# Near-linear Function aproximation

Hi,

I have a growing, near-linear function, that has some "noise" in its linearity. Is there any solution, how to approximate this function ? I tried Neural Network, but ist not best...

http://www.perry.cz/files/almost-linear.png

I'll assume that your data is discrete.

You can pick a class of function such as $g(x)=Ax+B+C\sin(Dx+E)$ and then solve for $A,B,C,D,E$ which minimize $\sum(f(x)-g(x))^2.$

later thoughts As I think about it, that might not be that easy to solve (at least using partial derivatives, perhaps a multi-dimensional Newtons method but that does not seem worthwhile).

Any modeling is a matter of judgement. I think you would in any case first just find the best linear fit $Ax+B$ (which is easy) and then work on the values $f(x)-(Ax+B).$ Now you have the cleaner problem:

Given something which looks like random noise with mean $0$ how would you model it?

Without any further information I might just find average absolute value $V$ of the error and then add a term $(2-{\sqrt{2}})\frac{V}{2}\sin(Mx+M)$ where $M$ is a huge constant. So this is essentially random sampling from a source with mean value $0$ and mean absolute value $V$. This would be expected (I would think) to be roughly no better or worse a fit to your actual data than the line $Ax+B$ but to have the right amount of noisiness.

Have U tried "least square method"?

• It gives me linear function, but without "noise" Feb 9 '13 at 13:02
• I don't understand your question... what do U means by approximation? Feb 9 '13 at 14:18
• Function, that is close to original one... but linear is ok only for couple of values, than it breaks due to "steps" in original function Feb 9 '13 at 15:06
• y it's not enough 4 u? It's will be a good approximation. If your goal is function with behavior similar the behavior of the input, u can use interpolation methods. Feb 9 '13 at 17:03
• Please don't use these horrible abbreviations!! If you can't write properly, just refrain from writing here at all. Feb 24 '13 at 10:35