# how to find the equation that best fit this curve

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Danilo NASCIMENTO on 24 Aug 2014
Commented: John D'Errico on 24 Aug 2014
Hello guys
I want to find a equation that best fit this curve I'm sending attached. I tried a polynomial approximation but it does not work

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Star Strider on 24 Aug 2014
What process produced these data? How best can you describe that process?

per isakson on 24 Aug 2014
Edited: per isakson on 24 Aug 2014
See SLM
Description: If you could only download one curve fitting tool to your laptop on a desert island, this should be it. [...] &nbsp SLM - Shape Language Modeling by John D'Errico

#### 1 Comment

John D'Errico on 24 Aug 2014
:)
Ok, I'd also make sure I had a spare battery for my laptop. Do they sell solar powered laptops now?

Roger Stafford on 24 Aug 2014
Edited: Roger Stafford on 24 Aug 2014
The right side of this curve looks as though it is asymptotically approaching a descending straight line. The left side doesn't look as if is approaching infinity, but more like y = 38 (or thereabouts) at x = 0 with an infinite derivative at x = 0. It looks very much like one branch of a hyperbola. Try adjusting constants A and B to make the hyperbola
y = 38 - sqrt(A*x+B*x^2)
fit the curve. (Just a guess.)

Star Strider on 24 Aug 2014
My guess was:
f = @(b,x) (b(1).*x + b(2)) + exp(b(3).*x);
Never got data to fit it, though.
John D'Errico on 24 Aug 2014
Star - that one has the wrong behavior at x == 0. It won't fit well.

John D'Errico on 24 Aug 2014
Edited: John D'Errico on 24 Aug 2014
There is an obvious singularity at x == 0. This suggests that a polynomial model will be useless, or at best poor. Polynomials do not have singularities, so you would need a high order polynomial. (When you try to fit a polynomial to a function with a singularity, it will do strange things. You will need to use a high order, and that in turn will be a problem.)
Likewise, my own SLM will have problems. Again, splines don't like singularities.
However, IF you swap the x and y axes, you will be able to gain a very nice fit, using a variety of tools. Essentially you will build a model of the form
x = f(y)
This model will have no singularities. A spline model (like SLM) will be trivial to fit. Polynomials will also work well enough in this inverse form since there will be no singularity.
To evaluate the model now is slightly harder, but still easy enough. For a polynomial mode, f(y), any given x to find y, use roots or fzero. Thus you would find the roots of the polynomial in y
f(y) - x == 0
If f(y) was built using SLM, then use SLMEVAL, which can invert a spline model.
If you insist on trying to fit a model of the form
y = f(x)
then you will need terms in the model that have a singularity in them. So a logical starting model might have terms like this:
y = a0 + a1/x + a2/x^2 + a2*x + ...
so a polynomial with some terms with negative exponents. My own polyfitn (also found on the file exchange) would work. Regardless, that model may be difficult to fit well using least squares because of the singularity.