10.3 Base and coherent forecasts
The aim is to generate a set of coherent forecasts across all the series of a hierarchical or a grouped structure.
Commonly used approaches are based on first generating forecasts for a single level of aggregation, and then producing coherent forecasts by aggregating these up using a bottom-up approach, disaggregating them down using a top-down approach, or a combination of the two using a middle-out approach. The details of these methods are presented in Sections 10.4 to 10.6. An alternative and a more advnanced approach is to first generate forecasts for all the time series and then reconcile these so that they become coherent. The details of this approach which we refer to as the optimal reconciliation approach are presented in Section 10.8.
In what follows in this section we introduce and formally define the concepts of base and coherent forecasts in order to build a unified framework that incorporates all of the approaches outlined above for both hierarchical and grouped time series.
Denote as \(\hat{y}_{h}\) the \(h\)-step-ahead forecast generated for the “Total” series having observed the time series up to time \(T\).19 Likewise denote as \(\yhat{j}{h}\) the \(h\)-step-ahead forecast generated for the series at node \(j\) having observed the time series up to time \(T\). We refer to these as base forecasts. These are forecasts generated for each time series in the aggregation structure using a suitable forecasting method. The hts package has three inbuit options to produce base forecasts. These are controlled by the fmethod method argument
forecast(..., fmethod = c("ets", "arima", "rw"), ...). as discussed in the Introduction of this Chapter 10.
Although the data follows exactly the aggregation structure (as reflected by the summing matrix \(\bm{S}\)) base forecasts will generally not. It is only under very special circumstances such as using a very simple method to forecasts all the time series (for example using naïve forecasts for all series) that the base forecasts will be coherent.
The base forecasts can however be combined, as briefly described above, to produce a set of forecasts that are coherent. We denote the set of coherent forecasts for the “Total” series and all series below the top-level as \(\tilde{y}_{h}\) and \(\ytilde{j}{h}\) respectively.
We have simplified the previously used notation of \(\hat{y}_{T+h|T}\) for brevity.↩