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I have time series where at each time step I have a bunch of real-valued points (e.g. individual purchases on a given day), and would like to produce a forecast of several quantiles.

One approach I'm thinking of is, for each quantile, compute the time-series of its empirical values, and forecast those. So if I want 10 quantile predictions, I would produce/forecast 10 separate time-series.

My only concern is that the above approach might yield quantiles that are not ordered. For example, 80th quantile might have a much larger trend than the 90th quantile, and the forecast of the former might be larger than the forecast of the latter.

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  • $\begingroup$ Do you have a fixed set of variables that you observe on different time points, or do your variables change at each time point? Regarding ordering, there is definitely some literature on that, but I cannot remember the relevant keywords at the moment (sorry). $\endgroup$ – Richard Hardy Sep 4 '19 at 19:46
  • $\begingroup$ The number of observations varies by day (it's basically whoever made the purchase at that point). $\endgroup$ – Shen Jenson Sep 5 '19 at 17:14
  • $\begingroup$ So you have a large number of variables that have either zeros (on days when no purchases of the particular good were made) or nonzeros (on days where at least one purchase of the particular good was made)? $\endgroup$ – Richard Hardy Sep 5 '19 at 20:14
  • $\begingroup$ It's more like, on every day I have a variable number of positive observations. I compress them into a single (mean) value. $\endgroup$ – Shen Jenson Sep 5 '19 at 20:41
  • $\begingroup$ It may be useful to first consider the original information without compressing it and see how you could use it in the most effective/efficient way. That is why I am trying to understand what data you actually have. $\endgroup$ – Richard Hardy Sep 5 '19 at 20:43
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why not use the observed data and generate a forecast and a probability distribution via montecarlo for each period in the future. Review the probability distributions and identify whatever "tile" that you wish.

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