High dimensional time series I'm not sure what words I should look for. I have an under determined dataset of 8000 correlated variables (sales) over 12 months (ie 12 observations for each variable). And I basically want to predict the future. Where should I start? PCA?
My question is, what are the techniques used to deal with lots of (correlated) variables and few observations in the case of time series.
I'm looking for orientations (but a full solution is of course welcome!).
I'm agnostic, so stats/econometrics/model-based are as fine as machine learning/AI/not-model-based, as long as they yield results that are useful and mean something. Repeatable/standardizable solutions are more than welcome.
 A: If you want to build any kind of casual model, you're going to run into problems if you try to forecast a time series in relation to other time series, because you'll need good forecasts of the independent time series before you can forecast the dependent time series.  This can turn into a recursive problem.
12 observations isn't a heck of a lot to work with, but one suggestion would be to check out the forecast package for R, specifically the auto.arima and ets functions, which will automatically fit univariate arima and exponential smoothing models to your data.  Fit some models to the 1st 11 points in each time series, and look at your aggregate error on point 12.  If you want to really do it right, cross-validate each time series model.
If building your own solution doesn't appeal to you, there's lots of proprietary software out there that is designed to help solve this sort of problem.  One good example is forecast pro, which is very fast and will automatically choose between exponential smoothing and arima models.
Keep in mind that no proprietary software is going to allow you to build causal models without first forecasting the independent variables.
