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See also the next section, however. A useful feature of the function is that it can be used in applications which require the continuity to be less than the normal continuity of the cubic spline.
For example, the fit may be required to have a discontinuous slope at some point in the range. This can be achieved by placing three coincident knots at the given point. Similarly a discontinuity in the second derivative at a point can be achieved by placing two knots there. Analogy with these discontinuous cases can provide guidance in more usual cases: for example, just as three coincident knots can produce a discontinuity in slope, so three close knots can produce a rapid change in slope.
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The closer the knots are, the more rapid can the change be. An example set of data is given in Figure 1. It is a rather tricky set, because of the scarcity of data on the right, but it will serve to illustrate some of the above points and to show some of the dangers to be avoided. Three interior knots indicated by the vertical lines at the top of the diagram are chosen as a start. We see that the resulting curve is not steep enough in the middle and fluctuates at both ends, severely on the right.
The spline is unable to cope with the shape and more knots are needed. In Figure 2 , three knots have been added in the centre, where the data shows a rapid change in behaviour, and one further out at each end, where the fit is poor. The fit is still poor, so a further knot is added in this region and, in Figure 3 , disaster ensues in rather spectacular fashion.
The reason is that, at the right-hand end, the fits in Figures , have been interpreted as poor simply because of the fluctuations about the curve underlying the data or what it is naturally assumed to be. But the fitting process knows only about the data and nothing else about the underlying curve, so it is important to consider only closeness to the data when deciding goodness-of-fit. Thus, in Figure 1 , the curve fits the last two data points quite well compared with the fit elsewhere, so no knot should have been added in this region.
In Figure 2 , the curve goes exactly through the last two points, so a further knot is certainly not needed here. Figure 4 shows what can be achieved without the extra knot on each of the flat regions. Remembering that within each knot interval the spline is a cubic polynomial, there is really no need to have more than one knot interval covering each flat region. What we have, in fact, in Figures , is a case of too many knots so too many coefficients in the spline equation for the number of data points.
The warning in the second paragraph of Section [Preliminary Considerations] was that the fit will then be too close to the data, tending to have unwanted fluctuations between the data points. The warning applies locally for splines, in the sense that, in localities where there are plenty of data points, there can be a lot of knots, as long as there are few knots where there are few points, especially near the ends of the interval. In the present example, with so few data points on the right, just the one extra knot in Figure 2 is too many!
The signs are clearly present, with the last two points fitted exactly at least to the graphical accuracy and actually much closer than that and fluctuations within the last two knot-intervals see Figure 1 , where only the final point is fitted exactly and one of the wobbles spans several data points. The situation in Figure 3 is different. The fit, if computed exactly, would still pass through the last two data points, with even more violent fluctuations.
However, the problem has become so ill-conditioned that all accuracy has been lost. Indeed, if the last interior knot were moved a tiny amount to the right, there would be no unique solution and an error message would have been caused.
Near-singularity is, sadly, not picked up by the function, but can be spotted readily in a graph, as Figure 3. B-spline coefficients becoming large, with alternating signs, is another indication. However, it is better to avoid such situations, firstly by providing, whenever possible, data adequately covering the range of interest, and secondly by placing knots only where there is a reasonable amount of data.
The example here could, in fact, have utilized from the start the observation made in the second paragraph of this section, that three close knots can produce a rapid change in slope. The example has two such rapid changes and so requires two sets of three close knots in fact, the two sets can be so close that one knot can serve in both sets, so only five knots prove sufficient in Figure 4.
It should be noted, however, that the rapid turn occurs within the range spanned by the three knots. This is the reason that the six knots in Figure 2 are not satisfactory as they do not quite span the two turns.mosfullkeenwind.gq
Curve and Surface Fitting
Some more examples to illustrate the choice of knots are given in Cox and Hayes You have to supply only a threshold for the sum of squares of residuals. Then, with the knot set decided, the final spline is computed to minimize a certain smoothing measure subject to satisfaction of the chosen threshold. Arbitrary weights are allowed. For choosing these knots, the advice given for cubic splines, in Section [Least squares cubic splines] above, applies here too see also the next section, however. If changes in the behaviour of the surface underlying the data are more marked in the direction of one variable than of the other, more knots will be needed for the former variable than the latter.
Curve and Surface Fitting
With some sets of data and some choices of knots, the least squares bicubic spline will not be unique. This choice of least squares solution tends to minimize the risk of unwanted fluctuations in the fit. The fit will not be reliable, however, in regions where there are few or no data points.
Again, this easier to use function is normally to be preferred, at least in the first instance. This kind of data allows a very much faster computation and so is to be preferred when applicable. Substantial departures from equal weighting can be ignored if you are not concerned with statistical questions, though the quality of the fit will suffer if this is taken too far.
For the general linear function 15 , functions are available for fitting in all three norms.
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