Quantifying nonlinearity from data

[BillKet]

Hello! I have a function of the form:

$$y = ax + b + f(x)$$
and I can measure experimentally only x and y. I also know that $f(x)<<ax,b$, where $f(x)$ is some non-linearity in x i.e. it can't be absorbed into the $ax+b$ part (for example $f(x) = cx^2$), but I don't know its form. Is there a way to extract $f(x)$, by measuring only $x$ and $y$? I am basically wondering if I can quantify the deviation of the expression above from linearity and connect that to the value of x. Thank you!

 

[Office_Shredder]

As a first pass, you could be in a lot of trouble if $f(x)=x^2+2x+1=(x+1)^2$, which is going to be literally indistinguishable from adding 2 to a, 1 to b, and $f(x)=x^2$.

That said, it sounds like maybe you don't care, and you are happy to just think of $f(x)$ as $x^2$ in this case. Is that right?

 

[BillKet]

As a first pass, you could be in a lot of trouble if $f(x)=x^2+2x+1=(x+1)^2$, which is going to be literally indistinguishable from adding 2 to a, 1 to b, and $f(x)=x^2$.

That said, it sounds like maybe you don't care, and you are happy to just think of $f(x)$ as $x^2$ in this case. Is that right?

Yes! I am fine with redefining a and b if needed (i.e. absorbing those terms you mentioned above). I am purely interested in any deviation from linearity, regardless of the actual value of a and b.

 

[Office_Shredder]

And the thing you care about specifically is trying to estimate y given a value of x? Are we assuming your measurements are perfect with no noise?

I think you would start with doing linear regression to get $y \approx ax+b$ for some $a$ and $b$. Then compute $y-ax-b$, and attempt to model it with your favorite parameterized function. If you have a specific example, just drawing a plot of that would probably be a good start for guessing the shape of the non linear piece