e02bac computes a weighted least squares approximation to an arbitrary set of data points by a cubic spline with knots prescribed by you. Cubic spline interpolation can also be carried out.
The function may be called by the names: e02bac, nag_fit_dim1_spline_knots or nag_1d_spline_fit_knots.
3Description
e02bac determines a least squares cubic spline approximation $s\left(x\right)$ to the set of data points $({x}_{\mathit{r}},{y}_{\mathit{r}})$ with weights ${w}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$. The value of $\mathbf{spline}\mathbf{\to}\mathbf{n}=\overline{n}+7$, where $\overline{n}$ is the number of intervals of the spline (one greater than the number of interior knots), and the values of the knots ${\lambda}_{5},{\lambda}_{6},\dots ,{\lambda}_{\overline{n}+3}$, interior to the data interval, are prescribed by you.
$s\left(x\right)$ has the property that it minimizes $\theta $, the sum of squares of the weighted residuals ${\epsilon}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,m$, where
The function produces this minimizing value of $\theta $ and the coefficients ${c}_{1},{c}_{2},\dots ,{c}_{q}$, where $q=\overline{n}+3$, in the B-spline representation
Here ${N}_{i}\left(x\right)$ denotes the normalized B-spline of degree 3 defined upon the knots ${\lambda}_{i},{\lambda}_{i+1},\dots ,{\lambda}_{i+4}$.
In order to define the full set of B-splines required, eight additional knots ${\lambda}_{1},{\lambda}_{2},{\lambda}_{3},{\lambda}_{4}$ and ${\lambda}_{\overline{n}+4},{\lambda}_{\overline{n}+5}$, ${\lambda}_{\overline{n}+6},{\lambda}_{\overline{n}+7}$ are inserted automatically by the function. The first four of these are set equal to the smallest ${x}_{r}$ and the last four to the largest ${x}_{r}$.
The representation of $s\left(x\right)$ in terms of B-splines is the most compact form possible in that only $\overline{n}+3$ coefficients, in addition to the $\overline{n}+7$ knots, fully define $s\left(x\right)$.
The method employed involves forming and then computing the least squares solution of a set of $m$ linear equations in the coefficients ${c}_{i}(i=1,2,\dots ,\overline{n}+3)$. The equations are formed using a recurrence relation for B-splines that is unconditionally stable (Cox (1972), de Boor (1972)), even for multiple (coincident) knots. The least squares solution is also obtained in a stable manner by using orthogonal transformations, viz. a variant of Givens rotations (Gentleman (1974) and Gentleman (1973)). This requires only one equation to be stored at a time. Full advantage is taken of the structure of the equations, there being at most four nonzero values of ${N}_{i}\left(x\right)$ for any value of $x$ and hence at most four coefficients in each equation.
Subsequent evaluation of $s\left(x\right)$ from its B-spline representation may be carried out using e02bbc. If derivatives of $s\left(x\right)$ are also required, e02bcc may be used. e02bdc can be used to compute the definite integral of $s\left(x\right)$.
4References
Cox M G (1972) The numerical evaluation of B-splines J. Inst. Math. Appl.10 134–149
Cox M G (1974) A data-fitting package for the non-specialist user Software for Numerical Mathematics (ed D J Evans) Academic Press
Cox M G (1975) Numerical methods for the interpolation and approximation of data by spline functions PhD Thesis City University, London
Cox M G and Hayes J G (1973) Curve fitting: a guide and suite of algorithms for the non-specialist user NPL Report NAC26 National Physical Laboratory
de Boor C (1972) On calculating with B-splines J. Approx. Theory6 50–62
Gentleman W M (1973) Least squares computations by Givens transformations without square roots J. Inst. Math. Applic.12 329–336
Gentleman W M (1974) Algorithm AS 75. Basic procedures for large sparse or weighted linear least squares problems Appl. Statist.23 448–454
Schoenberg I J and Whitney A (1953) On Polya frequency functions III Trans. Amer. Math. Soc.74 246–259
5Arguments
1: $\mathbf{m}$ – IntegerInput
On entry: the number $m$ of data points.
Constraint:
${\mathbf{m}}\ge \mathit{mdist}\ge 4$, where $\mathit{mdist}$ is the number of distinct $x$ values in the data.
On entry: the values ${w}_{\mathit{r}}$ of the weights, for $\mathit{r}=1,2,\dots ,m$. For advice on the choice of weights, see the E02 Chapter Introduction.
