – IntegerInput/Output
On entry: if the warm start option is used, the value of $\mathbf{nx}$ must be left unchanged from the previous call.
On exit: the total number of knots, ${n}_{x}$, of the computed spline with respect to the $x$ variable.
lamda – double *Input/Output
On entry: a pointer to which if ${\mathbf{start}}=\mathrm{Nag\_Cold}$, memory of size nxest is internally allocated. If the warm start option is used, the values $\mathbf{lamda}\left[0\right],\mathbf{lamda}\left[1\right],\dots ,\mathbf{lamda}\left[\mathbf{nx}-1\right]$ must be left unchanged from the previous call.
On exit: $\mathbf{lamda}$ contains the complete set of knots ${\lambda}_{i}$ associated with the $x$ variable, i.e., the interior knots $\mathbf{lamda}\left[4\right],\mathbf{lamda}\left[5\right],\dots ,\mathbf{lamda}\left[\mathbf{nx}-5\right]$ as well as the additional knots $\mathbf{lamda}\left[0\right]=\mathbf{lamda}\left[1\right]=\mathbf{lamda}\left[2\right]=\mathbf{lamda}\left[3\right]={x}_{\mathrm{min}}$ and $\mathbf{lamda}\left[\mathbf{nx}-4\right]=\mathbf{lamda}\left[\mathbf{nx}-3\right]=\mathbf{lamda}\left[\mathbf{nx}-2\right]=\mathbf{lamda}\left[\mathbf{nx}-1\right]={x}_{\mathrm{max}}$ needed for the B-spline representation (where ${x}_{\mathrm{min}}$ and ${x}_{\mathrm{max}}$ are as described in Section 3).
ny – IntegerInput/Output
On entry: if the warm start option is used, the value of $\mathbf{ny}$ must be left unchanged from the previous call.
On exit: the total number of knots, ${n}_{y}$, of the computed spline with respect to the $y$ variable.
mu – double *Input/Output
On entry: a pointer to which if ${\mathbf{start}}=\mathrm{Nag\_Cold}$, memory of size nyest is internally allocated. If the warm start option is used, the values $\mathbf{mu}\left[0\right],\mathbf{mu}\left[1\right],\dots ,\mathbf{mu}\left[\mathbf{ny}-1\right]$ must be left unchanged from the previous call.
On exit: $\mathbf{mu}$ contains the complete set of knots ${\mu}_{i}$ associated with the $y$ variable, i.e., the interior knots $\mathbf{mu}\left[4\right]$, $\mathbf{mu}\left[5\right]$, $\dots $, $\mathbf{mu}\left[\mathbf{ny}-5\right]$ as well as the additional knots $\mathbf{mu}\left[0\right]=\mathbf{mu}\left[1\right]=\mathbf{mu}\left[2\right]=\mathbf{mu}\left[3\right]={y}_{\mathrm{min}}$ and $\mathbf{mu}\left[\mathbf{ny}-4\right]=\mathbf{mu}\left[\mathbf{ny}-3\right]=\mathbf{mu}\left[\mathbf{ny}-2\right]=\mathbf{mu}\left[\mathbf{ny}-1\right]={y}_{\mathrm{max}}$ needed for the B-spline representation (where ${y}_{\mathrm{min}}$ and ${y}_{\mathrm{max}}$ are as described in Section 3).
c – double *Output
On exit: a pointer to which, if ${\mathbf{start}}=\mathrm{Nag\_Cold}$, memory of size $({\mathbf{nxest}}-4)\times ({\mathbf{nyest}}-4)$ is internally allocated. $\mathbf{c}\left[({n}_{y}-4)\times (i-1)+j-1\right]$ is the coefficient ${c}_{ij}$ defined in Section 3.
Note that when the information contained in the pointers $\mathbf{lamda}$, $\mathbf{mu}$ and $\mathbf{c}$ is no longer of use, or before a new call to e02ddc with the same spline, you should free this storage using the NAG macro NAG_FREE. This storage will have been allocated only if this function returns with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=\mathrm{NE\_NOERROR}$, NE_NUM_KNOTS_2D_GT_SCAT,
NE_NUM_COEFF_GT,
NE_NO_ADDITIONAL_KNOTS,
NE_SPLINE_COEFF_CONV or, possibly,
NE_ALLOC_FAIL.
