Curve fitting to step function

Question

If all that is known of a plot is that the curve passes through certain squares on a grid, is there any way, other than by estimation or trial & error, to work out what the equations to the curves may be?

Please see example below:

enter image description here

From the filled squares, I would like to work out the maximum curve, minimum curve, & the resultant step function.

(In the above example, the max & min curves are clearly $x^2$ & $(x-1)^2$. The blue curve enters the shaded squares on the LHS & the red curve exits the shaded squares on the RHS.)

In my opinion, I think that the problem is to have one curve; this could be done using standard polynomial regression for example. The second one could be mathematically deduced from the first (similarly to what you did in your post). — Claude Leibovici, Dec 04 '13 at 11:22
@ Claude Leibovici, many thanks for the reply - I'll give that a go :) — martin, Dec 04 '13 at 11:48

score 1 · Accepted Answer · edited Dec 20 '13 at 12:32

If the problem is to have a smooth "mean" curve, the equation is y(x)=(x-(1/2))²

If the problem is to have a stepped curve, the exact equation involves a Fourier series . An approximate is obtained with the the series limited to a number of terms depending on the expected accuracy.

$$y(x)=\left(x-\dfrac12+\dfrac1\pi\sum_{k=1}^\infty\dfrac1k\sin(2k\pi x)\right)^2$$

score 0 · Answer 2 · answered Dec 04 '13 at 13:32

I'd suggest curve fitting the bottom corners and the top corners of the shaded cells. One easy way to experiment with curve fitting is to use Excel. You can chart the data and then use the "Add Trendline" function to do fitting with various types of curves.

Alternatively, fit a curve through the centers of the cells, and then create the min and max curves by translating the mid curve upwards and downwards by half the cell height.

Curve fitting to step function

2 Answers2