a journal of a researcher

Wednesday, July 14, 2004

Curvature and the order of polynomial

I was wondering what the relation of curvature and the order of polynomial you select for regression. If the curvature is 0, it is better to use a straight line. If it is high, do we need to use higher order polynomial? I find the later statement does not hold. Very simple, the curvature depends on the point. Not every point on a higher order polynomial has higher curvature. Actually it is hard to tell where the curvature is high, it depends on the shape of polynomial. So there is no relation.

The regression method considers solely the error. Even if you do not have the best fitting curve, you still can get good fitting quality (if you are allowed to partition the domain).

What kind function to choose for approximating indeed depends on the data. If we can visualize the data, we can see that if it is a curve so that we can use polynomial or otherwise straight line. But if we can not, the only way is to try it out. I found a paper to decide the polynomial and the partition.

2 Comments:

  • I tend to agree with your statement. It is not merely a matter of looking at the curvature or slope... it is very much an application specific issue to decide how to best "model" the data.

    Say that you are interested in computing the second derivative. It could be, for example, that your diagnostic techniques lead you to believe that if the curvature is too high, then there is a problem. Well, clearly then, linear approximation is not good.

    Linear approximation is also not very good to approximate a curve like exp(x).

    But for many other applications, linear approximation is ideal. Think that computer 3D graphics are mostly done using linear approximation (small triangles).

    Also, consider that polynomials are not the only choice. You could certainly do reasonable regression using exponentials (though you have to worry about stability then) or sine and cosine...

    It really depends on your application.

    By Blogger Daniel Lemire, at 10:14 AM  

  • The regression method concerns only the error. What is the best regression function can not be determined by an alogrithm so far. If we allow segmentation, the finest segment is between two adjacent points. Straight line can be good function, no error. Actually that becomes interpolation. You can also use cubic spline for interpolation, no error, more smooth.

    In one word, only error matters. You need extra information about the data to select a best regression function.

    By Blogger flydragon, at 8:01 AM  

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