Constraint:
${w}_{r}>0$, for $\mathit{r}=1,2,\dots ,m$.
5: $\mathbf{ss}$ – double *Output
On exit: the residual sum of squares, $\theta $.
6: $\mathbf{spline}$ – Nag_Spline *
Pointer to structure of type Nag_Spline with the following members:
n – IntegerInput
On entry: $\overline{n}+7$, where $\overline{n}$ is the number of intervals of the spline (which is one greater than the number of interior knots, i.e., the knots strictly within the range ${x}_{1}$ to ${x}_{m}$) over which the spline is defined.
Constraint:
$8\le \mathbf{n}\le \mathit{mdist}+4$, where $\mathit{mdist}$ is the number of distinct $x$ values in the data.
lamda – double *Input/Output
On entry: a pointer to which memory of size $\mathbf{n}$ must be allocated. $\mathbf{lamda}\left[\mathit{i}-1\right]$ must be set to the $(\mathit{i}-4)$th interior knot, ${\lambda}_{\mathit{i}}$, for $\mathit{i}=5,6,\dots ,\overline{n}+3$.
On exit: the input values are unchanged, and $\mathbf{lamda}\left[i\right]$, $i=0,1,2,3\text{,}\mathbf{n}-4$, $\mathbf{n}-3$, $\mathbf{n}-2$, $\mathbf{n}-1$ contains the additional exterior knots introduced by the function.
On exit: a pointer to which memory of size $\mathbf{n}-4$ is internally allocated. $\mathbf{c}$ holds the coefficient ${c}_{\mathit{i}}$ of the B-spline ${N}_{\mathit{i}}\left(x\right)$, for $\mathit{i}=1,2,\dots ,\overline{n}+3$.
Note that when the information contained in the pointers $\mathbf{lamda}$ and $\mathbf{c}$ is no longer of use, or before a new call to e02bac with the same spline, you should free this storage using the NAG macro NAG_FREE.
7: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_INT_ARG_LT
On entry, $\mathbf{spline}\mathbf{\to}\mathbf{n}$ must not be less than 8: $\mathbf{spline}\mathbf{\to}\mathbf{n}=\u27e8\mathit{\text{value}}\u27e9$.
NE_KNOTS_DISTINCT_ABSCI_CONS
Too many knots for the number of distinct abscissae, $\mathit{mdist}$: $\mathbf{spline}\mathbf{\to}\mathbf{n}=\u27e8\mathit{\text{value}}\u27e9$, $\mathit{mdist}=\u27e8\mathit{\text{value}}\u27e9$.
These must satisfy the constraint $\mathbf{spline}\mathbf{\to}\mathbf{n}\le \mathit{mdist}+4$.
NE_KNOTS_OUTSIDE_DATA_INTVL
On entry, user-specified knots must be interior to the data interval, $\mathbf{spline}\mathbf{\to}\mathbf{lamda}\left[4\right]$ must be greater than ${\mathbf{x}}\left[0\right]$ and $\mathbf{spline}\mathbf{\to}\mathbf{lamda}\left[\mathbf{spline}\mathbf{\to}\mathbf{n}-5\right]$ must be less than ${\mathbf{x}}\left[{\mathbf{m}}-1\right]$: $\mathbf{spline}\mathbf{\to}\mathbf{lamda}\left[4\right]=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{x}}\left[0\right]=\u27e8\mathit{\text{value}}\u27e9$, $\mathbf{spline}\mathbf{\to}\mathbf{lamda}\left[\u27e8\mathit{\text{value}}\u27e9\right]=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{x}}\left[\u27e8\mathit{\text{value}}\u27e9\right]=\u27e8\mathit{\text{value}}\u27e9$.
NE_NOT_INCREASING
The sequence $\mathbf{spline}\mathbf{\to}\mathbf{lamda}$ is not increasing: $\mathbf{spline}\mathbf{\to}\mathbf{lamda}\left[\u27e8\mathit{\text{value}}\u27e9\right]=\u27e8\mathit{\text{value}}\u27e9$, $\mathbf{spline}\mathbf{\to}\mathbf{lamda}\left[\u27e8\mathit{\text{value}}\u27e9\right]=\u27e8\mathit{\text{value}}\u27e9$.
