# Search for universal monotoneous function for time-series trend removal

I have some time-series. They may be written as $$T(n)-T(n-1)=f(n)+\xi,$$ where T(n) - time of $n^{th}$ occurrence, $f(n)$ some monotonous function from number of event, $\xi$ random noise. These time series may be plotted in a following way: I am interested in $\xi$, thus I want to subtract $f(n)$. This function may differs from time-series to time-series. The only knowledge I have - it is monotonous (sign may vary). So, I want to find $f(n)$ just fiting some monotonous curve to data.

Could you help me and advice some universal class of equations to make such fit?

(May be something like Taylor or Fourier series?)

P.S. More generally speaking, I have some data, which may be well fitted by curves depicted above. The question is to find some class of curves, which will be good approximation to such a heterogenous curves.

• Your equation doesn't quite make sense to me. If you want $T$ to be monotonic in expectation, $f$ should be positive rather than monotonic. If $f$ is monotonic rather than positive, $T$ might not be monotonic (consider $f = -1/x$, which is monotonic increasing). – Glen_b Jun 18 '17 at 2:51
• I've edit question. I mean f(n). I have quite different trend of between events intervals, but they are monotonous. The picture demonstrates different trend of intervals, after reconstitution of time-series. – zlon Jun 18 '17 at 10:56