George Athanasopoulos (with Rob J. Hyndman, Nikos Kourentzes and Fotis Petropoulos) Forecasting hierarchical (and grouped) time series Total A AA AB C B AC BA BB BC CA CB CC 1 Outline 1 Hierarchical and grouped time series 2 Optimal forecasts 3 Approximately optimal forecasts 4 Temporal hierarchies 5 References Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 2 Summary 3 key developments: 1 2 3 We generalise the forecasting process in "properly" accounting for grouped data. (Empirical Application 1). We advance the “optimal combination” approach by proposing two new estimators based on WLS. Both now implemented in the hts package. We introduce temporal hierarchies. (Empirical Application 2). Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 3 Summary 3 key developments: 1 2 3 We generalise the forecasting process in "properly" accounting for grouped data. (Empirical Application 1). We advance the “optimal combination” approach by proposing two new estimators based on WLS. Both now implemented in the hts package. We introduce temporal hierarchies. (Empirical Application 2). Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 3 Summary 3 key developments: 1 2 3 We generalise the forecasting process in "properly" accounting for grouped data. (Empirical Application 1). We advance the “optimal combination” approach by proposing two new estimators based on WLS. Both now implemented in the hts package. We introduce temporal hierarchies. (Empirical Application 2). Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 3 Summary 3 key developments: 1 2 3 We generalise the forecasting process in "properly" accounting for grouped data. (Empirical Application 1). We advance the “optimal combination” approach by proposing two new estimators based on WLS. Both now implemented in the hts package. We introduce temporal hierarchies. (Empirical Application 2). Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 3 Hierarchical versus grouped Table: Geographical Hierarchy Level Australia States and Territories Total series per level 1 7 VIC, NSW, QLD, SA, WA, NT, TAS; Zones 27 VIC (5): Metro, West Coast, East Coast, Nth East, Nth West; NSW (6): Metro, Nth Coast, Sth Coast, Sth, Nth, ACT; QLD (4): Metro, Central Coast, Nth Coast, Inland; Regions 76 Metro VIC: Melbourne, Peninsula, Geelong; Metro NSW: Sydney, Illawarra, Central Coast; Metro QLD: Brisbane, Gold Coast, Sunshine Coast; Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 4 Australian domestic tourism Hierarchical: Australia (1) States (7) Zones (27) Regions (76) Total: 111 series Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 5 Australian domestic tourism Purpose of travel (PoT): Holiday Visiting Friends and Relatives Business Other Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 5 Australian domestic tourism Hierarchical: Australia (1) PoT (×4): AustraliaPoT (4) States (7) StatesPoT (28) Zones (27) ZonesPoT (108) Regions (76) RegionsPoT (304) Total: 111 series Forecasting hierarchical (and grouped) time series Total: 444 series Hierarchical and grouped time series 5 Australian domestic tourism Hierarchical: Australia (1) PoT (×4): AustraliaPoT (4) States (7) StatesPoT (28) Zones (27) ZonesPoT (108) Regions (76) RegionsPoT (304) Total: 111 series Total: 444 series Grouped Grand total: 555 series Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 5 Forecasting such structures Existing methods: ã Bottom-up ã Top-down ã Middle-out Key idea: forecast reconciliation å Ignore structural constraints and forecast every series of interest independently. å Adjust forecasts to impose constraints. Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 6 Forecasting such structures Existing methods: ã Bottom-up ã Top-down ã Middle-out Key idea: forecast reconciliation å Ignore structural constraints and forecast every series of interest independently. å Adjust forecasts to impose constraints. Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 6 Hierarchical data Total A B C Yt : observed aggregate of all series at time t. YX,t : observation on series X at time t. Bt : vector of all series at bottom level in time t. Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 7 Hierarchical data Total A B C Yt : observed aggregate of all series at time t. YX,t : observation on series X at time t. Bt : vector of all series at bottom level in time t. Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 7 Hierarchical data Total A B Yt = C Yt : observed aggregate of all series at time t. YX,t : observation on series X at time t. Bt : vector of all series at bottom level in time t. Yt YA,t YB,t YC,t Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 7 Hierarchical data Yt : observed aggregate of all series at time t. YX,t : observation on series X at time t. Bt : vector of all series at bottom level in time t. Total A B Yt = C Yt YA,t YB,t YC,t = 1 1 0 0 1 0 1 0 | {z S Forecasting hierarchical (and grouped) time series 1 Y A , t 0 YB,t 0 YC,t 1 } Hierarchical and grouped time series 7 Hierarchical data Yt : observed aggregate of all series at time t. YX,t : observation on series X at time t. Bt : vector of all series at bottom level in time t. Total A B Yt = C Yt YA,t YB,t YC,t = 1 1 0 0 1 0 1 0 | {z S Forecasting hierarchical (and grouped) time series 1 Y A , t 0 YB,t 0 YC,t 1 | {z } } Bt Hierarchical and grouped time series 7 Hierarchical data Yt : observed aggregate of all series at time t. YX,t : observation on series X at time t. Bt : vector of all series at bottom level in time t. Total A B Yt Yt = C Yt YA,t YB,t YC,t = SBt = 1 1 0 0 1 0 1 0 | {z S Forecasting hierarchical (and grouped) time series 1 Y A , t 0 YB,t 0 YC,t 1 | {z } } Bt Hierarchical and grouped time series 7 Hierarchical data Total A AX Yt = AY Yt YA,t YB,t YC,t YAX,t YAY ,t YAZ,t YBX,t YBY ,t YCX,t YCY ,t C B AZ BX 1 1 0 0 1 0 0 0 0 0 0 = | 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 {z S Forecasting hierarchical (and grouped) time series CX BY 1 0 0 1 0 0 0 0 0 1 0 CY 1 0 0 1 0 0 0 0 0 0 1 YAX,t Y AY,t YAZ,t YBX,t YBY,t Y CX,t YCY,t | {z } Bt } Hierarchical and grouped time series 8 Hierarchical data Total A AX Yt = AY Yt YA,t YB,t YC,t YAX,t YAY ,t YAZ,t YBX,t YBY ,t YCX,t YCY ,t C B AZ BX 1 1 0 0 1 0 0 0 0 0 0 = | 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 {z S Forecasting hierarchical (and grouped) time series CX BY 1 0 0 1 0 0 0 0 0 1 0 CY 1 0 0 1 0 0 0 0 0 0 1 YAX,t Y AY,t YAZ,t YBX,t YBY,t Y CX,t YCY,t | {z } Bt } Hierarchical and grouped time series 8 Hierarchical data Total A AX Yt = AY Yt YA,t YB,t YC,t YAX,t YAY ,t YAZ,t YBX,t YBY ,t YCX,t YCY ,t C B AZ BX 1 1 0 0 1 0 0 0 0 0 0 = | 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 {z S Forecasting hierarchical (and grouped) time series CX BY 1 0 0 1 0 0 0 0 0 1 0 CY 1 0 0 1 0 0 0 0 0 0 1 YAX,t Y AY,t YAZ,t YBX,t YBY,t Y CX,t YCY,t | {z } Bt } Yt = SBt Hierarchical and grouped time series 8 Grouped data Total Total A AX B AY BX Yt YA,t YB,t YX,t Yt = YY ,t = Y AX,t Y AY ,t YBX,t YBY ,t 1 1 0 1 0 1 0 0 0 | X BY 1 1 0 0 1 0 1 0 0 AX 1 0 1 1 0 0 0 1 0 Y BX AY BY 1 0 1 YAX,t 0 YAY ,t 1 YBX,t 0 YBY ,t 0 | {z } Bt 0 1 {z S Forecasting hierarchical (and grouped) time series } Hierarchical and grouped time series 