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I am trying to model a multivariate time series of percentages. And here's the kicker, at each point, each of my individual time series are bound between 0 and 1, and their Sum per period equals 1. An example of such time series would be, 

Y1={0.1,0.3,0.6}
Y2={0.2,0.3,0.5}
...

I am thinking of using Genetic Programming and compare it with ARIMA to forecast in and out of sample. I was looking for a while on how to model these issues (someone on this forum with a similar problem modeled ratios, which is not an option for me). If there is no way, I will just model the time series as is and then shave off or add the proportional difference, but this seems like a very ad-hoc procedure. Any ideas?

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You have what is known as compositional data. This comes up, e.g., in analysing market shares, or chemical and other composites. The problem typically is that your model fits and forecasts will obey neither the constraint of being between 0 and 1, nor that of summing to 1.

A standard approach is to transform your series in a way that these constraints are satisfied automatically, e.g., by mapping $y_{it}\mapsto \log(y_{it}/y_{0t})$ for some base series $y_{0t}$. If the transformation, analysis and forecast does not depend on your particular choice of base series, that is a Good Thing.

I recommend reading Snyder et al. (2017, International Journal of Forecasting).

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  • $\begingroup$ Thank you, Stephan. That mapping seems like a very interesting idea, and that way I need to model only n-1 series and obtain the remaining one by subtracting 1 from the sum of the others. Well thought. $\endgroup$
    – Arrebimbomalho
    Commented Dec 4, 2018 at 17:30
  • $\begingroup$ Just a quick question. If I have covariates, should I also apply the transformation to them? $\endgroup$
    – Arrebimbomalho
    Commented Dec 4, 2018 at 17:31
  • $\begingroup$ @StephanKolassa, when is modelling percentages "natural"... ? wouldn't it be better to model the underlying market quantities? $\endgroup$
    – seanv507
    Commented Dec 4, 2018 at 19:55
  • $\begingroup$ No, don't transform the covariates. (Unless they are composite data themselves.) $\endgroup$
    – Stephan Kolassa
    Commented Dec 5, 2018 at 12:48
  • $\begingroup$ @seanv507: it depends on the question you ask of the data. Sometimes you are truly more interested in the percentage market share, less than your share size. One example would be for antitrust regulation, for instance. $\endgroup$
    – Stephan Kolassa
    Commented Dec 5, 2018 at 12:50

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