# Convergence tests for experimental data

I ran a simulation for the critical density at which traffic flow transitions from laminar flow to start-stop-waves. The results for different road lengths are:

From the graph we can see that the critical density decreases as the road length increases. Is there any test I can do that guides me into a point where my data may converge?

All I can be sure is that it converges at least to 0 (which physically makes sense to me). How can I test that it does not converge to any other value greater than 0?

PS: It takes me about a week of the code running to get very high values (1 single point at $10^4$ took 5 days of the program running).

PPS: I add a link to the data Data in case someone wants to try something.

• The answer is likely no. Given the spread of your data, it seems like it should be easy to fit it with many different functions which converge to different asymptotes. Apr 24, 2018 at 20:28
• The plot you show involves a very high density of data points over quite a small range of the independent variable if you are worried about asymptotic behavior. You don't gain much by testing 125, 175 and 225 as well as 100, 150, 200, and 250. Spend some time to get small sets of data at higher values of the road length ($10^3$ and $3\times 10^3$, perhaps), which will put stronger constraints on the variation that @probably_someone talks about. I would be wary on including very small groups (like, say, a single data point at $10^4$): you need enough to reduce the odds of skewing the results. Apr 24, 2018 at 20:37
• But even that won't really solve your convergence problem; it'll just reduce the range of asymptotes that could work with your data, given a specific model. What you're dealing with here is the mother of all extrapolation problems, because you're trying to extrapolate infinitely far from your data; unless you have a specific model that you're trying to fit with, you can construct an arbitrary function (even assuming various smoothness criteria) that will fit all of your data and do literally anything at infinity. Apr 24, 2018 at 20:47
• You don't actually need infinity. Ideally, you'd like lengths long enough that they represent the longest sustained duration or region of fairly uniform traffic conditions, but it is sufficient that you can say with confidence that "over a wide range of cell lengths we can place a floor of XXX on the critical density at Y.Y sigma". Apr 24, 2018 at 20:58

You could try hypothesis testing. The first thing I would do is estimate the non-stationary mean of the data. Least square fit it with a model (e.g., exponential or something nonparametric). This is an estimate of the mean. You could also just apply a running (mean, say) filter over the data to get an estimate of the mean. Once you have this, subtract the mean from the data. This gives you an estimate of the population variance, assuming you've removed all the obvious structure. Of course you want to make sure that the mean and variance are insensitive to the model you use. Then you can compute asymptotes that are consistent with the uncertainty in the data. In other words, once your mean decays to the level of the noise then you stop. You can estimate the uncertainty of the asymptote by seeing how sensitive it is among all the mean-estimates that don't overfit the data. There is a vast statistics literature on this problem, and I've just given an oversimplified version. Talk to a statistician. But I think what I've said will give you a reasonably defensible start at your answer. It's also possible to pose the test: is the asymptote consistent with zero given the uncertainties in the data. You can also estimate confidence intervals on the asymptote itself. If you like, you could post a link to the data and I could give it a go, since I have a lot of statistical tools for doing similar calculations. Finally, a caveat. With enough degrees of freedom you can fit any model. So you need to make sure that only add degrees of freedom if they are required by the data. Occam's Razor.

• +1 "Talk to a statistician." Apr 24, 2018 at 21:01
• @Hector I learned this early on. I've gained a lot by hanging around with really good statisticians. They have a very open mindset since they are used to working on diverse problems. Every experimentalist should have at least one statistician on speed-dial. ;) Apr 24, 2018 at 21:09