9 Grouped data Total Total A AX B AY BX Yt YA,t YB,t YX,t Yt = YY ,t = Y AX,t Y AY ,t YBX,t YBY ,t 1 1 0 1 0 1 0 0 0 | X BY 1 1 0 0 1 0 1 0 0 AX 1 0 1 1 0 0 0 1 0 Y BX AY BY 1 0 1 YAX,t 0 YAY ,t 1 YBX,t 0 YBY ,t 0 | {z } Bt 0 1 {z S Forecasting hierarchical (and grouped) time series } Hierarchical and grouped time series 9 Grouped data Total Total A AX B AY BX Yt YA,t YB,t YX,t Yt = YY ,t = Y AX,t Y AY ,t YBX,t YBY ,t 1 1 0 1 0 1 0 0 0 | X BY 1 1 0 0 1 0 1 0 0 AX 1 0 1 1 0 0 0 1 0 S Forecasting hierarchical (and grouped) time series BX AY BY 1 0 1 YAX,t 0 YAY ,t 1 YBX,t 0 YBY ,t 0 | {z } Bt 0 1 {z Y Yt = SBt } Hierarchical and grouped time series 9 Outline 1 Hierarchical and grouped time series 2 Optimal forecasts 3 Approximately optimal forecasts 4 Temporal hierarchies 5 References Forecasting hierarchical (and grouped) time series Optimal forecasts 10 Optimal forecasts Key idea: forecast reconciliation å Ignore structural constraints and forecast every series of interest independently. å Adjust forecasts to impose constraints. ˆ n(h) be vector of initial h-step forecasts, Let Y made at time n, stacked in same order as Yt . Yt = SBt . ˆ n(h) So Y = Sβn(h) + εh . βn(h) = E[Bn+h | Y1, . . . , Yn]. εh has zero mean and covariance Σh. Estimate βn (h) using GLS? Forecasting hierarchical (and grouped) time series Optimal forecasts 11 Optimal forecasts Key idea: forecast reconciliation å Ignore structural constraints and forecast every series of interest independently. å Adjust forecasts to impose constraints. ˆ n(h) be vector of initial h-step forecasts, Let Y made at time n, stacked in same order as Yt . Yt = SBt . ˆ n(h) So Y = Sβn(h) + εh . βn(h) = E[Bn+h | Y1, . . . , Yn]. εh has zero mean and covariance Σh. Estimate βn (h) using GLS? Forecasting hierarchical (and grouped) time series Optimal forecasts 11 Optimal forecasts Key idea: forecast reconciliation å Ignore structural constraints and forecast every series of interest independently. å Adjust forecasts to impose constraints. ˆ n(h) be vector of initial h-step forecasts, Let Y made at time n, stacked in same order as Yt . Yt = SBt . ˆ n(h) So Y = Sβn(h) + εh . βn(h) = E[Bn+h | Y1, . . . , Yn]. εh has zero mean and covariance Σh. Estimate βn (h) using GLS? Forecasting hierarchical (and grouped) time series Optimal forecasts 11 Optimal forecasts Key idea: forecast reconciliation å Ignore structural constraints and forecast every series of interest independently. å Adjust forecasts to impose constraints. ˆ n(h) be vector of initial h-step forecasts, Let Y made at time n, stacked in same order as Yt . Yt = SBt . ˆ n(h) So Y = Sβn(h) + εh . βn(h) = E[Bn+h | Y1, . . . , Yn]. εh has zero mean and covariance Σh. Estimate βn (h) using GLS? Forecasting hierarchical (and grouped) time series Optimal forecasts 11 Optimal forecasts Key idea: forecast reconciliation å Ignore structural constraints and forecast every series of interest independently. å Adjust forecasts to impose constraints. ˆ n(h) be vector of initial h-step forecasts, Let Y made at time n, stacked in same order as Yt . Yt = SBt . ˆ n(h) So Y = Sβn(h) + εh . βn(h) = E[Bn+h | Y1, . . . , Yn]. εh has zero mean and covariance Σh. Estimate βn (h) using GLS? Forecasting hierarchical (and grouped) time series Optimal forecasts 11 Optimal forecasts Key idea: forecast reconciliation å Ignore structural constraints and forecast every series of interest independently. å Adjust forecasts to impose constraints. ˆ n(h) be vector of initial h-step forecasts, Let Y made at time n, stacked in same order as Yt . Yt = SBt . ˆ n(h) So Y = Sβn(h) + εh . βn(h) = E[Bn+h | Y1, . . . , Yn]. εh has zero mean and covariance Σh. Estimate βn (h) using GLS? Forecasting hierarchical (and grouped) time series Optimal forecasts 11 Optimal combination forecasts ˜ n(h) Y ˆ n (h ) = Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY = (S0Σ†hS)−1S0Σ†h. Σ†h is generalized inverse of Σh. Optimal P Revised forecasts unbiased: SPS = S. Revised forecasts minimum variance: ˜ n(h)|Y1, . . . , Yn] Var[Y Problem: = S(S0Σ†hS)−1S0. Σh hard to estimate. Forecasting hierarchical (and grouped) time series Optimal forecasts 12 Optimal combination forecasts ˜ n(h) Y ˆ n (h ) = Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY Initial forecasts = (S0Σ†hS)−1S0Σ†h. Σ†h is generalized inverse of Σh. Optimal P Revised forecasts unbiased: SPS = S. Revised forecasts minimum variance: ˜ n(h)|Y1, . . . , Yn] Var[Y Problem: = S(S0Σ†hS)−1S0. Σh hard to estimate. Forecasting hierarchical (and grouped) time series Optimal forecasts 12 Optimal combination forecasts ˜ n(h) Y ˆ n (h ) = Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY Initial forecasts = (S0Σ†hS)−1S0Σ†h. Σ†h is generalized inverse of Σh. Optimal P Revised forecasts unbiased: SPS = S. Revised forecasts minimum variance: ˜ n(h)|Y1, . . . , Yn] Var[Y Problem: = S(S0Σ†hS)−1S0. Σh hard to estimate. Forecasting hierarchical (and grouped) time series Optimal forecasts 12 Optimal combination forecasts ˜ n(h) Y ˆ n (h ) = Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY Revised forecasts Initial forecasts = (S0Σ†hS)−1S0Σ†h. Σ†h is generalized inverse of Σh. Optimal P Revised forecasts unbiased: SPS = S. Revised forecasts minimum variance: ˜ n(h)|Y1, . . . , Yn] Var[Y Problem: = S(S0Σ†hS)−1S0. Σh hard to estimate. Forecasting hierarchical (and grouped) time series Optimal forecasts 12 Optimal combination forecasts ˜ n(h) Y ˆ n (h ) = Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY Revised forecasts Initial forecasts = (S0Σ†hS)−1S0Σ†h. Σ†h is generalized inverse of Σh. Optimal P Revised forecasts unbiased: SPS = S. Revised forecasts minimum variance: ˜ n(h)|Y1, . . . , Yn] Var[Y Problem: = S(S0Σ†hS)−1S0. Σh hard to estimate. Forecasting hierarchical (and grouped) time series Optimal forecasts 12 Optimal combination forecasts ˜ n(h) Y ˆ n (h ) = Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY Revised forecasts Initial forecasts = (S0Σ†hS)−1S0Σ†h. Σ†h is generalized inverse of Σh. Optimal P Revised forecasts unbiased: SPS = S. Revised forecasts minimum variance: ˜ n(h)|Y1, . . . , Yn] Var[Y Problem: = S(S0Σ†hS)−1S0. Σh hard to estimate. Forecasting hierarchical (and grouped) time series Optimal forecasts 12 Optimal combination forecasts ˜ n(h) Y ˆ n (h ) = Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY Revised forecasts Initial forecasts = (S0Σ†hS)−1S0Σ†h. Σ†h is generalized inverse of Σh. Optimal P Revised forecasts unbiased: SPS = S. Revised forecasts minimum variance: ˜ n(h)|Y1, . . . , Yn] Var[Y Problem: = S(S0Σ†hS)−1S0. Σh hard to estimate. Forecasting hierarchical (and grouped) time series Optimal forecasts 12 Optimal combination forecasts ˜ n(h) Y ˆ n (h ) = Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY Revised forecasts Initial forecasts = (S0Σ†hS)−1S0Σ†h. Σ†h is generalized inverse of Σh. Optimal P Revised forecasts unbiased: SPS = S. Revised forecasts minimum variance: ˜ n(h)|Y1, . . . , Yn] Var[Y Problem: = S(S0Σ†hS)−1S0. Σh hard to estimate. Forecasting hierarchical (and grouped) time series Optimal forecasts 12 Outline 1 Hierarchical and grouped time series 2 Optimal forecasts 3 Approximately optimal forecasts 4 Temporal hierarchies 5 References