(with Rob J. Hyndman, Nikos Kourentzes and Fotis Petropoulos ... › wp-content › uploads ›...
61
Total A AA AB AC B BA BB BC C CA CB CC 1 George Athanasopoulos (with Rob J. Hyndman, Nikos Kourentzes and Fotis Petropoulos) Forecasting hierarchical (and grouped) time series
Transcript of (with Rob J. Hyndman, Nikos Kourentzes and Fotis Petropoulos ... › wp-content › uploads ›...
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
1
George Athanasopoulos(with Rob J. Hyndman, Nikos Kourentzes andFotis Petropoulos)
Forecasting hierarchical (andgrouped) time series
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 We generalise the forecasting process in"properly" accounting for grouped data.(Empirical Application 1).
2 We advance the “optimal combination”approach by proposing two new estimatorsbased on WLS.Both now implemented in the hts package.
3 We introduce temporal hierarchies.(Empirical Application 2).
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 3
Summary
3 key developments:
1 We generalise the forecasting process in"properly" accounting for grouped data.(Empirical Application 1).
2 We advance the “optimal combination”approach by proposing two new estimatorsbased on WLS.Both now implemented in the hts package.
3 We introduce temporal hierarchies.(Empirical Application 2).
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 3
Summary
3 key developments:
1 We generalise the forecasting process in"properly" accounting for grouped data.(Empirical Application 1).
2 We advance the “optimal combination”approach by proposing two new estimatorsbased on WLS.Both now implemented in the hts package.
3 We introduce temporal hierarchies.(Empirical Application 2).
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 3
Summary
3 key developments:
1 We generalise the forecasting process in"properly" accounting for grouped data.(Empirical Application 1).
2 We advance the “optimal combination”approach by proposing two new estimatorsbased on WLS.Both now implemented in the hts package.
3 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 Total series per levelAustralia 1States and Territories 7
VIC, NSW, QLD, SA, WA, NT, TAS;
Zones 27VIC (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 76Metro 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
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 5
Hierarchical:Australia (1)
States (7)
Zones (27)
Regions (76)
Total: 111 series
Australian domestic tourism
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 5
Purpose of travel (PoT):Holiday
Visiting Friends and Relatives
Business
Other
Australian domestic tourism
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 5
Hierarchical:Australia (1)
States (7)
Zones (27)
Regions (76)
Total: 111 series
PoT (×4):AustraliaPoT (4)
StatesPoT (28)
ZonesPoT (108)
RegionsPoT (304)
Total: 444 series
Australian domestic tourism
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 5
Hierarchical:Australia (1)
States (7)
Zones (27)
Regions (76)
Total: 111 series
PoT (×4):AustraliaPoT (4)
StatesPoT (28)
ZonesPoT (108)
RegionsPoT (304)
Total: 444 series
GroupedGrand total: 555 series
Forecasting such structures
Existing methods:
ã Bottom-up
ã Top-down
ã Middle-out
Key idea: forecast reconciliationå 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 reconciliationå 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
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 7
Yt : observed aggregate of allseries at time t.
YX,t : observation on series X attime t.
Bt : vector of all series atbottom level in time t.
Hierarchical data
Total
A B C
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 7
Yt : observed aggregate of allseries at time t.
YX,t : observation on series X attime t.
Bt : vector of all series atbottom level in time t.
Hierarchical data
Total
A B C
Yt =
YtYA,tYB,tYC,t
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 7
Yt : observed aggregate of allseries at time t.
YX,t : observation on series X attime t.
Bt : vector of all series atbottom level in time t.
Hierarchical data
Total
A B C
Yt =
YtYA,tYB,tYC,t
=
1 1 11 0 00 1 00 0 1
︸ ︷︷ ︸
S
YA,tYB,tYC,t
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 7
Yt : observed aggregate of allseries at time t.
YX,t : observation on series X attime t.
Bt : vector of all series atbottom level in time t.
Hierarchical data
Total
A B C
Yt =
YtYA,tYB,tYC,t
=
1 1 11 0 00 1 00 0 1
︸ ︷︷ ︸
S
YA,tYB,tYC,t
︸ ︷︷ ︸
Bt
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 7
Yt : observed aggregate of allseries at time t.
YX,t : observation on series X attime t.
Bt : vector of all series atbottom level in time t.
Hierarchical data
Total
A B C
Yt =
YtYA,tYB,tYC,t
=
1 1 11 0 00 1 00 0 1
︸ ︷︷ ︸
S
YA,tYB,tYC,t
︸ ︷︷ ︸
BtYt = SBt
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 7
Yt : observed aggregate of allseries at time t.
YX,t : observation on series X attime t.
Bt : vector of all series atbottom level in time t.
