How do you forecast a time series that is inherently uncertain? Most time series forecasting models take in fixed (so presumably deterministic) historical values, and then output either: 


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*A point (usually mean) forecast + forecast intervals, so $\hat{Y}_{t+1} = f_m(Y_t,...Y_{t_n})$ and and upper and lower intervals $\sigma^{U/L}_{t+1} = f_{\sigma^{U/L}}(Y_t,...Y_{t-n})$. 

*A quantile forecast: $\hat{Q}^{p}_{t+1} = f_p(Y_t,...Y_{t-n})$ with $p \in [0,0.99] $. 

*Or a full probability density forecast: $P(\hat{Y}_{t+1}) = f_\theta(Y_t,...Y_{t-n})$. (the density could be parametric or non parametric)


I have a situation where the input time series itself is uncertain, due to complications in the measurement process for the metric involved. My questions:


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*Are there any time series models that allow for uncertain or probabilistic inputs? 

*Presumably one could put together a monstrous LSTM that takes densities as inputs and densities as outputs, or represent the inputs as vectors of quantiles and then do something like VAR - but that seems like overkill, I'm hoping for something more elegant, but if not, are these approaches valid? 

 A: I am not aware of any forecasting methods that work off uncertain inputs, although it's an interesting question.
A simple (though possibly prohibitively expensive, performance-wise) method would be to draw a random time series, one observation from each corresponding past density (or a full series from the joint density, whatever you have). Fit a model to this series and forecast. Repeat this many times, say 1000 times. You now have 1000 point forecasts, or 1000 quantile forecasts, or 1000 predictive densities.


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*Simply average the point forecasts to get an overall point forecast.

*Similarly, average the upper and the lower ends of your quantile forecasts to get an overall quantile forecast.

*Finally, mix your density forecasts to get an overall density forecast.


Of course, I would do the full model selection step for each sampled time series, since we are especially uncertain about the data generating process.
Per above, this might be too expensive in terms of runtime. The upside is that it is easy to understand (and debug), you can use whatever time series forecasting method you want for each sample, and it is trivially parallelizable.
