What fields use modelling for analysis of a single time-series? What fields of science (social or physical) use modelling for time-series analysis? In particular, time-series that cannot be replicated? Two examples that I can think of are climate modelling, and econometrics. Are there any others?
I am asking because I am doing research on model independence in climate science, and am looking for references outside my field. I have had a look around, and couldn't find many, and was hoping to tap the knowledge resource here. I am reasonably sure that there is no "long list" to be requested, rather there are a few smallish research areas doing this kind of modelling. 
Clarification: By "cannot be replicated", I mean that only one time series can be relevant. eg. in climate science, we have only one earth, with which to validate the model. (in econometrics, there may be cases of both repeatable and not repeatable time-series)
 A: I think the key to how many answers you'll get is in how you define "model", and how strong your definition of "cannot be replicated" is.
If you drop the "especially for time series that cannot be replicated", the answer is every social and physical science models time series. From biology's predator-prey models to engineering "inventing" the Kalman Filter.
If "cannot be replicated" can mean "is difficult to replicate" or "cannot be replicated exactly", that opens a lot of doors, including psychology, etc. Or do you mean "intervention is impossible"? In that case, astronomy would be applicable. (Astronomers can't manipulate their targets, but they do have many examples to look at.)
In terms of "model", does curve-fitting apply? That is, if a paper fits a curve to certain data is that a model? Or are you insisting on a kind of first-principles kind of models?
A: In psychology, especially with single case study and more generally when modelling intra-individual variability (among others, the great works of Peter Molenaar (http://www.hhdev.psu.edu/hdfs/directory/bio.aspx?id=137)).
A: In neurophysiology, one records a time series from one or more individual neurons. Recordings from the brains of living animals cannot be exactly reproduced for several reasons: the state of the animal and the cell you're recording from is changing over time (may be highly nonstationary); even if you could record the same cell in the same state, it would be difficult to locate the same single neuron again; in some experiments, the cell is dying over the course of the experiment (especially if you're trying to record from within the cell body); in many experiments, the animal itself may be sacrificed. 
A: Marketing, which is a separate field from economics (just as psychology and sociology aren't the same thing). 
See, for example, Dominique M. Hanssens, Leonard J. Parsons and Randall L. Schultz, Market Response Models: Econometric and Time Series Analysis, 2nd ed. Kluwer Academic Publishers, 2001.