Hierarchical dataTotal
A
AX AY AZ
B
BX BY
C
CX CY
Yt =
YtYA,tYB,tYC,tYAX,tYAY,tYAZ,tYBX,tYBY,tYCX,tYCY,t
=
1 1 1 1 1 1 11 1 1 0 0 0 00 0 0 1 1 0 00 0 0 0 0 1 11 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1
︸ ︷︷ ︸
S
YAX,tYAY,tYAZ,tYBX,tYBY,tYCX,tYCY,t
︸ ︷︷ ︸
Bt
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 8
Hierarchical dataTotal
A
AX AY AZ
B
BX BY
C
CX CY
Yt =
YtYA,tYB,tYC,tYAX,tYAY,tYAZ,tYBX,tYBY,tYCX,tYCY,t
=
1 1 1 1 1 1 11 1 1 0 0 0 00 0 0 1 1 0 00 0 0 0 0 1 11 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1
︸ ︷︷ ︸
S
YAX,tYAY,tYAZ,tYBX,tYBY,tYCX,tYCY,t
︸ ︷︷ ︸
Bt
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 8
Hierarchical dataTotal
A
AX AY AZ
B
BX BY
C
CX CY
Yt =
YtYA,tYB,tYC,tYAX,tYAY,tYAZ,tYBX,tYBY,tYCX,tYCY,t
=
1 1 1 1 1 1 11 1 1 0 0 0 00 0 0 1 1 0 00 0 0 0 0 1 11 0 0 0 0 0 00 1 0 0 0 0 00 0 1 0 0 0 00 0 0 1 0 0 00 0 0 0 1 0 00 0 0 0 0 1 00 0 0 0 0 0 1
︸ ︷︷ ︸
S
YAX,tYAY,tYAZ,tYBX,tYBY,tYCX,tYCY,t
︸ ︷︷ ︸
Bt
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 8
Yt = SBt
Grouped dataTotal
A
AX AY
B
BX BY
Total
X
AX BX
Y
AY BY
Yt =
YtYA,tYB,tYX,tYY,tYAX,tYAY,tYBX,tYBY,t
=
1 1 1 11 1 0 00 0 1 11 0 1 00 1 0 11 0 0 00 1 0 00 0 1 00 0 0 1
︸ ︷︷ ︸
S
YAX,tYAY,tYBX,tYBY,t
︸ ︷︷ ︸
Bt
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 9
Grouped dataTotal
A
AX AY
B
BX BY
Total
X
AX BX
Y
AY BY
Yt =
YtYA,tYB,tYX,tYY,tYAX,tYAY,tYBX,tYBY,t
=
1 1 1 11 1 0 00 0 1 11 0 1 00 1 0 11 0 0 00 1 0 00 0 1 00 0 0 1
︸ ︷︷ ︸
S
YAX,tYAY,tYBX,tYBY,t
︸ ︷︷ ︸
Bt
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 9
Grouped dataTotal
A
AX AY
B
BX BY
Total
X
AX BX
Y
AY BY
Yt =
YtYA,tYB,tYX,tYY,tYAX,tYAY,tYBX,tYBY,t
=
1 1 1 11 1 0 00 0 1 11 0 1 00 1 0 11 0 0 00 1 0 00 0 1 00 0 0 1
︸ ︷︷ ︸
S
YAX,tYAY,tYBX,tYBY,t
︸ ︷︷ ︸
Bt
Forecasting hierarchical (and grouped) time series Hierarchical and grouped time series 9
Yt = SBt
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 reconciliationå Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial h-step forecasts,made at time n, stacked in same order as Yt.
Yt = SBt . So Yn(h) = 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 reconciliationå Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial h-step forecasts,made at time n, stacked in same order as Yt.
Yt = SBt . So Yn(h) = 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 reconciliationå Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial h-step forecasts,made at time n, stacked in same order as Yt.
Yt = SBt . So Yn(h) = 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 reconciliationå Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial h-step forecasts,made at time n, stacked in same order as Yt.
Yt = SBt . So Yn(h) = 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 reconciliationå Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial h-step forecasts,made at time n, stacked in same order as Yt.
Yt = SBt . So Yn(h) = 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 reconciliationå Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial h-step forecasts,made at time n, stacked in same order as Yt.
Yt = SBt . So Yn(h) = 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
Yn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Optimal P = (S′Σ†hS)−1S′Σ†h.
Σ†h is generalized inverse of Σh.
Revised forecasts unbiased: SPS = S.
Revised forecasts minimum variance:
Var[Yn(h)|Y1, . . . ,Yn] = S(S′Σ†hS)−1S′.
Problem: Σh hard to estimate.Forecasting hierarchical (and grouped) time series Optimal forecasts 12
Optimal combination forecasts
Yn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Initial forecasts
Optimal P = (S′Σ†hS)−1S′Σ†h.