Forecasting hierarchical (and grouped) time series Approximately optimal forecasts 13 Approx. optimal forecasts ˜ n(h) Y ˆ n(h) = S(S0Σ†hS)−1S0Σ†hY Solution 1: OLS Assume εh ≈ SεB,h where εB,h is the forecast error at bottom level. If Moore-Penrose generalized inverse used, then (S0 Σ† S)−1 S0 Σ† = (S0 S)−1 S0 . ˜ n(h) = S(S0S)−1S0Y ˆ n ( h) Y Forecasting hierarchical (and grouped) time series Approximately optimal forecasts 14 Approx. optimal forecasts ˜ n(h) Y ˆ n(h) = S(S0Σ†hS)−1S0Σ†hY Solution 2: Rescaling Suppose we approximate Σh by its diagonal. −1 ˆ1 Let Λ = diagonal Σ contain inverse one-step ahead in-sample forecast error variances. ˜ n(h) = S(S0ΛS)−1S0ΛY ˆ n ( h) Y Forecasting hierarchical (and grouped) time series Approximately optimal forecasts 15 Approx. optimal forecasts ˜ n(h) Y ˆ n(h) = S(S0ΛS)−1S0ΛY Solution 3: Averaging If the bottom level error series are approximately uncorrelated and have similar variances, then Λ is inversely proportional to the number of series contributing to each node. So set Λ to be the inverse row sums of S: Λ = diag(S × 1)−1 where 1 = (1, 1, . . . , 1)0 . Forecasting hierarchical (and grouped) time series Approximately optimal forecasts 16 Outline 1 Hierarchical and grouped time series 2 Optimal forecasts 3 Approximately optimal forecasts 4 Temporal hierarchies 5 References Forecasting hierarchical (and grouped) time series Temporal hierarchies 17 Temporal hierarchies: quarterly Annual Semi-Anual1 Q1 Semi-Anual2 Q2 Q3 Q4 Basic idea: å Forecast series at each available frequency. å Optimally combine forecasts within the same year. Forecasting hierarchical (and grouped) time series Temporal hierarchies 18 Temporal hierarchies: quarterly Annual Semi-Anual1 Q1 Semi-Anual2 Q2 Q3 Q4 Basic idea: å Forecast series at each available frequency. å Optimally combine forecasts within the same year. Forecasting hierarchical (and grouped) time series Temporal hierarchies 18 Temporal hierarchies: monthly Annual Semi-Anual1 Q1 M1 M2 Semi-Anual2 Q3 Q2 M3 M4 M5 M6 M7 M8 Q4 M9 M10 M11 M12 Aggregate: 3, 6, 12 Alternatively: 2, 4, 12. How about: 2, 3, 4, 6, 12? Forecasting hierarchical (and grouped) time series Temporal hierarchies 19 Temporal hierarchies: monthly Annual FourM1 BiM1 M1 M2 BiM2 M3 FourM3 FourM2 M4 BiM3 M5 M6 BiM5 BiM4 M7 M8 M9 BiM6 M10 M11 M12 Aggregate: 3, 6, 12 Alternatively: 2, 4, 12. How about: 2, 3, 4, 6, 12? Forecasting hierarchical (and grouped) time series Temporal hierarchies 19 Temporal hierarchies: monthly Annual FourM1 BiM1 M1 M2 BiM2 M3 FourM3 FourM2 M4 BiM3 M5 M6 BiM5 BiM4 M7 M8 M9 BiM6 M10 M11 M12 Aggregate: 3, 6, 12 Alternatively: 2, 4, 12. How about: 2, 3, 4, 6, 12? Forecasting hierarchical (and grouped) time series Temporal hierarchies 19 Monthly data = 0 1 0 A SemiA1 SemiA2 FourM1 FourM2 FourM3 Q1 .. . Q4 BiM1 .. . BiM6 M1 .. . M12 | {z (28×1) } 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 0 1 0 1 0 0 0 1 .. . 0 1 0 0 0 0 0 0 0 | Forecasting hierarchical (and grouped) time series I12 {z S M1 M2 M3 M 4 M5 M6 1 M7 0 M8 M9 1M10 M11 M 12 | {z } } Bt 1 0 1 0 0 1 0 Temporal hierarchies 20 Experimental setup: M3 forecasting competition (Makridakis and Hibon, 2000, IJF). In total 3003 series. 1,428 monthly series with a test sample of 12 observations each. 756 quarterly series with a test sample of 8 observations each. Forecast each series with ETS (ARIMA) models. Methods performed well in the actual competition. Forecasting hierarchical (and grouped) time series Temporal hierarchies 21 Experimental setup: M3 forecasting competition (Makridakis and Hibon, 2000, IJF). In total 3003 series. 