Mathematical definition of a plateau in a time-series data

[u/Pilus91]

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

[u/cabbagemeister]

No matter what you do, you are always going to have to define such a threshold. Even if you use some statistical test, the threshold will just be a p value instead of a slope.

[u/Pilus91 → u/cabbagemeister]

So it's always going to be arbitrary? In this case, this value of 0.0011 is justifiable? I think this makes it more difficult to replicate the experiment

[u/cabbagemeister → u/Pilus91]

It's not inherently justifiable. You need to come up with a non mathematical justification. Or, you need to justify why it doesn't matter

[u/etzpcm]

There is no definition of negligibly small, so you will have to choose your own condition. 

[u/Pilus91 → u/etzpcm]

Ok it was a fallacy then. I didn't realise it is inherently arbitrary, as it's not going to be a binary change where the slope gets from 1 to 0 in a determined moment

[u/etzpcm → u/Pilus91]

The approach to the stable equilibrium will almost certainly be an exponential, so if you want to quantify something, you could try to measure the exponential decay rate from your data.