I have some data (18 points) which are plotted on the attached image, f(x), the red line.
I do not know if this line can be described by a combination of known functions, but I am trying to find out.
It looks somewhat similar to sin(x), the blue line, and tan(x), the green line.
At this point I am trying random combinations of functions to see if I can find a function that accurately describes f(x):
1) f(x) = M*x + B + C*sin(x) + D*cos(x)
2) f(x) = M*x + B + sin(C*x) +cos(D*x)
3) f(x) = M*x + B + C*sin(x)*cos(x)
4) f(x) = M*x + B + C*sin2(x)*cos(x)
5) f(x) = M*x + B + C*sin(x)*cos2(x)
where M, B, C, and D are real constants.
I am pretty much just making random guesses hoping to get lucky.
However, since the function looks so close to a portion of a sine curve, I am wondering if a Fourier Transform might reveal something. Assuming f(x) is composed of a mix of sine and cosine terms, isn't the whole point of a Fourier Transform to reveal these separate components?
This is new territory for me; I was briefly exposed to Fourier Transforms during my undergraduate engineering studies but, since then, haven't touched them.
So . . . I have lots of questions.
If I perform a Fourier Transform on the data, even if it does not reveal the complete function, will it at least reveal if f(x) includes a linear term (i.e. - M*x + B)?
Does a Fourier Transform reveal only linear terms of cosine and sine? In other words, if the actual function were to take a form such as equations 4 and 5 above (which includes a power of 2), would a Fourier Transform reveal it?
Do I need more than 18 data points to make this work? If so, I will have to go back and get some more data points.
Any suggestions or advice is appreciated.