Different commands to do spline interpolation, are they the same?

Hello,

I have a question about (cubic) spline interpolation in MATLAB. For cubic spline interpolation I see several commands which 'seem' to do the same job but I am not 100% sure and this is why I am asking you. To be more clear consider the two commands pp = spapi(4,x,y) and pp = spline(x,y) (there are also the commands pp = csapi(x,y) and pp = interp1(x,v,'spline','pp') but let's skip these for simplicity). My question is : "Are these two commands equivalent?" If I trust my eyes, the answer is 'Yes' to me but, are they really the same?

One issue that makes me to doubt is that I see that the mentioned commands use rather different knot sequesnces. For instance, consider the following example:

x=linspace(-2*pi,2*pi,10);y=sin(x);
pp1 = spline(x,y);
pp2=spapi(4,x,y);
pp1.breaks
pp2.knots

ans =

-6.2832 -4.8869 -3.4907 -2.0944 -0.6981 0.6981 2.0944 3.4907 4.8869 6.2832

ans =

-6.2832 -6.2832 -6.2832 -6.2832 -3.4907 -2.0944 -0.6981 0.6981 2.0944 3.4907 6.2832 6.2832 6.2832 6.2832

Thanks in advance!

Babak

Answer 1

[John D'Errico]

Can two different functions exist that happen to do the same thing? Of course. MATLAB is a complicated language, that was created over the course of MANY years. For many of its users, MATLAB was around before they were born. Yet at the same time, MATLAB needs to support legacy code. So functions I wrote 35 years ago will often still work.

Consider these two examples, ones that you gave yourself:

x = 0:6;
y = sin(x);
spl1 = spline(x,y)

spl1 = struct with fields:
      form: 'pp'
    breaks: [0 1 2 3 4 5 6]
     coefs: [6×4 double]
    pieces: 6
     order: 4
       dim: 1

spl2 = spapi(4,x,y)

spl2 = struct with fields:
      form: 'B-'
     knots: [0 0 0 0 2 3 4 6 6 6 6]
     coefs: [0 0.7637 1.3839 0.1669 -1.2562 -1.0307 -0.2794]
    number: 7
     order: 4
       dim: 1

Yes, both create an interpolating cubic spline. But are they the same? HUH? Are you kidding? Did you look at what they created? Did you read the help?

Spline produces a pp form, so the piecewise polynomial form, whereas spapi produces a B-form. You can convert the B-form into a pp form.

pp2 = fn2fm(spl2,'pp')

pp2 = struct with fields:
      form: 'pp'
    breaks: [0 2 3 4 6]
     coefs: [4×4 double]
    pieces: 4
     order: 4
       dim: 1

Still an interpolating cubic spline, but do you see here they are on a different set of breaks? Breaks were actually not needed at locations 1 and 5, here because the not-a-knot end conditions were chosen by default. This can get a bit deep, so I'll stop at that point before I write 10 pages of explanation. Oh wait, see my comment below, and read the book. Actually a pretty good book. I have a copy on my shelves.

Even so, spapi is a more complex code, that can do more than just a simple cubic spline. For example, we can create a piecewise linear interpolating spline. A bit of overkill to do that, but we can.

spl0 = spapi(2,x,y)

spl0 = struct with fields:
      form: 'B-'
     knots: [0 0 1 2 3 4 5 6 6]
     coefs: [0 0.8415 0.9093 0.1411 -0.7568 -0.9589 -0.2794]
    number: 7
     order: 2
       dim: 1

fnplt(spl0)

And spapi is not limited to problems in one dimension. Spline is. Those things are outside of the scope of the function spline. spline is essentially a much simpler function.

As far as the functions csapi and interp1 go, they also have different uses, different behaviors, different abilities. Is there some overlap between them? Yes. But teaching the complete differences between them all would require writing a complete book on splines here. Oh. Wait. There are already books written on splines, one of which was written by de Boor, who was the source of many of these spline codes, at least long ago.

Remember that as MATLAB has grown over the years, new functions were introduced with more capability, that in some cases overlaps with the older ones. Yet they cannot just drop the older codes on a whim, as now too many legacy codes will fail to run for no good reason other than that someone wanted to make an improvement. And it is often the case that one cannot reasonably expand the functionality of an older tool, as that would make the interface and documentation far too complicated. So we end up with some degree of overlap. Such is life.

[Mohammad Shojaei Arani]

Helllo John,

Thanks for your kind explanation.

Well, I have studied some chapters of Carl de Boor. I know that these methods might use different end conditions, etc so they might differ from each other slightly, But, in my example I saw NO difference in the output results (for this, I do not care if the knot sequences are different. what matters is that all produce the same thing).

But, there was a different reason I decided to ask this question. After reading the book of Carl I learned that there are 'local' splines versus 'non-local' ones. For me, as a practical scientist, I am concerned whether the fitted spline is 'sensitive' to perturbations of data (y-values)? If the structure and geometry of the fitted spline changes for a slight change in a few data values then I would not be happy with it (In this reference "Sensitivity of Kriging and Spline interpolation to data perturbations" you can see that splines are, in general, sensitive to such perturbations, unfortunately). In the book of Carl it is mentioned that 'broken-line' interpolation is highly local, also the Hermite cubic splines are also, more or less, local, and finally the normal cubic splines (C^{2} continuous ones) have less local behaviour. It is also mentioned that if "the knot sequesnce is regular' then there is a garantee that the 'non-local' sensitivity of splines to perturbations vanish as we departure from the perturbed data point y(i) but such a garantee cannot be made if the knot sequence is irregular. So, this is the basis behind my question.

I am working on a problem where few y-values might be uncertain a bit and I do not like such uncertainties to dramatically change the structure of the fitted spline. Therefore, I need to know which of these methods respect the 'regular knot sequence I choose' and which do not respect them. In your example sp10 the knot sequence is [0 0 1 2 3 4 5 6 6] while in spl1 it is [0 1 2 3 4 5 6]. So, can I say that if I use spl1, instead of sp10, then I can be assured that the knot sequence is regular and therefore, I can have better hopes to get a spline interpolant which is less vulnerable (less sensitive) to the perturbations of y-values?

[Bruno Luong]

"[0 0 1 2 3 4 5 6 6] while in spl1 it is [0 1 2 3 4 5 6]. So, can I say that if I use spl1, instead of sp10, then I can be assured that the knot sequence is regular and therefore, I can have better hopes to get a spline interpolant which is less vulnerable (less sensitive) to the perturbations of y-values?"

They are different representation/coding of the same function, so there they have the exatct ame sensitive to y value.

In fact only the 2 boundary conditions are free to chose by different spline interpolation. As I undertsand the not-to-knots are universally used by various place in MATLAB, so there is not difference of sensitivity to perturbation of y.

The only bc that more or less more stable theoretically is natural spline, which is proved to be the one that minimizes the 2-norm of the second derivative. You can also enforce to be minimal other (semi) norm such as l2-norm of the first derivative or the fuction. But none of that is dorectly supported by MATLAB various spline functions.

[Mohammad Shojaei Arani]

ok. Thanks!