Estimating the standard deviation I have data of a function $f(t)$, where I always have just a single data point per $t$. It is known that the data is affected by noise, where the noise itself is also a function of $t$. Now I want to estimate the standard deviation of the noise for different $t$. What methods could I use to do this?
 A: What you describe appears to be a good scenario to estimate the "functional variable process" of the data. A formal treatment of the matter can be found in Mueller et al. (2006) Functional Variance Processes.
To succinctly describe how we would the above methodology here: We assume a smooth underlying mean function: $\mu_Y(t)$ and estimate it as $\hat{\mu}_Y(t)$, non-parametrically via local linear smoothing from our observed data $f(t)$. We then calculate the (observed) residuals: $R(t)$ as $f(t) - \hat{\mu}_Y(t)$,  and then square and log them such that we have the observed transformed residuals: $Z(t) = \log(R^2(t))$. Again assuming a smoothing underlying mean function for the variance we once estimate it as $\hat{\mu}_Z(t)$ based on the observed $Z(t)$. To quote the paper directly: "As in the case of regression residuals, the squared errors $R^2_{ij}$ can be expected to carry relevant information about the random variation, and the exponential factors convey the nonnegativity restriction" (of the variance). Notice here that, Mueller et al. expect to have multiple realisations of the underlying process, that's why they index by $i$ as well as $j$. In the case of a single process (i.e. just one $f(t)$) the $i=1$ but we still have $J$ measurement points. And that's about it! We have our variance function in: $\mu_Z(t)$ (or $\mu_V(t)$ to use the same notation as the paper). Taking the square root of it gives us our standard deviation along $t$.
