Probabilistic forecast combinations
1st April 2025 @ ACEFA Launch
Mitchell O’Hara-Wild, Monash University
Forecast combinations
There are many ways in which multiple forecasts can be combined to produce more better forecasts.
Ensemble forecasts
(this is what we currently do at ACEFA)
Decomposition forecasting
Forecast reconciliation
Ensemble forecasts
Ensemble forecasts average forecasts from:
- multiple models, of the
- same time-series.
Model uncertainty
This is well known to improve forecast accuracy.
Uncertainty over which model is closest to the data generating process is averaged over.
Australian ambulance costs
Future trend?
The trend changed in 2015 after a drop.
Will it change again following the 2024 drop?
Australian ambulance costs
Ensemble forecasts
Ensemble forecasts combine these forecasts.
Ensembling uses a weighted average of forecasts.
Finding the best weights is very hard.
Probabilistic ensemble forecasts
Combining probabilistic forecasts is done in two ways:
- Probability mixtures
- Quantile mixtures
Australian ambulance costs
Australian ambulance costs
Probabilistic ensemble forecasts
These forecast distributions can be combined mixture distributions which averages probabilities or quantiles.
Probability mixtures
\[ F(x) = \sum_{i=1}^{n} w_i \cdot F_i(x), \hspace{1em} f(x) = \sum_{i=1}^{n} w_i \cdot f_i(x) \]
Quantile mixtures
\[ F^{-1}(p) = \sum_{i=1}^{n} w_i \cdot F_i^{-1}(p) \]
Probability mixtures
Probability mixtures
Quantile mixtures
Quantile mixtures
Ensemble mixtures
Ensemble mixtures
Decomposition forecasts
Decomposition forecasts combine forecasts from:
- a single model, of each
- decomposed series.
Simpler forecasting
This can simplify the modelling challenge, since each pattern can be forecasted separately.
PBS Cost
Decomposition methods
There are many models available for separating patterns from a time series.
This process is commonly used for seasonal adjustment.
I’ll be showing decomposition via a STL model.
PBS Cost
Decomposition forecasting
Combine individual forecasts of the each decomposed series based on the decomposition structure.
For example,
\[ y_{T+h|T} = (T+R)_{T+h|T} + S_{T+h|T} \]
Seasonal adjustment
We usually model/forecast trend (\(T\)) and remainder (\(R\)) together with a single model.
PBS Cost
PBS Cost
Forecast reconciliation
Forecast reconciliation combine forecasts from:
- a single model, for
- each related series.
Improved accuracy
This improves forecasting accuracy by sharing information between related series by adhering to ‘coherency constraints’.
Coherency constraints
The relationships between each time series form constraints.
For example…
The total Australian infections must equal the sum of infections by jurisdiction.
Imposing constraints
Forecasts of each series (total infections, and infections in each jurisdiction) won’t add up!
We use reconciliation to adjust forecasts for coherency.
PBS scripts by ATC 1 classification
Produce forecasts on each series
Forecasts aren’t coherent
Adjust all forecasts for coherency
\[\tilde{\mathbf{y}}_h=\mathbf{S}(\mathbf{S}'\mathbf{W}_h^{-1}\mathbf{S})^{-1}\mathbf{S}'\mathbf{W}_h^{-1}\hat{\mathbf{y}}_h.\]
Where:
\(\mathbf{S}\) defines the coherency structure, and
\(\mathbf{W}_h\) are weights for each forecast
(usually based on in-sample fits)
Coherent (reconciled) forecasts
All forecasts are adjusted for coherency
Thanks for your time!
Summary
- Reduce model uncertainty with ensemble forecasts
- Decomposition forecasting combines simpler forecasts
- Leverage related information with reconciliation
