Forecasting a series that comes with uncertainty

Question

I have a time series resulting from a spatiotemporal aggregation on the spatial domain.

As a result, I have a central measurement (let's say mean average) and a dispersion (let's say standard deviation).

If I want to do a traditional time-series forecasting, such as ARIMA, how should I encode the central measurement and the dispersion so the final model incorporates such uncertainties?

Should I treat the central measurement as the main series and the dispersion as an exogenous covariate? Should I create a series of the lower bound (let´s say mean - 2 standard deviations)? and then forecast it as a lower bound forecast?

I would like to know more about what the options may be.

Answer 1

The answer depends much on what you want to forecast and also on the plausible models generating the data.

For the first aspect you may want to forecast the spatial average at a future point in time or the random value at a specific point in time and a specific point in the spatial domain, or even a quantity depending on several points in the domain, possibly a random number of these. Compared to the spatial average the resulting overdispersion has to be anticipated.

For the second aspect, it may help to think of a spatio-temporal process generating the data and to consider the effect of the spatial aggregation. The spatio-temporal process could involve a point process. Most likely you can use more than a single standard deviation. For instance if the the spatial aggregation arises from random events (storms, wildfires, accidents, ...) the intensity of these events may be described in parametric form.

Even without further details on the context, it can be anticipated that the quite general framework provided by state space models can help. In these models the hidden or partially observed state vector evolves in a Markovian fashion. The relation of the state to the (here aggregated) observation can change in time. Investigating the distribution of the state conditional on the observation is called filtering or smoothing. ARIMA models are special cases of state-space models but many appealing state space models can be derived from geographical or physical considerations possibly using stochastic differential equations.

Answer 2

It sounds like you may have entire probability distributions you would like to forecast. This is a case of functional-data-analysis. We don't have a lot on functional data forecasting; this thread may be interesting: Forecasting of density function. Apart from the Hyndman & Shang (2009) paper cited there, a more recent publication is Kokoszka et al. (2019). Unfortunately, I can't speak to the specific methods discussed there, but the papers and a search including "functional data forecasting" may be helpful.