14: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALL_ELEMENTS_EQUAL
On entry, all the values in the array x must not be equal.
On entry, all the values in the array y must not be equal.
${\mathbf{start}}=\mathrm{Nag\_Warm}$ at the first call of this function. start must be set to ${\mathbf{start}}=\mathrm{Nag\_Cold}$ at the first call.
NE_INT_ARG_LT
On entry, ${\mathbf{nxest}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nxest}}\ge 8$.
On entry, ${\mathbf{nyest}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nyest}}\ge 8$.
NE_NO_ADDITIONAL_KNOTS
No more knots added; the additional knot would coincide with an old one. Possibly an inaccurate data point has too large a weight, or s is too small. ${\mathbf{s}}=\u27e8\mathit{\text{value}}\u27e9$. A spline approximation is returned, but it fails to satisfy the fitting criterion (see (2) and (3)) – perhaps by only a small amount, however.
NE_NON_ZERO_WEIGHTS
On entry, the number of data points with nonzero weights $\text{}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: the number of nonzero weights $\ge 16$.
NE_NUM_COEFF_GT
No more knots can be added because the number of B-spline coefficients already exceeds m. Either m or s is probably too small: ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{s}}=\u27e8\mathit{\text{value}}\u27e9$. A spline approximation is returned, but it fails to satisfy the fitting criterion (see (2) and (3)) – perhaps by only a small amount, however.
NE_NUM_KNOTS_2D_GT_SCAT
The number of knots required is greater than allowed by nxest or nyest, ${\mathbf{nxest}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{nyest}}=\u27e8\mathit{\text{value}}\u27e9$. Possibly s is too small, especially if nxest, ${\mathbf{nyest}}>5+\sqrt{{\mathbf{m}}}$. ${\mathbf{s}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$. A spline approximation is returned, but it fails to satisfy the fitting criterion (see (2) and (3)) – perhaps by only a small amount, however.
NE_REAL_ARG_LE
On entry, s must not be less than or equal to 0.0: ${\mathbf{s}}=\u27e8\mathit{\text{value}}\u27e9$.
NE_SPLINE_COEFF_CONV
The iterative process has failed to converge. Possibly s is too small: ${\mathbf{s}}=\u27e8\mathit{\text{value}}\u27e9$. A spline approximation is returned, but it fails to satisfy the fitting criterion (see (2) and (3)) – perhaps by only a small amount, however.
7Accuracy
On successful exit, the approximation returned is such that its weighted sum of squared residuals fp is equal to the smoothing factor $S$, up to a specified relative tolerance of $0.001$ – except that if ${n}_{x}=8$ and ${n}_{y}=8$, fp may be significantly less than $S$: in this case the computed spline is simply the least squares bicubic polynomial approximation of degree $3$, i.e., a spline with no interior knots.
8Parallelism and Performance
e02ddc is not threaded in any implementation.
9Further Comments
9.1Timing
The time taken for a call of e02ddc depends on the complexity of the shape of the data, the value of the smoothing factor $S$, and the number of data points. If e02ddc is to be called for different values of $S$, much time can be saved by setting ${\mathbf{start}}=\mathrm{Nag\_Warm}$ after the first call.
It should be noted that choosing $S$ very small considerably increases computation time.
9.2Choice of $\mathit{S}$
If the weights have been correctly chosen (see the E02 Chapter Introduction), the standard deviation of ${w}_{r}{f}_{r}$ would be the same for all $r$, equal to $\sigma $, say. In this case, choosing the smoothing factor $S$ in the range ${\sigma}^{2}(m\pm \sqrt{2m})$, as suggested by Reinsch (1967), is likely to give a good start in the search for a satisfactory value. Otherwise, experimenting with different values of $S$ will be required from the start.
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 ${{\mathbf{fp}}}_{0}$, 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., $S={{\mathbf{fp}}}_{0}/10$, $S={{\mathbf{fp}}}_{0}/100$, and so on, and more carefully as the approximation shows more details.
To choose $S$ very small is strongly discouraged. This considerably increases computation time and memory requirements. It may also cause rank-deficiency (as indicated by the argument rank) and endanger numerical stability.
