[ Closed ] Cubic Spline Interpolation Bug (Page 1 of 2)

I have actually recently been made aware of the problem. I will fix it as soon as possible.

I have now been looking into this and there doesn't seem to be any bugs in the implementation of cubic splines in Graph. The difference in the image shown is because Graph uses 2-dimension cubic splines as opposed to 1-dimension cubic splines. The 2-dimension splines seem to work much better than 1-dimension and will therefore be kept as it is for now. More types of interpolation may be added in the future.

Here's another example. I sampled a sine wave at 0, 170, 190, and 360 deg. The red curve Graph generated to interpolate these four points goes backwards at 180 degrees, where it provides three values for the single-valued function. This is clearly wrong. The blue curve from the external cubic spline interpolator generates a reasonable curve even though it does not reach +1 at 90 deg and -1 at 270 deg.

Brian

As I explained, the function I took the four samples from was a sine wave. The blue curve is vastly more accurate at reconstructing the original function than the red curve. Not only does the red curve provide a very poor reconstruction, it yields an impossible three-valued function. That's not just a simple error. It is grossly wrong. It is completely out to lunch. It makes no sense whatsoever. Interpolation is not a matter of drawing a line through some points that looks esthetically pleasing. It is a matter of accurately approximating a smooth function given a limited number of samples of it.

Brian

One more example. Here I sampled a sine wave at 0, 90, 270, and 360 deg. The blue curve, which is an external cubic spline interpolator, yields an excellent approximation to the sampled function. The shape is symmetrical and the extreme values are correct. But the red curve, which is Graph's version of cubic spline interpolation, is skewed. Its peaks are in the wrong place and its magnitude exceeds 1, which is impossible for a sine wave. When Graph's interpolation behaves this way on a user-supplied data set where the underlying function is unknown, who would realize that it is in error? I often read interpolated points from a curve. Which curve yields the most accurate values?

Brian

It was never the intention to use point series to create sine waves. If you want a sine wave you should create it as a function. The cubic splines was intentioned to create a smooth line between the points as an alternative to linear interpolation. Maybe instead of calling it Linear, Cubic splines and Half cosine I should have called them Algorithm 1, Algorithm 2 and Algorithm 3, but that seemed a little silly.

Ivan, I spent several hours yesterday going through my collection of dozens and dozens of Graph plots. I compared Graph's interpolation with a standard cubic spline. Graph's interpolation works more or less OK when there are lots of samples and when the spacing isn't too nonuniform. But when there are one or more samples with significantly different spacing, it can yield funny plots that are not representative of the underlying process being sampled. This behaviour is insidious, allowing bad data to pass for good.

I use Graph to draw plots of measured data from my electronics hobby. I take manual readings from lab instruments, both analog and digital, and then use Graph to plot the points. Due to time constraints, I can't always collect as many samples as I'd like. Sometimes I sample outlier points just to complete a curve. I rely on smooth interpolation to fill in the missing data. The problem with Graph's interpolation is that you can't trust it to do a reasonable job of filling in missing points. It can generate funny curves that are nothing like the underlying process you've measured. But ordinary cubic spline interpolation works in these cases.

If you want to keep the current Graph interpolation, I strongly recommend that you add standard cubic spline interpolation that a user can invoke when necessary. Users will have to determine for themselves when to use one interpolation or another, but at least they will have the option of plotting correct, valid curves.

Cubic spline interpolation smooths a curve by making certain assumptions about the first and second derivatives of the underlying function that the data represents. Sometimes real data does not conform to those assumptions. For example, some of the electrical signals I measure may saturate. The plotted curve suddenly becomes a constant value. The circuits are operating correctly but they exhibit discontinuous or nonsmooth derivatives. Any cubic spline interpolation may generate nonconstant interpolated values over a region of constant data due to the influence of adjacent regions. Usually the error is small, but the deviation of a nonflat curve can obscure the underlying saturation, which may be entirely normal. The user is left to explain the unexpected behavior exhibited by the plot. I have added an option to my cubic spline interpolator that enables it to preserve saturated data values. See the next message. If you implement a standard cubic spline interpolation, you might consider adding such a feature. It is not often needed, but it can make the interpolation model more realistic for some data.

Brian

Brian, those exactly same 3 or  points  could have been from a different function, not sine function. Then what, would you expect that an interpolation method in one case will resemble to sine function, in other case--to another function??? Think about it! Interpolation methods are not intended to give the original function, but, as Ivan pointed, just to give a function that satisfies some smoothness conditions. That's all! Only in case you provide more and more points, the method will begin to give a graph close to the original particular function.
Read more about interp. on wiki...

Victor, you can draw any curve you want through a set of given points. That doesn't mean that all curves are equally valid or useful. The problem with Graph's interpolation method is that it can generate curves that are not representative of commonly encountered data, or that are not smooth, or that are impossible. When I say not representative, I mean that if you take more data samples, they will not lie on or anywhere near the curve Graph generated for the original samples. The curve has mislead you and given a false impression. When I say not smooth, I mean that Graph can insert kinks, abrupt changes of slope that are not warranted by the data, and it can generate straight line segments where a smoothly curved line is appropriate. When I say impossible, I am referring to the fact that Graph can generate curves that have multiple Y values for one X value. The curve is not just in error, it is nonsensical.

Brian

Ivan, I just made some little research about cubic spline interpolation. So, what do you mean by 1-dimensional and 2-dimensional cubic splines? Did you actually intend to interpolate a third variable, like z=f(x,y)? That could be the issue then.
Please see http://docs.scipy.org/doc/scipy/referen … olate.html , or other.
Also, some other simple cubic spline (called 'natural') do interpolate, indeed, in a more expected way. http://en.wikipedia.org/wiki/Spline_interpolation (I tried the example there in the end with 3 points); also read the 3rd external link in the same wiki page.

It was never a goal to make an interpolation function where every x-coordinate has exactly one y-coordinate. The goal was to make a line between the points that was more smooth than a linear interpolation, and I think that goal has been reached.

The normal 1-dimensional cubic spline does not work for my goal in all cases so I used the 2-dimensional variation instead. It is basically a parametric function where both x(t) and y(t) is interpolated as cubic splines.

I think I first saw it at MathWorld:
http://mathworld.wolfram.com/CubicSpline.html

There is also an interactive Java applet here:
http://www.cse.unsw.edu.au/~lambert/spl … cubic.html

But I understand that other people have other goals. It was always my intention to add more interpolation algorithms, but no matter how many I add there will always be some complaining that I have not implemented their favorite variation. However I will consider adding 1-dimensional cubic splines.

The picture in my last message should explain it. It creates a more smooth line between the points than just connecting them with a straight line. But I will try to implement 1-dimensional cubic spline to make you and Brian happy.