Σ†h is generalized inverse of Σh.
Revised forecasts unbiased: SPS = S.
Revised forecasts minimum variance:
Var[Yn(h)|Y1, . . . ,Yn] = S(S′Σ†hS)−1S′.
Problem: Σh hard to estimate.Forecasting hierarchical (and grouped) time series Optimal forecasts 12
Optimal combination forecasts
Yn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Initial forecasts
Optimal P = (S′Σ†hS)−1S′Σ†h.
Σ†h is generalized inverse of Σh.
Revised forecasts unbiased: SPS = S.
Revised forecasts minimum variance:
Var[Yn(h)|Y1, . . . ,Yn] = S(S′Σ†hS)−1S′.
Problem: Σh hard to estimate.Forecasting hierarchical (and grouped) time series Optimal forecasts 12
Optimal combination forecasts
Yn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Initial forecasts
Optimal P = (S′Σ†hS)−1S′Σ†h.
Σ†h is generalized inverse of Σh.
Revised forecasts unbiased: SPS = S.
Revised forecasts minimum variance:
Var[Yn(h)|Y1, . . . ,Yn] = S(S′Σ†hS)−1S′.
Problem: Σh hard to estimate.Forecasting hierarchical (and grouped) time series Optimal forecasts 12
Optimal combination forecasts
Yn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Initial forecasts
Optimal P = (S′Σ†hS)−1S′Σ†h.
Σ†h is generalized inverse of Σh.
Revised forecasts unbiased: SPS = S.
Revised forecasts minimum variance:
Var[Yn(h)|Y1, . . . ,Yn] = S(S′Σ†hS)−1S′.
Problem: Σh hard to estimate.Forecasting hierarchical (and grouped) time series Optimal forecasts 12
Optimal combination forecasts
Yn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Initial forecasts
Optimal P = (S′Σ†hS)−1S′Σ†h.
Σ†h is generalized inverse of Σh.
Revised forecasts unbiased: SPS = S.
Revised forecasts minimum variance:
Var[Yn(h)|Y1, . . . ,Yn] = S(S′Σ†hS)−1S′.
Problem: Σh hard to estimate.Forecasting hierarchical (and grouped) time series Optimal forecasts 12
Optimal combination forecasts
Yn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Initial forecasts
Optimal P = (S′Σ†hS)−1S′Σ†h.
Σ†h is generalized inverse of Σh.
Revised forecasts unbiased: SPS = S.
Revised forecasts minimum variance:
Var[Yn(h)|Y1, . . . ,Yn] = S(S′Σ†hS)−1S′.
Problem: Σh hard to estimate.Forecasting hierarchical (and grouped) time series Optimal forecasts 12
Optimal combination forecasts
Yn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Initial forecasts
Optimal P = (S′Σ†hS)−1S′Σ†h.
Σ†h is generalized inverse of Σh.
Revised forecasts unbiased: SPS = S.
Revised forecasts minimum variance:
Var[Yn(h)|Y1, . . . ,Yn] = S(S′Σ†hS)−1S′.
Problem: 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
Forecasting hierarchical (and grouped) time series Approximately optimal forecasts 14
Yn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Solution 1: OLSAssume εh ≈ SεB,h where εB,h is the forecasterror at bottom level.
If Moore-Penrose generalized inverse used,then (S′Σ†S)−1S′Σ† = (S′S)−1S′.
Yn(h) = S(S′S)−1S′Yn(h)
Approx. optimal forecasts
Forecasting hierarchical (and grouped) time series Approximately optimal forecasts 15
Yn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Solution 2: RescalingSuppose we approximate Σh by its diagonal.
Let Λ =[diagonal
(Σ1
)]−1contain inverse
one-step ahead in-sample forecast errorvariances.
Yn(h) = S(S′ΛS)−1S′ΛYn(h)
Approx. optimal forecasts
Forecasting hierarchical (and grouped) time series Approximately optimal forecasts 16
Yn(h) = S(S′ΛS)−1S′ΛYn(h)
Solution 3: AveragingIf the bottom level error series areapproximately uncorrelated and have similarvariances, then Λ is inversely proportional tothe number of series contributing to eachnode.
So set Λ to be the inverse row sums of S:
Λ = diag(S× 1)−1
where 1 = (1,1, . . . ,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 Temporal hierarchies 17
Temporal hierarchies: quarterly
Annual
Semi-Anual1
Q1 Q2
Semi-Anual2
Q3 Q4
Basic idea:å Forecast series at each available
frequency.
å Optimally combine forecasts within thesame year.
Forecasting hierarchical (and grouped) time series Temporal hierarchies 18
Temporal hierarchies: quarterly
Annual
Semi-Anual1
Q1 Q2
Semi-Anual2
Q3 Q4
Basic idea:å Forecast series at each available
frequency.