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 e02ddc is called with ${\mathbf{start}}=\mathrm{Nag\_Warm}$, 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 ${\mathbf{start}}=\mathrm{Nag\_Warm}$, a fit can finally be accepted as satisfactory, it may be worthwhile to call e02ddc once more with the selected value for $S$ but now using ${\mathbf{start}}=\mathrm{Nag\_Cold}$. Often, e02ddc then returns an approximation with the same quality of fit but with fewer knots, which is, therefore, better if data reduction is also important.
9.3Choice 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 e02ddc 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 $\left(y\right)$ direction. This may indicate that the value of nxest is too small. On the other hand, it gives you the option of limiting the number of knots the function locates in any direction. For example, by setting ${\mathbf{nxest}}=8$ (the lowest allowable value for nxest), you can indicate that you want an approximation which is a simple cubic polynomial in the variable $x$.
9.4Restriction of the Approximation Domain
The fit obtained is not defined outside the rectangle $[{\lambda}_{4},{\lambda}_{{n}_{x}-3}]\times [{\mu}_{4},{\mu}_{{n}_{y}-3}]$. The reason for taking the extreme data values of $x$ and $y$ for these four knots is that, as is usual in data fitting, the fit cannot be expected to give satisfactory values outside the data region. If, nevertheless, you require values over a larger rectangle, this can be achieved by augmenting the data with two artificial data points $(a,c,0)$ and $(b,d,0)$ with zero weight, where $[a,b]\times [c,d]$ denotes the enlarged rectangle.
9.5Outline of Method Used
First 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 $\theta $, the sum of squares of residuals, is computed. If $\theta >S$, a new knot is added to one knot set or the other so as to reduce $\theta $ at the next stage. The new knot is located in an interval where the fit is particularly poor. Sooner or later, we find that $\theta \le S$ and at that point the knot sets are accepted. The function then goes on to compute a spline which has these knot sets and which satisfies the full fitting criterion specified by 2 and $3$. The theoretical solution has $\theta =S$. The function computes the spline by an iterative scheme which is ended when $\theta =S$ within a relative tolerance of $0.001$. The main part of each iteration consists of a linear least squares computation of special form. The minimal least squares solution is computed wherever the linear system is found to be rank-deficient.
An exception occurs when the function finds at the start that, even with no interior knots $({n}_{x}={n}_{y}=8)$, the least squares spline already has its sum of squares of residuals $\le S$. In this case, since this spline (which is simply a bicubic polynomial) also has an optimal value for the smoothness measure $\eta $, namely zero, it is returned at once as the (trivial) solution. It will usually mean that $S$ has been chosen too large.
For further details of the algorithm and its use see Dierckx (1981b).
9.6Evaluation of Computed Spline
The values of the computed spline at the points $(\mathbf{tx}[\mathit{r}-1],\mathbf{ty}[\mathit{r}-1])$, for $\mathit{r}=1,2,\dots ,\mathbf{n}$, may be obtained in the array ff, of length at least n, by the following code:
e02dec(n, tx, ty, ff, &spline, &fail)
where spline is a structure of type Nag_2dSpline which is an output argument of e02ddc.
To evaluate the computed spline on a kx by ky rectangular grid of points in the $x$-$y$ plane, which is defined by the $x$ coordinates stored in $\mathbf{tx}[\mathit{q}-1]$, for $\mathit{q}=1,2,\dots ,\mathbf{kx}$, and the $y$ coordinates stored in $\mathbf{ty}[\mathit{r}-1]$, for $\mathit{r}=1,2,\dots ,\mathbf{ky}$, returning the results in the array fg which is of length at least $\mathbf{kx}\times \mathbf{ky}$, the following call may be used:
e02dfc(kx, ky, tx, ty, fg, &spline, &fail)
where spline is a structure of type Nag_2dSpline which is an output argument of e02ddc. The result of the spline evaluated at grid point $(q,r)$ is returned in element $[\mathbf{ky}\times (q-1)+r-1]$ of the array fg.
10Example
This example program reads in a value of m, followed by a set of m data points $({x}_{r},{y}_{r},{f}_{r})$ and their weights ${w}_{r}$. It then calls e02ddc 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.