This condition on $\mathbf{spline}\mathbf{\to}\mathbf{lamda}$ applies to user-specified knots in the interval $\mathbf{spline}\mathbf{\to}\mathbf{lamda}\left[4\right]$, $\mathbf{spline}\mathbf{\to}\mathbf{lamda}\left[\mathbf{spline}\mathbf{\to}\mathbf{n}-5\right]$.
The sequence x is not increasing: ${\mathbf{x}}\left[\u27e8\mathit{\text{value}}\u27e9\right]=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{x}}\left[\u27e8\mathit{\text{value}}\u27e9\right]=\u27e8\mathit{\text{value}}\u27e9$.
NE_SW_COND_FAIL
The conditions specified by Schoenberg and Whitney fail.
The conditions specified by Schoenberg and Whitney (1953) fail to hold for at least one subset of the distinct data abscissae. That is, there is no subset of $\mathbf{spline}\mathbf{\to}\mathbf{n}-4$ strictly increasing values, ${\mathbf{x}}\left[{r}_{0}\right]$, ${\mathbf{x}}\left[{r}_{1}\right],\dots ,{\mathbf{x}}\left[{r}_{\mathbf{spline}\mathbf{\to}\mathbf{n}-5}\right]$, among the abscissae such that
This means that there is no unique solution: there are regions containing too many knots compared with the number of data points.
NE_WEIGHTS_NOT_POSITIVE
On entry, the weights are not strictly positive: ${\mathbf{weights}}\left[\u27e8\mathit{\text{value}}\u27e9\right]=\u27e8\mathit{\text{value}}\u27e9$.
7Accuracy
The rounding errors committed are such that the computed coefficients are exact for a slightly perturbed set of ordinates ${y}_{r}+\delta {y}_{r}$. The ratio of the root-mean-square value for the $\delta {y}_{r}$ to the root-mean-square value of the ${y}_{r}$ can be expected to be less than a small multiple of $\kappa \times m\times $machine precision, where $\kappa $ is a condition number for the problem. Values of $\kappa $ for 20-30 practical datasets all proved to lie between $4.5$ and $7.8$ (see Cox (1975)). (Note that for these datasets, replacing the coincident end knots at the end-points ${x}_{1}$ and ${x}_{m}$ used in the function by various choices of non-coincident exterior knots gave values of $\kappa $ between 16 and $180$. Again see Cox (1975) for further details.) In general we would not expect $\kappa $ to be large unless the choice of knots results in near-violation of the Schoenberg–Whitney conditions.
A cubic spline which adequately fits the data and is free from spurious oscillations is more likely to be obtained if the knots are chosen to be grouped more closely in regions where the function (underlying the data) or its derivatives change more rapidly than elsewhere.
8Parallelism and Performance
e02bac is not threaded in any implementation.
9Further Comments
The time taken by e02bac is approximately $C\times (2m+\overline{n}+7)$ seconds, where $C$ is a machine-dependent constant.
Multiple knots are permitted as long as their multiplicity does not exceed $4$, i.e., the complete set of knots must satisfy ${\lambda}_{\mathit{i}}<{\lambda}_{\mathit{i}+4}$, for $\mathit{i}=1,2,\dots ,\overline{n}+3$, (see Section 6). At a knot of multiplicity one (the usual case), $s\left(x\right)$ and its first two derivatives are continuous. At a knot of multiplicity two, $s\left(x\right)$ and its first derivative are continuous. At a knot of multiplicity three, $s\left(x\right)$ is continuous, and at a knot of multiplicity four, $s\left(x\right)$ is generally discontinuous.
The function can be used efficiently for cubic spline interpolation, i.e., if $m=\overline{n}+3$. The abscissae must then of course satisfy ${x}_{1}<{x}_{2}<\cdots <{x}_{m}$. Recommended values for the knots in this case are ${\lambda}_{\mathit{i}}={x}_{\mathit{i}-2}$, for $\mathit{i}=5,6,\dots ,\overline{n}+3$.
10Example
Determine a weighted least squares cubic spline approximation with five intervals (four interior knots) to a set of 14 given data points. Tabulate the data and the corresponding values of the approximating spline, together with the residual errors, and also the values of the approximating spline at points half-way between each pair of adjacent data points.
The example program is written in a general form that will enable a cubic spline approximation with $\overline{n}$ intervals ($\overline{n}-1$ interior knots) to be obtained to $m$ data points, with arbitrary positive weights, and the approximation to be tabulated. Note that e02bbc is used to evaluate the approximating spline. The program is self-starting in that any number of datasets can be supplied.