How do I determine when a (non-independent) time series approaches a horizontal asymptote? I have time series data with many data points per subject over time. I want to determine the marginal time interval within which my dependent variable (dv) falls within given "equivalence" bounds around a (a priori unknown) horizontal asymptote.
I thought about fitting a linear mixed model with polynomials in time as fixed effects and the corresponding by-subject random intercept and slopes to get a (simultaneous) confidence region for the corresponding regression function and determine the desired time interval based on that confidence region. But I'm not exactly sure (1) whether this is a reasonable approach, and if so, (2) how I could construct the (simultaneous) confidence region and get the desired time interval, or (3) if there is another approach which is usually used to solve this tasks.
The functional form of the population relation between time and the dv could be something like
$\mathrm{dv}=\alpha\cdot\exp(\frac{\beta}{\mathrm{time}})+\epsilon$
but I'm more interested in a general approach that can be applied without assuming this functional form.
 A: So, if I may, I will attempt to formalize the problem you have described. There is an observable $dv(t)$ which behaves according to the following equation
$$dv(t) = f(t, \theta) + \epsilon$$
We know that:

*

*$f(t, \theta)$ is some deterministic (non-random) function of time that describes the transition of the hidden variable from some initial value to a known asymptotic value $f_0$.

*$\theta$ are the parameters of that function, namely, the initial value of the function, the timescale at which it reaches the asymptotic value to some pre-defined accuracy, as well as as potentially some shape parameters describing curvature, oscillations, etc.

*$\epsilon$ is some random variable due to e.g. imperfect observations. I assume it is known to be i.i.d and have zero mean. If it is not i.i.d, things get more complicated as you would need a noise model as well.

*We want to infer a certain property of $f$, namely, the place where it reaches its asymptotic value to a given precision. If $f$ is known, that value likely can be expressed as a function of the parameters $\theta$
IMHO, the problem you are trying to solve is fundamentally parametric (model-based), so there is no such thing as a non-parametric solution. In order to perform any estimation, you have to put constraints on how fast the function is expected to approach the asymptote. Namely, you must know the functional form of the derivative of the function within the time-region of interest. If you don't provide this information you essentially cannot prove convergence, because you cannot rely on relationships between consecutive datapoints. The best you can do is to perform hypothesis testing on each datapoint, testing if you can refute the hypothesis that this datapoint has the mean equal to the converged variable. But there are two problems: 1) failing to disprove convergence is not equivalent to convergence, so such estimation would be very imprecise 2) Since you will have to perform multiple tests, you will suffer from multiple comparisons problem, where you will need to adjust your p-value proportionally to the number of tests performed to guarantee significance, rendering the tests essentially useless for large number of observations.
Now that we have that out of the way, we can consider parametric approaches to the problem. Typical transitions that happen in many natural processes are described by a logistic function. It fits 4 parameters - asymptotic starting value, asymptotic finite value, transition midpoint time, and transition speed. Exponential function you have suggested is also a possibility if there is no initial plateau, although I am not quite sure why you place the time in the denominator. Unless you have a very good reason to use that exact function, I would suggest using a simpler function. I would have tried something like this:
$$dv(t) = A + B \exp \bigl(-\frac{t}{\tau} \bigr) + \epsilon$$
Here $A$ is the asymptotic value which you already know, so you only need to estimate $B$ and $\tau$.
Next, you will need to define the convergence time. For example, you may want to consider being within $5\%$ of the asymptotic value as convergence. We can solve for this time in the example of the exponential distribution:
$$\frac{dv(t)}{A} = 1 + \frac{B}{A} \exp \bigl(-\frac{t_c}{\tau} \bigr) = 1.05$$
$$t_c = -\tau \log \bigl [ 0.05 \frac{A}{B} \bigr ]$$
Finally, you will likely want to find a confidence interval for $t_c$. This post is already too long, so I won't go into too much detail. But your options are:

*

*derive exact solution for the confidence interval yourself as a function of data. I think an analytic expression for a function of the exponential distribution is possible.

*Find analytical or semianalytical confidence interval for your exact problem implemented somewhere and import it

*Calculate approximate confidence interval using a numerical method (e.g. bootstrapping). If performance is not an issue for you and you want to save time, I would in fact recommend this option to get you started

