I have several time series forecasts for a variable I'm working with, where I'm producing forecasts for a number of individual countries. In order to do the forecasting I am using the recently open-sourced Prophet algorithm from Facebook.

There is value in forecasting at the level of individual countries, because then each time-series model ends up with a country specific trend and seasonality components, as well as the model allowing for holidays to be included, which are of course also country dependent.

What I'd like to do is basically roll these forecasts up into a continental or worldwide forecast, but I'm not sure about the most rigorous way to treat the uncertainties. The naive approach is of course just to sum the maximum and minimum values of the uncertainty from each forecast to find the worldwide level uncertainty, but this doesn't seem like the correct approach.

Has anyone got any suggestions for a sensible approach to this (even if it requires certain assumptions about the data or the model)?

One suggestion in the comments was to look at the shape of the distributions in order to answer this question. The top plot that I show here are the raw distributions (for a single year) for the variable of interest, and the log-distributions are in the plot below.

Raw distributions Log distributions


1 Answer 1


Assuming the variables are normally distributed, adding the uncertainties is the right thing. You have to keep in mind two things:

  • if the distributions are e.g. log-normal, that does not work
  • you probably need to weigh the countries by something - population, GDP, or whatever is relevant.
  • $\begingroup$ Thanks a lot for your reply. What I'm forecasting isn't normally distributed unfortunately. It is analogous to predicting something like sales, where there is an underlying upward trend in volume, with fairly strong seasonal variations. Do you have a sense for how that affects the approach? $\endgroup$
    – anthr
    Apr 12, 2017 at 1:58
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    $\begingroup$ In my experience, a lot of things that are related to money are log-distributed. Ok, first, do a histogram of all sales for some country and some year, and see how does is look. Secondly, does the upward trend increase multiplicatively or additively? That should be related to the first thing, where multiplicative increases lead to lognormal distribution. Thirdly, something you need to keep in mind is, is there a natural limit of the sales? E.g. there is a finite number of people that would buy a car or whatever. Anyway, post some visualizations, those might help. $\endgroup$ Apr 12, 2017 at 2:03
  • $\begingroup$ Thanks for the suggestions - I have added some example distributions to the post. Interestingly, it seems as though the underlying distribution varies somewhat for each of the countries. In terms of the natural limit of sales, the situation is a very long way from being at that threshold, but it does exist. $\endgroup$
    – anthr
    Apr 12, 2017 at 16:37
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    $\begingroup$ Keep in mind that the variances should be added, not the standard deviations! $\endgroup$
    – Pieter
    Sep 12, 2017 at 10:36
  • $\begingroup$ This answer is almost entirely incorrect. The marginals being normal has no impact on whether you can add the variances or not (but the variance may not be sufficient to construct a prediction interval for the aggregate if it is not normal). Deriving a prediction interval for the aggregate requires modelling the full JOINT predictive distribution of the countries; the marginals are not enough. $\endgroup$
    – Chris Haug
    Mar 20, 2020 at 21:22

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