1,428 monthly series with a test sample of 12 observations each. 756 quarterly series with a test sample of 8 observations each. Forecast each series with ETS (ARIMA) models. Methods performed well in the actual competition. Forecasting hierarchical (and grouped) time series Temporal hierarchies 21 Experimental setup: M3 forecasting competition (Makridakis and Hibon, 2000, IJF). In total 3003 series. 1,428 monthly series with a test sample of 12 observations each. 756 quarterly series with a test sample of 8 observations each. Forecast each series with ETS (ARIMA) models. Methods performed well in the actual competition. Forecasting hierarchical (and grouped) time series Temporal hierarchies 21 Experimental setup: M3 forecasting competition (Makridakis and Hibon, 2000, IJF). In total 3003 series. 1,428 monthly series with a test sample of 12 observations each. 756 quarterly series with a test sample of 8 observations each. Forecast each series with ETS (ARIMA) models. Methods performed well in the actual competition. Forecasting hierarchical (and grouped) time series Temporal hierarchies 21 Results: Monthly sMAPE (obs) Forecast Horizon (h) Annual SemiA FourM (1) (2) (3) Q (4) BiM (6) M (12) Average ETS Initial 9.66 9.18 9.76 10.14 10.82 12.85 10.40 Bottom-up 8.38 9.14 9.78 10.06 11.04 12.85 10.21 OLS Scaling Averaging 7.80 7.64 7.51 8.64 8.44 8.31 9.39 9.15 9.05 9.72 10.68 12.68 9.49 10.45 12.40 9.38 10.34 12.30 9.82 9.60 9.48 Forecasting hierarchical (and grouped) time series Temporal hierarchies 22 Results: Quarterly sMAPE (obs) Forecast Horizon (h) Annual Semi-Ann Quart (2) (4) (8) Average ETS Initial 10.50 9.97 9.84 10.10 Bottom-up 8.87 9.35 9.84 9.35 OLS Scaling Averaging 9.31 8.75 8.81 9.78 9.19 9.26 10.28 9.70 9.78 9.79 9.21 9.28 Forecasting hierarchical (and grouped) time series Temporal hierarchies 23 Outline 1 Hierarchical and grouped time series 2 Optimal forecasts 3 Approximately optimal forecasts 4 Temporal hierarchies 5 References Forecasting hierarchical (and grouped) time series References 24 More information Vignette on CRAN Forecasting hierarchical (and grouped) time series References 25 References RJ Hyndman, RA Ahmed, G Athanasopoulos, and HL Shang (2011). “Optimal combination forecasts for hierarchical time series”. Computational Statistics and Data Analysis 55(9), 2579-2589. G Athanasopoulos, RA Ahmed, RJ Hyndman,(2009). “Hierarchical forecasts for Australian domestic tourism”. International Journal of Forecasting 25, 146-166. RJ Hyndman, et al., (2014). hts: Hierarchical time series. cran.r-project.org/package=hts. RJ Hyndman and G Athanasopoulos (2014). Forecasting: principles and practice. OTexts. www.otexts.org/fpp/. Forecasting hierarchical (and grouped) time series References 26 References RJ Hyndman, RA Ahmed, G Athanasopoulos, and HL Shang (2011). “Optimal combination forecasts for hierarchical time series”. Computational Statistics and Data Analysis 55(9), 2579-2589. å GEmail: Athanasopoulos, RA Ahmed, RJ Hyndman,(2009). “Hierarchical forecasts for Australian domestic tourism”. [email protected] International Journal of Forecasting 25, 146-166. Acknowledgments: Rob J Hyndman, Roman RJ Hyndman, al., (2014). hts: Hierarchical time series. Ahmed, HanetShang, Shanika Wickramasuriya, cran.r-project.org/package=hts. Nikolaos Kourentzes, Fotios Petropoulos. RJ Hyndman and G Athanasopoulos (2014). Forecasting: Acknowledgments: Excellent research principles and practice. OTexts. www.otexts.org/fpp/. assistance by Earo Wang. Forecasting hierarchical (and grouped) time series References 26

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