What is the best way to determine the convergence of data?

I'm doing some research with Matlab (materials science related), and a lot of my work involves statistical analysis, of which I'm not super familiar with (so I apologize in advance if I'm not using proper terminology). For this project I'm measuring some data, and I want to come up with a way to know when it converges. I have tried calculating the variance using binning in the past, but recently have discovered a moving variance function that is more accurate.

The idea is to take a moving variance of each data point, and once the variance is less than a certain amount, say 1e-6 for example, that's when I know my data has converged. However, the problem with using this method is that the convergence depends on the window size (the number of data points within the window) and the cut-off value. Specifically, it keeps increasing as I increase the window length, and keeps decreasing when I increase the cut-off value. I have been trying for the past few weeks to overcome this problem but can't come up with a solution.

Is there a way to know the best combination of these two variables? Or has anyone else run into this problem before and would be able to lend some tips? This is for a research project, so there is not a convergence that has already been found that I can compare it to, otherwise that would be the easiest method. Or has anyone used a better method for determining the convergence of something? Thank you for taking the time to help!


TS Contributor
Is it time series data? Do you have any idea of the underlying data generating process? Can you define convergence a little bit further? It has a very specific meaning in mathematics. Do you mean that the series is converging when the next data point is close to the previous one/s?


Fortran must die
It is common for statistical algorithms to either converge or not. And software has been developed to know when its not (you get nasty warnings when it does not). I don't know if that is useful for what you do or not - whether you can use that approach or not.

With time series I would think structural breaks would be a major issue for convergence.
Thanks for your input and questions- in layman's terms I am creating a microstructure of increasing size, and measuring a structural property each time I increase the size. I'm doing this to find a sample size for the whole microstructure. It starts out varying a lot, because my small microstructure doesn't account for the whole system, but eventually it gets to a put where the property converges and doesn't change, even when I keep increasing the size. I would like to know the x-value where I could say with a certain confidence interval that the data converges (where that property value would match that of a microstructure of infinite size). I've attached a screenshot of my data just to show what it going on (x axis is log-scale)- the x-axis is the number of grains in my microstructure, or the size, the y-axis is the property I am measuring. So in this case convergence means that its value would match (approximately) the limit of the data as the size goes to infinity. Does this help to clarify those questions?