å Optimally combine forecasts within thesame year.
Forecasting hierarchical (and grouped) time series Temporal hierarchies 18
Temporal hierarchies: monthly
Annual
Semi-Anual1
Q1
M1 M2 M3
Q2
M4 M5 M6
Semi-Anual2
Q3
M7 M8 M9
Q4
M10 M11 M12
Aggregate: 3, 6, 12Alternatively: 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 M4
FourM2
BiM3
M5 M6
BiM4
M7 M8
FourM3
BiM5
M9 M10
BiM6
M11 M12
Aggregate: 3, 6, 12Alternatively: 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 M4
FourM2
BiM3
M5 M6
BiM4
M7 M8
FourM3
BiM5
M9 M10
BiM6
M11 M12
Aggregate: 3, 6, 12Alternatively: 2, 4, 12.How about: 2, 3, 4, 6, 12?
Forecasting hierarchical (and grouped) time series Temporal hierarchies 19
Monthly data
ASemiA1
SemiA2
FourM1
FourM2
FourM3
Q1
...Q4
BiM1
...BiM6
M1
...M12
︸ ︷︷ ︸
(28×1)
=
1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 0 0 0 0 0 00 0 0 0 0 0 1 1 1 1 1 11 1 1 1 0 0 0 0 0 0 0 00 0 0 0 1 1 1 1 0 0 0 00 0 0 0 0 0 0 0 1 1 1 11 1 1 0 0 0 0 0 0 0 0 0
...0 0 0 0 0 0 0 0 0 1 1 11 1 0 0 0 0 0 0 0 0 0 0
...0 0 0 0 0 0 0 0 0 0 1 1
I12
︸ ︷︷ ︸
S
M1
M2
M3
M4
M5
M6
M7
M8
M9
M10
M11
M12
︸ ︷︷ ︸
Bt
Forecasting hierarchical (and grouped) time series Temporal hierarchies 20
Experimental setup:
M3 forecasting competition (Makridakisand Hibon, 2000, IJF). In total 3003 series.
1,428 monthly series with a test sample of12 observations each.
756 quarterly series with a test sample of8 observations each.
Forecast each series with ETS (ARIMA)models. Methods performed well in theactual competition.
Forecasting hierarchical (and grouped) time series Temporal hierarchies 21
Experimental setup:
M3 forecasting competition (Makridakisand Hibon, 2000, IJF). In total 3003 series.
1,428 monthly series with a test sample of12 observations each.
756 quarterly series with a test sample of8 observations each.
Forecast each series with ETS (ARIMA)models. Methods performed well in theactual competition.
Forecasting hierarchical (and grouped) time series Temporal hierarchies 21
Experimental setup:
M3 forecasting competition (Makridakisand Hibon, 2000, IJF). In total 3003 series.
1,428 monthly series with a test sample of12 observations each.
756 quarterly series with a test sample of8 observations each.
Forecast each series with ETS (ARIMA)models. Methods performed well in theactual competition.
Forecasting hierarchical (and grouped) time series Temporal hierarchies 21
Experimental setup:
M3 forecasting competition (Makridakisand Hibon, 2000, IJF). In total 3003 series.
1,428 monthly series with a test sample of12 observations each.
756 quarterly series with a test sample of8 observations each.
Forecast each series with ETS (ARIMA)models. Methods performed well in theactual competition.
Forecasting hierarchical (and grouped) time series Temporal hierarchies 21
Results: Monthly
Forecast Horizon (h)sMAPE Annual SemiA FourM Q BiM M Average(obs) (1) (2) (3) (4) (6) (12)
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 7.80 8.64 9.39 9.72 10.68 12.68 9.82Scaling 7.64 8.44 9.15 9.49 10.45 12.40 9.60Averaging 7.51 8.31 9.05 9.38 10.34 12.30 9.48
Forecasting hierarchical (and grouped) time series Temporal hierarchies 22
Results: Quarterly
Forecast Horizon (h)sMAPE Annual Semi-Ann Quart Average(obs) (2) (4) (8)
ETS
Initial 10.50 9.97 9.84 10.10
Bottom-up 8.87 9.35 9.84 9.35
OLS 9.31 9.78 10.28 9.79Scaling 8.75 9.19 9.70 9.21Averaging 8.81 9.26 9.78 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
Forecasting hierarchical (and grouped) time series References 25
Vignette on CRAN
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.
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
å Email:
Acknowledgments: Rob J Hyndman, Roman
Ahmed, Han Shang, Shanika Wickramasuriya,
Nikolaos Kourentzes, Fotios Petropoulos.
Acknowledgments: Excellent research
assistance by Earo Wang.
