Optimal combination forecasts for hierarchical time series

Transcription

1 Optimal combination forecasts for hierarchical time series Rob J. Hyndman Roman A. Ahmed Department of Econometrics and Business Statistics

2 Outline 1 Review of hierarchical forecasting 2 3 Simulation study 4 Summary

3 Introduction Review of hierarchical forecasting Total A B C AA AB AC BA BB BC CA CB CC Examples Manufacturing product hierarchies Pharmaceutical sales Net labour turnover

4 Hierarchical time series Review of hierarchical forecasting A hierarchical time series is a collection of several time series that are linked together in a hierarchical structure. Forecasts should be aggregate consistent, unbiased, minimum variance. Existing methods: Bottom-up Top-down Middle-out How to compute forecast intervals? Most research is concerned about relative performance of existing methods.

5 Top-down method Advantages Works well in presence of low counts. Single forecasting model easy to build Provides reliable forecasts for aggregate levels. Disadvantages Review of hierarchical forecasting Loss of information, especially individual series dynamics. Distribution of forecasts to lower levels can be difficult No prediction intervals

6 Bottom-up method Advantages No loss of information. Better captures dynamics of individual series. Disadvantages Review of hierarchical forecasting Laborious because of number of series to be forecast. Constructing forecasting models is harder because of noisy data at bottom level. No prediction intervals

7 We aim to develop a new statistical methodology for forecasting hierarchical time series which: 1 provides point forecasts that are consistent across the hierarchy; 2 allows for correlations and interaction between series at each level; 3 provides estimates of forecast uncertainty which are consistent across the hierarchy; 4 allows for ad hoc adjustments and inclusion of covariates at any level.

8 Notation Total A B C K : number of levels in the hierarchy (excl. Total). Y t : observed aggregate of all series at time t. Y X,t : observation on series X at time t. Y i,t : vector of all series at level i in time t. Y t = [Y t, Y 1,t,..., Y K,t ] Y t = [Y t, Y A,t, Y B,t, Y C,t ] = Y A,t Y B,t Y A,t Y B,t Y C,t Y C,t }{{}}{{} Y Y K,t t = SY K,t S

9 Notation Total A B C AA AB AC Y t = Y t Y A,t Y B,t Y C,t Y AA,t Y AB,t Y AC,t Y BA,t Y BB,t Y BC,t Y CA,t Y CB,t Y CC,t = BA BB BC }{{} S CA CB CC Y AA,t Y AB,t Y AC,t Y BA,t Y BB,t Y BC,t Y CA,t Y CB,t Y CC,t }{{} Y K,t

10 Forecasts Y t = SY K,t Let Ŷ n (h) be vector of independent (base) forecasts for horizon h, stacked in same order as Y t. Write where Ŷ n (h) = Sβ n (h) + ε h β n (h) = E[Y K,n+h Y 1,..., Y n ] is unknown mean of bottom level K ε h has zero mean and covariance matrix Σ h. Idea: Estimate β n (h) using regression.

11 Optimal combination forecasts ˆβ n (h) = (S Σ h S) 1 S Σ hŷn(h) where Σ h is generalized inverse of Σ h. Ỹ n (h) = S ˆβ n (h) = S(S Σ h S) 1 S Σ hŷn(h) Revised forecasts Base forecasts Revised forecasts unbiased if Ŷ n (h) unbiased: E[Ỹn(h)] = S(S Σ h S) 1 S Σ h Sβ n(h) = Sβ n (h). Minimum variance by construction: Var[Ỹn(h)] = S(S Σ h S) 1 S

12 Optimal combination forecasts Ỹ n (h) = S(S Σ h S) 1 S Σ hŷn(h) Revised forecasts Base forecasts Problem: Don t know Σ h and hard to estimate. Solution: Assume ε h Sε K,h where ε K,h is the forecast error at bottom level. Then Σ h SΩ h S where Ω h = Var(ε K,h ). If Moore-Penrose generalized inverse used, then (S Σ S) 1 S Σ = (S S) 1 S. Ỹ n (h) = S(S S) 1 S Ŷ n (h)

13 Optimal combination forecasts Ỹ n (h) = S(S S) 1 S Ŷ n (h) OLS solution. Optimal weighted average of base forecasts. Computational difficulties in big hierarchies due to size of S matrix. Optimal weights are S(S S) 1 S Weights are independent of the data! Total A B C Weights: S(S S) 1 S =

14 Optimal combination forecasts Total A B C AA AB AC Weights: S(S S) 1 S = BA BB BC CA CB CC

15 Variances Ỹ n (h) = S(S S) 1 S Ŷ n (h) Var[Ỹ n (h)] = SΩ h S = Σ h. Revised forecasts have the same variance matrix as original forecasts. Revised forecasts have the same variance matrix as bottom-up forecasts. The assumption Σ h SΩ h S results in loss of minimum variance property. Need to estimate Ω h to produce prediction intervals.

16 Adjustments and covariates Adjustments can be made to base forecasts at any level. These will automatically propagate through to other levels of the hierarchy. Covariates can be included in the base forecasts at any level. Point forecasts are always consistent across the hierarchy. Relationships between series at any level are handled automatically. However, some simplifying assumptions have been made. Estimates of forecast uncertainty are consistent across the hierarchy, but we need an estimate of Ω h.

17 Simulation design Simulation study Hierarchy of four levels with each disaggregation into four series. Bottom level: 64 series. Total series: 85. Series length: n = 100 (90 used for fitting, 10 for evaluation) Data generated for each series by ARIMA(p, d, q) process with random order and coefficients. Aggregate and three components generated. Fourth series obtained by subtraction. 600 hierarchies Base forecasts using Hyndman et al. (IJF 2002)

18 Simulation results Simulation study Average MAE by level Level Top-down Bottom-up Optimal combination 0 (Top) (Bottom) Average Top-down uses disaggregation by historical proportions

19 Summary Summary Very flexible method. Can work with any base forecasts. Method outperforms bottom-up and top-down, especially for middle levels. Conceptually easy to implement: OLS on base forecasts. Weights independent of the data. So calculate once and reapply when new data available. Remaining problems How to explain variance results compared to simulation? How to compute Ω h in order to obtain prediction intervals? Computationally it may be difficult for large hierarchies. (But solution coming up!)

20 For more information Summary Paper and computer code will appear at Paper will also appear on RePEc and in NEP-FOR report