Mathematical definition of a plateau in a time-series data
Hello, I'm a bioinformatician and I'm struggling with the current issue:
Given a time series y(t) that initially changes and eventually approaches a stable regime, how can I mathematically determine the earliest time t\* at which the rate of change dy/dt becomes negligibly small, using only the observed data and without defining an arbitrary threshold?
This is a collaboration I'm doing. My colleagues defined the plateau as the first time when a 101-point rolling mean of the relative increment (g' t+1 - g' t)/ g't falls below the arbitrarily chosen threshold of 0.0011. G' is the measure of material elastic-solid response btw. So the issues is that they used 2 arbitrary values because experimentally they know that a certain value of g' means that the gel is solid. But this doesn't hold for me. I tried using many statistical methods to define the threshold such as:
- exponential fitting
- change-point regression
- local slope analysis
But they all give me a plateau that is too early or too late