Analyzing an interpolated function

[ddddd28]

Hello,
Consider I interpolated some experimental data, and now I have a polynom. Knowing almost for sure that the funcion is not a polynom, but something else like root, trigonometry or combination. What steps do I need to do in order to distinguish the type of the funcion?

 

[micromass]

Are your data subject to experimental error?

 

[ddddd28]

Probably yes, but I think the error is not significant since the data is taken from a picture. Anyway, I asked theoretically what I should do in order to convert the polynom into another function.

 

[micromass]

Do you know the form of the function?

 

[ddddd28]

unfortunately no.

 

[micromass]

Then what I can think of is either
1) You plot the data in a suitable way and guess the form of the function or
2) You can use nonlinear regression techniques or neural networks to further aid you find the right form of the function or
3) You use nonparametric methods like splines to find a good interpolation
There's not much else I can say in such generalities without knowing anything of the data.

 

[ddddd28]

If I publish the data, it will help you to be more specific?

 

[ddddd28]

x y
2.71E-03 -3.93E-03
7.80E-02 1.84E-01
1.28E-01 3.31E-01
1.74E-01 4.65E-01
2.62E-01 6.32E-01
3.79E-01 8.41E-01
4.80E-01 9.71E-01
6.43E-01 1.15E+00
8.01E-01 1.32E+00
9.91E-01 1.50E+00
1.16E+00 1.62E+00
1.33E+00 1.75E+00
1.50E+00 1.86E+00
1.70E+00 1.97E+00
1.90E+00 2.09E+00
2.17E+00 2.20E+00
2.43E+00 2.30E+00
2.71E+00 2.39E+00
2.99E+00 2.48E+00
3.28E+00 2.54E+00
3.62E+00 2.61E+00
3.91E+00 2.65E+00
4.28E+00 2.68E+00
4.60E+00 2.70E+00
4.88E+00 2.70E+00
5.14E+00 2.69E+00
5.45E+00 2.70E+00
5.80E+00 2.66E+00
5.97E+00 2.66E+00
6.28E+00 2.60E+00

 

[micromass]

You definitely have measurement error in your data, even though it's slight. So a perfect fit with an interpolation will not be useful to you. I would look at regression, not interpolation.

If you obtained a good function that fits the data well, what are you going to do with it?

 

[ddddd28]

The data points are the coordinates of a water drop curve.One of my goals in that school project is to discover what is the geometrical curve of a drop, and then generalize it, so I can tell the figure if given all the parameters( surface tension and so on). I also know that in an ideal case, the curve is a perfect circle.

 

[micromass]

There is no way where you can input this data and magically the right geometrical curve pops up. You'll need to make a guess about what the curve is, then you'll need to perform a regression and see how well it fits. So unless you have a theoretical curve that it should fit, it's all very much trial and error where no model is correct. That said, your curve looks a lot like an ellipse, so you should try nonlinear regression with some parameters that'll give you an ellipse.

 

[ddddd28]

thank you, and I knew the right geometrical curve wouldn't pop up magically. this is an experiment

 

[Dale]

I also know that in an ideal case, the curve is a perfect circle.

When you know the ideal case then that is a good indication that you should use a related functional form for your regression.

 

[FactChecker]

If you want the function to go through the points, you should use a spline function (a polynomial would be a high order and would wiggle around too much).
If you want the function to prefer smooth behavior rather than going through the data points, you should use regression. In this case, using regression to determine a, b, c of y(x) = ax² + bx + c should give you a fairly good fit to the data.