# Quantifying nonlinearity from data

• A
• BillKet
In summary, the conversation discusses a function with the form y = ax + b + f(x), where f(x) is a non-linearity in x that cannot be absorbed into the ax+b part. The speaker is wondering if it is possible to extract f(x) by only measuring x and y and if it can be connected to the deviation from linearity. It is noted that without knowing the form of f(x), it may be difficult to distinguish it from adding values to a and b. The speaker is fine with redefining a and b and is only interested in any deviation from linearity. The other speaker suggests using linear regression to estimate y and then modeling the non-linear piece with a parameterized function. Noise in the

#### 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!

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?

Office_Shredder said:
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.

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