# Forecasting one time series with missing data with help of other time series

I have time series $R$, which shows, how something changes at the regional level.

I have several time series $U_i$, which show, how something changes at a special unit $I$ level.

There are many units in the region. $R$ has no missing data. Different $U_i$ have their own missing periods.

I want to forecast $U$ after a missing period using information of $R$ and information of $U$ when it was availible.

My thoghts till now:

Suppose $R$ is known on interval $[0, 365]$. Suppose $U_i$ is known on interval $[0,300]$. Let's take $R$ and $U_i$ both on interval $[0,300]$, take difference between them and trying to predict that difference with linear regression. So for interval $[301,365]$, I will have differences and to restore $U_i$, I will just have to take out my differences from $R$.

I don't like my solution, because:

1. We need a model for each $U_i$.
2. Because, sometimes data is more sparse and I don't even have a $[0,300]$ known interval, so not able to train regression properly.
• Is there a bias on why the data is missing? Or do you believe it to be completely random? – Jan van der Vegt Oct 12 '17 at 14:15
• @JanvanderVegt There's a bias - seasonality, but i wouldn't like to focus on it – Ladenkov Vladislav Oct 12 '17 at 14:57

Not a fully complete answer, but some inputs.

1. Your time series are correlated.
2. I assume that the measure you want to forecast for a region is an aggregation of units forecasts.

To address the first point, I usually use Vector Autoregressive Model (VAR) that forecast all time-series at once (each one being expressed as a regression using the others) The second point involves the concept of hierarchical forecasting and reconciliation. You can exploit the fact that the regional forecast should/must equal the unit- forecasts. There can be a process to adjust forecasts to take that into account.

There are both packages for VaR and hierarchical reconciliation in R but as far as I know no direct code to handle both at the same time...

You may find this paper providing some details on the proposed approach: https://mpra.ub.uni-muenchen.de/76556/1/MPRA_paper_76556.pdf

• Thanks, i'am going to study what VAR and hierarchical forecasting is. – Ladenkov Vladislav Oct 12 '17 at 16:53

Just a thought: Why wouldn't you do the training on the difference data for 1-250 and test using 251-300 to see if the underlying pattern actually is accurate. If that was the case, you can generalize from 1-300 to 301-365.

• Yes, but the solution is conceptually not really suitable for me. Even if the score is good, i won't bee really able to apply it – Ladenkov Vladislav Oct 12 '17 at 14:59

Looks like casual Impact library could help you with that http://google.github.io/CausalImpact/CausalImpact.html Or a at least principles from it